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Grade 2 Mathematics Frameworks

Unit 6Addition and Subtraction

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Georgia Performance Standards FrameworkSecond Grade Mathematics Unit 6 1st Edition

Unit Five: Addition and Subtraction (6 Weeks)

TABLE OF CONTENTS

Overview......................................................................................................................................3

Key Standards & Related Standards............................................................................................4 Enduring Understandings ............................................................................................................5 Essential Questions......................................................................................................................6

Concepts & Skills to Maintain.....................................................................................................6

Selected Terms and Symbols.......................................................................................................7

Classroom Routines.....................................................................................................................8

Strategies for Teaching and Learning..........................................................................................14 Evidence of Learning...................................................................................................................14 Tasks............................................................................................................................................15

Place Value Play..............................................................................................................16 Building Base Ten Numbers............................................................................................20 Shake, Rattle, and Roll.....................................................................................................24 Take 100...........................................................................................................................30 Multi-digit Addition Strategies........................................................................................36 Addition with Regrouping...............................................................................................40 Grocery Store Math..........................................................................................................45 Sale Flyer Shopping.........................................................................................................48 Subtraction: Modeling with Regrouping.........................................................................53 Subtraction Story Problems.............................................................................................59 Menu Math.......................................................................................................................63 Counting Mice.................................................................................................................66 Every Picture Tells a Story..............................................................................................70 Mental Mathematics.........................................................................................................73 Culminating Task: How Many Buses?..........................................................................78

Georgia Department of EducationKathy Cox, State Superintendent of Schools

MATHEMATICS GRADE 2 UNIT 6: ADDITION AND SUBTRACTIONMay 2010 of 88

Copyright 2010 © All Rights Reserved


OVERVIEW

In this unit students will: correctly add and subtract two whole numbers two- and three-digits each with regrouping solve problems and check solutions by using the inverse relation between addition and

subtraction with two- and three-digit numbers solve problems using mental math strategies with one- and two-digit numbers use the commutative, associative, and identity properties to simplify problems involving

one- and two-digit numbers Although the units in this instructional framework emphasize key standards and big ideas at specific times of the year, routine topics such as counting, time, money, positional words, patterns, and tallying should be addressed on an ongoing basis through the use of calendar, centers (tubs), and games. It is important that routines be established allowing students to gradually understand and build on the concepts of number and time.

To assure that this unit is taught with the appropriate emphasis, depth, and rigor, it is important that the tasks listed under “Evidence of Learning” be reviewed early in the planning process. A variety of resources should be utilized to supplement, but not completely replace, the textbook. Textbooks not only provide much needed content information but excellent learning activities as well. The tasks in this units illustrate the types of learning activities that should be utilized from a variety of sources.





KEY STANDARDS AND RELATED STANDARDS

KEY STANDARDS M2N2. Students will build fluency with multi-digit addition and subtraction.

a. Correctly add and subtract two whole numbers up to three digits each with regrouping.

b. Understand and use the inverse relation between addition and subtraction to solve problems and check solutions.

c. Use mental math strategies such as benchmark numbers to solve problems. d. Use basic properties of addition (commutative, associative, and identity) to simplify

problems (e.g. 98 + 17 by taking two from 17 and adding it to the 98 to make 100 and replacing the original problem by the sum 100 + 15).

e. Estimate to determine if solutions are reasonable for addition and subtraction.

RELATED STANDARDS M2N1. Students will use multiple representations of numbers to connect symbols to quantities.

a. Represent numbers using a variety of models, diagrams, and number sentences (e.g., 4703 represented as 4,000 + 700 + 3, and units, 47 hundreds + 3, or 4,500 + 203).

b. Understand the relative magnitudes of numbers using 10 as a unit, 100 as a unit, or 1000 as a unit. Represent 2-digit numbers with drawings of tens and ones and 3-digit numbers with drawings of hundreds, tens, and ones.

c. Use money as a medium of exchange. Count back change and use decimal notation and the dollar and cent symbols to represent a collection of coins and currency.

M2D1. Students will create simple tables and graphs and interpret their meaning.

a. Organize and display data using picture graphs, Venn diagrams, bar graphs, and simple charts/tables to record results.

b. Know how to interpret picture graphs, Venn diagrams, and bar graphs. M2P1. Students will solve problems (using appropriate technology).

a. Build new mathematical knowledge through problem solving. b. Solve problems that arise in mathematics and in other contexts. c. Apply and adapt a variety of appropriate strategies to solve problems. d. Monitor and reflect on the process of mathematical problem solving.

M2P2. Students will reason and evaluate mathematical arguments.

a. Recognize reasoning and proof as fundamental aspects of mathematics. b. Make and investigate mathematical conjectures. c. Develop and evaluate mathematical arguments and proofs. d. Select and use various types of reasoning and methods of proof.





M2P3. Students will communicate mathematically. a. Organize and consolidate their mathematical thinking through communication. b. Communicate their mathematical thinking coherently and clearly to peers, teachers,

and others. c. Analyze and evaluate the mathematical thinking and strategies of others.

d. Use the language of mathematics to express mathematical ideas precisely. M2P4. Students will make connections among mathematical ideas and to other disciplines.

a. Recognize and use connections among mathematical ideas. b. Understand how mathematical ideas interconnect and build on one another to produce

a coherent whole. c. Recognize and apply mathematics in contexts outside of mathematics.

M2P5. Students will represent mathematics in multiple ways.

a. Create and use representations to organize, record, and communicate mathematical ideas.

b. Select, apply, and translate among mathematical representations to solve problems. c. Use representations to model and interpret physical, social, and mathematical

phenomena.

ENDURING UNDERSTANDINGS

When one quantity is joined or added on to another quantity, the result is greater than or equal to the initial quantity.

When one quantity is removed from another quantity, the result is less than or equal to the initial quantity.

When one quantity is compared to another quantity, the initial quantity is either equal to, greater than, or less than the second quantity.

The sum of the parts is equal to the whole. Joining, removing, part-part-whole, and comparing problems can be modeled. The order of addends may be changed and the result will not change. However this is not

true for subtraction. The grouping of addends may be changed and the result will not change. However this is

not true for subtraction. Addends can be decomposed and regrouped differently to simplify adding. Regrouping may be used to add and subtract whole numbers. Solutions may be solved and checked by using the inverse relation between addition and

subtraction of numbers. Mental math strategies may be used to solve problems involving numbers. Problems involving numbers may be simplified by using the commutative, associative,

and identity properties. Reasonableness of addition and subtraction problems may be determined by using

estimation.

ESSENTIAL QUESTIONSGeorgia Department of Education

Kathy Cox, State Superintendent of SchoolsMATHEMATICS GRADE 2 UNIT 6: ADDITION AND SUBTRACTION

May 2010 of 88Copyright 2010 © All Rights Reserved


How can we represent numbers using place value? How do the value of digits change when their position in a number changes? How can we tell which numbers are larger or smaller than others? How can we compare the values of two- and three-digit numbers? What is an effective way to estimate numbers? How can estimation strategies help us build our addition skills? When will estimating be helpful to us? How can benchmark numbers help us add and subtract? How can we model and solve addition and subtraction problems with and without

regrouping? What is a number sentence and how can I use it to solve word problems? How do we use addition to tell number stories? Can we change the order of numbers when we add (or subtract)? Why or why not? How can subtraction help us to know if we added correctly? How can addition help us know if we subtracted correctly? How can we solve problems mentally? What strategies help us with this? What strategies will help me add multiple numbers quickly and accurately? How can mental math strategies, for example estimation and benchmark numbers, help us

when adding and subtracting with regrouping? Why would these strategies be useful in everyday activities? What is mental math? How does mental math help us calculate more quickly? How does a Venn diagram organize information? How do we count quantities using a Venn diagram?

CONCEPTS/SKILLS TO MAINTAIN It is expected that students will have prior knowledge/experience related to the concepts and skills identified below. It may be necessary to pre-assess in order to determine if time needs to be spent on conceptual activities that help students develop a deeper understanding of these ideas.

Fluency with single digit addition/subtraction facts to 18 Fair trades with coins or bills Duration and sequence of events Number patterns-skip count, odd/even Fact families Fractions: halves, fourths Picture graphs, bar graphs, tally marks Estimation: closest group of ten Commutative Property

SELECTED TERMS AND SYMBOLS

The following terms and symbols are often misunderstood. These concepts are not an inclusive list and should not be taught in isolation. However, due to evidence of frequent difficulty and





misunderstanding associated with these concepts, instructors should pay particular attention to them and how their students are able to explain and apply them. The definitions below are for teacher reference only and are not to be memorized by the students. Teachers should present these concepts to students with models and real life examples. Students should understand the concepts involved and be able to recognize and/or demonstrate them with words, models, pictures, or numbers.

+: symbol that represents addition

–: symbol that represents subtraction

Associative Property for Addition: no matter how the numbers are grouped, the answer will always be the same. For example, (2 + 3) + 5 = 2 + (3 + 5).

Associative Property for Multiplication: no matter how the numbers are grouped, the answer will always be the same. For example (2 x 3) x 5 = 2 x (3 x 5).

Commutative Property for Addition: changing the order of two numbers to be added does not change the sum. For example 3 + 4= 4 + 3.

Commutative Property for Multiplication: changing the order of two numbers to be multiplied does not change the product. For example 6 x 4 = 4 x 6.

Difference: the amount that remains after one quantity is subtracted from another

Identity Property for Addition: when zero is added to a number the result is the number itself

Identity Property for Multiplication: when a number is multiplied by one the result is the number itself

Join: to put together (adding)

Re-Grouping: strategy used to simplify addition and subtraction problems using either place value or expanded form of a number.

Remove: to take apart (subtracting)

Thousands Place: in place value, the place that represents the number of thousands; the place to the left of the hundreds place.

CLASSROOM ROUTINES

The importance of continuing the established classroom routines cannot be overstated. Daily routines must include obvious activities such as taking attendance, doing a lunch count,





determining how many items are needed for snack, lining up in a variety of ways (by height, age, type of shoe, hair color, eye color, etc.), daily questions, and calendar activities. They should also include less obvious routines, such as how to select materials, how to use materials in a productive manner, how to put materials away, how to open and close a door, how to do just about everything! An additional routine is to allow plenty of time for children to explore new materials before attempting any directed activity with these new materials. The regular use of the routines are important to the development of students’ number sense, flexibility, and fluency, which will support students’ performances on the tasks in this unit. See Unit 1 for suggestions concerning specific ideas for classroom routines.

