5thgrade lessons 4learning 22616

GRADE

5

PUBLIC SCHOOLS OF NORTH CAROLINAState Board of Education | Department of Public Instruction

FOR THE COMMON CORE STATE STANDARDS IN MATHEMATICSLESSONS FOR LEARNING

K-12 MATHEMATICShttp://www.ncpublicschools.org/curriculum/mathematics/

STATE BOARD OF EDUCATIONThe guiding mission of the North Carolina State Board of Education is that every public school student will graduate from high school, globally competitive for work and postsecondary education and prepared for life in the 21st Century.

NC DEPARTMENT OF PUBLIC INSTRUCTIONJune St. Clair Atkinson, Ed.D., State Superintendent301 N. Wilmington Street :: Raleigh, North Carolina 27601-2825

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WILLIAM COBEYChair :: Chapel Hill

A.L. COLLINSVice Chair :: Kernersville

DAN FORESTLieutenant Governor :: Raleigh

JANET COWELLState Treasurer :: Raleigh

JUNE ST. CLAIR ATKINSONSecretary to the Board :: Raleigh

BECKY TAYLORGreenville

REGINALD KENANRose Hill

KEVIN D. HOWELLRaleigh

GREG ALCORNSalisbury

OLIVIA OXENDINELumberton

JOHN A. TATE IIICharlotte

WAYNE MCDEVITTAsheville

MARCE SAVAGEWaxhaw

PATRICIA N. WILLOUGHBYRaleigh

M0414

operations and algebraiC thinKingWrite and interpret numerical expressions.5. oa.1 Use parentheses, brackets, or braces in numerical expressions, and

evaluate expressions with these symbols.5. oa.2 Write simple expressions that record calculations with numbers, and

interpret numerical expressions without evaluating them. For example, express the calculation “add 8 and 7, then multiply by 2” as 2 × (8 + 7). Recognize that 3 × (18932 + 921) is three times as large as 18932 + 921, without having to calculate the indicated sum or product.

analyze patterns and relationships.5. oa.3 Generate two numerical patterns using two given rules. Identify

apparent relationships between corresponding terms. Form ordered pairs consisting of corresponding terms from the two patterns, and graph the ordered pairs on a coordinate plane. For example, given the rule “Add 3” and the starting number 0, and given the rule “Add 6” and the starting number 0, generate terms in the resulting sequences, and observe that the terms in one sequence are twice the corresponding terms in the other sequence. Explain informally why this is so.

number and operations in base tenunderstand the place value system.5. nbt.1 Recognize that in a multi-digit number, a digit in one place

represents 10 times as much as it represents in the place to its right and 1/10 of what it represents in the place to its left.

5. nbt.2 Explain patterns in the number of zeros of the product when multiplying a number by powers of 10, and explain patterns in the placement of the decimal point when a decimal is multiplied or divided by a power of 10. Use whole-number exponents to denote powers of 10.

5. nbt.3: Read, write, and compare decimals to thousandths.a. Read and write decimals to thousandths using base-ten

numerals, number names, and expanded form, e.g., 347.392 = 3 × 100 + 4 × 10 + 7 × 1 + 3 × (1/10) + 9 × (1/100) + 2 × (1/1000).

b. Compare two decimals to thousandths based on meanings of the digits in each place, using >, =, and < symbols to record the results of comparisons.

5. nbt.4 Use place value understanding to round decimals to any place.

perform operations with multi-digit whole numbers and with decimals to hundredths.5. nbt.5 Fluently multiply multi-digit whole numbers using the standard algorithm.5. nbt.6 Find whole-number quotients of whole numbers with up to four-digit

dividends and two-digit divisors, using strategies based on place value, the properties of operations, and/or the relationship between multiplication and division. Illustrate and explain the calculation by using equations, rectangular arrays, and/or area models.

5. nbt.7 Add, subtract, multiply, and divide decimals to hundredths, using concrete models or drawings and strategies based on place value, properties of operations, and/or the relationship between addition and subtraction; relate the strategy to a written method and explain the reasoning used.

number and operations – FraCtionsuse equivalent fractions as a strategy to add and subtract fractions.5. nF.1 Add and subtract fractions with unlike denominators (including mixed

numbers) by replacing given fractions with equivalent fractions in such a way as to produce an equivalent sum or difference of fractions with like denominators. For example, 2/3 + 5/4 = 8/12 + 15/12 = 23/12. (In general, a/b + c/d = (ad + bc)/bd.)

5. nF.2 Solve word problems involving addition and subtraction of fractions referring to the same whole, including cases of unlike denominators, e.g., by using visual fraction models or equations to represent the problem. Use benchmark fractions and number sense of fractions to estimate mentally and assess the reasonableness of answers. For example, recognize an incorrect result 2/5 + 1/2 = 3/7, by observing that 3/7 < 1/2.

apply and extend previous understandings of multiplication and division to multiply and divide fractions.5. nF.3 Interpret a fraction as division of the numerator by the denominator

(a/b = a ÷ b). Solve word problems involving division of whole numbers leading to answers in the form of fractions or mixed numbers, e.g., by using visual fraction models or equations to represent the problem. For example, interpret 3/4 as the result of dividing 3 by 4, noting that 3/4 multiplied by 4 equals 3, and that when

Fifth Grade – Standards 1. developing fluency with addition and subtraction of fractions,

developing understanding of the multiplication of fractions and of division of fractions in limited cases (unit fractions divided by whole numbers and whole numbers divided by unit fractions) – Students apply their understanding of fractions and fraction models to represent the addition and subtraction of fractions with unlike denominators as equivalent calculations with like denominators. They develop fluency in calculating sums and differences of fractions, and make reasonable estimates of them. Students also use the meaning of fractions, of multiplication and division, and the relationship between multiplication and division to understand and explain why the procedures for multiplying and dividing fractions make sense. (Note: this is limited to the case of dividing unit fractions by whole numbers and whole numbers by unit fractions.)

2. extending division to 2-digit divisors, integrating decimal fractions into the place value system and developing understanding of operations with decimals to hundredths, and developing fluency with whole number and decimal operation – Students develop understanding of why division procedures work based on the meaning of base-ten numerals and properties of operations. They finalize fluency with multi-digit addition, subtraction, multiplication, and division. They apply their understandings of models for decimals, decimal notation, and properties of operations to add and subtract decimals to hundredths. They develop fluency in these computations, and make reasonable estimates of their results. Students use the relationship between decimals and fractions, as well as

the relationship between finite decimals and whole numbers (i.e., a finite decimal multiplied by an appropriate power of 10 is a whole number), to understand and explain why the procedures for multiplying and dividing finite decimals make sense. They compute products and quotients of decimals to hundredths efficiently and accurately.

3. developing understanding of volume – Students recognize volume as an attribute of three-dimensional space. They understand that volume can be quantified by finding the total number of same-size units of volume required to fill the space without gaps or overlaps. They understand that a 1-unit by 1-unit by 1-unit cube is the standard unit for measuring volume. They select appropriate units, strategies, and tools for solving problems that involve estimating and measuring volume. They decompose three-dimensional shapes and find volumes of right rectangular prisms by viewing them as decomposed into layers of arrays of cubes. They measure necessary attributes of shapes in order to solve real world and mathematical problems.

mathematiCal praCtiCes1. make sense of problems and persevere in solving them. 2. reason abstractly and quantitatively. 3. Construct viable arguments and critique the reasoning of others. 4. model with mathematics. 5. use appropriate tools strategically. 6. attend to precision. 7. look for and make use of structure. 8. look for and express regularity in repeated reasoning.

3 wholes are shared equally among 4 people each person has a share of size 3/4. If 9 people want to share a 50-pound sack of rice equally by weight, how many pounds of rice should each person get? Between what two whole numbers does your answer lie?

5. nF.4 Apply and extend previous understandings of multiplication to multiply a fraction or whole number by a fraction.

a. Interpret the product (a/b) × q as a parts of a partition of q into b equal parts; equivalently, as the result of a sequence of operations a × q ÷ b. For example, use a visual fraction model to show (2/3) × 4 = 8/3, and create a story context for this equation. Do the same with (2/3) × (4/5) = 8/15. (In general, (a/b) × (c/d) = ac/bd.)

b. Find the area of a rectangle with fractional side lengths by tiling it with unit squares of the appropriate unit fraction side lengths, and show that the area is the same as would be found by multiplying the side lengths. Multiply fractional side lengths to find areas of rectangles, and represent fraction products as rectangular areas.

5. nF.5: Interpret multiplication as scaling (resizing), by: a. Comparing the size of a product to the size of one factor on

the basis of the size of the other factor, without performing the indicated multiplication.

b. Explaining why multiplying a given number by a fraction greater than 1 results in a product greater than the given number (recognizing multiplication by whole numbers greater than 1 as a familiar case); explaining why multiplying a given number by a fraction less than 1 results in a product smaller than the given number; and relating the principle of fraction equivalence a/b = (n×a)/(n×b) to the effect of multiplying a/b by 1.

5. nF.6 Solve real world problems involving multiplication of fractions and mixed numbers, e.g., by using visual fraction models or equations to represent the problem.

5. nF.7 Apply and extend previous understandings of division to divide unit fractions by whole numbers and whole numbers by unit fractions. (Note: Students able to multiply fractions in general can develop strategies to divide fractions in general, by reasoning about the relationship between multiplication and division. But division of a fraction by a fraction is not a requirement at this grade.)

a. Interpret division of a unit fraction by a non-zero whole number, and compute such quotients. For example, create a story context for (1/3) ÷ 4, and use a visual fraction model to show the quotient. Use the relationship between multiplication and division to explain that (1/3) ÷ 4 = 1/12 because (1/12) × 4 = 1/3.

b. Interpret division of a whole number by a unit fraction, and compute such quotients. For example, create a story context for 4 ÷ (1/5), and use a visual fraction model to show the quotient. Use the relationship between multiplication and division to explain that 4 ÷ (1/5) = 20 because 20 × (1/5) = 4.

c. Solve real world problems involving division of unit fractions by non-zero whole numbers and division of whole numbers by unit fractions, e.g., by using visual fraction models and equations to represent the problem. For example, how much chocolate will each person get if 3 people share 1/2 lb of chocolate equally? How many 1/3-cup servings are in 2 cups of raisins?

measurement and dataConvert like measurement units within a given measurement system. 5. md.1 Convert among different-sized standard measurement units within

a given measurement system (e.g., convert 5 cm to 0.05 m), and use these conversions in solving multi-step, real world problems.

represent and interpret data. 5. md.2 Make a line plot to display a data set of measurements in fractions

of a unit (1/2, 1/4, 1/8). Use operations on fractions for this grade to solve problems involving information presented in line plots. For example, given different measurements of liquid in identical beakers, find the amount of liquid each beaker would contain if the total amount in all the beakers were redistributed equally.

geometric measurement: understand concepts of volume and relate volume to multiplication and to addition.5. md.3 Recognize volume as an attribute of solid figures and understand

concepts of volume measurement. a. A cube with side length 1 unit, called a “unit cube,” is said to have

“one cubic unit” of volume, and can be used to measure volume. b. A solid figure which can be packed without gaps or overlaps

using n unit cubes is said to have a volume of n cubic units.5. md.4 Measure volumes by counting unit cubes, using cubic cm, cubic in,

cubic ft, and improvised units.5. md.5 Relate volume to the operations of multiplication and addition and

solve real world and mathematical problems involving volume. a. Find the volume of a right rectangular prism with whole-number

side lengths by packing it with unit cubes, and show that the volume is the same as would be found by multiplying the edge lengths, equivalently by multiplying the height by the area of the base. Represent threefold whole-number products as volumes, e.g., to represent the associative property of multiplication.

b. Apply the formulas V = l × w × h and V = b × h for rectangular prisms to find volumes of right rectangular prisms with whole-number edge lengths in the context of solving real world and mathematical problems.

c. Recognize volume as additive. Find volumes of solid figures composed of two non-overlapping right rectangular prisms by adding the volumes of the non-overlapping parts, applying this technique to solve real world problems.

geometrygraph points on the coordinate plane to solve real-world and mathematical problems.5. g.1 Use a pair of perpendicular number lines, called axes, to define a

coordinate system, with the intersection of the lines (the origin) arranged to coincide with the 0 on each line and a given point in the plane located by using an ordered pair of numbers, called its coordinates. Understand that the first number indicates how far to travel from the origin in the direction of one axis, and the second number indicates how far to travel in the direction of the second axis, with the convention that the names of the two axes and the coordinates correspond (e.g., x-axis and x-coordinate, y-axis and y-coordinate).

5. g.2 Represent real world and mathematical problems by graphing points in the first quadrant of the coordinate plane, and interpret coordinate values of points in the context of the situation.

Classify two-dimensional figures into categories based on their properties.5. g.3 Understand that attributes belonging to a category of two-dimensional

figures also belong to all subcategories of that category. For example, all rectangles have four right angles and squares are rectangles, so all squares have four right angles.

5. G.4 Classify two-dimensional figures in a hierarchy based on properties.

NC DEPARTMENT OF PUBLIC INSTRUCTION FIFTH GRADE

FIFTH GRADE LESSONS FOR LEARNING

Table of Contents Spin and Race ....................................................................................................................................................... 1

Standard: 5.NF.1 | Additional /Supporting Standard(s): 5.NF.2 Mathematical Practice: 2, 6, 8 Student Outcomes: I can use common denominators to add fractions including mixed numbers. I can compare fractions to determine which is largest.

Building Powers of Ten .................................................................................................................................. 5 Standard: 5.NBT.2 | Additional /Supporting Standard(s): 5.NBT.1, 5.NBT.3, 5.NBT.5 Mathematical Practice: 2, 3, 4, 7 Student Outcomes: I can make models of several powers of ten. I can demonstrate the meaning of exponential form. I can make conjectures about patterns in powers of ten

Decimal Clue Conundrum ............................................................................................................................. 9

Standard: 5.NBT.3 | Additional /Supporting Standard(s): 5.NBT.4 Mathematical Practice: 2, 3, 4, 7 Student Outcomes: I can compose decimals that are in between two numbers. I can use comparison and place value to compose decimals. I can represent decimals in many ways.

Rounding Decimal Number #1 ................................................................................................................. 16

Standard: 5.NBT.4 | Additional /Supporting Standard(s): 5.NBT.3 Mathematical Practice: 1, 2, 3, 5, 7 Student Outcomes: I can reasonably round a decimal to the nearest tenth, hundredth, thousandth, or whole number. I can use models to justify my solutions when rounding decimal numbers to the nearest tenth, hundredth, thousandth, or whole number.

Rounding Decimal Numbers #2 ................................................................................................................ 24

Standard: 5.NBT.4 | Additional /Supporting Standard(s): 5.NBT.3 Mathematical Practice: 1, 2, 3, 5, 7 Student Outcomes: I can reasonably round a decimal to the nearest tenth, hundredth, thousandth, or whole number. I can use models to justify my solutions when rounding decimal numbers to the nearest tenth, hundredth, thousandth, or whole number.

Multiplying Multi-Digit Whole Numbers Using the Standard Algorithm #1: Background .............................................................................. 30

Standard: 5.NBT.5 | Additional /Supporting Standard(s): 5.NBT.2, 4.NBT.5 Mathematical Practice: 2, 4, 7 Student Outcomes: I can use multiple strategies to find the product of multi-digit whole numbers.


Multiplying Multi-Digit Whole Numbers Using the Standard Algorithm #2: Background – Decomposing Numbers .................................................................................................. 34

Standard: 5.NBT.5 | Additional /Supporting Standard(s): 5.NBT.2, 4.NBT.5 Mathematical Practice: 1, 2, 4, 7 Student Outcomes: I can use multiple strategies to find the product of multi-digit whole numbers.

Multiplying Multi-Digit Whole Numbers Using the Standard Algorithm #3: Developing the Standard Algorithm ....................................................................................................... 37

Standard: 5.NBT.5 | Additional /Supporting Standard(s): 5.NBT.2, 4.NBT.5 Mathematical Practice: 1, 2, 4, 5, 6, 7, 8 Student Outcomes: I can use multiple strategies to find the product of multi-digit whole numbers. I can use the standard algorithm to find the product of multi-digit whole numbers.

Multiplying Multi-Digit Whole Numbers Using the Standard Algorithm #4: Estimation ................................................................................. 45

Standard: 5.NBT.5 | Additional /Supporting Standard(s): 5.NBT.2 Mathematical Practice: 3, 2, 4, 7, 8 Student Outcomes: I can use multiple strategies to find the product of multi-digit whole numbers. I can use the standard algorithm to find the product of multi-digit whole numbers.

Division of Whole Numbers: Dealing with Remainders .............................................................. 49Standard: 5.NBT.6 | Additional /Supporting Standard(s): 5.NBT.1, 4.NBT.6 Mathematical Practice: 1, 2, 4, 7 Student Outcomes: I can use multiple strategies to find the quotient of two whole numbers. I can illustrate and explain a division calculation in a variety of ways. I can divide up to four-digit whole numbers by one- and two-digit divisors. I can solve problems involving division, including dealing with remainders.

Multiplying Whole Numbers by Decimals .......................................................................................... 53Standard: 5.NBT.7 | Additional /Supporting Standard(s): 5.NBT.3 Mathematical Practice: 1, 4, 5, 7 Student Outcomes: I can reasonably estimate the product of a whole number and a decimal. I can use multiple strategies to find the product of a whole number and a decimal.

Window Time ..................................................................................................................................................... 57Standard: 5.NBT.7 | Additional /Supporting Standard(s): 5.NBT.5 Mathematical Practice: 1, 2, 5, 6 Student Outcomes: I can use my understanding of decimal operations to solve problems. I can use clear notations to show my solution strategies


Introduction to Dividing with Fractions .............................................................................................. 60Standard: 5.NF.7 | Additional /Supporting Standard(s): 5.NBT.6 Mathematical Practice: 1, 2, 4, 6, 7 Student Outcomes: I can create a visual representation for division. I can relate division with whole numbers to division with fractions. I can justify my reasoning when developing a strategy for division.

Measurement Mania ....................................................................................................................................... 63Standard: 5.MD.1 | Additional /Supporting Standard(s): 5.NBT.1, 5.NBT.2 Mathematical Practice: 6, 7, 8 Student Outcomes: I can convert measures within the same system. I can explain why my answers make sense.

Filling Boxes ........................................................................................................................................................ 69Standard: 5.MD.3 | Additional /Supporting Standard(s): 5.MD.4, 5.MD.5 Mathematical Practice: 1, 4, 5, 6, 8 Student Outcomes: I can develop and implement a strategy for determining volume. I can determine the volume of a figure by using only a small number of cubes. I can make observations about the structure of cubic units.

Candy Boxes ........................................................................................................................................................ 72Standard: 5.MD.4 | Additional /Supporting Standard(s): 5.MD.3, 5.MD.5 Mathematical Practice: 2, 4, 5, 6 Student Outcomes: I can demonstrate the meaning of volume. I can determine the dimensions of a rectangular prism. I can relate the dimensions of a figure to its volume

Exploring Volume as Additive ................................................................................................................... 75

Standard: 5.MD.5 | Additional /Supporting Standard(s): 5.MD.3, 5.MD.4 Mathematical Practice: 2, 4, 5, 7 Student Outcomes: I can use addition and/or multiplication to find volume. I can describe how to find the volume of a right rectangular prism. I can find the volume of a figure made from two right rectangular prisms.

Finding Volume Using a Formula ........................................................................................................... 79Standard: 5.MD.5 | Additional /Supporting Standard(s): 5. MD.3, 5. MD.4 Mathematical Practice: 2, 4, 5, 6, 8 Student Outcomes: I can use addition and/or multiplication to find volume. I can describe how to find the volume of a right rectangular prism. I can use a formula to find the volume of a right rectangular prism.

Representing and Finding Volume .......................................................................................................... 86Standard: 5.MD.5 | Additional /Supporting Standard(s): 5. MD.3, 5. MD.4 Mathematical Practice: 2, 4, 5, 6, 8 Student Outcomes: I can use addition and/or multiplication to find volume. I can describe how to find the volume of a right rectangular prism. I can use a formula to find the volume of a right rectangular prism.