I Spy a NumberThis is a whole group game that should be introduced in this unit. The teacher can use an overhead projector to make a simple chart like this:

Set a range for student guesses such as 100 to 999. However, the first few times you play, you may want to use a 2-digit number. Decide on a target number and write it in a place where you can keep it covered. For example, use the number 75. Have a student guess the target number and record the guess in the first column. Are any digits in the guess correct? If not, put a “0” in both the “Digits Correct” and “Places Correct” columns. As you model the game, verbalize your own strategy, step by step, so that students who need to build reasoning skills can hear the processes that result in discovering the target number.

If any digits are correct, record how many digits and how many places are correct in the appropriate columns. Students will use deductive reasoning and knowledge of place value to find the target number.

An example is shown below:

Guess Digits Correct

Places Correct

12 0 0

36 0 0Georgia Department of Education




Places Correct


45 1 1

73 1 1

98 0 0

75 2 2

The chart shows that none of the digits in 12 and 36 will be found in the target number. The next guess of 45 is a wise guess because the digits 4 and 5 had not yet been used. We do not know if the 4 or the 5 is correct at this point, but we do know that one digit and one place is correct. The task now is to find out if the 4 or the 5 is the digit to keep.

Next in the chart, 73 is a crucial guess because we already know that the target number has no three. Therefore, the correct digit must be the 7. Since the place is also correct, the number has to be seventy-something. The table confirms that no digits in 98 are correct. So, the only remaining digits possible in the ones place of the target number are 4 and 5. We know 0 is not a consideration because the target number has only two digits. There is a 7 in the tens place and the chart shows us that either 4 or 5 in 45 is in the correct place. We can eliminate the 4 because it is in the tens place and we already know there is a 7 in the tens place. Therefore, the five in forty-five must be correct because it is in the ones place. So, “seventy-something” is definitely 75.

Even though this task yields one correct answer for the target number, it is important for you to see the thought processes and reasoning that students use in the game. Students should play this game many times before they are given the “I Spy a Number” Recording Sheet to play on their own or with a partner.

Students must have numerous experiences with the game as a large group before they can complete this task in small cooperative groups or with a partner. A clear understanding of place value names and values is also essential.

Game DirectionsStudents will follow the directions below from the “I Spy a Number” recording sheet.1. This game should be played with a partner. Player one will choose a target number. The

player will write it down secretly on the back of this paper and give player two a range of numbers from which to choose (Examples: between 0 and 99 or between 100 and 999).

2. Player two will then give his/her first guess. Player one will write the guess in the chart below and use the correct columns to write how many digits and the number of places that are correct. Use a zero to show that neither the digit nor the place value is correct.

3. After each guess, Player two should explain why each guess was made. Continue playing until the target number has been determined.

4. Then Player two will choose the target number and repeat the game.





5. Both players must explain the strategy for the guesses they make.

Sample Questions/Prompts for Formative Student Assessment What are your strategies for determining the target number? How does knowing that the digit and/or the place value are correct help you figure out the

target number? What can you conclude if your partner tells you that you didn’t correctly guess the digit

or the place value? How would you explain the best way to win this game to another student? What strategy do you think works best for finding the target number? Why did you choose this number?

Questions for Teacher Reflection Which students are guessing haphazardly, demonstrating lack of a reasonable strategy? Are students able to record their answers in order to reflect on the implications of them? How much automaticity are students demonstrating in their understanding of place value?

Alternative Chart:




Guess Tens Digit

Ones Digit

Digits Correct Guess Tens

DigitOnes Digit

Digits Correct

47 1 0 1

53 0 0 1

45 1 1 2


Name ______________________________________ Date _________________________

I Spy a NumberDirections:

1. This game should be played with a partner. Player One will choose a target number. The player will write it down secretly on the back of this paper and give Player Two a range of numbers from which to choose (Examples: between 0 and 99 or between 100 and 999).

2. Player Two will then give his/her first guess. Player One will write the guess in the chart below and use the correct columns to write how many digits and the number of places that are correct. Use a zero to show that neither the digit nor the place value is correct.

3. After each guess, Player Two should explain why each guess was made. Continue playing until the target number has been determined.

4. Then Player Two will choose the target number and repeat the game.

5. Both players must explain the strategy for the guesses they make.





Places Correct


Places Correct


Addition and Subtraction Strategies

The strategies that students use to solve problems provide important information concerning number sense, and place value.

It is important to look at more than answers students get. The strategies used provide useful information about what problems to give the next day, and how to differentiate instruction.

It is important to relate addition and subtraction. Student-created strategies provide reinforcement of place value concepts. Traditional

algorithms can actually “unteach” place value. Student created strategies are built on a student’s actual understanding, instead of on what

the book says or what we think/hope they know! Students make fewer errors with invented strategies, because they are built on

understanding rather than memorization.

Addition Strategies with Two-Digit Numbers Add Tens, Add Ones, Then Combine

46 + 38

40 and 30 is 70.6 and 8 is 14.70 and 14 is 84.

Move Some to Make Tens

46 + 38

Move 2 from the 46 and put it with the 38 to make 40.Now you have 44 and 40 more and that makes 84.

Add on Tens, then Add Ones

46 + 38

46 and 30 is 76.Then I add on the other 8.76 + 4 is 80 and 4 more is 84.

Use a Nice Number and Compensate.

46 + 38

46 and 40 is 86.That’s 2 extra (more than what I am supposed to have), so it’s 84.





Strategies for Subtraction by Counting Up Add Tens to Get Close, Then Ones

73 – 46

46 and 20 is 66.(30 more is too much)Then 4 more is 70 and 3 more is 73. That’s 20 and 7 or 27.

Add Tens and Overshoot, Then Come Back

73 – 46

46 and 30 is 76.That’s 3 too much, so it’s 27.

Add Ones to Make Ten, Then Tens and Ones

73 – 46

46 and 4 is 50.50 and 20 moremakes 70 and 3 moremakes 73. The 4 and the 3 is 7 and 20 is 27.

Similarly,46 and 4 is 50.50 and 23 is 73.23 and 4 is 27.

Strategies for Take-Away Subtraction Take Tens from Tens, then Subtract

Ones

73 – 46

70 minus 40 is 30.Take away 6 more is 24.Now add in the 3 ones 27.

Or

70 minus 40 is 30.I can take those three away,but I need 3 more from the 30 to make 27.

Take Away Tens, Then Ones

73 – 46

73 minus 40 is 33.Then take away 6:3 makes 30 and 3 more makes 27.

Add to the Whole if Necessary

73 – 46

Give 3 to 73 to make 76.76 take away 46 is 30.Now give 3 back 27.

Take Extra Tens, Then Add Back

73 – 46

73 take away 50 is 23.That’s 4 too many.23 and 4 is 27.





BridgingEach of the examples above involved bridging a ten (if done with the traditional algorithm, they would require regrouping). For most of the strategies, it is easier to add or subtract when bridging is not required. Easier problems instill confidence and permit you to challenge students with “a harder one.” There is also the issue of bridging 100 or 1000. Try 57 + 68 with different strategies. Bridging across 100 is also an issue for subtraction. Problems such as 128 – 50 or 128 – 45 are more difficult than ones that do not bridge 100.

Larger NumbersTry seeing how you would do these without using the traditional algorithms: 487 + 235 and 623 – 247. For subtraction, a counting up strategy is usually the easiest. Occasionally, other strategies appear with larger numbers. For example, “chunking off” multiples of 50 or 25 is often a useful method. For 462 + 257, pull out 450 and 250 to make 700. That leaves 12 and 7 more 719. Students could also use open number lines as they work with larger numbers, as well as with smaller numbers.

STRATEGIES FOR TEACHING AND LEARNING

Students should be actively engaged by developing their own understanding. Mathematics should be represented in as many ways as possible by using graphs, tables,

pictures, symbols, and words. Appropriate manipulatives and technology should be used to enhance student learning. Students should be given opportunities to revise their work based on teacher feedback,

peer feedback, and metacognition which includes self-assessment and reflection. Math journals are an excellent way for students to show what they are learning about a

concept. These could be spiral bound notebooks that students could draw or write in to describe the day’s math lesson. Second graders love to go back and look at things they have done in the past so journals could also serve as a tool for a nine week review

EVIDENCE OF LEARNING

By the conclusion of this unit, students should be able to demonstrate the following competencies:

Add and subtract two whole numbers up to three digits each with regrouping. Use the inverse operation to check that they have correctly solved the problem. Solve problems using mental math strategies. Simplify problems using the commutative, associative, and identity properties.





TASKS

The following tasks represent the level of depth, rigor, and complexity expected of all second grade students. These tasks or a task of similar depth and rigor should be used to demonstrate evidence of learning. It is important that all elements of a task be addressed throughout the learning process so that students understand what is expected of them. While some tasks are identified as a performance task, they also may be used for teaching and learning (learning task).

Task Name Task Type/ Grouping Strategy Content Addressed

Place Value Play Learning TaskLarge Group Place value understanding

Building Base Ten Numbers Learning TaskPartners Building 3-digit numbers

Shake, Rattle, and Roll Learning TaskPartners Estimation, Mental math strategies

Take 100 Learning TaskPartners Addition to 100

Multi-digit Addition StrategiesLearning Task

Whole group, Partners, Small group, Individual

Multi-digit addition with regrouping

Addition with Regrouping Performance TaskWhole group, Partners Multi-digit addition with regrouping

Grocery Store Math Learning TaskWhole group, Partners Modeling addition with money

Sale Flyer Shopping Performance TaskIndividual Addition with money

Subtraction: Modeling with Regrouping

Learning TaskIndividual

Multi-digit subtraction with regrouping

Subtraction Story Problems Performance TaskWhole group, Partners

Multi-digit subtraction with regrouping

Menu Math Performance TaskSmall group

Addition and subtraction with money

Counting Mice Performance TaskSmall group Multi-digit addition and subtraction

Every Picture Tells a Story Performance TaskSmall group Multi-digit addition and subtraction

Mental Mathematics Performance TaskSmall group Multi-digit addition and subtraction

Culminating Task:How Many Buses?

Performance TaskPartners

Multi-digit addition and subtraction, Venn diagrams and charts





Learning Task: Place Value Play

STANDARDS ADDRESSED

M2N1. Students will use multiple representations of numbers to connect symbols to quantities.