Volume as Additive .......................................................................................................................................... 92

Standard: 5.MD.5 | Additional /Supporting Standard(s): 5.MD.3, 5.MD.4 Mathematical Practice: 1, 4, 5, 7 Student Outcomes: I can use addition and/or multiplication to find volume. I can describe how to find the volume of a right rectangular prism. I can use a formula to find the volume of a right rectangular prism. I can find the volume of a figure made from two right rectangular prisms.

Growing Sumandas ......................................................................................................................................... 96Standard: 5.G.2 | Additional /Supporting Standard(s): 5.G.1, 5.OA.3 Mathematical Practice: 2, 4, 5, 7 Student Outcomes: I can model and extend a pattern. I can represent a pattern using models, drawings, words, tables, and graphs. I can make connections between a pattern and a graph.


Spin and Race

Common Core Standard: Use equivalent fractions as a strategy to add and subtract fractions 5.NF.1 Add and subtract fractions with unlike denominators (including mixed numbers) byreplacing given fractions with equivalent fractions in such a way as to produce an equivalent sumor difference of fractions with like denominators. For example, 2/3 + 5/4 = 8/12 + 15/12 = 23/12.(In general, a/b + c/d = (ad + bc)/bd.)

Additional/Supporting Standards: 5.NF.2 Use equivalent fractions as a strategy to add and subtract fractions

Standards for Mathematical Practice: 2. Reason abstractly and quantitatively6. Attend to precision8. Look for and make use of repeated reasoning

Student Outcomes: • I can use common denominators to add fractions including mixed numbers• I can compare fractions to determine which is largest

Materials: • Spin and Race Fraction Spinners (1 spinner sheet per group)• Paper clips (1 per group)• Spin and Race Recording Sheet (one half sheet per student)

Advance Preparation: • Copy the Spin and Race Fraction Spinners• Gather paper clips• Copy the Spin and Race Recording Sheet• Consider how you will group students• Students should be able to add fractions with unlike denominators, including mixed numbers

Directions: 1. This game provides practice for adding fractions.2. Students should play in pairs or groups of 3.3. To play each student spins both spinners and records the fractions in the chart. To complete

their turn, they record the sum of the two fractions.4. After each player takes five turns, each student should add their five sums to get a Final

Score. Most Final Scores can be represented as mixed numbers.5. The student with the largest Final Score wins the game.

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Questions to Pose: Before:

• Which strategy for adding fractions with unlike denominators do you prefer? Why?

During: • Which strategy are you using? Why did you choose that strategy? • Why do you need to use common denominators when adding fractions? • How did you decide which denominator to use?

After:

• What did you do when your answer was greater than one? Explain your reasoning. • Which denominators did you use frequently? Why do you think these denominators were

more common? Possible Misconceptions/Suggestions: Possible Misconceptions Suggestions Student has trouble finding equivalent fractions and/or common denominators

Students should work with models such as pattern blocks, fraction strips, fraction circles, and number lines to explore equivalent fractions. For example 1 trapezoid ( 1/2 ) is equivalent to 3 triangles ( 3/6 ). Record many examples and help the student observe how multiplication is related to equivalent fractions.

Student adds both the numerator and denominator. For example ⅓ + ⅖ = ⅜

Using a fraction model such as bars, strips, or circles demonstrate the following:

Therefore ⅓ + ⅖ ≠ ⅜ Ask the student to explain how this model proves that adding both the numerators and denominators does not accurately determine the sum.

Special Notes:

• To practice subtracting fractions, students should subtract fractions 1 and 2 from the Recording Sheet. At the end they will still need to add their five differences to receive a Final Score.

• Games should be played multiple times to increase fluency • Discussion with students during and after the game should focus on why it is important to

use common denominators, how common denominators can be quickly determined, and strategies for efficiently adding fractions, including mixed numbers.

Solutions: NA

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Spin and Race Recording Sheet Fraction 1 Fraction 2 Total for this turn

FINAL SCORE: _________

Spin and Race Recording Sheet Fraction 1 Fraction 2 Total for this turn

FINAL SCORE: _________

Directions: On each turn spin both spinners. Then add both fractions. After 5 turns, add all your totals to get your Final Score. The winner is the person with the largest Final Score.

Directions: On each turn spin both spinners. Then add both fractions. After 5 turns, add all your totals to get your Final Score. The winner is the person with the largest Final Score.

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Spin and Race Fraction Spinners

Fraction 1 Fraction 2

To use the spinners, you will need a paper clip and a pencil. Put the paper clip down with one end on the center of the spinner. Put the point of the pencil inside the paper clip at the center. Use your fingers to spin the paper clip.

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Building Powers of Ten Common Core Standard:

Understand the place value system. 5.NBT.2 Explain patterns in the number of zeros of the product when multiplying a number by powers of 10, and explain patterns in the placement of the decimal point when a decimal is multiplied or divided by a power of 10. Use whole-number exponents to denote powers of 10.

Additional/Supporting Standards:

5.NBT.1, 5.NBT.3 Understand the place value system. 5.NBT.5 Perform operations with multi-digit whole numbers and with decimals to hundredths.

Standards for Mathematical Practice:

2. Reason abstractly and quantitatively 3. Construct viable arguments and critique the reasoning of others 4. Model with mathematics 7. Look for and make use of structure

Student Outcomes:

• I can make models of several powers of ten • I can demonstrate the meaning of exponential form • I can make conjectures about patterns in powers of ten

Materials:

• Base-ten blocks • Powers of Ten Sheet (one per student)

Advance Preparation:

• Make copies of the Powers of Ten Sheet • Gather base-ten blocks • Consider how you will pair students • Students should be familiar with base-ten blocks and have a strong understanding of place

value to 100,000 • Students should have experience multiplying with tens • Students should have experience building square arrays such as 32 • Students should be familiar with square numbers

Directions:

1. “How could you represent 10 x 10?” Allow many student suggestions. If no one suggests a base-ten block, ask how these blocks could be used to show 10 x 10

2. Show students the flat base-ten block (sometimes called a “hundreds block”). “What shape is this block?” “Why does it make sense that it is a square?”

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3. Distribute base-ten blocks to each student. Have the students use their blocks to prove that 10 x 10 = 100 and that this array forms a square. Students make need to share materials and work together.

4. On the board write the following: 10 x 10 = 102

Tell students that 102 is read “ten squared.” Ask the class to explain how it might have gotten that name (the array for 10 x 10 is a square).

5. Distribute the Powers of Ten chart to each student. Point out that they have just worked on the first row.

6. Ask students to use their blocks to build 10 x 10 x 10. Have students share their strategies. Did they recognize that this could be rewritten as 10 x 100 and could be built as ten hundreds blocks stacked into a cube?

7. On the board write the following: 10 x 10 x 10 = 103 Tell the class that 103 is read “ten cubed.” Ask the class to explain how it might have gotten that name (the base-ten blocks form a cube of 10 flats).

8. Continue the chart by working with 101. Allow students to debate with one another how this

might be written and built. Help the students make connections between the number of tens being multiplied and the number used in the exponent. 10 x 10 multiplies two tens, so the exponent is a 2 10 x 10 x 10 multiplies three tens so the exponent is a 3 101 has an exponent of one, so it must have only one 10

9. Explain to the students that 101 tells us that we are working with one ten. On the board write the following: 10 = 101

Students can use a tens block (rod) to show 101

10. Challenge students to try the next row in their chart with a neighbor. Allow the students ample time to struggle with the work. As you circulate, listen to see if students are making connections to the exponents that they have already explored.

11. When most have finished invite students to share their thinking about 104. The students are likely to have a variety of answers. Engage a lively debate by asking students to justify their thinking and explain their reasoning. Students may question one another. They may use blocks, observations, patterns, and logic to prove their answer.

12. At a reasonable stopping point, help students see that 104 is written as 10 x 10 x 10 x 10 This might be written in words as “ten to the fourth power.” A picture is not necessary here. Write on the board: 10 x 10 x 10 x 10 = 104

13. For the final row of the chart, have students work with their partner. Then engage the class in another discussion of reasoning.

14. To summarize, ask students to complete the “I Discovered That…” section of the recording sheet. Ask students, “What pattern do you notice?” Students should see that each the number of zeros is changing. Ask students to explain why this pattern makes sense.

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Questions to Pose: As Students Work

• How did you figure out that…? • How could you represent that with blocks? • How could you represent that with multiplication?

In Class Discussion

• Why might some people think that 104 = 40? Where is the error in their reasoning? • What patterns do you notice in the standard notations? • Why does this pattern make sense?

Possible Misconceptions/Suggestions: Possible Misconceptions Suggestions Students think of the exponents as factors. For example: 103 = 10 x 3 101 = 10 x 1

• Students should build square arrays and record them as exponents. For example:

• Return to 10 x 10 = 100. Use the flat base-ten block to show that 10 x 10 creates a square array. Explain that mathematicians shortened 10 x 10 to 102. Observe that 102 is not the same as 10 x 2, but that it represents 10 x 10. Make note of the location and size of the exponent. Connect these observations with other exponents.

Special Notes:

• Before this lesson, students should conduct an investigation of square numbers using arrays. See the suggestions in above table.

• A follow up lesson should support students as they apply their understanding of powers of ten to create scientific notations such as 4 x 103 = 4,000

Solutions: Note – Students may also use the phrase “ten to the power of ____”

Multiplication Expression Words and/or Pictures Exponential

Notation Standard Notation

10 x 10 x 10 ten cubed 103 1,000

10 101 10

10 x 10 x 10 x 10 ten to the fourth power 104 10,000

10 x 10 x 10 x 10 x 10 ten to the fifth power 105 100,000

3 x 3 = 32

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Building Powers of 10 Multiplication

Expression Words and/or Pictures Exponential Notation

Standard Notation

10 x 10

102 100

10 x10 x 10

101

104

105

I Discovered That:

101 = _______

102 = _______

103 = _______

104 = _______

105 = _______

ten squared

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Decimal Clue Conundrum Common Core Standard:

Understand the place value system. 5.NBT.3 Read, write, and compare decimals to thousandths.

• Read and write decimals to thousandths using base-ten numerals, number names, and expanded form, e.g., 347.392 = 3 × 100 + 4 × 10 + 7 × 1 + 3 × (1/10) + 9 × (1/100) + 2 × (1/1000).

• Compare two decimals to thousandths based on meanings of the digits in each place, using >, =, and < symbols to record the results of comparisons.

Additional/Supporting Standard: 5.NBT.4 Understand the place value system


2. Reason abstractly and quantitatively 3. Construct viable arguments and critique the reasoning of others 4. Model with mathematics 7. Look for and make use of structure

Student Outcomes:

• I can compose decimals that are in between two numbers • I can use comparison and place value to compose decimals • I can represent decimals in many ways

Materials:

• Number cards 0-9 (one set per student) • Decimal point and direction cards (one of each per student) • Decimal Clue cards (one set per group)


• Copy and cut number, direction, and decimal cards. • Consider how you will group students • Students should have ample experience representing decimals in many ways • Students should have some experience comparing decimals

Directions:

1. Students work in groups of 3. 2. Each student needs number cards 0-9, a decimal point card, and a directions card. 3. Each group of students needs a stack of Decimal Clue cards to share. 4. Groups should set the Decimal Clue cards face down in the center of their group. 5. Each student should take one Decimal Clue card. The student should use their number cards

to Decimal Clue that according to the clue on the card. For example: “Build a number between 0.1 and 0.2”

1 0 3

9


6. When everyone in the group has built their decimal, they should take turns sharing their Decimal Clue card and explaining how their decimal fits the clue on the card. Group members should discuss whether they agree or disagree.

7. Now the students should put their 3 clues together and try to create one decimal that fits all 3 clues. 8. Once they find a decimal that fits the clues, they should write their decimal on a record sheet

or white board using 3 additional representations. Possible representations include: as a fraction, in words, as a picture, on a number line, or in expanded form.

9. When the group has finished, the students should raise their hands to indicate that they are ready to share their work with the teacher. As you examine student work, pose the questions listed below.

10. When the teacher is finished examining the student work, the group may draw 3 new cards and repeat the process: 1. Build a number that fits your clue 2. Share with your group 3. As a group build a number that fits all 3 clues 4. Represent the decimal in 3 additional ways (If it is impossible to build a number that meets all 3 clues, explain why.)

Questions to Pose: • How did you figure out that this decimal fits the clue? • How can you prove that this decimal makes sense? • How could your answer be different if this clue was removed? • Create a clue you could add to the set that would make the decimal impossible.

Possible Misconceptions/Suggestions: Possible Misconceptions Suggestions Student has difficulty comparing decimals Student should have ample experience building

decimals with blocks, shading on decimal grids, and using number lines. Use one of these models to represent two decimals. Now the student can make a visual comparison.

Student has difficulty representing the decimal in 3 ways

Make a list of possible representations on the board for the students to refer to during the task Suggest that the student focus on three models: shade on a grid, write in expanded form, and place on a number line. These representations are especially helpful for strengthening student understanding of decimals.

Special Notes:

• This task can be adapted to include cards with scientific notation • During this task, students will work more cooperatively if they are grouped with others of

similar ability. Solutions: NA

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5

0

6

1

7

2

8

3

9

4

11


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Directions Cards

D

ecimal C

lue Conundrum

D

irections

1. Build a num

ber that fits your clue

2. Share with your group

3. A

s a group build a number that

fits all 3 clues

4. Represent the decim

al 3 other ways

If it is im

possible to build a number

that meets all 3 clues, explain w

hy.

D

ecimal C

lue Conundrum

D

irections

1. B

uild a number that fits your clue

2. Share w

ith your group

3. As a group build a num

ber that fits all 3 clues

4. R

epresent the decimal 3 other w

ays

If it is impossible to build a num

ber that m

eets all 3 clues, explain why.

D

ecimal C

lue Conundrum

D

irections

1. Build a num

ber that fits your clue

2. Share with your group

3. A

s a group build a number that

fits all 3 clues

4. Represent the decim

al 3 other ways

If it is im

possible to build a number

that meets all 3 clues, explain w

hy.

D

ecimal C

lue Conundrum

D

irections

1. B

uild a number that fits your clue

2. Share w

ith your group

3. As a group build a num

ber that fits all 3 clues

4. R

epresent the decimal 3 other w

ays

If it is impossible to build a num

ber that m

eets all 3 clues, explain why.

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Decimal Clue Cards

Build a number between zero and one-half

Build a number between 0.2 and 0.4

Build a number with 5 in the tenths place

Build a number with 3 in the hundredths place

Build a number greater than the value of two dimes

Build a number less than the value of one nickel

Build a number less than the value of a quarter

and two dimes

Build a number greater than the value of three nickels

and one penny

Build a number between the value of a quarter

and two dimes

Build a number between the value of three nickels

and one dime

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Build a number greater than 0.75

Build a number where the digit in the ones

place is half the size of the digit in the tenths place.

Build a number with an 8 in the thousandths place

Build a number that is between one and two

Build a number in which the digit in the tenths place

is larger than the digit in the thousandths place

Build a number where the digit in the hundredths

place is three times the digit in the ones place

Build a number with an odd digit in

the hundredths place

Build a number with a digit in the tenths place

that is less than 5

Build a number less than two where the sum

of the digits is 9

Build a number with an even digit in the thousandths place

15


Rounding Decimal Numbers #1 Common Core Standard:

Understand the place value system. 5.NBT.4 Use place value understanding to round decimals to any place.

Additional/Supporting Standard: Understand the place value system. 5.NBT.3 Read, write, and compare decimals to thousandths. a. Read and write decimals to thousandths using base-ten numerals, number names, and expanded

form, e.g., 347.392 = 3 × 100 + 4 × 10 + 7 × 1 + 3 × (1/10) + 9 × (1/100) + 2 × (1/1000). b. Compare two decimals to thousandths based on meanings of the digits in each place, using >, =,

and < symbols to record the results of comparisons. Standards for Mathematical Practice:

1. Make sense of problems and persevere in solving them. 2. Reason abstractly and quantitatively. 3. Construct viable arguments and critique the reasoning of others. 5. Use appropriate tools strategically. 7. Look for and make use of structure.

Student Outcomes:

• I can reasonably round a decimal to the nearest tenth, hundredth, thousandth, or whole number. • I can use models to justify my solutions when rounding decimal numbers to the nearest tenth,

hundredth, thousandth, or whole number. Materials:

• Decimal Grids Sheet (one for each student), both 10x10 grids and 100x10 grids • base-10 blocks, meter sticks, (optional: empty number lines that can be used to show decimals) • Paper or journals for student responses


• Copy Decimal Grids Sheet. • Gather base-10 blocks and meter sticks. • Gather paper or student journals. • Consider how you will to pair or group students. • Students should be familiar with decimal notation and decimal place value. • Students should be familiar with the 10-to-1 relationship between adjacent digits in our

number system (each place has a value that is 10 times the value of the digit to its right, whether to the left or the right of the decimal point).

• Students should be familiar with placing decimals on a number line. • Students should be able to use a decimal grid to model a decimal to the hundredths and

thousandths places. • Students should be familiar with metric measurements as based on powers of 10, and familiar

with the meter stick. • Students should be comfortable struggling with a task and explaining their reasoning.

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Directions: Note: If your students have had a lot of experience comparing and ordering decimals, you may choose to skip steps 1 and 2 below. Do so only if you are comfortable with your students’ understanding of comparing and ordering decimals.

1. Pose the following question and write it on the board or show it on a document camera: Mr. Bowman’s fifth grade class was growing sunflowers, which grow to be quite tall. They measured the heights of the plants weekly. Last week the heights of five of their plants in meters were: 1.25, 1.64, 1.052, 1.064, and 1.6. Which of the plants is tallest? Which is shortest? Put the plant heights in order from shortest to tallest. Give the children a partner to work with. Ask them to make their decisions about the order of the heights and to use a model of their choice to defend their decision.

Note: It will be very important that students have had previous experience using various models to represent decimal numbers: base-10 blocks, 10x10 and 100x10 grids, meter sticks, number lines. Allow plenty of time for students to make their decisions and prepare their models. Monitor their activity. As pairs finish, have them share with another pair who used a different model to compare their solutions and models. As the students work, choose particular solutions and models to share in the follow-up whole group discussion.

2. As students share their solutions and models, allow other students to ask questions to clarify any misunderstandings or differences in solutions. Guide the conversation so that students have the opportunity to learn different strategies from each other.

3. Pose a new question: When Mr. Bowman’s students measured the sunflowers this week, they were asked to record their measurements to the nearest hundredth of a meter. The tallest plant measured 1.827m. How would they record this measurement to the nearest hundredth of a meter? Have pairs of children discuss their ideas, find a solution, and use a model to prove the correctness of their solution, as they did in the previous problem. Note: You are asking the children to solve a problem for which they have not been taught a method for solving (at least not by you!) Allow them to grapple with the problem. The follow-up discussion will provide an opportunity for students to learn methods from each other, and follow-up problems will provide an opportunity for students to try out different methods.

4. When students have completed their work, have a few pairs share their ideas. Be sure these points are emphasized:

• The height will be rounded to either 1.82m or 1.83m. The hundredths place in the given measurement is 2 hundredths; the 7 thousandths means that the height is more than 1.82, but less than 1.83.

• In this instance, 5 thousandths would be half way between 1.82 and 1.83. Since 7 thousandths is more than 5 thousandths, the height will be over half way between 1.82m and 1.83m, so it will be closer to 1.83m.

Be sure that a variety of models are shared. If no students used one or more useful models, ask the students how they might have used a number line or whatever model was not highlighted.

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Examples of models Meter Stick: Using a meter stick, students may see 1.827 meters as 1 meter, 827 millimeters. 827 mm could be shown on the meter stick to be between 82cm (or .82m) and 83cm (or .83m). The extra 7 millimeters would then be closer to 1.83 m than to 1.82 m. Number Line: Students might model 1.827 on a number line as in the figure below, and reason in a similar way to that described for using the meter stick (which is also a type of number line model.)

1.82 1.83 1.827

Continue Discussion:

After several student pairs have shared their ideas/models, ask this question: One of the sunflowers was 1.715 meters high. How would you round that measurement to the nearest hundredth of a meter? Give the students a minute or so to think about this question individually, then have them talk with their partner about what they think. Then have some students share their thinking. The important thing to be sure that students understand after this discussion is that the convention is to always round up when the digit to the right of the place to which the number is to be rounded is 5 or more, but this is not a hard and fast “rule.” In reality, there are occasions when the number should be rounded down when the digit to the right is 5, particularly if there are many numbers to be rounded.

5. Tell the students that they are to pretend to be Mr. Bowman’s students. They have to round the remaining heights to the nearest hundredth. Give them the following list with the directions to 1) decide how to record each measurement, 2) choose 3 of the measures for which they will defend their decision using a model and 3) explain in words why they made that decision.

Sunflower Measures (Round to the nearest hundredth) 1.034, 1.495, 1.13, 1.607, 1.8, 1.503, 1.777, 1.098

Have available blank paper, 100x10 decimal grids, meter sticks, base-10 blocks, and any other materials your students are likely to choose as models. Allow pairs to work together, but each student should complete his/her own recording. Monitor as pairs work. Choose a few examples of models and explanations to share with the whole class. Pay particular attention to solutions and strategies for rounding 1.495 and 1.098.

Notes:

a) Some students will be confused by the 1.8 measurement. To the nearest hundredth, 1.8m would actually be 1.80m. Let them grapple with how to deal with this. Likewise, 1.13m is already named in hundredths, so it should be left as is. Not having to make a change in the number will be confusing to some students. Do not tell them to leave it as is, but ask questions to help them figure this out on their own; e.g., “Why is that confusing?” “If you wrote this number including a thousandths place, what would the thousandths digit be?” (0) “If you rounded the number written with the zero in the thousandths place, what would be the number rounded to the nearest hundredth?”

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b) Rounding 1.495 to the nearest hundredth changes the 9 hundredths to 10 hundredths which is equivalent to one more tenth, so the solution is 1.50 meters. Monitor the students as they deal with this problem and troubleshoot as necessary, but let the students come up with their own solutions and share their ideas with each other. Likewise, 1.098 requires that the 9 hundredths be rounded to 10 hundredths or 1 tenth, so the solution is 1.10 meters. Notice what kind of models the students are using. Which ones are most helpful?

6. Continue to practice this concept with problems that require rounding to the nearest tenth and nearest whole number as well as the nearest hundredth.

Possible problems: a) Mr. Bowman had his students measure a number of items in the classroom in meters

and centimeters, then write the measures as meters, with the centimeters written as hundredths of a meter. They then made a graph showing the measures of the items, but the graph was designed to show only measures rounded to the nearest tenth of a meter. Tony and Juan measured the items in the chart. Write the measures in meters and hundredths of a meter; then round these measures to the nearest tenth of a meter.

Item Measure in Meters and Centimeters

Measure in Meters and Hundredths

Measure to Nearest Tenth of a Meter

Height of door 2m, 23cm

Width of white board 3m, 75cm

Width of classroom 11m, 36cm Height of

student desk 0m, 80cm

Length of counter 6m, 97cm

b) Georgia was growing tomatoes for a gardening club contest. She had a scale that

showed weights to thousandths of a kilogram. She had to report the weights to the nearest hundredth of a kilogram. Below are the weights she recorded for 5 of her largest tomatoes. Help her by rounding them to the nearest hundredth of a kilogram. Tomato A: 0.739kg Tomato B: 0.888kg Tomato C: 1.007kg Tomato D: 1.032kg Tomato E: 0.905kg

c) Bella was making lemonade for the fall carnival. She had ten different sized pitchers to use. They held these amounts in liters: 2.358, 3.74, 5.916, 1.5, 4.44, 8.08, 3.373, 4.682, 9.903, 12.621. She wanted to estimate the total amount of lemonade these pitchers would hold, so she could use your help. Round each amount to the nearest whole number of liters. Then write her a note telling her about how much lemonade she should make to fill the pitchers and how you figured it out.

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Questions to Pose: As students work together:

• What place are you rounding to? • What are some possible answers? • Why did you decide to use this model? • Is there a different model that might help you?

During class discussion:

• How did your model help you? • Did someone else use a model or have a strategy that you think might be more useful than the

one you used? • Was there anything difficult about solving these problems? How did you work through the

difficulty? Possible Misconceptions/Suggestions: Possible Misconceptions Suggestions • Students are not sure what the

concept of rounding means • For these students you may need to provide some

practice rounding whole numbers to various places. • They may need help seeing what two hundredths (or

whatever place is being rounded to) the given number is exactly between. Monitor these students carefully as they use models and be sure they are identifying the location of the given decimal correctly, then identifying the nearest decimals of the place to be rounded to.

• Be sure these students have ample opportunity to hear the explanations from other students about how they have thought about these problems and how they used the models.

• Students are unsure how to use the various tools to model the decimal number

• For a problem involving metric measurements, suggest that students use a meter stick and find the decimal number on the meter stick. You might ask (if rounding to hundredths), “What two hundredths of a meter is this measurement between?” Then “Which of those measures in hundredths is the actual number closest to?”

• If students have difficult modeling decimals on a grid, have them practice building the decimal number with base-10 blocks and then transfer their concrete model to the grid.

• If students have difficult knowing how to use a number line, provide practice drawing number lines (or using prepared number lines without numbers) to show given decimals.

• Students have difficulty rounding up when the digit being rounded to is a 9, for example, rounding 1.397 to the nearest hundredth requires rounding to 10 hundredths or 1 tenth.

• These students may need more work with regrouping decimal numbers from one place to the next, developing the understanding that 10 tenths are 1 whole, 10 hundredths make 1 tenth, and 10 thousandths are equivalent to 1 hundredth.

• Be sure students are using models that more easily show these relationships.

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Special Notes:

• Students should continue to practice rounding decimal numbers during the weeks and months following this and similar lessons to keep their skills sharp. Practice can be in a problem context or quick practice rounding a few numbers without a context.

Solutions for Problems Used in This Lesson: Activity #1: 1.052, 1.064, 1.25, 1.6, 1.64. Models used will vary. Activity #3: 1.83m Activity #4: 1.72m (An argument could be made that 1.71m is also an acceptable answer.) Activity #5: 1.03, 1.50, 1.13, 1.61, 1.8, 1.50, 1.78, 1.10 Models and explanations will differ. Activity #6a:

Item Measure in Meters and Centimeters

Measure in Meters and Hundredths

Measure to Nearest Tenth of a Meter

Height of door 2m, 23cm 2.23m 2.2m Width of white

board

3m, 75cm

3.75m

3.7m or 3.8m Width of classroom

11m, 36cm

11.36m

11.4m Height of student

desk

0m, 80cm

0.80m

0.8m Length of counter 6m, 97cm 6.97m 7.0m

Activity #6b: Tomato A: 0.73kg

Tomato B: 0.89kg Tomato C: 1.01kg Tomato D: 1.03kg Tomato E: 0.91kg

Activity #6c: Original amounts in liters: 2.358, 3.74, 5.916, 1.5, 4.44, 8.08, 3.373, 4.682, 9.903, 12.621 Rounded amounts in liters: 2, 4, 6, 2, 4, 8, 3, 5, 10, 13

Total of rounded amounts: 57 liters Explanations will vary. Look for how students thought about the rounding. Some students may tell her 57 liters. Others may round up to 58 liters since the rounding created approximate amounts rather than exact amounts. Either solutions is fine as long as the students make a reasonable case for their solution.

21


Decimal Grids

22


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Rounding Decimal Numbers #2 Common Core Standard:

Understand the place value system. 5.NBT.4 Use place value understanding to round decimals to any place.

Additional/Supporting Standard: Understand the place value system. 5.NBT.3 Read, write, and compare decimals to thousandths. a. Read and write decimals to thousandths using base-ten numerals, number names, and expanded

form, e.g., 347.392 = 3 × 100 + 4 × 10 + 7 × 1 + 3 × (1/10) + 9 × (1/100) + 2 × (1/1000). b. Compare two decimals to thousandths based on meanings of the digits in each place, using >, =,

and < symbols to record the results of comparisons. Standards for Mathematical Practice:

1. Make sense of problems and persevere in solving them. 2. Reason abstractly and quantitatively. 3. Construct viable arguments and critique the reasoning of others. 5. Use appropriate tools strategically. 7. Look for and make use of structure.

Student Outcomes:

• I can reasonably round a decimal to the nearest tenth, hundredth, thousandth, or whole number. • I can use models to justify my solutions when rounding decimal numbers to the nearest

tenth, hundredth, thousandth, or whole number. Materials:

• Decimal Grids Sheet, both 10x10 grids and 100x10 grids • base-10 blocks, meter sticks, (optional: empty number lines that can be used to show decimals) • Paper or journals for student responses


• Copy Decimal Grids Sheet. • Gather base-10 blocks and meter sticks (optional: empty decimal number lines). • Gather paper or student journals. • Consider how you will to pair or group students. • Students should be familiar with decimal notation and decimal place value . • Students should be familiar with the 10-to-1 relationship between adjacent digits in our

number system (each place has a value that is 10 times the value of the digit to its right, whether to the left or the right of the decimal point).

• Students must be familiar with the concept of rounding a decimal number to the nearest tenth, hundredth, thousandth, or whole number.

• Students should be comfortable struggling with a task and explaining their reasoning.

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Directions: 1. Begin the lesson with a review of rounding decimal numbers. Ask the students to round

45.837 to the nearest hundredth. Have a few students respond and tell how they thought about the rounding. (Hopefully the responses are 45.84 because the 7 thousandths is closer to 45.84 than to 45.83.) Then ask them to round the number (45.837) to the nearest tenth. Again, solicit responses and explanations. (Solution: 45.8, because the 3 hundredths is closer to 45.8 than to 45.9). Finally ask them to round the number to the nearest whole number (46, because 8 tenths is closer to 46 than to 45.)

2. Pose this question to the class: What numbers could be rounded to 4.8? Try to give several answers. Can you find them all? Have the children work with a partner to make a list of numbers. This activity gives the children an opportunity to think about rounding in a different way. It emphasized the significance of place value. The numbers can include or be bigger than 4.75 but must be less than 4.85 (using the convention of rounding up when the digit to the right is 5). Some children may come up with only one number. Encourage them to think of others. Some children may be able to come up with the whole range of possible numbers (but will not be able to list them all because there is an infinite number. Between any two decimal numbers, another decimal number can be identified. For example, between 3.82 and 3.83, you can identify 3.821, 3.822, etc. Since we are working with decimal numbers up to the thousandths place, all of the numbers involving hundredths or thousandths could be identified, but numbers involving place values smaller than thousandths would be too many to count.)

As the students complete the task, have them write an explanation of their process. You might also have them make a visual representation of their solution. Look for evidence that students understand the range of possible numbers. These students show they have a complete understanding of the concept of rounding.

You might choose to have pairs of students who found a lot of possible numbers or figured out the range of numbers to join a pairs of students who found only a few and discuss how they came up with their list of numbers.

3. Hold a whole group discussion of the solutions and strategies used. Be sure that students who found only a few numbers have the opportunity to hear other strategies and solutions and to ask questions. If any pairs used a number line or other model to help them answer the question, give them the opportunity to share their strategy. This will help students who could not “see” the range of numbers for themselves.

4. Following the discussion, ask pairs to choose from the following problems: a. What numbers could be rounded to 8.5? b. What numbers could be rounded to 16.61? c. What numbers could be rounded to 26.09 d. What numbers could be rounded to 37?

Make as complete a list as you can. Try to find the range of possible numbers. Students who had a hard time finding several solutions to the previous problem may want to try problem a. If students find a reasonable list of solutions for the problem they choose, and particularly if they identify the range of numbers, ask them to try a second question.

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As the students work, choose a few examples of solutions and strategies that you want to have shared in the whole class discussion.

5. When all pairs have completed at least one problem, hold a class discussion. Have the pairs you selected share their solutions and the strategies they used to find those solutions. Be sure that all children have the opportunity to ask questions to clarify their own thinking.

Extension: This problem requires similar thinking to those posed in the lesson, but adds practice adding and/or subtracting decimal numbers in a more challenging problem. You may use this kind or problem with all students or with students who readily understood the concept of a range of numbers that could be rounded to a given number while others work with more problems like those in the lesson. Pose this question: Two numbers, each having four digits, are added and the result is rounded off to 3.8. What might those numbers be? Note: The two numbers being added must total in the range 3.750 to 3.849 in order to be rounded off to 3.8. As students work, observe to see if any children think about this relationship before trying to find the two numbers? Questions to Pose: As students work together:

• What is the smallest number that would round to [the given number]? • What is the largest number that would round to [the given number]? • How does this model help you? • Tell me about what you are thinking.


• How did your model help you? • How did you decide on the range of numbers you have chosen? • How did knowing how to round a given number help you decide what numbers might be

rounded to this number [the number identified in the problem]? • Was there anything difficult about solving these problems? How did you work through the difficulty?

Possible Misconceptions/Suggestions: Possible Misconceptions Suggestions • Students are unsure

about how to start thinking about this kind of problem, which is requiring thinking that is opposite to thinking about rounding a given number

• Ask questions like: “What is a decimal number smaller than [the given number] that would round to [the given number]? Given the number 5.7, ask questions like: “Would 5.71 round to 5.7? Would 5.60 round to 5.7? Would 5.79 round to 5.7?”

• Suggest a model that the student(s) have found helpful in rounding decimals. Have them find the rounded number in that model; then consider what numbers smaller than the rounded number would round to it and what numbers larger than the rounded number would round to it. (Be careful not to lead their thinking, but guide them by asking questions as in the above bullet.)

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• Allow unsure students to work with more sure students who will share their thinking about this kind of problem.

• In class discussions, be sure these students have ample opportunity to hear the explanations from other students about how they have thought about these problems and how they used the models.

Special Notes:

• This kind of problem is more difficult than the process of rounding a number to the nearest given place. You may choose to do this kind of problem with those students who have mastered the skill of rounding and understand its processes, while you continue practicing problems requiring rounding decimal numbers with those students who do not yet have a strong understanding of rounding.

• Students should continue to practice rounding decimal numbers during the weeks and months following this and similar lessons to keep their skills sharp. Practice can be in a problem context or quick practice rounding a few numbers without a context. Problems like the ones in this lesson can be used with all or some students for this practice.

Solutions for Problems Used in This Lesson: Activity #4:

a. … rounded to 8.5? any numbers from 8.45 up to (but not including) 8.55 b. … rounded to 16.61? any numbers from 16.605 up to (but not including) 16.615 c. … rounded to 26.09 any numbers from 26.085 up to (but not including) 26.095 d. … rounded to 37? any numbers from 36.5 up to (but not including) 37.5 Make as complete a list as you can. Try to find the range of possible numbers. For problem a, students might list the range as 8.45 – 8.54 or 8.450 – 8.549. Since we are dealing with decimal numbers only to the thousandths, you may accept this, but be sure to indicate that there are actually more numbers smaller than 8.55 such as 8.5499 and 8.54999. Similar answers could be given for the remaining problems.

27


Decimal Grids

28


29


Multiplying Multi-Digit Whole Numbers Using the Standard Algorithm #1: Background Common Core Standard:

Perform operations with multi-digit whole numbers and with decimals to hundredths. 5.NBT.5 Fluently multiply multi-digit whole numbers using the standard algorithm.

Additional/Supporting Standard:

Understand the place value system. 5.NBT.2. Explain patterns in the number of zeros of the product when multiplying a number by powers of 10, …. 4.NBT.5 Multiply a whole number of up to four digits by a one-digit whole number, and multiply two two-digit numbers, using strategies based on place value and the properties of operations. Illustrate and explain the calculation by using equations, rectangular arrays, and/or area models.


2. Reason abstractly and quantitatively. 4. Model with mathematics. 7. Look for and make use of structure.

Student Outcomes: • I can use multiple strategies to find the product of multi-digit whole numbers.

Materials:

• Paper or journals for student responses • Base-10 blocks • Base-10 cutouts (optional) • Grid paper


• Gather paper or student journals; have available base-10 blocks and grid paper. • Consider how you will pair or group students. • Students should be familiar with models and non-standard methods of multiplying whole

numbers. • Students should have some fluency with multiplying whole numbers using methods other

than the standard algorithm, particularly the array and area models. • Students should have some understanding of the concept of area and how to calculate the

area of a rectangular region. • Students should be comfortable struggling with a task and explaining their reasoning.

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Directions: Part I: Note: Assess your students’ proficiency with solving equations involving multiples of 10 and 100. Begin with equations like 10 x 12 and 100 x 34; followed by equations like 40 x 30; 200 x 8; 90 x 300. Be sure to ask the students to explain how they get their answers. Some (perhaps many) students will say something like, “to multiply by 10, just add a zero on the end.” Pay particular attention to these students. This shortcut rule works for whole number equations, but will not work when they begin to solve equations like 3.5 x 10. It is important that students understand the process involved in multiplying with multiples of 10 and 100. The following lesson will help students see how and why their shortcut works for whole numbers. If your students do not need this review, you may skip this part.

1. Pose the following problem: A thirsty camel drinks about 20 gallons of water. How many gallons will 80 thirsty camels drink? Ask students how they might model this problem. There may be a variety of ideas. Since the numbers are larger, making 80 groups of 20 counters or a picture of 80 groups with 20 dots or x’s in each would not be efficient. If anyone suggests an array model, say that an array can be helpful, but you do not want to draw 20 rows of 80 squares. Tell them that you can use what we know about using 10’s to draw the array in a more efficient way. Then draw this model on the board or using a document camera. Tell the students that each row will represent 10 gallons of water and that each section of the row represents 10 camels. Label the array like this: 10 10 Ask the students how many gallons are represented by one of the small sections of the array. They should see that it is 10 gallons drunk by 10 camels, or 0 x 10 = 100 gallons. Ask how many small sections there are in the array. (16) Ask how they know. Hopefully someone will suggest that there are 2 rows with 8 sections in each; 2 x 8 = 16 small sections. Begin to record this information like this:

20 × 80 (the original problem)

(2 × 10) × (8 × 10) (the original numbers decomposed)

(2 × 8) × (10 × 10) (2x8 = number of small sections; 10 x 10 = number of gallons in each small section)

Ask the students what these numbers represent. Then do the multiplication. Record:

16 × 100 Then multiply for the solution of:

1600 So the 80 camels would drink 1600 gallons in one day.

10 10 10 10 10 10 10 10

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This kind of representation is very important so that children see that shortcuts work for a reason.

If you have students who are not skilled with multiplying with powers of 10, be sure to give them plenty of practice using models such as this one so they can develop this skill that is so foundational to understanding and accurately solving multiplication equations with non-zero numbers with 2 or more digits.

2. Continue the practice with problems involving multiples of 10 and 100 such as the following: a) The Girl Scout troop received their delivery of cookies. They had 400 cartons with 20

boxes of cookies in each. How many boxes of cookies did they have to sell? b) A full grown elephant eats 40 pounds of food each day. How many pounds of food

would be needed to feed the elephant for 30 days? c) Thirty squirrels each stored 200 acorns in the forest. How many acorns were stored in all?

Part II: Note: Students will have been exposed to a variety of methods and models for multiplication in fourth grade. This activity will help you see what kinds of methods the students choose to use.

3. Pose this problem:

The circus came to town with 27 clowns. In the parade, each clown carried 16 balloons. How many balloons were there altogether? Tell the students to solve the problem in any way that makes sense to them. Monitor as students work to see what kinds of strategies they use. Are any looking for smaller products such as 4 x 27, then adding that result 4 times? Are any drawing some kind of array? How are they labeling it? Did anyone do 30 x 16, then subtract 3 x 16? Did any find partial products and add them: 10 x 20 = 200, 6 x 20 = 120, 10 x 7 = 70, and 6 x 7 = 42? Is anyone adding up a string of 27’s or 16’s? Did anyone use the standard algorithm? (If so, do not highlight it as “the right way” to solve this problem. You may just include it as another strategy at this point.)