M2N2. Students will build fluency with multi-digit addition and subtraction. a. Correctly add and subtract two whole numbers up to three digits each with

regrouping.

M2P1. Students will solve problems (using appropriate technology). a. Build new mathematical knowledge through problem solving. b. Solve problems that arise in mathematics and in other contexts. c. Apply and adapt a variety of appropriate strategies to solve problems.



d. Use the language of mathematics to express mathematical ideas precisely. M2P5. Students will represent mathematics in multiple ways.



phenomena.

ESSENTIAL QUESTIONS

How can we represent numbers using place value? How do the value of digits change when their position in a number changes?





MATERIALS

Base Ten Blocks (each pair needs at least 8 Hundreds, 20 Tens, and 10 Ones) Place Value Mat String, rope, or thin pieces of fabric in lengths that will go around a group of 10 students

GROUPING

Large group

TASK DESCRIPTION, DEVELOPMENT, AND DISCUSSION

Background KnowledgeStudents should have had prior experiences and/or instruction with place value. This task will review and expand on this understanding. Students should also have experience using base-ten blocks.

Part I Create a large place value board on the floor (using butcher paper, masking tape, or you could even draw it on the sidewalk with chalk and play the game outside). Make sure that the area that you create for both the ones and tens columns can hold at least 9 students comfortably in each. With a large die, or a deck of cards with the numbers 1 – 9, roll/draw a number. Ask the class for volunteers and have that many students stand in the ones column.

Ask questions like: Why are you standing in the ones column? Is there room for any more students in your column? How many more students could join you in the ones column? What would happen if

more than that joined you?

Roll/draw another number. Ask for more volunteers and add that many more students to the ones area. The area might be getting a little more crowded now.

Ask questions like: How many students are now in the ones column? Do we now have enough students to make a group of ten? How do you know? What happens to the students that aren’t linked with the group of ten?

If the group consists of ten or more then have ten students link their arms, or “tie” them into a group of 10, and move to the tens place. If there are any extra students they will remain in the ones column.

Ask questions like: Why have we moved this group to the tens place? What number is now represented on the place value board?





Is that the same number of children standing who are standing on the place value board?

What digits do we use to write this number? Why do we use these particular digits? What do they stand for?

Continue with the game by rolling the die/drawing cards until all the students are standing on the board. Have students explain what is happening and why groups are moving. *Encourage them to use the terms place and value as they are explaining the answers to the questions!

Part II After several rounds of the game ask:

“How would this game change if we used all of the second grade students in our school?”

“How would it change if we used all of the students in our school?”

CommentsThese questions are designed to get them to talk about how many bundles it will take to make 100. These questions are also designed to help students understand how and why the digits 0-9 are used to represent a particular amount. If you want to have some straws or sticks to represent more students so they can physically move the 10 bundles to the hundreds spot, then do so. Making a connection to money at this point is a natural medium for students to use. Asking the question like, “If we thought about this place value mat in terms of money and we were building up to dollar, what would be the place value positions represent?” Ones (pennies), tens (dimes) and hundreds (dollars). Listen for students to make the connection that 10 pennies equal a dollar and 10 dimes equal a dollar. Once it is clear that the students understand it takes 10 groups of ten to make 100, then students are prepared to complete the next portion of the task.

Give student pairs a copy of a place value board and a baggie with at least 8 hundred blocks, 20 ten blocks, and 10 ones. Call out various 2 and 3 digit numbers and students should create these using their manipulatives. It is important for them to be able to use these manipulatives to model these 2 and 3 digit numbers, as well as explain what the digit 2 represents in the number 285, what the 8 represents and what the 5 represents. Using the base ten blocks helps students see and create what the number 285 means. Before students can understand how to combine 2 and 3 digit numbers, they must first be able to explain numbers in an expanded notion form, for example, 200+80+5. Once this understanding is established, they will use this knowledge as they expand their understanding of how numbers are combined.

After students have worked with a variety of 2 and 3 digit numbers, allow several students to demonstrate their understanding of the base ten blocks.

Questions for Teacher Reflection Are students able to use the terms place and value correctly? Are students able to solve addition problems with and without regrouping? Are students able to describe how to use a place value board?





Can students articulate the difference between the ones column, tens column, and hundreds column?

DIFFERENTIATION

Extension Have students represent 4-digit numbers with expanded notation. (Note: All students

will be responsible for representing 4-digit numbers by the end of this unit.)

Intervention Provide students with a ten-frame to help them recognize when to group items to move to

the next column.





Learning Task: Building Base Ten Numbers

STANDARDS ADDRESSED










phenomena.

ESSENTIAL QUESTIONS

How do the value of digits change when their position in a number changes?

How can we tell which numbers are larger or smaller than others?

MATERIALS

Three 6-sided dice per pair Base ten blocks “Building Base Ten Numbers” recording sheet





Place Value Charts (optional)GROUPING

Partners

TASK DESCRIPTION, DEVELOPMENT AND DISCUSSION

In this task, students will roll dice and make three-digit numbers from the results. Students will then draw or use base ten blocks to build models of the numbers created and explain how they know which numbers are larger or smaller. CommentsBecause there are several steps involved in this task, you may want to model this activity while “thinking aloud.” Include a discussion about assigning a place value to the number rolled on a given die. Ask questions such as, “Will you make the 3 on this die represent 3, 30, or 300? How will each of those numbers look if we use base ten blocks?” Some students may need extra support with this concept.

When using dice, base ten blocks, or any other manipulatives, students need to understand procedures and routines associated with using them. You may want to give students a few minutes to explore with the pieces and make observations before moving into the task.

Background Knowledge Students need to understand how to order digits from largest to smallest and vice versa. They also need to understand place value concepts and how they relate to the base ten pieces.

Task DirectionsGather students in the meeting area. Model with students the “Building Base Ten Numbers” recording sheet. Then allow students to work with a partner to complete the task.

Your task is to build large numbers and identify them. Use dice and base ten blocks or models to complete this exercise.

1. Roll all 3 dice at once. 2. Record the number formed when you place the digits on the dice in order from

smallest to largest. 3. Record the new number formed when you order them from largest to smallest. 4. Using the smallest number of base ten pieces possible, draw (or build) a model of

each number you recorded. Have the flats represent hundreds, the rods represent tens, and the unit cubes represent ones. Use words to write how each of the two numbers is spoken.

5. Repeat the same exercise two more times and record. 6. When you have completed all your rolls, put a star beside the greatest number you

rolled. Then put a check beside the smallest number you rolled. Explain how you know what the largest and smallest numbers are possible when using three dice.





Questions/Prompts for Formative Student Assessment How did you decide in which order to place your dice? What would happen if you changed the order of your dice? What would happen to the size of your numbers if you used more or fewer dice? Which representation of your numbers makes the most sense to you?

Questions for Teacher Reflection Are student representations of numbers accurate? Which representation of the numbers are most students using? Which representation

might need more modeling? How ready are students to move away from using the manipulatives and utilize

internalized place value concepts?

DIFFERENTIATION

Extension Students may use four dice instead of three (or use three dice and always have one digit

be zero). If you continue to have them draw/build base ten models, you will need to provide the large cube in the base ten blocks to represent the thousands place. If students have access to base ten stamps, those may be used instead of or in addition to the actual manipulatives.

Intervention Students may need to use a place value chart to align their digits. Have students complete the task in small groups with direct instruction or modeling for

additional support. For students who have difficulty counting the dots on a die, dice with numbers printed on

them may be used.





Name ______________________________________ Date __________________________

Building Base Ten NumbersYour task is to build large numbers and identify them. Use dice and base ten blocks or models to complete this exercise.

1. Roll all 3 dice at once.

2. Record the number formed when you place the digits on the dice in order from smallest to largest.

3. Record the new number formed when you order them from largest to smallest.

4. Using the smallest number of base ten pieces possible, draw (or build) a model of each number you recorded. Have the flats represent hundreds, the rods represent tens, and the unit cubes represent ones. Use words to write how each of the two numbers is spoken. Repeat the same exercise two more times and record.

5. When you have completed all your rolls, put a star beside the greatest number you rolled. Then put a check beside the smallest number you rolled. Explain how you know what the largest and smallest possible numbers are when using three dice.

Smallest Number Model Largest

Number Model

1 ___ ___ ___ ___ ___ ___

2 ___ ___ ___ ___ ___ ___

3 ___ ___ ___ ___ ___ ___





Learning Task: Shake, Rattle, and Roll

STANDARDS ADDRESSED




M2N2. Students will build fluency with multi-digit addition and subtraction.c. Use mental math strategies such as benchmark numbers to solve problems.







phenomena.

ESSENTIAL QUESTIONS

What is an effective way to estimate numbers? How can estimation strategies help us build our addition skills? When will estimating be helpful to us? How can benchmark numbers help us add?





MATERIALS

Two six-sided dice “Shake, Rattle, and Roll” recording sheet

GROUPING

Partners


In this task, students play a game with dice that enables them to build mental math concepts as they practice addition skills and strategies and determine to which multiple of ten a given number is nearest.

CommentsThis task is designed to provide addition practice and mental math/estimation skills. You may want to use a book like Mental Math in the Primary Grades by Jack Hope, R. Reys, Larry Leutizinger, Barbara Reys, and Robert Reys to practice mental math with the class as a whole group.

Use all available opportunities during the day to incorporate the use of estimation, for example, determining to which multiple of ten a given number is nearest. This skill was originally introduced in Grade 1, supported with the use of a number line 0-99 chart and/or a hundreds chart. Students should have these tools available for this task. Alternatively, students can create a number line to determine the closest multiple of ten. A student sheet with open number lines could be provided. An example of an open number line is shown below.

For the number 36, students can fill in the numbers around 36, including the two closest multiples of ten as shown below. Then looking at the number line, students can determine the multiple of ten that is the closest to 36. In this case 40 is 4 away, but 30 is 6 away, so 40 is the closest multiple of ten.

For the number 63, students can follow the same procedure to determine the multiple of ten that is the closest to 63. In this case 60 is 3 away, but 70 is 7 away, so 60 is the closest multiple of ten.

Estimating skills will help students determine reasonableness of answers, a vital skill for everyday living.




30 31 32 33 34 35 36 37 38 39 40

60 61 62 63 64 65 66 67 68 69 70


If you incorporate calendar activities into your instruction, many opportunities present themselves for activities with estimation. Also, be sure students make connections between counting by tens, multiplying by ten, and estimating to the nearest ten before adding or multiplying.

Background KnowledgeStudents should be proficient in determining to which multiple of ten any given two-digit number is nearest. They should also be comfortable adding two-digit multiples of ten (For example, 20 + 60 = 80).

Task DirectionsGather students to the meeting area and model with the students how to play “Shake, Rattle, and Roll.” Provide each pair with a recording sheet and allow students time to complete the task.

This is a two player game that will help you practice your estimation skills. The goal of the game is to be the person with the most points at the end of ten turns.

1. Play with a partner. You will need 2 dice and a recording sheet for each player.

2. Roll two dice. Form the two possible numbers as shown below.

Example:

Using the digits 3 and 6, make the numbers 36 and 63. Find the nearest multiple of 10 for each number, and then using mental math, add to find an estimate.

Estimated sum = 40 + 60 = 100

3. Player one records the estimate on the game recording sheet to end round 1. Your partner must agree with your estimation.

4. Player two takes a turn, following steps 2 and 3 above.

5. Players take turns for a total of ten rounds.

6. After ten rounds, each player finds the sum of their estimates. The player with the higher sum wins the game.





Questions/Prompts for Formative Student Assessment Explain how you found the closest multiple of ten. Do you think your estimated sum is higher or lower than the actual sum? Why? How

could you check? What kinds of situations in life might be easier if you knew how to estimate and add

numbers like this?

Questions for Teacher Reflection What estimation strategies are students in my class using most? Least? Can students explain the process they used to estimate?

DIFFERENTIATION

Extension Use one or more additional dice.

Intervention Use number lines, hundreds charts, and models to help students who are having difficulty

determining to which multiple of ten their number is nearest. Use counting up/counting back to the nearest multiple of ten and compare the results to determine which multiple of ten a number is closest.





Name _________________________________________ Date __________________________

Shake, Rattle, and Roll Game Directions

This is a two player game that will help you practice your estimation skills. The goal of the game is to be the person with the most points at the end of ten turns.

Directions:

1. Play with a partner. You will need 2 dice and a recording sheet for each player.

2. Player one rolls two dice and forms the two possible numbers as shown below.

Example:

Using the digits 3 and 6, make the numbers 36 and 63. Find the nearest multiple of 10 for each number, and then using mental math, add to find an estimate.

Estimated sum = 40 + 60 = 100

3. Player one records the estimate on the game recording sheet to end round 1. Your partner must agree with your estimation, using a calculator to check if needed.

4. Player two takes a turn, following steps 2 and 3 above.

5. Players take turns for a total of ten rounds.

6. After ten rounds, each player finds the sum of their estimates. The player with the higher sum wins the game.




63 36


Shake, Rattle, and Roll GamePlayer 1 ________________________________

RoundDice Numbers Smaller Number Larger Number

EstimateDie 1 Die 2 ActualNearest Multiple

of 10Actual

Nearest Multiple

of 10123456789

10

Player 2 ________________________________

RoundDice Numbers Smaller Number Larger Number

EstimateDie 1 Die 2 ActualNearest Multiple

of 10Actual

Nearest Multiple

of 10123456789

10

Learning Task: Take 100 STANDARDS ADDRESSED





M2N2. Students will build fluency with multi-digit addition and subtraction.c. Use mental math strategies such as benchmark numbers to solve problems.e. Estimate to determine if solutions are reasonable for addition and subtraction.




d. Use the language of mathematics to express mathematical ideas precisely. ESSENTIAL QUESTIONS

How can we solve problems mentally? What strategies will help me add multiple numbers quickly and accurately? How can mental math strategies, for example estimation and benchmark numbers, help us when adding?

MATERIALS

A deck of cards containing two of each of the following numbers: 10, 20, 30, 40, 50, 60, 70, 80, 90, 50, 5, 95, 15, 85, 25, 75, 35, 65, 45, 55. (Copy 2 game cards sheets for each deck of cards)

“Take Ten Game” Student Recording Sheet

GROUPING

Partners






CommentsStudents should have had practice developing strategies to make combinations of one hundred easily using mental math.

Task DirectionsGather students in the meeting area to model the “Take 100” game. Use the think-aloud strategy to model ways students can think about pairs to 100. Then allow time for students to complete the game in pairs. They should select only 2 cards that total 100.

Players place a shuffled deck of cards (see attached cards) between them. Player 1 turns over the top card and lays it to one side of the deck so that the number shows.

Player 2 turns the next card, laying it to the other side of the overturned deck so that the numbers are showing. If the 2 cards total one hundred, the first student to say “one hundred!” gets those two cards. If the cards do not total 100, each player turns over another card placing it beside the cards turned over previously. This allows students to choose 2 cards that total 100 from a set of numbers. Both players look to see if a sum of 100 can be made. The first player to find a pair of cards that totals 100 and says “one hundred!”, gets the two cards. Play continues until all the cards have been used.

As students play, ask them to record their pairs of 100 as an addition number sentence. This gives students an opportunity to focus on the pairs that make 100 and provides a record of the game.

This game can be adapted to eliminate the speed aspect to the game. Students can take turns turning over two cards and placing them face up next to the deck of cards. If the sum of the numbers is 100 the student gets to take those cards and any others that have been turned over. If the numbers do not equal 100, then the cards are left face up and the student’s turn ends. Play continues until all of the cards have been turned over. The player with the most cards at the end of the game wins.

Questions/Prompts for Formative Student Assessment What do you do to help you remember the number combinations that make up one

hundred? What can you do to find the answer quicker than your partner? Why doesn’t 63 + 47 equal 100?

Questions for Teacher Reflection Do I need to review strategies for making 100? Are my students evenly matched for this activity according to ability level? Which students seem to have a more difficult time remembering the combinations of ten?

How will I assist them?Georgia Department of Education




DIFFERENTIATION

Extension To determine a winner have each student take all the cards he or she won and add them.

Students will trade cards and let their partner add the cards with a calculator. When the amounts agree, the student with the larger total wins the game.

To determine a winner have each student estimate to the nearest 100 the sum of the cards earned. The student with the larger total wins the game.

Have students play game again, but this time have them look for combinations of 3 cards or more.

Intervention Play a “Pairs to Twenty Game” using two of each of the following cards: 1, 19, 2, 18, 3,

17, 4, 16, 5, 15, 6, 14, 7, 13, 8, 12, 9, 11, 10, 10.





5 10 15 20

25 30 35 40

45 50 55 60





65 70 75 80

85 90 95 50





Name __________________________________ Date _________________Take 100 Game

Number of Players: 2

Materials: Deck of 40 Cards

Directions:1. Shuffle the cards well and lay them face down in a pile on the desk.

2. Turn the top card over and set it to the side where both partners can see it. Now turn the next card over and set it to the side of the first overturned card.

3. Your goal in this game is to make sets of one hundred.

4. If the first two overturned cards equal one hundred when added together, try to be the first one to say, “One hundred!” loudly enough for your partner to hear you. If you are first to notice, you may take the cards. If your partner is the first to notice, he or she gets to take the cards.

5. If the first two cards do not make a set of one hundred, keep turning cards over and setting them next to the first overturned cards. When someone spots a combination of one hundred, they can take the two cards that total 100. Keep playing this way until all cards have been claimed or until no cards are left and the overturned cards do not make a set of one hundred.

6. The player with the most cards at the end of the game is the winner.





Learning Task: Multi-digit Addition Strategies

STANDARDS ADDRESSED





regrouping. c. Use mental math strategies such as benchmark numbers to solve problems. d. Use basic properties of addition (commutative, associative, and identity) to simplify








phenomena.

ESSENTIAL QUESTIONS

How can we solve addition problems with and without regrouping?





Can we change the order of numbers when we add (or subtract)? Why or why not? How can we solve problems mentally? How can strategies help us when adding and subtracting with regrouping?

MATERIALS

Various manipulatives (counters, base-ten blocks, unifix cubes) Chart paper for class recording sheets

GROUPING

Large group, Partners, Individual, Small group


Background KnowledgeStudents should have had prior experiences and/or instruction with addition and subtraction of two-digit numbers without regrouping. Students should also have experience solving various story problems with the use of manipulatives.

Part IIntroduce task with this story problem:

Mrs. Brown and Mrs. White are going to join their classes together for a popsicle party. Mrs. Brown has 18 students in her class and Mrs. White has 19 students in her class. They plan on getting one popsicle for each student in their classes. How many popsicles do Mrs. Brown and Mrs. White need to buy?

Have several students retell the story problem to you and discuss what is happening in the problem.

Part IISplit students into pairs and give each student a half sheet of chart paper to use in solving the problem. Also, have various manipulatives available for students to use as they work to solve the problem. Walk around and observe students as they are problem solving. Ask questions such as:

What are you trying to find out? How many students are in Mrs. Brown’s class? How many students are in Mrs. White’s class? Can you explain the strategies you are you using to solve this problem? Are there other ways you could solve the problem? Is there a way you can check your answer?





CommentsAs you are walking around, find students who are using a variety of strategies. Some students may draw a picture, solve the problem with manipulatives, or use benchmark numbers. All of these strategies demonstrate a solid foundation of number sense. If you notice students using the traditional algorithm for regrouping, it is imperative that you ask them to explain their reasoning for using this method. The idea that numbers can be “carried” is not a natural progression when numbers are combined. Algorithms are a short cut method that makes recording numbers more convenient and efficient. Students need to explore many different strategies for combining numbers before they can understand the idea of an amount being “carried” from one place value position to another. Moving to the standard algorithm too early will often prevent students from continuing to make sense of the numbers that work within a given situation. Part IIILet several students share their different strategies and answers to the problems. Allow the students to call on their peers to ask questions or make comments about their strategy and the answer that was found. After students have shared various strategies, spend some additional time discussing the different strategies students have used. Some students may have broken the numbers into smaller pieces to simplify the addition problem.

For example, in 18 + 19 you can begin by pulling out the tens and add 10 +10 = 20. You then have 8 + 9. You can then break up the 9 into 7 and 2. Next, add 8 + 2 to get 10. You will then have 7 more to add. 10 +10 +10 +7 = 37.

Other students may have used benchmark numbers to help add. For example 18 + 19 could have been solved by keeping the 18 and taking 2 from 19. You can have 20 and 17. You can then add 20 + 17 to get 37.

Another Strategy may be to add 1 to 19 to make a group of 20 then add 17 to 20 by grouping 10 more, equally 30 then add 7.