4. Choose a variety of methods to share. Consider how you want to order the sharing of these methods. Have these selected students share their methods for solving the problem. Discuss with the class how each represents the situation of 27 groups of 16 balloons. Ask the students to look for similarities and differences in the different methods. Discuss the efficiency of the different methods.

5. To close the lesson write the following journal prompt on the board. Before students write in their journal, ask them to briefly talk with a partner about the two strategies they plan to describe. Describe two strategies you can use when multiplying two whole numbers. Tell which you prefer and why.


• What equation do you need to solve this problem? • What did you think about first? • Which method works best for you? • Why did you decide to…..? • Tell me why you think that is a reasonable answer.

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During class discussion: • How are these two methods alike? …different? • Which strategy worked best for you? Why do you think that is?

Possible Misconceptions/Suggestions: Possible Misconceptions Suggestions • Students have difficulty

calculating accurately due to slow or inaccurate recall of the multiplication facts.

• Allow the students to use a multiplication facts chart to insure accuracy of solutions as they work with these problems. Work individually or in a small group with these students to develop strategies for remembering the facts. Show them how to use known facts to figure out others; e.g., 2 x 8 = 16, so 4 x 8 would be double 16 or 32.

Special Notes:

• This lesson should precede lessons that are geared to developing any standard multiplication algorithm.

• It is extremely important that students not be exposed to the standard multiplication algorithm until they have had numerous opportunities to explore a variety of strategies, models, and methods first. The area/array model is particularly advantageous in meaningfully developing the multiplication algorithm.

Solutions for Problems Used in This Lesson:

Note that some solutions are within the text. Part I. #2

a. 400 x 200 = 8,000 b. 40 x 30 = 1,200 c. 200 x 30 = 6,000 a. 22,660 b. 403,541 c. 1,743 d. 809,900 e. 2,401

Part II. #3 432 balloons

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Multiplying Multi-Digit Whole Numbers Using the Standard Algorithm #2:

Background – Decomposing Numbers Common Core Standard:





1. Make sense of problems and persevere in solving them. 2. Reason abstractly and quantitatively. 4. Model with mathematics. 7. Look for and make use of structure.

Student Outcomes:

• I can use multiple strategies to find the product of multi-digit whole numbers. Materials:

• Paper or journals for student responses • Base-10 blocks • Grid paper for students; large grid paper for demonstration (optional)


• Gather paper or student journals; have available base-10 blocks and grid paper. • On grid paper (large or small depending on how you will display it to students), draw a 28 x

14 rectangle. • Consider how you will pair or group students. • Students should have some understanding of the concept of area and how to calculate the

area of a rectangular region. • Students should be comfortable struggling with a task and explaining their reasoning.

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Directions: Part I: If your students have shown difficulty in decomposing the numbers being multiplied, the following activity may be particularly helpful. You may choose to use it with a small group or the whole class depending on the need.

1. Tell the students that you want them to explore a model of the problem, 28 x 14 with you. Use either large grid paper that can be posted on the board or grid paper that can be shown on the document camera (or with a small group gathered around a table, small grid paper will suffice.) On the grid paper, draw four or more 28 x 14 rectangles. Tell the students that you want them to help you find different ways to “slice” the array into pieces, particularly ways that would make multiplying the numbers easier. Give one example: Show how the array might be cut into a piece 28 x 10 and one 28 x 4. Draw a line in your array to show how you “sliced” it. 28 (note: gridlines are not shown) 14 “slice” Record your division as (28 x 10) + (28 x 4) or (10 x 28) + (4 x 28). Be sure the students see that you are decomposing or breaking up the 14 into 10 and 4 and multiplying both parts by 28, which is using the distributive property. Ask the students to suggest other “slices” (horizontal or vertical) into which the rectangle might be cut and demonstrate them on the rectangles you have drawn. Possibilities include:

1) Slice the 28 into 20 + 8, record the division as (20 x 14) + (8 x 14); 2) Slice the 28 into 10 + 10 + 8, record the division as (10 x 14) + (10 x 14) + (8 x 14); 3) Slice the 14 into 9 + 5, record the division as (9 x 28) + (5 x 28); 4) Someone may suggest slicing the array both horizontally and vertically; for example: Slice

the 28 into 20 + 8 and the 14 into 10 + 4, record as (20 x 10) + (20 x 4) + (8 x 10) + (8 x 4). Many other divisions are possible. Have only 3-4 people share their ideas at this point if you are working with the whole class.

2. Give groups of four base-10 blocks for building an array and grid paper to use individually or with a partner to draw an array. Tell the students that you want them to look for as many different ways to “slice” the array for 37 x 25 into two or more pieces. Students may use either model to create the array, but ask them to show and record the different ways they find on the grid paper, both by “slicing” the rectangle on the grid and by recording the numerical expression for the divisions. Some students may need the concrete model of the Base-10 blocks before transferring the array to the grid paper. Tell the students to look particularly for divisions that would help make the multiplication easier. Monitor as the students work. Note any students who need to build with the blocks first. Watch for students who have trouble finding more than one or two divisions. Which students are finding efficient “slices” and which may be making random divisions which would not be particularly helpful when multiplying the original problem? Are any students “slicing” both horizontally and vertically? As you monitor the students, make some choices about which methods you want to have students share and in what order you want to share them. (If working with a small group, all students may be able to share, but carefully choose the order of sharing.)

10 10 x 28 4 4 x 28

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3. When all students have been able to find a few ways to slice the rectangle, bring the class back together. Have the selected students share their divisions. Discuss which “slices” are the most helpful in finding a solution to the problem. Hopefully, students will see that divisions using multiples of 10 are the most helpful. Ask: Does the total amount change when it is sliced into different parts? Be sure that students see that the total is not changed by slicing the rectangle into different sized pieces.


• How did you decide how to “slice” your array? • Which divisions would make the problem easier? • Why did you decide to…..?


• Which “slices” make the problem easier to solve? Why? • Do the different ways to slice the rectangle change the total amount of squares in

the rectangle? Possible Misconceptions/Suggestions: Possible Misconceptions Suggestions • Students have difficulty

finding a way to “slice” the rectangle

• Ask the students for one way to separate one of the factors into two parts. Remind them of how you separated 28 into 20 and 8. Ask them how they could separate the other factor into two parts.

Special Notes:

• If your students need this work with decomposing numbers in order to multiply, this lesson should precede lessons that are geared to developing any standard multiplication algorithm.

• It is extremely important that students not be exposed to the standard multiplication algorithm until they have had numerous opportunities to explore a variety of strategies, models, and methods first.

Adapted from Van de Walle, et.al., Teaching Student-Centered Mathematics, Grades 3-5, 2nd Edition. Pearson, 2014.

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Multiplying Multi-Digit Whole Numbers Using the Standard Algorithm #3:

Developing the Standard Algorithm Common Core Standard:





1. Make sense of problems and persevere in solving them. 2. Reason abstractly and quantitatively. 4. Model with mathematics. 5. Use appropriate tools strategically. 6. Attend to precision. 7. Look for and make use of structure. 8. Look for and express regularity in repeated reasoning.

Student Outcomes:

• I can use multiple strategies to find the product of multi-digit whole numbers. • I can use the standard algorithm to find the product of multi-digit whole numbers.

Materials:

• Paper or journals for student responses • Base-10 blocks • Base-10 cutouts • Grid paper


• Gather paper or student journals; have available base-10 blocks and grid paper. • Consider how you will pair or group students. • Students should be familiar with models and non-standard methods of multiplying

whole numbers. • Students should have some fluency with multiplying whole numbers using methods other

than the standard algorithm, particularly the array and area models.

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• Students should have some understanding of the concept of area and how to calculate the area of a rectangular region.

• Students should be familiar with the vocabulary of multiplication: factor, product, partial product.

• Students should be familiar with estimating. • Students should be comfortable struggling with a task and explaining their reasoning.

Directions: Part I. In this part students explore the array models which they probably encountered in fourth grade. The array model is very helpful in developing meaningful recordings of algorithms without the pictorial representation.

1. Tell the students that you want to explore the array model for multiplication as a class. The students should be familiar with this model from their work in fourth grade. If you have noticed students using the array model in previous lessons when given their choice of methods, you may have one of these students help you begin this lesson. Pose this problem: What is the area in square centimeters of a desktop that is 79 cm by 45 cm? If possible, ask a student who used the array method to draw and label an array for this problem. Draw it yourself if no student used this method.

70 9

40 40 × 70 40 × 9 5 5 × 70 5 × 9

Ask the students how this is similar to the “slicing” they did in multiplication lesson #2, if you chose to use it. Hopefully they see how you are decomposing both numbers into tens and ones. Ask the students what to do next. (Find the partial products and then add them.) Ask the students to find the partial products and check their answers with a friend before adding them. Their results should be some order of these:

Draw the array again, but label it differently, with the 40 + 5 on top and the 70 + 9 on the side. Ask the students to help you label the inside of each section. It should look something like this:

40 5

70 70 × 40 70 × 5 9 9 × 40 9 × 5

Ask the students what they notice. Hopefully they notice that the partial products are the same but in a different order.

40 × 70 = 2800 40 × 9 = 360 5 × 70 = 350 5 × 9 = 45 3555

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2. Do a few more examples with the class involving 1-digit by 2- or 3- or 4-digit numbers and 2-digit by 3-digit numbers as well as 2-digit by 2-digit numbers. Examples: You need to cut a rope into 6 pieces for sailing cord. Each cord must be 538 meters long. How many meters of rope do you need?

500 30 8 6 6 x 500 6 x 30 6 x 8

Solution: 3,228 meters The Sunny Vale Orchard had 79 rows of peach trees with 346 trees in each row. How many peach trees were in the Sunny Vale Orchard?

300 40 6 70 70 x 300 70 x 40 70 x 6 9 9 x 300 9 x 70 9 x 6 Solution: 27,334 trees

Have your students try this 3-digit by 3-digit situation using the array. The Grow ‘Em Right Nursery specialized in azalea bushes. They had a field with 342 rows with 657 azalea bushes in each row. How many azalea bushes did they have to sell? 600 50 7 300 300 x 600 300 x 50 300 x 7 40 40 x 600 40 x 50 40 x 7 2 2 x 600 2 x 50 2 x 7 Solution: 224,964 azalea bushes

3. Provide as much practice with the array as your students need. Here are some possible problems that would provide more practice. a. George and Sahil finished working a giant jigsaw puzzle. They saw that the pieces fit into

rows of 37 pieces and that there were 28 rows in all. How many pieces did the puzzle have? b. Mara and Jessi were covering the bulletin board by the school office with different colored

papers that were each one square foot in area. They wondered how many they would need. If the bulletin board is 5 feet high and 17 feet long, how many sheets of the paper will they need?

c. Mr. Parker grows corn on his farm. His silo can hold 2,500 bushels of corn. He hauled 39 truckloads to the farmyard. Each load held 63 bushels. Will his silo hold that much corn?

d. The ship Great Eastern laid the first telegraph cable across the Atlantic Ocean in 1866, about a century and a half ago. If the ship laid 3,246 meters of cable each day, how many meters of cable did it lay in a week?

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For the following problems, ask the students to provide the story problem context for the problem. Students often enjoy creating the problem, and it gives you an opportunity to be sure they know the meaning of division – how it is used in a real life situation. e. 37 x 48 f. 26 x 374 g. 87 x 63 h. 169 x 52

Part II. In this part, students build on the array model and the partial products it models to solve multiplication equations without the pictorial representation the array provides.

4. Write the following equation on the board: 26 x 47 = Ask a student to provide the story to go with the equation or provide a context yourself. Then ask the students for their ideas about how to find the partial products without drawing the array as they were doing in the previous activities. Accept any possibilities that work. Hopefully the work they have done with the array will lead them to see that they can decompose 26 into 20 and 6, and decompose 47 into 40 and 7. The partial products should then be the results of 20 x 40, 20 x 7, 6 x 40, and 6 x 7. The order of the partial products is not important in this instance. Help the students see that each part of the 26 (20 and 6) must be multiplied by each part of the 47 (40 and 7), so that the partial products are: 20 x 40 = 800 20 x 7 = 140 6 x 40 = 240 6 x 7 = 42 for a product of: 1222 It may help some students to write the problem in expanded form, like this: 47 = 40 + 7 x 26 = x 20 + 6 This recording may help them more easily see the decomposed parts of the number that would have been represented in the array model and which must be multiplied whether using the array or not. Do a few more examples with the class, including some 1- by 2-digit equations and 1- by 3-digit equations as well as 2- by 2-digit equations. Possible examples (answers in parentheses): 86 x 7 264 x 9 38 x 47 567 x 3 92 x 45 (602) (2376) (1786) (1701) (4140) Partial products for 264 x 9 would be the following, possibly in a different order: 9 x 200 = 1800 9 x 60 = 540 9 x 4 = 36

for a product of: 2376

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After students have had practice finding the partial products without using the array, pose this equation and have the students help you find the partial products.

85 x 43 = As they give you the partial products, be sure they tell you which parts of the two factors they are

multiplying to get the partial products. Record the partial products in the order in which the students give you the partial products, and have them add to find the product. Then rewrite the problem like this:

85 x 43 15 240 200 3200 3655 Ask the students how this recording is different (if it is) from the order they gave you. In this

recording of partial products, you have started multiplying by the 3 ones (3 x 5 and 3 x 80) before multiplying by the 4 tens (40 x 5 and 40 x 80). Record the factors for each partial product. (See below.) Beside this recording of the partial products, write the shorter way of recording the solution (that we know as the standard algorithm).

85 85 x 43 x 43 15 (3 x 5) 255 240 (3 x 80) 3400 200 (40 x 5) 3655 3200 (40 x 80) 3655

Ask the students to consider the two recordings and figure out where the two partial products (255 and 3400) came from. Hopefully they will see that the 255 is 15 + 240 and is the product of 3 x 85, and that 3400 is 200 + 3200 and is the product of 40 x 85. You might draw the arrows to show these equivalences after the students see the connection. Then ask what the 1 and the 3 above the 8 in 85 stand for. If no students see that these are the regrouped 1 ten from 15 (3 x 5) and the regrouped 2 hundreds from 200 (40 x 5), show them yourself how you did this by going through this problem again or doing another example. This is much like regrouping when adding, which works here because one meaning of multiplication is repeated addition recorded more efficiently. Do several more examples with the class. Initially, you may have students solve a few problems by first writing out the partial products, then solving using the shorter standard form. Have volunteers come to the board to show their solutions. Monitor as students work to see which students are having difficulty. Students may work in pairs if that is helpful.

2 1

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Do at least one example with a zero, so that students have one partial product of zero. For example:

507 X 3 21 (3 x 7) 0 (3 x 0) 1500 (3 x 500) 1521

Any students who understand that you do not need to write the zero since it does not add anything to the final product should be allowed to leave out the zero when listing the partial products.

Problems to use: Contextualized Problems (Story Problems):

a. Miguel packages the coffee that grew on his farm. The packages of coffee are put into cases to ship to market. If he puts 25 bags of coffee into each case, how many bags will fill 45 cases?

b. Joseph had a bumper crop of pears this year. He gave each of his 15 cousins a gift of 3 dozen pears. How many pears did he give his cousins in all? (Note that this problem has 2 steps and that students must know how many are in a dozen.)

c. Mr. Stephens’ fifth grade class helped to stuff envelopes advertising their school’s fund-raising carnival event. If the 23 children in his class each stuffed 69 envelopes, how many envelopes did they stuff?

d. Malin is reading a book with 357 pages. If she reads 49 pages each day for a whole week, will she be able to finish the book? (Note: The equation will be 49 x 7 = ? and the product will be compared to 357.)

e. Seth traveled to Europe with his family last summer. He took lots of pictures, an average of 72 each day. If he did this each day for the 19 days they traveled, how many pictures did he take?

f. Gustav, a friendly giant, was having his apartment carpeted. Find out how many square meters of carpet he needs for each room:

i. Living Room: 98 meters by 76 meters ii. Kitchen: 53 meters by 86 meters

iii. Bedroom: 67 meters by 69 meters iv. Hall: 7 meters by 135 meters

g. Mr. Sethi can drive his car 23 miles on each gallon of gas. How far can he drive when he fills up his gas tank with 18 gallons of gas?

h. Carlos drives at an average rate of about 57 miles per hour. At this rate, how far would Carlos drive in 5 hours?

i. If an airplane flies at 645 miles per hour, would it travel 2,000 miles in 3 hours? Decontextualized Problems: For the following examples, the students can solve the problems without a context or you can have them make up a story problem to go with some of the examples. (Answers given in parentheses.)

36 x 7 56 x 8 304 x 9 258 x 4 967 x 5 (252) (448) (2736) (1032) (4835) 48 x 67 29 x 18 56 x 89 72 x 35 83 x 69 (3216) (522) (4984) (2520) (5727)

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NOTE: This recording of the regrouped digit in the standard algorithm is the source of many errors. It often gets added in before the subsequent multiplication, or is forgotten altogether. Because the standard algorithm is part of the Common Core standards, it is important for students to know how it works and how to use it, but that does not mean that they cannot use other methods to solve multiplication problems as well. As students work with problems involving multiplication, be sure they have experience with the standard algorithm as it relates to the array model and recording all partial products, then allow them to continue recording the partial products if they are more comfortable with this approach. If students need to continue using the array model to accurately find solutions, allow this; then continue to work with these students to make the transition from the array to recording the partial products without the array.


• What equation do you need to solve this problem? • What did you think about first? • Which method works best for you? • Why did you decide to…..? • Tell me why you think that is a reasonable answer.


• Which strategy worked best for you? Why do you think that is? • How do you decide if your solution is reasonable?


calculating the partial products accurately.

• Determine if the difficulty lies in misunderstanding the process or in problems with the multiplication facts, themselves. If the problem is with the facts, allow students to use a multiplication facts chart to insure accuracy of solutions; then work individually or in a small group with these students to develop strategies for remembering the facts. If the problem lies in understanding how to find the partial products, work with individuals or a small group with the array model until they “see” where these numbers come from. If necessary, use base-10 blocks or grid paper to create or draw the arrays so that these students can see the actual number of blocks or squares in each part of the array.

• If the problem lies in multiplying with multiples of ten, use the ideas in the lesson “Multiplying Multi-Digit Whole Numbers Using the Standard Algorithm #1: Background” Part 1 to help students understand this process.

• In some cases, students may be adding partial products incorrectly. These students may just need encouragement to take more care and time in the process, as long as they understand the underlying procedures.

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• When multiplying a problem like 36 x 45, students treat the 4 tens as 4 ones, so that the second partial product in the standard algorithm is given as 144 rather than 1440.

• Have these students write out all the partial products and compare them to the two in the standard algorithm. Ask them what the 4 in 45 stands for. (40, not just 4).

• Encourage students to estimate the product before calculating and to compare their final answer to their estimate. In the example 36 x 45, an estimate would be either 30 x 40 = 1200 or 40 x 40 = 1600. A product much outside that range would not be reasonable, so the student should re-check his/her work.

Special Notes:

• It is extremely important that students not be exposed to the standard multiplication algorithm until they have had numerous opportunities to explore a variety of strategies, models, and methods first. The area/array model is particularly advantageous in meaningfully developing the multiplication algorithm.