Creating groups of 10 with the numbers to 20 is another strategy. Adding 1 to 19 and 2 to 18 to make them both 20. Then adding 20 and 20 to get 40 then taking off the 3 (from the 1 and 2) to get 37.

This may sound convoluted to adults, but students who have strong number sense will tend to think in this way. When we teach just the algorithm we discourage the students from using a more natural strategy. When they are allowed to develop strategies that make sense to them then they are developing better number sense of addition… and subtraction! Create a list with students of the various strategies they can use when solving addition problems. Some students may have also mentioned the traditional algorithm for addition with regrouping. As long as they can explain what they are doing and why it works then it is okay to include this as a strategy. It is more beneficial to encourage students to utilize the various other strategies at this time; then move towards the algorithm when they can demonstrate true number sense.

CommentsIf no student describes using the number line or number chart as a strategy, then this is a good time to bring up this tool for combining amounts. Students should be able to use the number line or number chart as a tool for adding numbers. For example, students could find 18 on the number line or number chart and count on 19 or vice versa. This use of these models also elicits a natural discussion about the commutative property of addition.





Part IVGive students this problem: Max had an ant collection with 38 ants. His friend Lily had an ant collection with 24 ants. How many ants did the two friends have?

Allow students to attempt to solve this problem on their own. As students work, walk around asking questions about the students’ strategy use. Look to see if students are using the strategies mentioned above.

After students have completed solving the problem, allow students to take turns sharing their strategy with people at their table or other small groups of students. The task should be closed with the teacher selecting students to highlight various strategies used in the classroom and again referring to the number line or number chart if it is not one of the strategies presented by students.

Questions for Teacher Reflection What strategies were students using to solve addition with regrouping problems? Could students explain their thinking? Could students solve addition with regrouping problems in more than one way? Did students listen to and understand various strategies from their peers?

DIFFERENTIATION

Extension Give students this problem to supplement problem 1: If popsicles come in boxes of 10,

how many boxes do Mrs. Parkerson and Mrs. Young need to get for their classes of 24 students? If each student gets one popsicle, how many popsicles will be left over?

Give students this problem to supplement problem 2: If Max and Lily join their ants together in Lily’s ant farm, and the ant farm can hold up to 100 ants, will there be enough room for both Max and Lily’s ants? How do you know? How many more ants could Max and Lily place into the ant farm before it reaches its maximum capacity?

Write a problem involving either the ants or the popsicles, and ask a partner to solve it. What strategy was used?

Intervention Some students may need to work on the second problem with partner groups. They may

not be ready to utilize the addition strategies independently in this lesson. They may also need to use manipulatives to physically act out the problem.





Performance Task: Addition with Regrouping

STANDARDS ADDRESSED





regrouping.







phenomena.

ESSENTIAL QUESTIONS

How can we solve addition problems with and without regrouping? What is a number sentence and how can I use it to solve word problems? How do we use addition to tell number stories?





MATERIALS

Base Ten Blocks (Hundreds, Tens, and Ones) Place Value Mat “Addition with Regrouping” task sheet

GROUPING

Large group


Background KnowledgeStudents should have had prior experiences and/or instruction with addition and subtraction of two-digit numbers without regrouping. Students should also have experience regrouping using base-ten blocks.

Please see “Special Comments” section in the introduction of the unit for various addition strategies.

Part I Give students this problem: Lucy has 28 green candies and 46 blue candies. How many candies does she have?

Allow the children to utilize various strategies discussed in earlier task. Have their base ten blocks, place value mats, as well as number lines and number charts available to help them find the answer to the problem. While students are working, circulate and ask these kinds of questions:

What are you trying to find out? How many red candies does Lucy have? How many blue candies does Lucy have? Are you using a picture, number, or words to organize your thinking? How is this problem similar to problems we solved before? Can you explain the strategies you are you using to solve this problem? Could you use a different strategy to combine these amounts? Does it matter which number you begin with? Is there a way you can check your answer?

If students are using the place value map and blocks, these are questions you may ask: Does it matter which number you begin with? How many ones do you need in the ones place to begin with? How did you figure that

out? Which digit did you look at? What is its value in that place? How many tens do you need in the tens place to begin with? How did you figure that out?

Which digit did you look at? What is its value in that place?





What did you notice about the ones? What will you have to do with some of your ones? Can you regroup them in any way?

How many tens do you have now? Why did it change? How did you figure that out? Which digit did you look at? What is its value in that place?

Can you explain the strategies you are using to solve this problem? Could you use a different strategy to combine these amounts? Is there a way you can check your answer?

After students have solved the problem, pull the class together for a class discussion. Allow several students to demonstrate their strategy using the board or the overhead projector.

Part IIIGive students the task sheet to solve in partner groups. Have base ten blocks, place value mats, number lines or number charts available for student use.

While students are working, circulate and question students: What are you trying to find out? How can you use base-ten blocks to help you solve this problem? How many ones do you need in the ones place to begin with? How did you figure that

out? Which digit did you look at? What is its value in that place? How many tens do you need in the tens place to begin with? How did you figure that out?

Which digit did you look at? What is its value in that place? What did you notice about the ones? What will you have to do with some of your ones?

Can you regroup them in any way? How many tens do you have now? Why did it change? How did you figure that out?

Which digit did you look at? What is its value in that place? Can you explain the strategies you are you using to solve this problem? How can you solve this problem using a number sentence? Is there a way you can check your answer?

Part IVAfter students have completed the problems, allow them to take turns sharing the strategy they used for each problem. Allow other classmates to make comments on the strategies and/or ask questions.

Questions for Teacher Reflection What strategies were students using to solve addition with regrouping problems? Could students explain their thinking? Could students solve addition with regrouping problems using base-ten blocks? Did students gain various strategies from their peers? Were students using the terms place and value to talk about the numbers they were

making?

DIFFERENTIATION





Extension Allow students to make up their own three-digit addition story problems. They can solve

their own problems or trade with a partner. Provide students with number cubes to create their own problems.

Intervention Some students may need additional support during the problem solving through

additional questioning and scaffolding. Use pennies, dimes, and dollars to help with the understanding of regrouping and still

have the same amount.





Name ______________________ Date_____________

Addition with Regrouping Problems

1. Jody and Cindy are playing basketball. Jody makes 25 shots in a row and Cindy makes 37 shots in a row. How many shots did Jody and Cindy make altogether?

2. It rained 14 inches last month and 18 inches this month in Seattle. How many inches of rain has Seattle had in the past two months?

3. Ellen has 46 baseball cards. She buys another pack with 12 baseball cards. How many baseball cards does she have in all?





Learning Task: Grocery Store Math

STANDARDS ADDRESSED


a. Represent numbers using a variety of models, diagrams, and number sentences (e.g., 4703 represented as 4,000 + 700 + 3, and units, 47 hundreds + 3, or

4,500 + 203). c. Use money as a medium of exchange. Count back change and use decimal notation

and the dollar and cent symbols to represent a collection of coins and currency.


regrouping. b. Understand and use the inverse relation between addition and subtraction to solve

problems and check solutions. c. Use mental math strategies such as benchmark numbers to solve problems. d. Use basic properties of addition (commutative, associative, and identity) to simplify


M2P1. Students will solve problems (using appropriate technology).



a. Recognize reasoning and proof as fundamental aspects of mathematics.b. Make and investigate mathematical conjectures.c. Develop and evaluate mathematical arguments and proofs.d. Select and use various types of reasoning and methods of proof.



d. Use the language of mathematics to express mathematical ideas precisely.





M2P4. Students will make connections among mathematical ideas and to other disciplines. a. Recognize and use connections among mathematical ideas. b. Understand how mathematical ideas interconnect and build on one another to produce


M2P5. Students will represent mathematics in multiple ways. a. Create and use representations to organize, record, and communicate mathematical

ideas. b. Select, apply, and translate among mathematical representations to solve problems. c. Use representations to model and interpret physical, social, and mathematical

phenomena.

ESSENTIAL QUESTIONS

How can we model and solve addition problems with and without regrouping?

MATERIALS

Lemonade for Sale by Stuart Murphy or similar book Bag containing 5-7 grocery items

GROUPING

Individual


Background KnowledgeStudents should have had prior experiences and/or instruction with addition and subtraction of two-digit numbers with and without regrouping. Students should also have had experience working with money amounts. In the first task, Place Value Play, the connection between pennies, dimes, and dollars was made to place value. This task extends on this important connection.

CommentThe purpose of this task is to develop students’ understanding of the concept of addition with money. Students should NOT be using the traditional addition algorithm. The addition of decimals with tenths and hundredths is a Fourth Grade concept. Students in Second Grade should be modeling addition with money.

Task DirectionsGather students together on the meeting area. Read Lemonade for Sale by Stuart J. Murphy or similar book. Discuss scenario from story. Then show students your bag of groceries. The bag





should contain 5-7 items, which could include an empty box of cereal, empty carton of juice or milk, bag of peanuts, can of soup, etc. You want to include items that cost between $1.00 and $10.00.

Create an interactive story about your last visit to the grocery store such as “I went to the grocery store to buy a box of cereal for $3.79 and a can of soup for $1.59. How much did I spend?” Then model for students the strategy of drawing a picture for each amount. The model should reinforce the place value of each digit. For example, cereal would cost $3.79; therefore it would be represented by drawing 3 rectangles labeled with $1 on each, 7 circles labeled with 10¢ on each and 9 larger circles label with 1¢ on each.

Once you have modeled the box of cereal, allow students to model the can of soup. Allow students to share their model. Record it with the model of the cereal price. This is building on the understanding of place value in the Play Value Play task at the beginning of this unit. Students may attempt to create a pictorial model in a different way for the change. Acknowledge that there are many different ways to make the same amount; however, using the example which was described creates fluency with the number 10.

Discuss how to solve the problem and find out how much money was spent to purchase the items. Continue grocery store shopping by having another pair of students select two items to purchase. Share the items and prices with the class and allow all students to find the total amount of money spent. Have the two “shoppers” present their model and solution to the class and discuss the strategies used. Repeat this process with additional pairs of students to allow all students to have an opportunity to shop, solve, and share their purchases with the class.

Questions for Teacher Reflection Are students able to represent item amounts with pictures of dollars, dimes, and pennies? Are they able to make the connection between place value and money? What addition strategy were they using to determine the total cost? How are students showing and checking their work?

DIFFERENTIATION

Extension Have students choose more than two items. Have students create their own grocery store story problem and solve.