• Research analyzing sixth graders strategies for solving multiplication problems showed that only 20% chose to use the standard algorithm, with less than half of that group solving accurately. The array or area model was the most frequently selected approach for these students. (Van de Walle, et.al., Teaching Student-Centered Mathematics, Grades 3-5, 2nd Edition. Pearson, 2014, p. 185)


Note that some solutions are within the text. Part I. #3

a. 37 x 28 = 1,036 b. 5 x 17 = 85 c. 39 x 63 = 2,457 Yes, the silo will hold that amount. d. 3,246 x 7 = 22,722 e. 37 x 48 = 1776 f. 26 x 374 = 9724 g. 87 x 63 = 5481 h. 196 x 52 = 8788

Part II. #4 Contextualized Problems

a. 25 x 45 = 1125 b. 15 x 36 = 540 c. 23 x 69 = 1587 d. 49 x 7 = 343 343 < 357 She will not finish the book in a week. e. 19 x 72 = 1368 f. i. 98 x 76 = 7448 sq m

ii. 53 x 86 = 4558 sq m iii. 67 x 69 = 4623 sq m iv. 7 x 135 = 9045 sq m

g. 23 x 18 = 414 miles h. 57 x 5 = 285 i. 645 x 3 = 1935 1935 < 2000 No

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Multiplying Multi-Digit Whole Numbers Using the Standard Algorithm #4: Estimation

Common Core Standard: Perform operations with multi-digit whole numbers and with decimals to hundredths. 5.NBT.5 Fluently multiply multi-digit whole numbers using the standard algorithm.




1. Make sense of problems and persevere in solving them. 2. Reason abstractly and quantitatively. 4. Model with mathematics. 7. Look for and make use of structure. 8. Look for and express regularity in repeated reasoning.

Student Outcomes:

• I can use multiple strategies to find the product of multi-digit whole numbers. • I can use the standard algorithm to find the product of multi-digit whole numbers.

Materials:

• Paper or journals for student responses • Base-10 blocks • Base-10 cutouts • Grid paper


• Gather paper or student journals; have available base-10 blocks and grid paper. • Consider how you will pair or group students. • Students should have some fluency with multiplying whole numbers using a variety of

methods, including array and area models and the standard algorithm. • Students should be familiar with estimating. • Students should be comfortable struggling with a task and explaining their reasoning.

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Directions: Part I. Making Preliminary Estimations of Products

1. Pose this problem: The owner of a large furniture store wants to buy 290 chairs that cost her 28 dollars each. Will this purchase cost as much as $10,000?

Ask the students: Do you think the cost will be as high at $10,000? Why do you think that? (Depending on their answers to that question, continue as follows.)

- What numbers close to the numbers in the problem would be easy to multiply? (If no one suggests using numbers rounded to a multiple of 10 or 100 closest to the given numbers, suggest it yourself. The number 28 is close to 30 and the number 290 is close to 300. So the product for 28 x 290 would be close to 30 x 300 or 9,000).

- Would the exact product be more or less than 30 x 300? (Since both given numbers are less than the estimates, the exact product should be less.)

Students should conclude that the cost will be less than $10,000. Have the students carry out the multiplication using a method of their choice and compare the exact answer (8,410) to the estimates.

2. Tell the students that someone wrote the equation 31 x 49 = 3,619. Ask how they can quickly tell whether the product is correct or not. Hopefully, they will use estimation to see that the answer should be close to 30 x 50 or 1,500, much smaller than the product of 3,619. Ask whether some other answers such as 1,539; 1,499; and 1,936, appear to be reasonable answers. They may agree that the first two are close to the estimate of 1,500 and may be correct, but the 1.936 appears to be too large. Ask them how estimating like this can help them check their answers when multiplying. Discuss how estimation can help reject wildly incorrect answers, but does not help detect close but still incorrect answers. Have them find the correct answer (1,519).

3. Pose another problem: For a class game, each student in our class needs 28 counting chips.

If counting chips come in boxes of 200, how many boxes would I need to have on hand?Ask the students first to give you the equation needed to solve the problem (28 x the number of children in the class = ?) and then how they would estimate the answer. Most likely the number of students in the class would round to 20 or 30, and the 28 would round to 30. Ask: Would 3 boxes be too few? Are 30 boxes too many? Would 5 boxes be enough? Discuss: Would this be a situation where you would need an exact number of counters? Would having an estimate give you enough information to know how many boxes to have?

4. Provide more examples for practice as needed. Note: If a number is halfway between two possible

estimates (for example: 35 is halfway between 30 and 40), let the students use either the higher or the lower estimate. Do not make or use any automatic “round up” or “round down” rule. Possible examples: Have the students estimate the product of each equation below, record the estimate, then choose the correct answer from the choices given. a. 28 x 110 = ? 2,080 3,080 30,800 b. 61 x 420 = ? 246,2000 2,420 25,620 c. 103 x 220 = ? 2,260 260,600 22,660 d. 67 x 6,023 = ? 403,541 360,541 43,541 e. 83 x 21 = ? 1,463 1,743 17,433 f. 890 x 910 = ? 81,080 710,800 809,900 g. 49 x 49 = ? 24,001 2,401 1,681

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Part II. Assessment After your students have had numerous experiences to practice using the different models and methods of multiplication within these lessons (“Multiplying Multi-Digit Whole Numbers Using the Standard Algorithm” #1-4), pose this problem: Washington, D.C., is 39 miles from Baltimore, Maryland. It is 69 times as far from Baltimore to Los Angeles. How far is it from Baltimore to Los Angeles?

a. Make an estimate of the solution. Explain how you arrived at your estimate. b. Solve the problem in two different ways. Show all your work. c. Explain how your two methods are similar and how they are different. Tell which method you

prefer to use and why. Questions to Pose: As students work and during class discussion:

• How does estimating help you? • Which strategy worked best for you? Why do you think that is? • Tell me why you think that is a reasonable answer.


rounding numbers to appropriate numbers for estimating

• Be sure that the students understand the purpose of rounding the numbers in order to estimate a solution. Help them see that you are rounding 392 to 300 rather than 390 because 300 is so much easier to think about and to use.

• Meet with individuals or small groups to explore rounding numbers to the nearest ten or hundred or thousand. Be sure they understand that the object is to make the computation easier.

• Students hesitate to estimate to find an “about” answer because they are so used to needing the exact right answer.

• Changing this mindset will take time. As students learn that you are interested in their thinking, their strategies, their processes, they will become more willing to find an approximate answer when that is appropriate.

• Encourage students to see that their estimations help them evaluate solutions to see if they are reasonable and therefore more likely to be correct.

Special Notes: Estimation is an extremely important skill. Be sure students have plenty of opportunity to estimate solutions to multiplication and other computations. If students begin to estimate solutions reasonably, they will be able to more accurately judge their solutions as sensible or not.

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Solutions for Problems Used in This Lesson: Note that some solutions are within the text. Part I. #4

a. 3,080 b. 25,620 c. 22,660 d. 403,541 e. 1,743 f. 809,900 g. 2,401

Part II. 2691 miles Look for:

• Reasonable estimate and explanation of how student arrived at estimate • Accurate solution • Two different methods of finding the solution • Explanation of how the two methods are both similar and different, including • statement of which method student prefers and why

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NC DEPARTMENT OF PUBLIC INSTRUCTION FIFTH GRADE NC DEPARTMENT OF PUBLIC INSTRUCTION FIFTH GRADE

Division of Whole Numbers: Dealing with Remainders

Common Core Standard: Perform operations with multi-digit whole numbers and with decimals to hundredths. 5.NBT.6 Find whole number quotients of whole numbers with up to four-digit dividends and two-digit divisors, using strategies based on place value, the properties of operations, and/or the relationship between multiplication and division. Illustrate and explain the calculation by using equations, rectangular arrays, and/or area models.


Understand the place value system. 5.NBT.1 Recognize that in a multi-digit number, a digit in one place represents 10 times as much as it represents in the place to its right and 1/10 of what it represents in the place to its left. Use place value understanding and properties of operations to perform multi-digit arithmetic. 4.NBT.6 Find whole-number quotients and remainders with up to four-digit dividends and one-digit divisors, using strategies based on place value, the properties of operations, and/or the relationship between multiplication and division. Illustrate and explain the calculation by using equations, rectangular arrays, and/or area models.


1. Make sense of problems and persevere in solving them. 2. Reason abstractly and quantitatively. 4. Model with mathematics. 7. Look for and make use of structure.

Student Outcomes:

• I can use multiple strategies to find the quotient of two whole numbers. • I can illustrate and explain a division calculation in a variety of ways. • I can divide up to four-digit whole numbers by one- and two-digit divisors. • I can solve problems involving division, including dealing with remainders.

Materials:

• Paper or journals for student responses • Copies of division problems, one for each student • Base-10 blocks (optional)


• Copy the division problems, one per child. • Gather paper or student journals. • Consider how you will pair or group students. • Students should be familiar multiplying whole numbers. • Students should be familiar with the concept of division. • Students should be familiar with estimating. • Students should be comfortable struggling with a task and explaining their reasoning.

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Directions: Option: Rather than doing these first three problems separately, you could have individuals or partners work on all three problems and then discuss them together, with particular emphasis on how the treatment of the remainder is different in each context.

1. Pose the following problem: All the third, fourth, and fifth grade students at Sunshine Elementary School were going to attend the ACC women’s basketball tournament at the coliseum. The 228 students will be riding on activity buses which can hold 30 students each. How many buses will be needed to transport the students to the game? Ask: What equation do we need to solve this problem? Someone should suggest 228 ÷ 30 =

Have the students work individually or with a partner to find a way to solve the problem. They may use a variety of strategies. The students should have some experience with division from their earlier work in fourth and fifth grades. Do not expect them to necessarily use the standard algorithm. Monitor as the students work. Look for different strategies, but also look for which students are considering their quotient and how to use it to answer the question in the problem. The quotient will be 7 with a remainder of 18. Students must realize that they will need an extra bus to accommodate the extra 18 students, so the solution to the problem is 8 buses.

Have selected students share their methods of solving the equation and deciding on the answer to the question. Be sure students understand why the answer to the question is not “7 r.18”, which is the answer to the division problem, not the answer to the question asked in the problem.

2. Pose a second problem: Shana took 239 photos with her digital camera when she was on vacation in the mountains last summer. She arranged them into a digital book with 4 pictures on each page. How many pages in the book would have had exactly 4 photos? Again ask for the equation that will help solve the problem. The division equation is 239 ÷ 4 = . It is possible that someone may suggest 4 x = 239. If so, ask the students if this equation can represent the situation. (Yes, it is an equivalent equation to the division equation and actually represents a way to think about the problem: “What number times 4 is equal to or comes close to 239?”)

Have the students work individually or with a partner to find a way to solve the problem. Monitor as they work to see what strategies they are using and whether or not they are thinking about how to use their solution to answer the question in the problem. Select a few solution strategies to be shared. The quotient is 59 with a remainder of 3. So the solution to the problem is 59 pages will be full. In this instance, the remainder of 3 would not make another full page, so it does not add another page to the 59. Be sure that students understand how this solution requires different thinking about whether and how to use the remainder to answer the question stated in the problem. In the bus problem, an extra bus was required to accommodate the extra students. In this photo case, the extra photos do not add an extra page since the question is about full pages.

Ask: What if the question had been “How many pages would the book need to have if each page can hold 4 photos?” In this case the answer to the question would be 60 pages to include a page for the remaining 3 photos.

3. Pose a third problem: Rohan had a piece of rope that was 188 inches long. If he cut it into 8 equal lengths, how long would each piece be if he used the entire length of rope?

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Ask for the equation needed to solve the problem: 188 ÷ 8 = . Have individuals or partners find a way to solve the problem. 188 is not a multiple of 8, so there is a remainder. Since in this case, all the rope is to be used, students have to decide how to divide up the remaining inches so that each new length is equal to the others and all the rope is used. 188 ÷ 8 gives a quotient of 23 with a remainder of 4 inches. The extra 4 inches can be divided between the 8 lengths with each length getting 4 ÷ 8 or 4/8 of an inch. Let students grapple with this. Many students may recognize that 4 inches divided into 8 equal parts would give ½ inch to each length. Some students will want to “drop” the extra inches. Ask them if that fits what is said in the problem. (It does not.) Monitor as students work, noting how different students deal with the remaining 4 inches. Choose work to share that will show different methods of dealing with dividing the extra inches. After students have shared a variety of ways of thinking about this problem, lead a discussion about why the remainder in this problem could be divided into fractional parts, but in the previous two problems this was not appropriate. (One might argue that the answer to the photo album problem might be 59 ¾ pages, which is a mathematically correct answer.) Ask them to suggest other scenarios where the remainder might be stated as a fraction. Possible suggestions: cookies, pizza, ribbon, almost anything that is dealing with a unit of measure and an object that can realistically be divided into fractional parts.

4. Provide students with a few problems for which they must decide how to deal with the remainder. Ask them to show their work, state their answer clearly, and to explain how and why they dealt with the remainder as they did. You can differentiate this activity by giving the students different numbers to use, depending on the level of their mastery of division concepts.

Possible problems: a. Your teacher is organizing a field trip to Gerber’s Pet Shop. If all the parents who are driving

have cars which hold 6 passengers, how many cars are needed to transport 64 fifth graders? b. Jenny has a piece of ribbon 252 inches long. She is tying ribbons around some gift

packages. If she needs 8 equal pieces of ribbon, how long can she make each piece if she uses all 252 inches of ribbon?

c. Jean had a spool of string with 294 inches of ribbon left on it. If she cut it into pieces to share equally with 8 friends and herself, how many inches of string would each person get?

d. A square made of string has a perimeter of 657 cm. If the same string is used to form a 6-sided figure with all sides the same length, how long will each side be?

e. How many pages of my photo album will it take to display the 254 pictures from my last vacation if each page holds 4 pictures?

f. Aerostar Airlines seats 7 passengers in each row of seats in its airplanes. On a flight across the U.S., a plane had 331 passengers. What is the minimum number of rows of seats that would be needed to seat all those passengers?

g. Twelve children found a huge burlap bag with 2,788 pieces of candy in it. If the candy is divided equally, how many pieces will each child receive?

h. The zoo printed 7,390 free passes and gave them to 6 schools. How many passes did each school receive if each school received the same number?

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5. Optional: Challenge all or some of your students to write their own division story problems for which a decision has to be made about how to deal with the remainder. They should solve their problems and show the solution. These problems could be shared with the class as extra practice.


• What is the problem asking you to find? • What equation will help you solve the problem? • Why did you decide to…..?


• Why did you round up your quotient to answer the question? … ignore the remainder to answer the question? … change the remainder to a fraction to answer the question?

Possible Misconceptions/Suggestions: Possible Misconceptions Suggestions • Students ignore the remainder

and give the answer to the story problem as the quotient without the remainder and without reference to the context of the problem

• Ask students what the problem is asking. For example in #1 above the question is: “How many buses?” If students say 7 buses, ask them what does the remainder of 18 students mean? If they are not sure, ask them if the 18 students are going to the game and how they will get there. With questioning like this, students should realize that they will need another bus. (Students often think they have solved a problem if they have done the requisite computation, without realizing that they need to consider what question has been asked in the problem and how their computation solution relates to that question.)

• Students state the solution as the quotient with remainder, e.g., 8 r.5, with no reference to the context of the problem or the question being asked

• Ask students what the problem is asking. For example in #1 above the question is: “How many buses?” If students say 7 r.18, ask them how that answers the question asked in the problem. (See note in italics above.)

Special Notes:

• This lesson should follow other lessons dealing with the process and methods of division. • Students will need continued practice with division problems for which the remainder

affects the solution to the problem. Solutions for Problems Used in This Lesson: Section #4 Quotient Solution to problem

a. 10 r.4 11 cars b. 31 r.4 31 4/8 or 31 ½ inches c. 32 r.6 32 6/9 or 32 2/3 inches d. 109 r.3 109 3/6 or 109 ½ cm e. 63 r.2 64 pages f. 47 r.2 48 rows g. 232 r.4 232 4/12 or 232 1/3 pieces of candy or 232 each with 4 left over h. 1262 r.3 1262 passes

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Multiplying Whole Numbers by Decimals Common Core Standard:

Perform operations with multi-digit whole numbers and with decimals to hundredths. 5.NBT.7 Add, subtract, multiply, and divide decimals to hundredths, using concrete models or drawings and strategies based on place value, properties of operations, and/or the relationship between addition and subtraction; relate the strategy to a written method and explain the reasoning used.

Additional/Supporting Standard: 5.NBT.3 Standards for Mathematical Practice:

1. Make sense of problems and persevere in solving them 4. Model with mathematics 5. Use appropriate tools strategically 7. Look for and make use of structure

Student Outcomes:

• I can reasonably estimate the product of a whole number and a decimal • I can use multiple strategies to find the product of a whole number and a decimal

Materials:

• Decimal Grids Sheet (one for each student) • Paper or journals for student responses


• Copy Decimal Grids Sheet • Gather paper or student journals • Consider how you will to pair or group students • Students should be familiar with adding and subtracting decimals • Students should be familiar with estimating • Students should be comfortable struggling with a task and explaining their reasoning • Students should be able to use a decimal grid to model a decimal to the hundredths place

Directions:

1. Pose the following question and write it on the board: A farmer fills each jug with 1.4 liters of cider. If you buy 3 jugs of cider, how much cider will you have?

2. Begin with a class discussion of student estimates. Students explain their reasoning for their estimations (It is bigger than 3. It is less than 6.).

3. Allow students to individually solve the problem using strategies that make sense to them. When most have finished, students should share their strategy and reasoning to a partner or small group.

4. Initiate a whole group discussion of the different strategies. Some will use repeated addition: 1.4 + 1.4 + 1.4. Others will multiply 3 x 1, and then add in 0.4 + 0.4 + 0.4. A wide range of other reasonable strategies may be used. As students explain their thinking, record their methods for all to see.

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5. Repeat this process with problems using nice fractional parts such as 2.5 x 4 and 2 x 3.25 • Ask a student to verbalize a word problem for the expression • Conduct a class discussion of student estimates • Students solve the problem independently • Students share their strategies with a partner or small group • Teacher facilitates whole group discussion

6. Distribute decimal grids. Students choose one of the previous multiplication problems to solve using the decimal grid. Students should working in pairs or small groups

7. Allow students to struggle with this task. Students may not know how to use the grids to solve the problem. Encourage students to use the grids in a way that makes sense to them.

Possible solutions for 3 x 1.4: Solution 1

Solution 2

8. Invite a few pairs or groups to share their solution with the class. Challenge the students to

defend their reasoning. Why does the method you used work? 9. As students share their solutions, ask students make connections between the different

strategies. How are these strategies alike? How are they different? Do they all make sense? 10. Now have students make connections across problems. Could each of these solutions work

for all three problems?

3 x 1

3 x .4

3 x 1

3 x .4

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11. To close the lesson write the following journal prompt on the board. Before students write in their journal, ask them to briefly talk with a partner about the two strategies they plan to describe.

Describe two strategies you can use when multiplying a whole number by a decimal. Questions to Pose: As students work together:

• What did you think about first? • Why did you decide to…..? • What could you do it you did not have enough grids?


• What connections do you see with ways you multiply whole numbers? • How did estimating help your thinking? • Which strategy worked best for you? Why was this one the best?

Possible Misconceptions/Suggestions: Possible Misconceptions Suggestions • Students make unreasonable

estimates • Students should use various representations to model

decimals. They should make generalizations about their magnitude. For example, “I can tell that seven tenths is more than one half.”

• Students can practice estimating decimal sums and differences since addition and subtraction are more familiar.

• Students have difficulty modeling decimals on a grid

• Relating decimals as parts of a whole

• Practice building decimals with base ten blocks and transferring these representations to a grid. Use number lines to explore the magnitude of decimals and how they connect to whole numbers.

• Connect these models with numerical representations of decimals.

Special Notes:

• This lesson should be used first in a study of multiplication with decimals. • Follow-up lessons should consider multiplying a decimal by another decimal.


3 x 1.4 = 4.2 2.5 x 4 = 10 2 x 3.25 = 6.5

Adapted from Teaching Student-Centered Mathematics, Grades 3-5. Van de Walle and Lovin (2006).

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Decimal Grids

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Window Time Common Core Standard:

Perform operations with multi-digit whole numbers and with decimals to hundredths. 5.NBT.7 Add, subtract, multiply, and divide decimals to hundredths, using concrete models or drawings and strategies based on place value, properties of operations, and/or the relationship between addition and subtraction; relate the strategy to a written method and explain the reasoning used.