Intervention Use prices that do not require much regrouping. Provide fake bills and coins for students to use to find the total of their purchase.





Performance Task: Sale Flyer Shopping

STANDARDS ADDRESSED


a. Represent numbers using a variety of models, diagrams, and number sentences (e.g., 4703 represented as 4,000 + 700 + 3, and units, 47 hundreds + 3, or

4,500 + 203). c. Use money as a medium of exchange. Count back change and use decimal notation

and the dollar and cent symbols to represent a collection of coins and currency.



problems and check solutions. c. Use mental math strategies such as benchmark numbers to solve problems. d. Use basic properties of addition (commutative, associative, and identity) to simplify




M2P2. Students will reason and evaluate mathematical arguments.a. Recognize reasoning and proof as fundamental aspects of mathematics.b. Make and investigate mathematical conjectures.c. Develop and evaluate mathematical arguments and proofs.d. Select and use various types of reasoning and methods of proof.








M2P4. Students will make connections among mathematical ideas and to other disciplines.

a. Recognize and use connections among mathematical ideas. b. Understand how mathematical ideas interconnect and build on

one another to produce a coherent whole. c. Recognize and apply mathematics in contexts outside of

mathematics.


ideas. b. Select, apply, and translate among

mathematical representations to solve problems. c. Use representations to model and interpret

physical, social, and mathematical phenomena.

ESSENTIAL QUESTIONS

How can we solve addition problems with and without regrouping?

MATERIALS

“Shopping Flyer” student sheet “Shopping Flyer” student task sheet

GROUPING

Individual


Background KnowledgeThis task builds on the understanding developed in the previous task, Grocery Store Math. Students should have had prior experiences and/or instruction with addition and subtraction of two-digit numbers with and without regrouping. Students should also have had experience working with money amounts. In the first task, Place Value Play, the connection between pennies, dimes and dollars was made to place value. This task extends on this important connection.

CommentThe purpose of this task is to develop students’ understanding of the concept of addition with money. Students should NOT be using the traditional addition algorithm. The addition of





decimals with tenths and hundredths is a Fourth Grade concept. Students in Second Grade should be modeling addition with money.





Task DirectionsShow students a sale flyer with various toys. Explain to the students that they are to choose two items that they would like to buy. They have to determine the total cost of the two items. Remind students to draw pictures of dollars, dimes, and pennies to represent the prices.

Give each student a shopping recording. Have students choose two items with their prices from the sale flyers for each shopping page. Glue these on the page. Instruct students to find the total price of the two items by modeling each amount and finding the total. Students should use the “$” and “¢” symbols to represent the prices and total. Once students finish solving the problem, encourage them to write a word problem for each page of their shopping book.

After students have created several pages for their shopping books, allow a few students to share their problems and strategies with the class. Allow other students to make questions and comments about the work.

Questions for Teacher Reflection How did students choose their items for purchase? What addition strategy were they using to determine the total cost? How are students showing and checking their work?

DIFFERENTIATION

Extension Have students purchase more than two items and find the total. You may also allow students to attempt to total the sum of all pages in their shopping

book to see how much they have spent altogether.

Intervention Provide fake bills and coins for students to use to find the total of their purchase. For students having difficulty with adding the two items, encourage them to use a 100-

chart or money number line. Students can also be encouraged to break the amounts apart, e.g., buy the toy radio ($8.27) and the toy phone ($5.68). The dollars are $8 + $5 = $13. The cents are 27¢ + 68¢. Students can make 68¢ + 2¢ + 25¢ and get 70¢ + 25¢ = 95¢. So, the total is $13 + 95¢ = $13.95.





______________’s Cool Stuff Store Shopping Flyer for the Week of ____________

Name Date Georgia Department of Education



$3.54 $4.76 $1.86

$1.79 $9.75 $2.98

$ 6.57 $3.49 $16.99

$8.29

$5.89

$7.25


Items I Bought: (glue pictures and prices here)

Picture Model of my Addition:

Addition Problem:

$________________ + $__________________ = ________________

My Story Problem:

__________________________________________________________________________________

__________________________________________________________________________________

__________________________________________________________________________________

__________________________________________________________________________________

__________________________________________________________________________________

__________________________________________________________________________________

__________________________________________________________________________________





Learning Task: Subtraction: Modeling with Regrouping

STANDARDS ADDRESSED


regrouping.

M2P1. Students will solve problems (using appropriate technology). a. Build new mathematical knowledge through problem solving. b. Solve problems that arise in mathematics and in other contexts. c. Apply and adapt a variety of appropriate strategies to solve problems. d. Monitor and reflect on the process of mathematical problem solving.


a. Recognize reasoning and proof as fundamental aspects of mathematics. d. Select and use various types of reasoning and methods of proof.







ideas. b. Select, apply, and translate among mathematical representations to solve problems. c. Use representations to model and interpret physical, social, and mathematical

phenomena.

ESSENTIAL QUESTIONS

How can we model and solve subtraction problems with and without regrouping? Can we change the order of numbers we subtract? Why or why not? How can we solve problems mentally? What strategies help us with this?





How can mental math strategies, for example estimation and benchmark numbers, help us when adding and subtracting with regrouping?

MATERIALS

Bags of Base Ten Blocks (at least 8 Hundreds, 20 Tens, and 10 Ones per pair of students)

Place Value Mat “Subtraction with Regrouping” student task sheet

GROUPING

Large group, Partners


Background KnowledgeStudents should have had prior experiences and/or instruction with addition and subtraction of two-digit numbers without regrouping. Students should also have experience regrouping using base-ten blocks.

Please see “Special Comments” section in the introduction of the unit for various addition and subtraction strategies.

Part IGive student pairs a copy of a place value board and a plastic bag with at least 8 hundreds blocks, 20 ten blocks, and 10 ones. Have students play the game “Bust One Hundred” to review regrouping skills and prepare for subtraction with regrouping.

Have students place a hundred block on their place value mat. Each partner takes a turn rolling the dice. The student has to take away the number of ones that matches their roll. Demonstrate for students how to regroup their hundred for tens and their tens for ones in order to subtract ones. Allow students to play until they reach zero ones. While students are playing, walk around and ask questions like:

How many (hundreds, tens, ones) do you have? What digits would be in those places? What is their value?

How will you regroup your hundreds for tens? What is your new number? What digit would be in the hundreds place now? What is its

value? How do you know? What is happening to your number? Why? How will you regroup your tens for ones? What digit would now be in the tens place?

What would be the value of that digit? How about the ones place? Which place would have the largest digit in it right now? Does it also have the greatest

value? How do you know?





Part IIOnce students have played for a while, present them with this problem:Julie has 43 math problems to complete for homework. She has already finished 38 of the problems. How many more math problems does she need to solve? Ask questions such as:

What are you trying to find out? How many problems did Julie have to solve for homework? How many problems has Julie already solved? What number do you need to represent on your place value mat first? Why? How many will you take away/remove? Why? What did you notice about the ones? What will you have to do in order to subtract? Can

you regroup them in any way? How will this help? How many tens do you have now? What digit is now in the tens place? What is the value

of that digit? How many ones do you have now? What digit is now in the ones place? What is the

value of that digit? Can you explain the strategies you are using to solve this problem? Is there a way you can check your answer?

After students have solved the problem, pull the class together for a class discussion. Allow several students to demonstrate their processes using the smart board or the overhead projector. If it has not been done so already then demonstrate the regrouping algorithm for students. Solve several more problems with the students using both the base ten blocks and the numerical algorithm simultaneously. Make sure the students can explain/articulate what is going on when you are performing the algorithm. Invite discussion about whether they “like it” or not and have them explain their thinking.

Part IIIGive students the “Subtraction: Modeling with Regrouping” student task sheet to solve in partner pair using base-ten blocks and the regrouping algorithm. While students are working, circulate and question students:

What are you trying to find out? What number do you need to represent on your place value mat first? Why? How many will you take away/remove? Why? What did you notice about the ones? What will you have to do in order to subtract? Can

you regroup them in any way? How will this help? How many tens do you have now? What digit is now in the tens place? What is the value

of that digit? How many ones do you have now? What digit is now in the ones place? What is the

value of that digit? Can you explain the strategies you are you using to solve this problem? Is there a way you can check your answer?

Part IVGeorgia Department of Education




After students have completed solving the problems, allow students to take turns sharing the strategy they used to solve each problem. Allow other classmates to make observations and ask questions.

Parts V-VII of the task should be completed the following day.

Part VGather students in the class meeting area and present students with this story problem:

Mr. Lundquist has a very large family. One evening he brought home 36 potatoes for his family of sixteen children and his wife. Each person had a potato for dinner that evening. How many potatoes are left?

Have several students retell the story problem and discuss what is happening in the problem.

Part VISplit students into pairs and give each student a half sheet of chart paper to use in solving the problem. Also, have various manipulatives available for students to use as they work to solve the problem. Walk around and observe students as they are problem solving. Ask questions such as:

What are you trying to find out? How many potatoes did Mr. Lundquist have at the beginning of the story? How many potatoes did his family eat? How do you know? Can you explain the strategies you are using to solve this problem? Are there other ways you can solve this problem? Is there a way you can check your answer?

As you are walking around, find students who are using a variety of strategies. Some students may draw a picture, solve the problem with manipulatives, use benchmark numbers, or use the traditional algorithm for subtraction with regrouping.

Part VIILet several students share their different strategies and answers to the problems. Allow the students to call on their peers to ask questions or make comments about their strategy, and the answer that was found. After students have shared various strategies, spend some additional time discussing the use of benchmark numbers. Demonstrate for students how the problem could have been solved by breaking the larger numbers into number combinations that are easier to subtract. For example with 36-18, you can take 2 from 36 and give it to 18 to change the problem to 34-20. You can then subtract 30-20 to get 10 and 4-0 to get 4. The difference would be 14. Create a list with students of various strategies students can use when solving subtraction problems.

Questions for Teacher Reflection What strategies were students using to solve subtraction with regrouping problems? Could students explain their thinking?





Could students solve subtraction with regrouping problems using base-ten blocks? Could students solve subtraction with regrouping problems using the algorithm? Were students able to explain how/why the algorithm works? Did students understand the various strategies from their peers?

DIFFERENTIATION

Extension Allow students to attempt regrouping problems with three-digits. Allow students to make up their own three digit subtraction story problems. They can

solve their own problems or trade with a partner.

Intervention Some students may need additional support during the problem solving through

additional questioning and scaffolding. Having them work with a partner who is very articulate about their mathematical thinking will also help.





Name: _____________________________________________________

Subtraction with Regrouping Problems

1. Moby the dog buried 62 bones in the backyard. His friend KJ dug up 28 of the bones. How many of Moby’s bones are still buried in the backyard?

2. Mrs. Hardy’s class is reading a chapter book with 196 pages. They have already read 57 pages. How many more pages does Mrs. Hardy’s class have to read in order to finish the book?

3. Emily began collecting stickers when she was five. She now has 346 stickers. She decided to give her little sister Madelyn 138 of her stickers. Now how many stickers does Emily have?





Performance Task: Subtraction Story Problems

STANDARDS ADDRESSED





regrouping. c. Use mental math strategies such as benchmark numbers to solve problems.



a. Recognize reasoning and proof as fundamental aspects of mathematics. d. Select and use various types of reasoning and methods of proof.







ideas. Georgia Department of Education





phenomena.

ESSENTIAL QUESTIONS

How can we model and solve subtraction problems with and without regrouping? Can we change the order of numbers if we subtract? Why or why not? How can we solve problems mentally? What strategies

help us with this? How can mental math strategies, for example estimation

and benchmark numbers, help us when adding and subtracting with regrouping?

MATERIALS

Various manipulatives (counters, base-ten blocks, etc.) “Subtraction Story Problems” student task sheet

GROUPING

Partners


Background KnowledgeStudents should have had prior experiences and/or instruction with addition and subtraction of two digit numbers without regrouping. Students should also have experience solving various story problems with the use of manipulatives.

Please see “Special Comments” section in the introduction of the unit for various addition and subtraction strategies.

Task Directions Give students the “Subtraction Story Problems” task sheet and allow them to solve the problems independently. As students work, walk around asking questions about the students’ strategy use. Look to see if students are using the strategy of benchmark numbers or other strategies. Ask students if changing the order of numbers in subtraction matters or not.

After students have completed solving the problems, gather the class in the meeting area. Allow students to share the strategies they used to solve the problems on the task sheet.





After students have completed solving the problem, allow students to take turns sharing their strategy use with people at their table or other small groups of students in order for students to hear strategies from their classmates.

Questions for Teacher Reflection What strategies were students using to solve addition with regrouping problems? Could students explain their thinking? Could students solve addition with regrouping problems in more than one way? Did students understand the various strategies from their peers?

DIFFERENTIATION

Extension Give students this two-step addition and subtraction with regrouping problem to attempt

to solve.

Olivia picked 15 yellow daisies and 17 white daisies. She gave 26 of the daisies to her mother and kept the rest. How many daisies did Olivia keep for herself?

Intervention Have students build a subtraction problem with base ten blocks and then draw pictures to

represent their problem. Example: 24 – 16




Since 6 ones cannot be taken from 4 ones, regroup a ten into 10 ones.

Subtract 6 ones (show this by crossing out 6 boxes).


Name: __________________________________ Date: _______________

Subtraction Story Problems

1. Mr. Holte had 23 crickets in his bug collection. He allowed 17 to hop away. How many crickets does Mr. Holte still have?

2. Mary had a large stuffed animal collection. She had 27 animals in her collection. She decided to share 9 of them with needy children. How many animals does she have now?

3. Ben loves jellybeans. He has a bag with 32 jellybeans and he eats 18 during the day. How many jellybeans does Ben have to eat tomorrow?





Performance Task: Menu Math

STANDARDS ADDRESSED





problems and check solutions. c. Use mental math strategies such as benchmark numbers to solve problems.


M2P3. Students will communicate mathematically.

a. Organize and consolidate their mathematical thinking through communication. b. Communicate their mathematical thinking coherently and clearly to peers, teachers,

and others. d. Use the language of mathematics to express mathematical ideas precisely.


a. Recognize and use connections among mathematical ideas. c. Recognize and apply mathematics in contexts outside of mathematics.

ESSENTIAL QUESTIONS

How can we solve subtraction problems with and without regrouping? How can addition help us know we subtracted two numbers correctly?

MATERIALS

Gift cards with money amounts (Examples: $20, $50, $80) Class menu Calculators Restaurant recording sheet





GROUPING

Small group


Background KnowledgeStudents should have had prior experiences and/or instruction with addition and subtraction of three digit numbers with and without regrouping. Students should also have experience adding and subtracting money.

Task DirectionsAsk students if they have ever been to a restaurant. Have them list some things they would see at a restaurant. Tell the students that today they will create a class restaurant.

Let the class decide what kind of restaurant to create. Brainstorm a name for the restaurant. Vote and take tally marks to determine the class’ favorite name. Have students list the types of drinks, appetizers, entrees, sides, and desserts they would like to serve at their restaurant. This list will serve as the class menu (you may make paper menus if desired). Assign prices to each of the items available on the menu. Separate the children into pairs. One of the two children in each group will be the customer and the other student in each group will be the waiter/waitress. The customer should begin by choosing a gift card. The student then chooses one to two items from the menu. The waiter/waitress takes the order and tallies the bill. Meanwhile, the customer should estimate his/her bill. The waiter/waitress gives the bill to the customer. Using estimation, the customer should decide if the bill seems correct. The customer then pays with his/her gift card and figures out how much change he/she should get (using subtraction with regrouping). After the customer writes down the total and the change, the waiter/waitress may check the math on a calculator (serves as the register). All work should be saved and turned in to be double checked by the teacher. Students should switch roles and play again.

Questions for Teacher Reflection What strategies were students using to solve subtraction with regrouping problems? What strategies were students using to solve addition with regrouping problems? Could students explain their thinking?

DIFFERENTIATION

Extension Students have a gift card of $100 and can choose three items from the menu.

Intervention Give the student a $20 gift card. The student should choose one item from the menu.

Name __________________________________ Date _______________Georgia Department of Education




Menu Math

How much is your gift card?

What would you like to order?

What do you estimate as the sum of your meal?

$___________

+ $___________

_________________

What is the actual sum of your meal?

$___________

+ $___________

_________________

Do you have enough money to purchase this meal?

How much change will you get back?

$___________

- $___________

_________________

Performance Task: Counting Mice

STANDARDS ADDRESSED

M2N2. Students will build fluency with multi-digit addition and subtraction.a. Correctly add and subtract two whole numbers up to three-digits each with

regrouping.Georgia Department of Education




b. Understand and use the inverse relation between addition and subtraction to solve problems and check solutions.

c. Use mental math strategies such as benchmark numbers to solve problems.d. Use basic properties of addition (commutative, associative, and identity) to simplify


e. Estimate to determine if solutions are reasonable for addition and subtraction.

M2P1. Students will solve problems (using appropriate technology).a. Build new mathematical knowledge through problem solving.b. Solve problems that arise in mathematics and in other contexts.c. Apply and adapt a variety of appropriate strategies to solve problems.d. Monitor and reflect on the process of mathematical problem solving.


and others. c. Analyze and evaluate the mathematical thinking and strategies of others.d. Use the language of mathematics to express mathematical ideas precisely.

M2P5. Students will represent mathematics in multiple ways.a. Create and use representations to organize, record, and communicate mathematical

ideas.b. Select, apply, and translate among mathematical representations to solve problems.c. Use representations to model and interpret physical, social, and mathematical

phenomena.

ESSENTIAL QUESTIONS

How can we solve subtraction problems with and without regrouping? How can addition help us know if we subtracted two numbers correctly?

MATERIALS

Mouse Count by Ellen Stoll Walsh or similar counting book “Counting Mice” recording sheet

GROUPING

Partners


Background Knowledge





Students should have had prior experiences solving a variety of problem types. Some examples are in the chart below.

Task DirectionsRead the book Mouse Count by Ellen Stoll Walsh to the class. After reading the story, explain to the children that Snake was very hungry one day and put 25 mice in the jar before he took a nap. He put 17 more in after the nap. Their job is to figure out how many mice are in the jar all together.

Have paper, pencils, and crayons available for the children to use to help them solve their problem. Have students work alone, but sitting in groups to observe and discuss each other’s strategies. As students work, look for a variety of strategies and/or solutions. Before asking students to share, plan which solutions should be shared and in which order they should be shared. In this way students are exposed to strategies/solutions that would add to students’ understanding of numbers and solving problems.

After students have had an opportunity to finish their work, let several children share the strategies they used. Create a class chart to document the strategies.





CommentsReading the book helps children visualize this combination situation as they try to solve the problem. The actual numbers in the problem (25 + 17) can be adjusted to meet the needs of your children or to differentiate instruction within the class.

Questions/Prompts for Formative Student Assessment How did you solve the problem? Can you write a number sentence (make a picture, make a model) that shows your

thinking? How do you know your answer is correct? Can you solve this problem in a different way?

Questions for Teacher Reflection What strategies are students using to solve the problems? Are students able to explain their thinking?

DIFFERENTIATION

Extension Have students write story problems and describe strategies they could use to solve

them.

Intervention Give students more experiences with regrouping using base ten blocks, pennies, dimes,

and dollars. For problems that require regrouping, create problems with money that can be solved by

regrouping pennies and dimes.





Name Date

Counting Mice

1. Snake was very hungry one day and put 25 mice in the jar before he took a nap. He put 17 more mice in the jar after his nap. How many mice are in the jar all together?

2. Snake had 52 mice. His cousin Rattles had 64 mice. They put all their mice in one big jar. 17 of the mice escaped. How many mice did Snake and Rattles have now?





Performance Task: Every Picture Tells a Story

STANDARDS ADDRESSED





problems and check solutions.c. Use mental math strategies such as benchmark numbers to solve problems. d. Use basic properties of addition (commutative, associative, and identity) to simplify








phenomena.

ESSENTIAL QUESTIONS

How can subtraction help us to know if we added correctly? How can addition help us know if we subtracted correctly?





MATERIALS

Math Appeal by Greg Tang or similar addition riddle story Pictures from magazines Paper

GROUPING

Small group


Background KnowledgeStudents should have had prior experiences solving a variety of problem types. Students should be familiar with problem solving using various manipulatives, and be able to explain their strategies.

Part I Gather students together on the meeting area. Read Math Appeal by Greg Tang or similar addition riddle story. Discuss addition pictures with class and have students answer riddles as you read the story.

Show students a picture from a magazine. Make sure the magazine picture offers several possible groups or combinations and would be interesting. Discuss the picture with the whole class. The children may want to ask questions that can be answered by the picture, but do not require an operation and can be found by counting. Some examples of questions the students might make up:

There are 6 groups of 2 dogs pulling the sled. How many dogs are pulling the sled? There are 23 mice in the picture. 15 are under the table. How many are on top of the

table?