5.NBT.5 Perform operations with multi-digit whole numbers and with decimals to hundredths. Standards for Mathematical Practice:

1. Make sense of problems and persevere in solving them 2. Reason abstractly and quantitatively 5. Use appropriate tools strategically 6. Attend to precision

Student Outcomes:

• I can use my understanding of decimal operations to solve problems • I can use clear notations to show my solution strategies

Materials:

• Window Time task sheet (one per student) • Base-ten blocks, decimal grids, and grid paper


• Copy the Window Time sheet • Set aside a place in the classroom for base-ten blocks, decimal grids, and grid paper in case

students want to use them • Consider how you will pair students • Students should be able to add and multiply decimals to the hundredths place • Students should be familiar with area and perimeter

Directions:

1. This task provides practice with adding and multiplying decimal numbers: Your neighbor, Mrs. Jones, has asked for your help with a project. She wants to put a new window in her living room. Her new window will be 3.4 feet wide and 4.65 feet tall. Mrs. Jones needs your help figuring out how many feet of framing she should buy for the window. She also needs to know how many square feet of glass she should purchase.

2. Distribute the task to students. Read the task together and ensure that students understand the directions.

3. Dismiss students to work on the task in pairs. 4. Provide any materials that might be helpful including blank decimal grids, base-ten blocks,

and grid paper. Encourage the students to use these materials to support their thinking.

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5. Circulate as students work. Observe student strategies and pose the questions listed below. 6. Allow students to struggle. Resist the urge to step in and help to quickly. 7. When most students have finished the task, lead a class discussion about the following

questions: “What mathematics did you use today?” “How did you decide what to do?” “What strategies worked best for you today? Why were they so helpful?”


• How could drawing a picture help you solve this problem? • How might you use the base-ten blocks, grid paper, or decimal grids?

During: • Which part of the problem are you working on? How are you approaching this part? • What challenges are you running into? What are you doing to help you with these

challenges? • How did you decide to use….?

After: • What mathematics did you use today? • How did you decide what to do? • What strategies worked best for you today? Why were they so helpful?

Possible Misconceptions/Suggestions: Possible Misconceptions Suggestions Students have difficulty applying their knowledge of area and perimeter

Set up an array to reinforce the concepts of area and perimeter. Relate these to the window frame and glass. Allow the student to show how area and perimeter would be found using the array. Relate this process to the window task.

Students have difficulty multiplying and/or adding decimals

Students need ample practice using the four operations with decimals. Use grid paper, blocks, and other models to make connections with meaning and procedure.

Special Notes:

• If many students are struggling with the task, you may want to ask the students to pause and share their strategies with the entire class.

• This problem can be extended by asking the students to write letters to Mrs. Jones explaining their reasoning and mathematics.

• Students may also wish to find the cost of multiple windows. Solutions: Amount of framing needed: 16.1 feet

Amount of glass needed: 15.81 square feet Extra Challenge: Cost of framing and glass: $27.02

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Window Time

Your neighbor, Mrs. Jones, has asked for your help with a project. She wants to put a new window in her living room. Her new window will be 3.4 feet wide and 4.65 feet tall. Mrs. Jones needs your help figuring out how many feet of framing she should buy for the window. She also needs to know how many square feet of glass she should purchase. Extra Challenge: Each foot of framing costs $0.50. Window glass costs $1.20 per square foot. How much will Mrs. Jones pay for her new window?

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Introduction to Dividing with Fractions Common Core Standard:

Apply and extend previous understandings of multiplication and division to multiply and divide fractions. 5.NF.7 Apply and extend previous understandings of division to divide unit fractions by whole numbers and whole numbers by unit fractions


5.NBT.6 Perform operations with multi-digit whole numbers and with decimals to hundredths. Standards for Mathematical Practice:

1. Make sense of problems and persevere in solving them 2. Reason abstractly and quantitatively 4. Model with mathematics 6. Attend to precision 7. Look for and make use of structure

Student Outcomes:

• I can create a visual representation for division • I can relate division with whole numbers to division with fractions • I can justify my reasoning when developing a strategy for division

Materials:

• Paper and markers for student posters

Advance Preparation: • Consider how you will pair students • Students should have abundant experience dividing with whole numbers • Students should have a strong understanding of fractions as parts of a whole

Directions:

1. The focus of this lesson should be on exploring the meaning of division by fractions. At no time should students use the “invert and multiply” method or find the reciprocal.

2. Stimulate student thinking about division by asking them to draw their own models for these problems:

• Jimmy has 16 inches of ribbon. He is making bows for his sister’s birthday gifts. If each bow requires 8 inches of ribbon, how many bows can he make?

• Mary and her 2 friends go on a hike together. They plan to collect 21 leaves for a scrapbook. How many leaves should each person collect so they share the work equally?

3. After students have time to create their drawings, invite them to share their work with a neighbor. Then, ask a few students to share their representations with the whole class. Be sure to include models such as:

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21 ÷ 3

16 ÷ 8

4. Use student work from these problems to facilitate a whole group discussion about the meaning of division. When we see a problem like 12 ÷ 4, today we will think about “How many groups of 4 can be found in 12?”

5. Tell students that today they will be applying their knowledge of division to fractions. Share

this task: You are going to a birthday party. From the Sweet Tooth Ice Cream Factory, you order 6 pints of ice cream. If you serve ¼ of a pint of ice cream to each guest, how many guests can be served?

6. Allow students to struggle with this task since dividing with fractions is brand new to them.

Encourage them to use whatever method makes the most sense to them. Circulate as they work independently and observe how students approach the problem.

7. Now pair students and instruct them to share their strategies. Encourage them to explain the

mathematics they used and the reasoning behind their approaches. Each pair should agree on one strategy that they want to display for the class.

8. Each pair should create a poster on blank paper, chart paper, or poster board. Their poster

should clearly show a representation of how they solved the problem. For example:

9. When students finish, display the posters and facilitate a discussion about the strategies students used. Do all the strategies make sense? How are the strategies similar to one

7 7 7

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another? How are they different? How do these representations connect to dividing with whole numbers?

10. Finally, ask students why we would write this problem as 6 ÷ ¼. Students should see that we

are asking “How many groups of ¼ can be found in 6?” Note that: When we divide with whole numbers, the answer is smaller. However, when we divide a whole number by a fraction, the answer is larger. How can this be?


• What do we mean when we say “divide?”

During: • How did you decide on this strategy? • How did you show the 6 pints of ice cream? How did you show fourths? • What answer did you get? How can you tell if your answer makes sense?

After:

• Did everyone get the same answer? • Do all the strategies make sense? • How are the strategies similar to one another? How are they different? • How do these representations connect to dividing with whole numbers? • Why is our answer a larger number even though we divided these numbers?

Possible Misconceptions/Suggestions: Possible Misconceptions Suggestions Students attempt to divide by 4 rather than ¼ Refer to the context. Remind students that each

party guest receives ¼ of a pint, rather than 4 pints. Student attempts to use “invert and multiply” Encourage the student to draw a representation

of the problem. Suggest that the student examine the visual models used in the division problems with whole numbers.

Student has trouble understanding how the answer to a division problem could be a larger number.

Relate 6 ÷ ¼ to the context, “How many ¼ portions are found in 6 pints?” Refer to the visual models to show that each whole has 4 ¼ portions.

Special Notes:

This is an introductory lesson to be used when students are unfamiliar with division by fractions. At no time should students use the “invert and multiply” method or find the reciprocal. The focus of this lesson should be on exploring the meaning of division by fractions.

Solutions:

Students will use many different strategies to show that 6 ÷ ¼ = 24. Adapted from Teaching Student-Centered Mathematics, Grade 3-5. Van de Walle and Lovin (2006).

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Measurement Mania Common Core Standard

Convert like measurement units within a given measurement system. 5.MD.1 Convert among different-sized standard measurement units within a given measurement system (e.g., convert 5 cm to 0.05 m), and use these conversions in solving multi-step, real world problems.

Additional/Supporting Standard: 5.NBT.1, 5.NBT.2 Understand the place value system


6. Attend to precision 7. Look for and make use of structure 8. Look for and express regularity in repeated reasoning

Student Outcomes:

• I can convert measures within the same system • I can explain why my answers make sense

Materials:

• One game board, set of cards, and answer sheet for each group of 3 students • One die for each group of 3 students • Items to use as markers for each group of students


• Copy the game board and set of cards. Laminating them on cardstock will help the materials last longer. Cut out each set of 20 cards and store in a baggie.

• Copy an answer sheet for each group • Place markers for each group in the baggie of cards • Consider how you will group students • Students should have ample experience exploring the meanings of cm, m, km, g, kg, ml, L,

oz, lb, in, ft, and yd. • Students should be familiar with strategies for converting measures within the same system.

Directions:

This game provides practice for converting measures. There are two options for game play:

Option 1 3 students play the game with one serving as the Answer Person and the other two solving the problems. The answer person uses the Measurement Mania Answers sheet and acts as supervisor. The Answer Person does not roll the die or move a marker.

Option 2

3 students play the game. The students take turns being the Answer Person and using the Answer Person, switching to a new person on each turn.

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• What strategies can you use for converting measures? • How are cm, m, and km related? How are oz and lb related? • How can you make sure that your answer makes sense?

During:

• What strategy did you use to convert the measures? • How can you tell that your answer makes sense? • What error did you make? How can you prevent this error in the future?

After:

• What strategies worked especially well for you? Why were they so effective? • What errors did you make frequently? How can you prevent these errors in the future? • How might it help to visualize the unit in your mind? • How might it help to use a benchmark? • How might it help to think about 10s, 100s, and 1000s?

Possible Misconceptions/Suggestions: Possible Misconceptions Suggestions Students have difficulty remembering how the units are related to one another.

Use objects and measuring tools to explore the base units. For instance, on a meter stick how many centimeters are equivalent to a meter? Engage the students in activities to establish benchmarks. What can they use to remember the size of a gram? Through classroom tasks, develop a chart to explore how measures are related. For instance 2 m = 200 cm and 3 m = 300 cm, therefore meters are 100 times the size of centimeters.

Special Notes:

This game is intended for practice. Students should not practice the procedures for conversion until they have had ample opportunities to explore measurement units and relationships.

Solutions:

See Measurement Mania Answers

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Measurement Mania

STA

RT

WIN

!

On Your Turn: 1. Draw a card. 2. Solve the problem 3. Check with the Answer Person in your group 4. If you got it correct, roll the dice and move forward 5. If you got it incorrect, you lose your turn

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#1.

1 cm = ____ m

#2.

1 kg = _____ g

#3.

1 m = _____ cm

#4

1 L = _____ ml

#5

100 cm = ____ m

#6

1 km = ____ m

#7

1000 m = ____ km

#8

1 lb = ____ oz

#9

3 ft = ____ in

#10

9 yd = ____ ft

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#11

4.2 m = ____ cm

#12

3000 ml = ____ L

#13

32 oz = ____ lb

#14

5 cm = ____ m

#15

200 m = ____ km

#16

500 g = ____ kg

#17

45.6 cm = ____ m

#18

36 in = ____ ft

#19

150 ft = ___ yd

#20

3.2 kg = ____ g

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Measurement Mania Answers 1. 1 cm = 0.01 m 2. 1 kg = 1000 g 3. 1 m = 100 cm 4. 1 L = 1000 ml 5. 100 cm = 1 m 6. 1 km = 1000 m 7. 1000 m = 1 km 8. 1 lb = 16 oz 9. 3 ft = 36 in

10. 9 yd = 27 ft 11. 4.2 m = 420 cm 12. 3000 ml = 3 L 13. 32 oz = 2 lb 14. 5 cm = 0.05 m 15. 200 m = 0.2 km 16. 500 g = 0.5 kg 17. 45.6 cm = 0.456 m 18. 36 in = 3 ft 19. 150 ft = 30 yd 20. 3.2 kg = 3200 g

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Filling Boxes Common Core Standard:

Geometric measurement: understand concepts of volume and relate volume to multiplication and to addition. 5.MD.3 Recognize volume as an attribute of solid figures and understand concepts of volume measurement.

• A solid figure which can be packed without gaps or overlaps using n unit cubes is said to have a volume of n cubic units.

Additional/Supporting Standards:

5.MD.4, 5.MD.5 Geometric measurement: understand concepts of volume and relate volume to multiplication and to addition.


1. Make sense of problems and persevere in solving them 4. Model with mathematics 5. Use appropriate tools strategically 6. Attend to precision 8. Look for and express regularity in repeated reasoning

Student Outcomes:

• I can develop and implement a strategy for determining volume • I can determine the volume of a figure by using only a small number of cubes • I can make observations about the structure of cubic units

Materials:

• A variety of small empty boxes such as shoe boxes and tissue boxes (at least one per pair) • Small same-size cubes (enough for each pair to complete only 1-3 layers of a box)


• Gather a collection of small boxes such as shoe boxes and tissue boxes. The boxes should be a variety of sizes.

• Gather small cubes. Unifix or Snap Cubes work best. Each pair of student will need enough cubes to make 1-3 layers in a box, but not enough to fill the whole thing. Test a few boxes to determine how many cubes they will need.

• Consider how you will pair students • Students should understand the concept of volume, but not the formula l x w x h • Students should have some experience creating solid figures with cubes

Directions:

1. Display your collection of boxes. Ask the students to predict which box has the largest volume. Ask which has the least. Have students justify their reasoning. Think about the height, width, and length of each box.

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2. Tell the students that they will be determining each box’s volume by using cubic blocks. Their task is to figure out how many cubes will fill each box completely. Note that they will not be given all the cubes that will fit. For instance, they may only have 30 cubes even though their box can hold 100. They will need to develop a strategy for finding the total number of cubes that will completely fill their box.

What is the total number of cubes that will completely fill your box?

3. Give each pair of students one box and a collection of same-size cubes. This is a great

opportunity to ask students why it is important to use same-size cubes.

4. Circulate as students work. Some students will determine how many cubes fit in one layer of the box and then determine the number of layers. Others may line up the blocks along the edges and multiply. Still others will try to physically move the blocks so that they fill in the space. Be sure to ask questions including those listed below.

5. About half way through the work time ask the students to pause. Invite several groups to share a strategy that did not work for them. Then invite a few pairs to share a strategy that is working. This discussion will especially support groups that are struggling with the task.

6. When a group has finished working with one box, they should trade boxes with another group that has finished and start again. Some groups may finish 2 or 3 boxes, while others will only complete one.

7. When each pair has completed at least one box bring the group back together to discuss their strategies and observations.

8. Discuss the following questions: What strategies worked best for your group? What strategies did not work? What made one strategy more effective than another? What problems did you run into?

9. If students do not point out the potential problems of accidentally leaving gaps or

overlapping the blocks, be sure to bring this topic up in discussion.

10. Ask the students how they might get the most precise measure of volume for each box. Many students will point out that you could fill the box completely with cubes. Guide the students to see that the layers of the box can be seen as arrays. These arrays are repeated with each layer.

If students make the connection to multiplication, guide their thinking by asking “What numbers would you multiply to find the volume?” If students do not make the connection to multiplication, help them see how we can use multiplication or repeated addition to find the precise volume.

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11. To summarize the lesson, show students one additional box. Ask the students to describe to a neighbor the steps they would take to find the volume if they only had 20 blocks.


• Predict which box will have the largest volume? Explain your reasoning. • Predict which box will have the smallest volume? Explain your reasoning. • How could you use the cubes to figure out the volume of your box, even though you will not

have enough to fill up the whole box?

During: • What strategy are you using? Is it working well? How can you tell? • What problems are you running into? How could you adjust your strategy so you have fewer

problems? • What will you do next? • How can you tell if you are leaving any gaps? How can you tell if you are overlapping the cubes?

After:

• What strategy worked best for you? Why did it work so well? • What problems did you encounter? How did you address them? • How could thinking about layers help you determine a figure’s volume? • How could multiplication be related to a figure’s volume?

Possible Misconceptions/Suggestions: Possible Misconceptions Suggestions Students have difficulty recognizing that a strategy is problematic. Their strategy may overlap the cubes, leave gaps, or be incomplete.

Form a cube with 8 blocks. Guide the student to recognize that the volume of this cube is 8 cubic units and that there are no gaps or overlaps. Reinforce the concept that the current task requires the student to find out how many cubic units would completely fill the entire box.

Students have difficulty keeping track of their progress and/or how many cubes they have used.

Provide paper for students to record the number of cubes used. Suggest that student make marks on their box to keep track of their progress.

Special Notes:

• The focus of this lesson is to explore concepts of volume and help students discover volume’s connection to multiplication.

• Rather than using pre-made boxes, you might have the students create several boxes from nets. Be sure to use large nets that will accommodate many cubes. You may want to copy the nets on cardstock so that they stay together as they are manipulated.

Solutions: NA

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Candy Boxes Common Core Standard:

Geometric measurement: understand concepts of volume and relate volume to multiplication and to addition. 5.MD.4 Measure volumes by counting unit cubes, using cubic cm, cubic in, cubic ft, and improvised units.

Additional/Supporting Standard(s):

5.MD.3, 5.MD.5 Geometric measurement: understand concepts of volume and relate volume to multiplication and to addition.


2. Reason abstractly and quantitatively 4. Model with mathematics 5. Use appropriate tools strategically 6. Attend to precision

Student Outcomes:

• I can demonstrate the meaning of volume • I can determine the dimensions of a rectangular prism • I can relate the dimensions of a figure to its volume

Materials:

• Unifix cubes, connecting cubes, 1” blocks • Yummy Candy Boxes Recording sheet


• Gather 36 cubes per student • Copies of recording sheet per student • Consider partnering students for sharing of strategies

Directions:

1. Teacher presents the task to the class: The Yummy Candy Company makes fudge cut into pieces that are one cubic inch in volume. That is, each piece is one inch long, one inch wide, and one inch deep. The company wants to make a package that holds 36 pieces of fudge. Investigate the different dimensions for three different boxes the company could use to package the fudge.

2. Using 36 cubes each (fudge pieces), students work to create multiple representations of organizing the cubes into rectangular prisms ‘box’. After completing a representation of a box, students record the dimensions with a visual, words or numbers on the recording sheet.

3. Students continue to build ‘boxes’. Once students have built and recorded 3 ‘boxes’, they share work with a partner justifying their thinking.

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4. Teacher facilitates a whole class discussion. Students share box dimensions. Teacher or

student records their representation for class to see. As students share, bring out vocabulary such as length, width, and height of their box. What do you notice about how the height, width and length help you with the volume? If students cannot connect to L x W x H, ask follow up questions such as: How does the length, width, height relate to addition? Multiplication?

5. Students discuss which box the Yummy Candy Company would choose to package their fudge pieces. There is not one right answer, but students need to justify their thinking.

6. Students record which box they chose and explain their thinking in their journal.

Questions to Pose: As students work individually or with a partner:

• What is the area of the bottom of your box? • If the area of the bottom of your box is 18, how many layers would you have? • If you have a bottom area of 5, what would your box look like? • What would your box look like if you had 50 cubes?


• What do you notice about how the height, width and length help you with the volume? • Which size box would the Yummy Candy Company choose? Explain your thinking. • What is volume? • How does the length, width, height relate to addition? Multiplication?

Possible Misconceptions/Suggestions: Possible Misconceptions Suggestions • Student has difficulty constructing a box

that is a rectangular prism. • Provide multiple examples of boxes for

student to examine then build with cubes. Make a connection to rows and columns.

Special Notes:

This is an introductory lesson for exploring the meaning of volume. Students need time to create their own prisms and explore their properties. Students develop their own way of finding the volume.

Solutions: Multiple solutions

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Yummy Candy Boxes

Box Dimensions

Box 1

Box 2

Box 3

Box 4

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Exploring Volume as Additive Common Core Standard:

Geometric measurement: Understand concepts of volume and relate volume to multiplication and to addition. 5. MD.5 Relate volume to the operations of multiplication and addition and solve real world andmathematical problems involving volume. 5. MD.5a Find the volume of a right rectangular prism with whole number side lengths by packing itwith unit cubes, and show that the volume is the same as would be found by multiplying the edge lengths, equivalently by multiplying the height by the area of the base. Represent threefold whole-number products as volumes, e.g., to represent the associative property of multiplication. 5. MD.5c Recognize volume as additive. Find volumes of solid figures composed of two non-overlapping right rectangular prisms by adding the volumes of the non-overlapping parts, applying this technique to solve real world problems.

Additional/Supporting Standard: 5. MD.3 Recognize volume as an attribute of solid figures and understand concepts ofvolume measurement.

a. A cube with side length 1 unit, called a “unit cube,” is said to have “one cubic unit”of volume, and can be used to measure volume.

b. A solid figure which can be packed without gaps or overlaps using n unit cubes issaid to have a volume of n cubic units.

5. MD.4 Measure volumes by counting unit cubes, using cubic cm, cubic in, cubic ft, andimprovised units.

5. MD.5b Apply the formulas V= l x w x h and V = B x h for rectangular prisms to find volumes ofright rectangular prisms with whole-number edge lengths in the context of solving real world and mathematical problems.

Standards for Mathematical Practice: 2. Reason abstractly and quantitatively.4. Model with mathematics.5. Use appropriate tools strategically.7. Look for and make use of structure.

Student Outcomes: • I can use addition and/or multiplication to find volume.• I can describe how to find the volume of a right rectangular prism.• I can find the volume of a figure made from two right rectangular prisms.

Materials: • Centimeter cubes, preferably interlocking; or other interlocking cubes

NC DEPARTMENT OF PUBLIC INSTRUCTION FIFTH GRADE 75

Advance Preparation: • Use the interlocking cubes to assemble these rectangular solids. Each solid should be a

different color.

4 x 3 x 5 4 x 3 x 1

1 x 3 x 6

5 x 1 x 6

• Consider how you will pair or group students• Students should be familiar with processes of multiplication• Students should be comfortable with estimating• Students should be comfortable explaining their reasoning

Directions: Note: This lesson can be done along with the “Representing and Finding Volume” lesson, which develops the basic concept of volume measurement as filling a shape with cubic units (5.MD.5a); but it also develops the idea of volume as additive (5.MD.5c). It helps students to see how adding one more layer to any dimension makes a difference to the total volume, illustrating the difference one or more units in each dimension makes. It may also help students make better estimates of leftover space if the cubes do not exactly fill up the boxes in the “Representing and Finding Volume” lesson. This lesson is a whole group lesson. The class discussion is key to students understanding the concepts you are presenting.

1. Pose this problem to the class: Juanita measured a box’s dimensions to the nearest wholenumber and its dimensions were: 4 centimeters long, 3 centimeters wide, and 5 centimetershigh. What was its volume? Show the 4 x 3 x 5 prism you made with the centimeter (orother) interlocking cubes. (If you have used some other unit cube rather than centimetercubes, change the problem to reflect the units you used.) Have students consider the volumeof the prism in pairs, then report their conclusions. They should figure out that the prism has60 cubes in it, so Juanita’s box would have a volume of 60 cubic centimeters. Ask how theyfigured it out. Some students will have multiplied 4 x 3, then multiplied 12 by 5. Others mayhave used the associative and commutative properties to multiply 5 x 4 first, then multiply20 by 3. Here is another good opportunity for children to see that in multiplication, the orderof the factors does not change the result.

When all agree that Juanita’s box would hold 60 cubic centimeters, ask How much largerwould the volume be if it were one centimeter higher? Have the children make estimatesof how many more cubic centimeters the box would hold. Then show them the originalprism and the 4 x 3 x 1 prism you made (in a different color) and add it to the top of the4 x 3 x 5 prism.


4 x 3 x 1 (12 cu cm)

4 x 3 x 6 (72 cu cm)

4 x 3 x 5 (60 cu cm)

Students should see that adding one centimeter to the height of 5 cm, making the height 6 cm, adds another layer of 4 x 3, or 12 cubic centimeters, making the total volume of the new prism (box) 72 cubic centimeters.

2. Now ask the children to predict how much larger Juanita’s box would be if not only theheight were larger by one centimeter, but the length were also larger by one centimeter.Make sure they understand that a vertical layer as well as the top layer would be added.After the children have made predictions (and given some explanations of why they aremaking the predictions), show them the new 4 x 3 x 6 prism and the 1 x 3 x 6 prism whichwill be added to the length dimension of the prism.

1 x 3 x 6 4 x 3 x 6 5 x 3 x 6 (18 cu cm) + (72 cu cm) = (90 cu cm)

3. Next ask for estimates of how much larger Juanita’s box would be if all three dimensionswere one centimeter longer. After the students give estimates and explanations for them,repeat the demonstration, adding the 5 x 1 x 6 prism to the “box.”

5 x 4 x 6 (120 cu cm)


The discussion following this demonstration should emphasize that adding just one more unit to each of the three dimensions can make quite a difference in the volume of a box. In this case, the volume grew from 60 cu cm to 120 cu cm, doubling the original volume. It is also important to emphasize that increasing a dimension by one unit means adding another layer of cubes to the volume, which demonstrates the “additive” nature of volume (5.MD.5c).

4. Students may make the conjecture that increasing each dimension by one unit would alwaysdouble the volume. Provide pairs or groups a set of cubes and ask them to create a rectangularprism, find its volume, and then add one unit to each dimension as you did in the demonstration.Suggest that for each extra unit in one dimension, they use a different color for the cubes as youdid in the demonstration. Have them record their results. Make a class chart and follow up witha class discussion of the results. They will find that adding one unit to each dimension will notalways (or even often) double the total volume. Be sure they make note of any originaldimensions which do result in a doubling of the total volume when each dimension increases byone unit. Are there patterns that exist between those prisms?

Questions to Pose: During class discussion:

• How do you use multiplication when finding volume?• How do you use addition when finding volume?• Why does changing the measurement of a dimension by only one unit change the total

volume by more than one?

As students work together on #4: • What did you do to get started?• What do you think the volume of your new shape will be after changing each dimension by

one unit?• Which dimension did you decide to change first?• Why did you decide to…..?

Extension: As students work in other lessons to find the volumes of various boxes or spaces (See lesson: “Finding Volume Using a Formula”), have them figure out the volume of some of the boxes or spaces if each dimension were increased by one unit.

Possible Misconceptions/Suggestions: Possible Misconceptions Suggestions • Students see the volume as

only the cubes they can seeon the surface of the prism

• Take the interlocking cubes apart if necessary so thatstudents see the cubes that are “inside” the shape.

Special Notes: • This lesson may help students make better estimates of leftover space when placing cubes in

boxes whose dimensions are not whole numbers of the unit being used, but do not expect this one activity to greatly increase their ability to make appropriate estimates.



Finding Volume Using a Formula Common Core Standard:

Geometric measurement: Understand concepts of volume and relate volume to multiplication and to addition. 5.MD.5 Relate volume to the operations of multiplication and addition and solve real world and mathematical problems involving volume. 5.MD.5b Apply the formulas V=l x w x h and V = B x h for rectangular prisms to find volumes of right rectangular prisms with whole-number edge lengths in the context of solving real world and mathematical problems.


5. MD.3 Recognize volume as an attribute of solid figures and understand concepts of volume measurement.

a. A cube with side length 1 unit, called a “unit cube,” is said to have “one cubic unit” of volume, and can be used to measure volume.

b. A solid figure which can be packed without gaps or overlaps using n unit cubes is said to have a volume of n cubic units.

5. MD.4 Measure volumes by counting unit cubes, using cubic cm, cubic in, cubic ft, and improvised units.

Standards for Mathematical Practice: 2. Reason abstractly and quantitatively. 4. Model with mathematics. 5. Use appropriate tools strategically. 6. Attend to precision. 8. Look for and express regularity in repeated reasoning.

Student Outcomes:

• I can use addition and/or multiplication to find volume. • I can describe how to find the volume of a right rectangular prism. • I can use a formula to find the volume of a right rectangular prism.

Materials:

Parts 1 and 2: • At least one larger box, appropriate to measure dimensions in inches; several other larger

boxes and/or rectangular prism shaped objects found in the classroom (possibilities: cartons, small bookcases, tissue boxes, cereal boxes, etc.)

• Rulers and/or measuring tapes Part 2: • String, tape, construction paper or other material at least 12 inches on one side, cm graph

paper, scissors, yardstick, meter stick

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Advance Preparation: • Consider how you will pair or group students. • Prepare a list of rectangular prism shapes in the classroom for use in Part 1 (to be measured

in cubic centimeters or cubic inches) and a list of rectangular prism shapes large enough to be measured in larger units (cu dm, cu ft., cu m, cu yd.) to be used in Part 2

• Students should be familiar with processes of multiplication. • Children should be familiar with the processes of measurement, especially length and area,

and the units used for each. • Students should be comfortable explaining their reasoning. • Students should be familiar with estimating.

Directions: NOTE: This lesson should follow the lesson “Representing and Finding Volume” (5.MD.5a). The activities in this lesson should be done over the course of at least 2-3 days. Part 1

1. Begin by reminding the students of their work filling boxes with centimeter cubes in the previous lesson (“Representing and Finding Volume”). Ask them what they discovered about finding the volume of the boxes: units needed to be cubes in order to fill the space with no gaps; knowing the number of cubes that match the length of the box, the width of the box, and the height of the box allowed one to figure out the volume, the volume could be found by multiplying the length by the width by the height, represented by the formula V= l x w x h or V = B x h. Have someone share how these two formulas are alike and how they are different. Students should recognize that in the second formula, the B stands for the area of the base, found by multiplying the length by the width (1 x w).

2. Show the students the larger box. Tell them that you do not want to fill this box with centimeter cubes – that you may not have enough cubes even if you wanted to take the time to fill the box. Ask what you need to know about the box in order to find the volume. Someone should respond that you need to know the length, width, and height of the box, in other words, the dimensions of the box. Ask them how you could find the dimensions without filling the box with cubes. Someone should suggest measuring with a ruler. Ask if there is a unit that might be more appropriate for the size of the box than the centimeter. Hopefully someone will suggest using inches. (Someone might suggest decimeters. Acknowledge that this would be appropriate, but in this case the smaller inches will be more precise.) Have a student help you do the measurements. Record the measurements of the dimensions; e.g. length = 10 inches, width = 7 inches, height = 5 inches. Have the students suggest the equation to use to find the volume of this box. They should give you V = 10 x 7 x 5, computed as either 70 x 5 or 10 x 35 or even 50 x 7 (again using the associative property). The volume of this box would be 350 cubic inches. Emphasize the importance of using the correct unit, in this case cubic units for volume. Measuring volume is measuring a 3-dimensional attribute, so the unit used must have 3 dimensions. Show them the cubic inch you made from the “network for cubic inch.” (See attached sheet.) Or if you have commercial blocks that are a cubic inch in size, you may use them as a model. Students need to see models of the cubic units in order to be able to visualize the size of the units they are using.

3. Have the students in pairs find the volumes of a variety of boxes and/or rectangular prism shapes in the classroom in square inches. They may record on the “Finding Volume” sheet from the previous lesson or on other paper. Be sure they identify the object measured, record

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the equation and solution, including the unit. Circulate as pairs work to be sure they are measuring as accurately as possible, measuring the correct dimensions, recording the equation correctly, and finding accurate solutions to the equations, including the appropriate unit.

4. As closure for this activity, show the students a box that has not been used in any of the previous activities. Ask the students to write a description of how to find the volume of the box. They should include the units they are using as well as the process of measuring.

Part 2

1. Begin the class by suggesting to the students that you would like to find the volume of the classroom (or other room that is a rectangular prism in shape.) Ask if the cubic centimeter or cubic inch would be sensible units to use. Hopefully someone will suggest that a bigger unit would be wiser because the number of units for each dimension would be smaller. If no one suggests this, do so yourself. Remind the students that one edge of a centimeter cube is one centimeter in length. Ask for suggestions of a cube with a longer edge that would be useful in finding the volume of the classroom. Suggestions may (and should) include cubic decimeters, cubic meters, cubic feet, and cubic yards. List these suggestions on the board, reminding the students that each of these units is a cube whose edges are the given length. Tell them that before they figure out the volume of the classroom, you would like them to have a mental picture of the size of each of these units. The students will construct at least one example of each. Students could work in groups of from 2-4 students, depending on the unit being created and the materials available. Have these materials available: For cubic decimeter: centimeter graph paper, scissors, tape For cubic meter: string, meter stick, tape (masking or painters tape) For cubic foot: construction paper (at least 12 inches on each side; e.g., 12x18 inches) or other large paper, poster board, or cardboard; rulers, tape For cubic yard: string, yardstick, tape (masking or painters tape) Directions:

Cubic decimeter: Students cut square decimeter (10 x 10 centimeter) panels from the centimeter graph paper and tape them together to make a cube. (The thousands cube from Base-10 materials is also an example of a cubic decimeter.) Groups making the cubic decimeter might also make more examples of the cubic inch from the network. Cubic foot: Students cut square feet from construction paper (or other appropriate material) and tape the panels together to form a cube with edges one foot in length. Cubic meter or Cubic yard: In a corner of the room, have the students measure and mark with tape a square meter (or square yard) on the floor. Place the meter stick (or yardstick at the corner of the square opposite the corner of the room. Tape it so it will stand up. Then run a string from the top of the stick to the wall so that it is parallel to the floor and the other wall.

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strings wall wall meter stick or yardstick floor

2. After the students have made the models of the cubic units, bring them back to the original question of finding the volume of the classroom. Ask them which unit would be the most sensible. Hopefully someone will suggest either the cubic meter or cubic yard. Have several students provide an estimate of the volume of the classroom in cubic meters (or cubic yards) and share how they came up with the estimate. Record these for comparison with the actual result. Ask what needs to be done in order to find the volume. Have a few students use meter or yard sticks or tapes to measure the length, width, and height. (It may be necessary to estimate the height in some cases.) Have students (perhaps in pairs) write and solve the equation to find the volume of the space; then have a volunteer share his/her equation and solution. Be sure to have students with different equations share. Discuss whether or not both/all give the correct solution. Compare the solution with the estimates. Did a particular strategy (or strategies) seem to help students make estimates closer to the actual volume?

3. Put the children in groups of 3 or 4. Have appropriate measuring tools available. Post the list of rectangular prism objects they may choose to measure. Tell them they should choose at least 4 of the objects, one each to be measured in the four larger units (cu dm, cu ft., cu m, cu yd.) For each task, the group should record the name of the object, an estimate of the object in the chosen unit, the equation used to find the volume, and the volume labeled with the appropriate unit. When a group has finished one task, they should choose another object which is not being measured by another group. Try to provide enough time for each group to be able to find at least one volume in each of the four larger units. **Have each of the four larger unit models measured in smaller units by at least three groups to aid in the follow-up discussion, e.g., have the cubic meter measured in cubic decimeters, or the cubic foot measured in cubic inches.

4. When the groups have had enough time to complete the measurements and calculations, call them together again to discuss results. Ask groups who measured the same objects to compare the capacities they found. If there are differences, ask why these occurred (probably because of differing estimates of heights when these were needed or because of differences in rounding lengths to the nearest unit). Ask questions related to the suitability of the new units to certain tasks. For example: Would you measure a wastebasket’s capacity in cubic meters? (No; all dimensions are less than one meter.)

Would you measure the volume of the room in cubic decimeters? (Probably not; the numbers that would result would be quite large.)

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Pay special attention to the measurements of the four larger units in smaller units. The students do not need to memorize these equivalences, but should understand them.

1 cubic meter = 1,000 cubic decimeters (10 x 10 x 10)

1 cubic decimeter = 1,000 cubic centimeters (10 x 10 x 10) 1 cubic yard = 27 cubic feet (3 x 3 x 3)

1 cubic foot = 1,728 cubic inches (12 x 12 x 12)


• What did you think about first? • Show me how you are using this tool (ruler, measuring tape, etc.). • Why did you decide to use that unit? • How is finding volume different from finding area? How is it the same?


• Does it matter which dimensions you count as the length, the width, and the height? Would that change your result?

• What strategies did you use for estimating the volumes? • Why is choosing an appropriate unit important when measuring an object?

Extension When the students have had lots of experience measuring actual objects to find the volume in various cubic units, provide challenging questions such as the following:

1. The Renews-it Recycling Company filled a back room with cubes of compacted metal. Harold counted 11 cubes along the side wall and 9 cubes along the back wall. He knew that the workers had put 1188 cubes in the back room and that it was completely full. How many layers of cubes did the room hold? (length, width, and volume given; need to find height)

2. Jeneen’s backpack was 18 inches high, 12 inches wide, and 8 inches deep. She needed to carry her social studies project to school in the backpack. It was packed in a box with a volume of 1800 cubic inches that was 12 inches wide and 8 inches deep. Would it fit into her backpack? Why or why not? (find volume of backpack and compare to volume of box)

3. The carpenter’s toolbox was 5 decimeters by 9 decimeters by 10 decimeters. He needed to put in it 225 cu dm of nails, 150 cu dm of screws, 50 cu dm of washers of different sizes, and 75 cu dm of twine. Does he have enough room for these items? Why or why not? (find volume of toolbox; add other volumes and compare)

Special Notes Placing cubes in boxes as the students did in the previous activity is actually measuring capacity, generally used to refer to the amount that a container will hold. Volume typically refers to the amount of space that an object takes up. Both volume and capacity are terms for measures of the size of three-dimensional regions. These are not distinctions to be concerned about. The term volume can also be used to refer to the capacity of a container, and that is how it is used in these lessons. (See Teaching Student-Centered Mathematics, Grades 3-5. Van de Walle and Lovin (2006).)

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Possible Misconceptions/Suggestions: Possible Misconceptions/Problems Suggestions • Students confuse linear units or

units of area with units of volume or leave out the unit altogether when recording volume

• Having models of the different units available may be helpful. It will be important to focus on the attribute being measured, i.e., for volume the attribute is how much space an object takes up, so the unit must take up space. The discussion during these activities should keep a focus on the units for volume as cubic units. A chart posted in the classroom outlining the linear, area, and volume units for each base unit may be helpful.

• Be sure that students always record the unit for any kind of measurement. Emphasize that the number alone doesn’t tell you anything about the size of the object. You need to know if “3” means 3 cubic centimeters, 3 miles, 3 gallons, etc. It is meaningless as a measurement without the unit.

• Students have difficulty performing the calculations involving multiplication

• If the measures of the dimensions are larger, students could use a calculator to find the volume. (The focus of these lessons is on finding volume, not on accurate computation, although this is a perfect opportunity to use multiplication skills in context.)

• You may need to have some mini-lessons for any students who continue to show difficulty with accurate multiplication.

• Students have difficulty using measuring tools appropriately; e.g., not lining up ruler appropriately beginning at zero units; counting hash marks rather than spaces, etc.

• By fifth grade, most students should have had experience using rulers and other measurement tools for length, but some may need a refresher. Troubleshoot with individuals and small groups as they work on the activities. For example, be sure they recognize whether or not their tool has a leader before the zero mark. Be sure they understand that one unit on a ruler is the space between the hash marks, not the first hash mark which denotes zero units.

Solutions for Problems Used in This Lesson: Parts 1-2: Solutions will vary with the objects being used in the activities.

Extension: 1) 12 layers 2) No; Volume of backpack is only 1728 cubic inches, less than the 1800 cubic inches needed for the box 3) No; Volume of toolbox is 450 cu dm; he needs 225 + 150 + 50 + 75 = 500 cu dm

Activities adapted from Developing Mathematical Processes, Topic 75, Standard Cubic Units, The Wisconsin Research and Development Center for Cognitive Learning, University of Wisconsin-Madison. 1974. Most recently published by Delta Education, Nashua, NH. Currently out of print.

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Net for cubic inch Cut along the outer lines, then fold along the interior lines to form a cubic inch. Tape the sides together to make a model of a cubic inch.

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Representing and Finding Volume Common Core Standard:

Geometric measurement: Understand concepts of volume and relate volume to multiplication and to addition. 5.MD.5 Relate volume to the operations of multiplication and addition and solve real world and mathematical problems involving volume.

5.MD.5a Find the volume of a right rectangular prism with whole number side lengths by packing it with unit cubes, and show that the volume is the same as would be found by multiplying the edge lengths, equivalently by multiplying the height by the area of the base. Represent threefold whole-number products as volumes, e.g., to represent the associative property of multiplication.




5. MD.4 Measure volumes by counting unit cubes, using cubic cm, cubic in, cubic ft, andimprovised units.

Standards for Mathematical Practice: 2. Reason abstractly and quantitatively.4. Model with mathematics.5. Use appropriate tools strategically.6. Attend to precision.8. Look for and express regularity in repeated reasoning.

Student Outcomes: • I can use addition and/or multiplication to find volume.• I can describe how to find the volume of a right rectangular prism.• I can use a formula to find the volume of a right rectangular prism.

Materials: • Approximately 16 relatively small rectangular boxes such as those used for powdered

sugar, cookies, toothpicks, animal crackers, tea bags, paper clips, small puzzles, and frozen vegetables

• Centimeter cubes, enough to fill each of the 16 boxes about half full• A few marbles or other small spheres• Finding Volume recording sheet


Advance Preparation: • Gather at least 16 different rectangular boxes, each with a base less than 100 square

centimeters in area and opened so that the largest faces are the top and bottom. (The children could help bring in boxes.) Label a smallish box, like a paper clip box as box A. Label each other box with different letters.

• Except for Box A, put enough centimeter cubes in each box to make a little more than alayer at the bottom

• Copy Finding Volume recording sheet, one for each student or one for each pair• Consider how you will pair students• Students should be familiar with processes of multiplication• Children should be familiar with the processes of measurement, especially length and area,

and the units used for each• Students should be comfortable explaining their reasoning

Directions:

1. Begin the lesson with this story: The Renews-it Recycling Company recycles cans, autoparts, machine parts, and other metal products. They store the metal in bins in a warehouseuntil they have enough to ship to a factory that makes new products from the recycled metal.But they have run into a problem: They have run out of space in the warehouse before theyare scheduled to ship the metal to the factory. They have no room to build a new warehouse,so they have decided to compact the metal.One person suggested that the metal be compacted into brick shapes; another suggestedcompacting the metal into balls. Some other shapes were suggested, too. They have todecide into what shape they should use to compact the metal. Because their space is limited,no space should be wasted.Then ask: What shape should they choose? Which shape will fill the space with no gaps?What do you think? Why?

Hopefully, the students will suggest cubes or rectangular box shapes (like bricks). If thisidea is not forthcoming or some children seem unsure about it, do the following:

Show them Box A and the marbles. Have a child begin to fill the box with the marbles. If you have a document camera, show this process so the whole class can get a good view. Ask the class what they are noticing. They should notice that the marbles can’t fill the whole space because of their round shape. Show them the centimeter cubes and ask if they would fill the space better? They should realize that the cubes can fit together with no gaps.

Have another child fill the box with centimeter cubes. At this point, the students should realize that the metal compacted into cubes would be an efficient way to store the recycled metal without wasted space.

Ask the class how to find out how many cubes were needed to fill the box. Some may suggest counting them. Ask if they want you to dump the cubes out and count them one at a time – or if there might be a more efficient way. Listen to whatever suggestions are made. Hopefully someone will suggest that you could find out how many are in one layer and then count the number of layers. Ask if there is an efficient way to find out how many are in one


layer. Students should recognize that the cubes are arranged in an array so that they could multiply the length by the width to find out the number in one layer. Be sure they recognize this as the area of the base of the box. Ask students if you need to completely fill in the rest of the layers in order to know how many layers are needed to fill the box. Hopefully someone will suggest stacking a column of cubes in one corner of the box and counting to see how many layers will fit. They may have other suggestions. Be sure that stacking cubes in one corner is one of the suggestions and that students recognize this as the most efficient strategy when using the actual cubes.

2. Ask the students to suggest an equation to describe the situation. For a box with length of 8cm, width of 6 cm, and height of 5 cm, they might suggest 8 x 6 x 5 or 48 x 5. Ask studentswhat these numbers describe. The first example describes the length x the width x the heightin centimeters. The second describes the area of the base (in square centimeters) times theheight in centimeters. Be sure they understand that the 48 is the result of multiplying thelength (8cm) by the width (6cm). Ask what unit is being used to measure the volume of thebox. Be sure that students understand the unit as cubic centimeters, differentiated fromsquare centimeters or linear centimeters.

3. Tell the children they will use centimeter cubes to find the volumes of various boxes. Showthem the 16 (or more) small boxes. Tell them they will use the cubes inside each to find thevolume, but there are not enough cubes for each to completely fill the box. Tell them to usea strategy that would not require completely filling the box. Suggest that they look for a wayto use the fewest possible cubes.NOTE: The centimeter cubes will most likely not fit exactly into each of the boxes. Tell thestudents to use as many of the cubes as possible and ignore the extra space for the purposesof this lesson. Their answers will then be approximate volumes. They can give wholenumber answers in this situation, but you may want to have students use “about” languageas they discuss their solutions; e.g., “The volume of this box is about 45 cubic centimeters.”(Measurements are by nature approximations because the units we use to measure canalways be made more precise. Each smaller unit or subdivision of a unit produces a greaterdegree of precision, but since there is mathematically no “smallest unit,” any measurementwill include some error in precision. It is important to develop the idea that allmeasurements include some degree of error.)

4. Put the children in pairs and give each pair a copy of the “Finding Volume by FindingCapacity” record sheet. Give each pair a box with cubes. They record the letter of the box,use the cubes to find the volume, either by filling the bottom layer, then stacking cubes in acorner, OR by filling one row and one column along the bottom and stacking cubes in acorner to show length, width, and height. They then record the equation that represents howthey found the volume. After students have had time to work with at least 2 boxes, call theclass back together. If any pairs have used the more efficient strategy of filling only one rowand one column of the bottom layer and the stack in the corner, have them share theirstrategy with the class. If no one has used that strategy, ask how they could use less cubesthan needed to fill the entire bottom layer. Discuss how knowing one row and one column ofthe array that forms the bottom layer is enough to find the number of cubes in the bottomlayer, which also represents the area of the base of the box.


5. Circulate as students work, asking how they are finding the volumes, what their numbersmean, and how they came up with the equations. Troubleshoot as needed. Allow time foreach pair to find the volume of at least six boxes if possible.

6. Call the class back together. Have one pair share how they found the volume of one of the boxes.Focus particularly on the equation they wrote. For example: Say Box E had a length of 9 cm, awidth of 3 cm, and a height of 4 cm. The equation might be 9 x 3 x 4 = 108 cu cm. Ask what thesenumbers represent. Students should recognize that they are multiplying the length of the box by thewidth of the box by the height of the box. Write this as the formula V = l x w x h. If they wrote theequation as 27 x 4 = 108, ask where they got the 27 (9 x 3) and what it represents (the area of thebase). Turn the box on its smaller end. Ask what the base is with the box turned in this direction (3cm x 4 cm). The equation would then be V = 3 x 4 x 9. Ask if this would change the volume (thesolution). Students should recognize that it is the same box with the same dimensions and that thesame number of centimeter cubes would fit, so that changing the order of the dimensions in theequation does not change the result. (Note that this is an example of the associative property ofmultiplication.) Likewise, in this case, the area of the base would be 3 x 4 = 12 cu cm, so we couldsay V = 12 x 9 = 108 cu cm. The formula in this case would be V = B x h, or Base x height. Capitalletters are used to designate variables that are the result of another computation, so B in thisformula represents the Area of the Base, found by multiplying the length of the base by the widthof the base. Have this kind of discussion for at least a couple more examples of the students’ work.


• How did you decide to find the number of cubes when you didn’t have enough to fill the box?• Do you think the volume of this box will be larger or smaller than the one you just measured

(or compared to one you hold for comparisons)?

During class discussion: • Does it matter which dimensions you count as the length, the width, and the height? Would

that change your result?

Possible Misconceptions/Suggestions: Possible Misconceptions/Problems Suggestions • Students record solutions

without unit labels• Be sure that students always record the unit for any

kind of measurement. Emphasize that the numberalone doesn’t tell you anything about the size of theobject. You need to know if “3” means 3 cubiccentimeters, 3 miles, 3 gallons, etc. It is meaninglessas a measurement without the unit.

• Students have difficultyperforming the calculationsinvolving multiplication

• If the measures of the dimensions are larger, studentscould use a calculator to find the volume. (The focusof these lessons is on finding volume, not on accuratecomputation, although this is a perfect opportunity touse multiplication skills in context.)

• You may need to have some mini-lessons for anystudents who continue to show difficulty withaccurate multiplication.


Special Notes: When the children are placing cubes in the boxes in Part 1, they are actually measuring capacity, which is generally used to refer to the amount that a container will hold. Volume typically refers to the amount of space that an object takes up. Both volume and capacity are terms for measures of the size of three-dimensional regions. These are not distinctions to be concerned about. The term volume can also be used to refer to the capacity of a container, and that is how it is used in these lessons. (See Teaching Student-Centered Mathematics, Grades 3-5. Van de Walle and Lovin (2006).)

Solutions for Problems Used in This Lesson: Solutions will vary with the objects being used in the activities.

Adapted from Developing Mathematical Processes, Topic 75, Standard Cubic Units, The Wisconsin Research and Development Center for Cognitive Learning, University of Wisconsin-Madison. 1974. Most recently published by Delta Education, Nashua, NH. Currently out of print.


Finding Volume by Finding Capacity Your job is to find the volume of several boxes in cubic centimeters by finding the capacity of each box. Use the centimeter cubes and an efficient method to find the dimensions that you need. Fill in the chart with the equation you use to find the volume. Solve on the back of this paper any sentences that you cannot solve mentally. Be sure to record the unit of volume with your answer.

Letter of the Box Equation

Describe your process. Why does this process help you find the volume of the box?



Volume as Additive Common Core Standard:

Geometric measurement: Understand concepts of volume and relate volume to multiplication and to addition. 5. MD.5 Relate volume to the operations of multiplication and addition and solve real world andmathematical problems involving volume.

5. MD.5c Recognize volume as additive. Find volumes of solid figures composed of two non-overlapping right rectangular prisms by adding the volumes of the non-overlapping parts, applyingthis technique to solve real world problems.




5. MD.4 Measure volumes by counting unit cubes, using cubic cm, cubic in, cubic ft, andimprovised units.5. MD.5a Find the volume of a right rectangular prism with whole number side lengths by packing itwith unit cubes, and show that the volume is the same as would be found by multiplying the edgelengths, equivalently by multiplying the height by the area of the base. Represent threefold whole-number products as volumes, e.g., to represent the associative property of multiplication.5. MD.5b Apply the formulas V=l x w x h and V = B x h for rectangular prisms to find volumes ofright rectangular prisms with whole-number edge lengths in the contest of solving real world andmathematical problems.

Standards for Mathematical Practice: 1. Make sense of problems and persevere in solving them.4. Model with mathematics.5. Use appropriate tools strategically.7. Look for and make use of structure.

Student Outcomes: • I can use addition and/or multiplication to find volume.• I can describe how to find the volume of a right rectangular prism.• I can use a formula to find the volume of a right rectangular prism.• I can find the volume of a figure made from two right rectangular prisms.

Materials: • Paper or journals for student responses; and/or chart paper for group responses• Measuring tools: rulers, measuring tapes (Be sure both standard and metric unit tools are available)• Boxes of various sizes (not larger than about 1 foot by 1 foot), tape (masking, painters, or packaging)

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• Gather boxes - see Materials above. Tape pairs of boxes together to create new figures composed of two rectangular prisms. For example: If possible, make enough for pairs or small groups of children to each have a shape to work with simultaneously. Label each with a letter, A, B, …

• Consider how you will pair or group students. • Students should be familiar with processes of multiplication. • Children should be familiar with the processes of measurement, especially length and area,

and the units used for each. • Students should be familiar with the process and formula for finding volume of rectangular prisms. • Students should be comfortable explaining their reasoning.

Directions: NOTE: This lesson should follow the lessons “Representing and Finding Volume” (5.MD.5a) and “Finding Volume Using a Formula” (5.MD.5b).

1. Ask the students to tell you what they have learned so far about finding the volume of a box shaped figure (rectangular prism). When appropriate descriptions of the process have been given, show them one of the figures you made composed of two rectangular prisms. Tell them that you want them to work with a partner to find the volumes of some of these shapes. Do not tell them how to do it.

2. Assign partners or small groups and give each pair or group one of the prepared shapes. Tell them to find a way to figure out the volume of the shape, to record all their work, and to then explain the process they used. Each child should record their own work and their own explanation. (Or groups could create a common recording on chart paper. Be sure that everyone in the group is contributing to the final product.) The pairs or groups could work with a second figure if there is time.

3. Monitor the pairs/groups as they work, noting different strategies. Most may measure the two prisms separately, figure the volume of each, and then add the two (which mirrors the standard 5.MD.5c). Some might visualize the figure with the “cut-out” part included, find the volume of the larger figure, then subtract the “cut-out” part.

Choose 2 or 3 groups to share their process and solution. If more than one group finds the volume of the same figure and does it in different ways or finds different solutions, be sure to include those in the discussion. If mistakes are made, give the class an opportunity to discover what the mistake is, how it was made and how it can be corrected - in the safe environment of a learning community where all students (and the teacher) are learning together.


4. If students have not completed the explanations of their work, allow time for them to finish.If students worked as a group, have each student work individually to respond to the prompt:What strategies did you use to find the volume of your shape? Describe your process.


• What did you think about first?• What unit of measurement makes sense for you to use with this shape?• Why did you decide to…..? • Make an estimate of the volume of your shape. Was your estimate a reasonable one

compared to the actual volume?

During class discussion: • What strategies did you use to find the volume of your shape?• How did our previous work with volume help you find the volume of your shape?• How did you decide which unit of measurement to use?

Extension: Have the students suggest spaces in the school building that are made up of two (or more) rectangular prisms. Perhaps your classroom is not a single rectangular prism. An L-shaped hallway would be another example. Have the class discuss how they might find the volume of one or more of these spaces, including appropriate units of measure to use. You might have pairs or groups find the volume of some of these spaces. Have them make estimates first. Record these to compare to the results. Be sure to have the pairs/groups record their work including an explanation of their process and their result.

Possible Misconceptions/Suggestions: Possible Misconceptions Suggestions • Students have trouble seeing

that their shape is made oftwo rectangular prisms whosevolume they already knowhow to find

• Ask the students if they see any part of the shape is like theboxes whose volumes they have been finding in previousclasses. Hopefully they will see that the new shape iscomposed of two of these box shapes. Then ask, “How canyou use that information to find the volume of this newshape?” Continue to ask such questions until they see thatthey can find the separate volumes of the two parts of thenew shape and add them to find the total volume.

• Students do not record theunit for volume or record anincorrect unit, e.g., squareinches or inches rather thancubic inches

• Ask students what they are measuring when they arefinding volume. They should answer that they arefinding the amount of 3-D space the shape takes up. Askthem if the square unit or linear unit measures 3-Dspace. They should recognize that they need a unit thatis also 3-dimensional.

• Be sure that students always record the unit for any kindof measurement. Emphasize that the number alonedoesn’t tell you anything about the size of the object.You need to know if “3” means 3 cubic centimeters, 3miles, 3 gallons, etc. The number alone is meaninglessas a measurement without the unit.

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• Students have difficultyperforming the calculationsinvolving multiplication

• If the measures of the dimensions are larger, studentscould use a calculator to find the volume. (The focus ofthese lessons is on finding volume, not on accuratecomputation, although this is a perfect opportunity touse multiplication skills in context.)

• You may need to have some mini-lessons for anystudents who continue to show difficulty with accuratemultiplication.

• Students have difficulty usingmeasuring toolsappropriately; e.g., not liningup ruler appropriatelybeginning at zero units;counting hash marks ratherthan spaces, etc.

• By fifth grade, most students should have hadexperience using rulers and other measurement tools forlength, but some may need a refresher. Trouble shootwith individuals and small groups as they work on theactivities. For example, be sure they recognize whetheror not their tool has a leader before the zero mark. Besure they understand that one unit on a ruler is the spacebetween the hash marks, not the first hash mark whichdenotes zero units.

Special Notes: • This lesson should follow previous work with understanding the concept of volume and

discovering the formula for volume.

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Growing Sumandas

Common Core Standard: Graph points on the coordinate plane to solve real-world and mathematical problems. 5.G.2 Represent real world and mathematical problems by graphing points in the first quadrantof the coordinate plane, and interpret coordinate values of points in the context of the situation.

Additional/Supporting Standards: 5.G.1 Graph points on the coordinate plane to solve real-world and mathematical problems5.OA.3 Analyze patterns and relationships

Standards for Mathematical Practice: 2. Reason abstractly and quantitatively4. Model with mathematics5. Use appropriate tools strategically7. Look for and make use of structure

Student Outcomes: • I can model and extend a pattern• I can represent a pattern using models, drawings, words, tables, and graphs• I can make connections between a pattern and a graph

Materials: • Pattern blocks (at least 15 squares and 5 triangles per student or pair of students)• Growing Sumandas sheet (1 for each student)

Advance Preparation: • Gather pattern blocks• Copy the Growing Sumandas sheet• Consider whether students will work in pairs or as a class for this task• If students are working in pairs, they should have some experience with patterns, including

working with tables• If students are working pairs they should have experience using a table to plot points on

a coordinate grid.

Directions: 1. Distribute pattern blocks.2. Present the following scenario to the students:

During your scientific exploration to a newly discovered rainforest you encounter a newcreature called a sumanda.

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3. Invite students to build the sumanda with their blocks. Tell them that this is the sumandaat 1 year old.

4. Tell students that they will observe the sumanda’s growth. Demonstrate and have studentsbuild years 2, 3, and 4 beneath year one:

2 years old

3 years old

4 years old

5. Use the Growing Sumandas sheet to guide your instruction. Ask students to complete onesection at a time, then elicit student answers and model the component. However: If studentshave experience with this process, have them work in pairs with much less teacher guidance.

6. To close, extend student thinking by asking the questions below.

Questions to Pose: As students work in pairs:

• How did you decide to draw your table? How does your table relate to the blocks?• What surprised you about the graph? Why was it surprising?• What challenges are you running into? What are you doing to help you face them?

After: • What do you notice about the shape of the graph? Why does this shape make sense?• How does the graph help you describe the sumanda’s growth?• How can the graph help you predict the sumanda’s future growth?• What does this point on the graph tell you about the sumanda?

Possible Misconceptions/Suggestions: Possible Misconceptions Suggestions

Students have difficulty placing the attribute on the axis (number of years on the x-axis and number of blocks on the y-axis).

Have the students examine a collection of simple line graphs. Ask them to draw conclusions about the type of data listed on each axis. Note that the x-axis shows the data you are changing. The y-axis should show the data being measured.

Special Notes: A follow up task should follow this process with 2 patterns. The patterns should be graphed on the same coordinate grid and analyzed (5.OA.3).

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Solutions: Extend the Pattern:

5 years old

6 years old

Words: Student words will vary. One example is - Each year, the sumanda grows one square on each side.

Table: Function tables may be set up horizontally or vertically. One example is – Years old (n) 1 2 3 4 5 6 Number of blocks 4 6 8 10 12 14

Number of blocks at the 10th stage: 22

Pattern Rule: Student answers may vary. Some examples are – • There are twice as many squares as the age of the sumanda. There are always 2 triangles.• Add 2 blocks to the previous year• Multiply the age by 2 then add 2 more• Number of blocks = 2 n + 2• Number of blocks = (2 x n) + 2

Coordinate points: (1,4) (2,6) (3,8) (4,10) (5,12) (6,14)

Adapted from Partners for Mathematics Learning, 2008

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Growing Sumandas, side 1

Based on this pattern, draw and extend the sumanda for 2 more years:

5 years old

6 years old

Use words to describe the pattern Create a table to describe the pattern

Draw and describe the 10th stage of the pattern

Write a rule for this pattern


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Growing Sumandas, side 2

Use the table to write the coordinate points for this pattern:

Plot the coordinate points on the graph below. Should these points be connected to form a line graph? How do you know?

Title ______________________________

______________________________


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