Part IIGive students a picture from a magazine or a coloring book. Try to make sure the pictures have several number combinations for two-digit numbers. Have students create number sentences to describe their pictures. Encourage students to check their answers using the inverse operations. As students are working on their number sentences, walk around and ask questions like:

What numbers do you see within your picture? How can you use these numbers in an equation? How are you describing your picture with number sentences? How can you check your answer to see that it is correct? Is there another number sentence you could write with those numbers?





Part IIILet students share their story problems and pictures with the class. Allow other students to ask questions or make comments about the student’s work. See if students can come up with additional story problems for each picture. Questions for Teacher Reflection

Were students able to find numbers within their pictures? Were students able to use those numbers to create a number sentence? What strategies were students using to solve the problem? Could students explain their thinking? Did students use inverse relationships between addition and subtraction to check their

answers? Were students able to ask questions when their peers shared work?

DIFFERENTIATION

Extension Students create their own pictures and write story problems based on their creation.

Intervention

Provide story frame for students to complete and illustrate. Example: Joey has balloons. ballons popped. How many balloons does Joey have now?





Learning Task: Mental Mathematics

STANDARDS ADDRESSED



problems and check solutions.c. Use mental math strategies such as benchmark numbers to solve problems. d. Use basic properties of addition (commutative, associative, and identity) to simplify








phenomena.

ESSENTIAL QUESTIONS

What is mental math? How does mental math help us calculate more quickly? How can mental math strategies, for example estimation and benchmark numbers, help us

when adding and subtracting with regrouping?

MATERIALS

Chalkboard, overhead projector, or Interactive whiteboard “Mental Mathematics” recording sheet





GROUPING

Large group


CommentsDiscussions should move beyond whether or not the answers are correct. The goal here is to develop efficient ways to group numbers and/or develop compensation strategies for mental addition and subtraction. The value of group discussions and modeling is evident when students gather insights from their classmates that will reinforce basic number sense and develop strategies that will help them become better at mental computation.

Students should be encouraged to solve problems in ways that make sense to them. If students have never been encouraged to solve problems mentally and share their own strategies with others, they may be reluctant to share or may feel that their strategy is inappropriate. Establish ground rules in your classroom about sharing ideas and how to appropriately respond to each other.

Background Knowledge Students should have some prior experiences with basic computation strategies allowing them to calculate quickly and reliably. Examples include counting on, doubling, making tens, and using benchmark numbers.

Task DirectionsBegin this activity by placing one problem at a time on the board, preferably horizontally. Be aware that students may initially need individual time to solve these problems mentally, so encourage students to be patient and quiet during this time.

After allowing enough time for students to consider the problem, lead a discussion by asking several students to share their solution and/or strategy. Simply stating an answer is not enough to make this a rich activity. Encourage students to share different strategies, asking them to try to make sense of each solution as it is presented. Remind students that the goal is to become efficient and flexible in their thinking and strategies.

Have students follow the directions below:Solve the following problems as they are placed on the board using no paper or manipulatives. Use your mental math strategies. Be prepared to share your solutions and strategies.

15 + 7Students may solve this problem in a variety of ways. Examples are: 15 + 5 is 20 and 2 more is 22. 5 + 7 is 12 and 10 more is 22. 10 + 7 is 17, 3 more is 20 and 2 more is 22.





24 + 16Students may solve this problem in a variety of ways. Examples are: 20 + 10 is 30 and 4 + 6 is 10, so 30 + 10 is 40. 4 + 6 is 10 and 20 + 10 is 30, so 10 + 30 is 40. 24 + 6 is 30 and 10 more is 40.

99 + 17Students may solve this problem in a variety of ways. Examples are: 99 and 1 more is 100, 100 + 17 is 117, but take 1 away that was added to the 99 to get

100, so the answer is 116. Some may attempt a traditional algorithm, but should notice that this is more

cumbersome that examining the numbers and using the ideas above to compute.

50 - 12Students may solve this problem in a variety of ways. Examples are: 50 – 10 is 40, then 40 – 2 is 38. 50 – 2 is 48, them 48 – 10 is 38. You need 8 more to get to 20 from 12, then 30 more to get to 50, so the answer is 8 +

30 or 38. Note: Students who use this method are actually finding the difference between the two numbers and not simply “taking away.” This is a wonderful opportunity to discuss different approaches to subtraction.

Questions/Prompts for Formative Student Assessment What is one strategy you could use to solve the problem quickly? How can you verify your solution? Could this problem be solved another way? How? Which problem solving strategy works best for you?

Questions for Teacher Reflection What routines do I have in place that will encourage students to verbalize their math

thinking? Were students able to develop additional mental math strategies? How effectively did students explain their thinking? How will I be sure to listen to each student’s explanations during a given math lesson? What mental math strategies do my students seem to use most? Least?

DIFFERENTIATION

Extension When you are presenting problems to students, vary the problems you use. Include

various operations and numbers. Have students develop their own mental math problems, solve them, and explain their

solution strategies.

Intervention Georgia Department of Education




Have students work with smaller, single-digit numbers initially. Have students work with a partner to develop strategies. Students who struggle with math reasoning often have difficulty communicating their

thinking. Extra sensitivity and encouragement must be shown for these students as they develop and strengthen these sets of process skills. Questioning can scaffold students who are challenged by discussing their math thinking.





Name Date

Mental MathematicsWhen your teacher gives you an addition problem, solve it using

mental mathematics and then record your thinking in the correct box below. During student sharing, if you like a strategy used by another student, record it in the same box.

Problem #1 Problem #2

Problem #3 Problem # 4





CULMINATING TASK: Planning a Field Trip

STANDARDS ADDRESSED






regrouping.

M2D1. Students will create simple tables and graphs and interpret their meaning. a. Organize and display data using picture graphs, Venn diagrams, bar graphs, and

simple charts/tables to record results. b. Know how to interpret picture graphs, Venn diagrams, and bar graphs.




a. Recognize reasoning and proof as fundamental aspects of mathematics. b. Make and investigate mathematical conjectures. c. Develop and evaluate mathematical arguments and proofs. d. Select and use various types of reasoning and methods of proof.











M2P5. Students will represent mathematics in multiple ways.



phenomena.

ESSENTIAL QUESTIONS

How can we solve addition problems with and without regrouping? How can subtraction help us to know if we added two numbers correctly? How can addition help us know if we subtracted two numbers correctly? What strategies can help us when adding and subtracting with regrouping? How does a Venn diagram organize information? How do we count quantities using a Venn diagram?

MATERIALS

School Population Documents

Any of the Magic School Bus books Large Sheet of Chart Paper and

markers Brown paper lunch bags or other small containers “Planning a Field Trip” picture cards “Planning a Field Trip” Venn Diagram “Planning a Field Trip” Cafeteria Order





GROUPING

Large group, Partners






Background KnowledgeIn order to complete this culminating activity, students should have prior experiences solving two digit addition and subtraction problems. Students should also feel comfortable analyzing their strategies for problem solving and sharing these strategies with the class. Students should have experience working with Venn diagrams and tallying and making simple charts. Part IGather students together on the class meeting area. Read one of the Magic School Bus books. Tell students we are going to plan for our upcoming field trip. (If you are not going on a field trip, tell the students they are going to plan for the field trip in the story.) Ask the students to help figure out how many buses they would need for a school field trip and what is the least expensive way to order them. Have a list of all the classes in the school and the number of students in each class. The actual number of classes you include in the problem will depend on the size of your school. It is recommended that the total be a three-digit number. If the school is very small you may include all the classes. If the school is large you may want to include only a few grade levels. Pass out “Planning a Field Trip” student task sheet. Have the students work with a partner to figure out how many of each kind of bus the school should rent and how to spend the least amount of money. Have each pair make a poster of their strategy for determining the total number of students and the recommended number of buses. By this time, most students should be more fluent in multi-digit addition. After the students have found the total number of students, they will need to find ways to makes groups of 30 (or 12) to determine how many buses are needed. Some students may guess and check multiples of 30 (or 12). Encourage a variety of methods. Have the students make posters of their processes (for example, drawing pictures and decomposing and composing the numbers to make groups of 30). Have the partners share their posters with the class. Compare the various options. Have students write in their math journals which were the top three options in their opinion and tell why.

Part IIGather students back together on the class meeting area. Tell them that we also need to plan our lunch for the field trip because we will not be eating in the cafeteria. They will have the choice of a sandwich, bags of carrot sticks, and/or juice box for lunch. Every student will be able to choose what items will put in their lunch bags. (Possible choices could include: only a sandwich, a sandwich and juice, carrot sticks and sandwich, OR all three – ONE, TWO, OR ALL THREE DIFFERENT ITEMS.) Explain to the students that the cafeteria will need to know the correct quantity of each item to prepare for our trip.





Students then go back to their desks and cut out the item(s) to put in their paper bag. Students will place 1, 2, or all 3 items in the paper bags. Students return to the carpet to graph their data on a large 3 circle Venn diagram. Students can come to the board and write their names in the correct section of the Venn diagram. Teacher will need to support students’ thinking about what region of the Venn that they will write their name inside. Have students explain to classmates why they placed their names in specific areas of the Venn diagram. Discuss the results.

Questions for Teacher Reflection What strategies were students using to solve the regrouping problems? Could students explain their thinking?

DIFFERENTIATION

Extension Students can create other Venn diagrams:

sandwich choices: PBJ, cheese, or meat juice choices: apple, cranberry, orange field trip choices: zoo, aquarium, river

Give students a price per student lunch and have them find the total cost.

Intervention Students who are having difficulty with the operations may complete the task for one or

two grades. Students can the appropriate place clip art pictures on each student’s name on the Venn to

assist with counting and tallying.





Name Date

Planning a Field Trip

Directions: Work with your partner to determine how many of each kind of bus the school should rent and spend the least amount of money. Show your work.

Large bus holds 30 people Rental $10.00 Small bus holds 12 people Rental $ 5.00





Planning a Field Trip Picture Cardssandwich carrot sticks juice box








Planning a Field Trip Venn Diagram

Carrot sticks

sandwich juice box


Planning a Field Trip Cafeteria Order

Please order the following quantities for our field trip.Date:Sandwiches:

Bags of carrot sticks:

Juice boxes:


