grade 5 mathematics curriculum document...

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Grade 5 Mathematics Curriculum Document

2016-2017

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Table of Contents Cover Page Pg. 1 Table of Contents Pg. 2 Trouble Shooting Guide Pg. 3 Best Practices in the Math Classroom Pg. 4 Problem Solving 4-Square Model Pg. 6 Problem Solving with Pictorial Modeling/ Strip Diagrams Pg. 7 Number Sense/ Number Talks Pg. 8 Year at a Glance Pg. 17 TEKS Spiraling Checklist Pg. 18 Mathematics Process Standards Pg. 21 Math Instructional Resources Pg. 22 Bundle 1: Representing and Comparing Positive Rational Numbers Pg. 24 Bundle 2: Whole Number Operations Pg. 28 Bundle 3: All Operations with Decimals Pg. 38 Bundle 4: All Operations with Fractions Pg. 49 Bundle 5: Graphing on Coordinate Planes Pg. 58 Bundle 6: Geometry and Measurement Pg. 65 Bundle 7: Data Analysis Pg. 72 Bundle 8: Personal Financial Literacy Pg. 78 Bundle 9: STAAR Review and Testing Pg. 83 Bundle 10:Extended Learning Pg. 84 Bundle 11: Continued Extended Learning Pg. 86

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Trouble Shooting Guide

• The 2015-2016 Mathematics Curriculum Document for Grade 5 includes the following features:

• The NISD Curriculum Document is a TEKS-Based Curriculum.

• Year at a Glance Indicating Bundle Titles and Number of Days for Instruction

• Color Coding: Green- Readiness Standards, Yellow- Supporting Standards, Blue- Process Standards,

Purple- ELPS, Strike-Out- Portion of TEKS not Taught in Current Bundle

• NISD Math Instructional Focus Information

• The expectation is that teachers will share additional effective resources with their campus Curriculum &

Instructional Coach for inclusion in the document.

• The NISD Curriculum Document is a working document. Additional resources and information will be

added as they become available.

• **Theresourcesincludedhereprovideteachingexamplesand/ormeaningfullearningexperiencestoaddresstheDistrictCurriculum.InordertoaddresstheTEKStotheproperdepthandcomplexity,teachersareencouragedtouseresourcestothedegreethattheyarecongruentwiththeTEKSandresearch-basedbestpractices.Teachingusingonlythesuggestedresourcesdoesnotguaranteestudentmasteryofallstandards.Teachersmustuseprofessionaljudgmenttoselectamongtheseand/orotherresourcestoteachthedistrictcurriculum.

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NISD Math Focus

Best Practices in the Math Classroom • Teaching for Conceptual Understanding: Math instruction should focus on developing a true understanding of the math concepts being

presented in the classroom. Teachers should avoid teaching “quick tricks” for finding the right answers and instead focus on developing student understanding of the “why” behind the math. Math is not a list of arbitrary steps that need to be memorized and performed, but is, rather, a logical system full of deep connections. When students see math as a set of disconnected steps to follow they tend to hold many misconceptions, make common mistakes, and do not retain what they have learned. However, when students understand the connections they have fewer misconceptions, make less errors, and tend to retain what they have learned.

• Developing Student Understanding through the Concrete-Pictorial-Abstract Approach: When learning a new math concept, students should be taken through a 3-step process of concept development. This process is known as the Concrete-Pictorial-Abstract approach. During the concrete phase, students should participate in hands-on activities using manipulatives to develop an understanding of the concept. During the pictorial phase, students should use pictorial representations to demonstrate the math concepts. This phase often overlaps with the concrete phase as students draw a representation of what they are doing with the manipulatives. During the abstract phase, students use symbols and/or numbers to represent the math concepts. This phase often overlaps with the pictorial phase as students explain their thinking in pictures, numbers, and words. If math concepts are only taught in the abstract level, students attain a very limited understanding of the concepts. However, when students go through the 3-step process of concept development they achieve a much deeper level of understanding.

• Developing Problem Solving Skills through Quality Problem Solving Opportunities: Students should be given opportunities to develop their problem solving skills on a daily basis. One effective approach to problem solving is the think-pair-share approach. Students should first think about and work on the problem independently. Next, students should be given the opportunity to discuss the problem with a partner or small group of other students. Finally, students should be able to share their thinking with the whole group. The teacher can choose students with different approaches to the problem to put their work under a document camera and allow them to talk through their thinking with the class. The focus of daily problem solving should always be Quality over Quantity. It is more important to spend time digging deep into one problem than to only touch the surface of multiple problems.

• Developing Problem Solving Skills through Pictorial Modeling: One of the most important components of students’ problem solving development is the ability to visualize the problem. Students should always draw a pictorial representation of the problem they are trying to solve. A pictorial model helps students to better visualize the problem in order to choose the correct actions needed to solve it. Pictorial modeling in math can be done with pictures as simple as sticks, circles, and boxes. There is no need for detailed artistic representations. One of the most effective forms of pictorial modeling is the strip diagram (or part-part-whole model in lower grades). This type of model allows students to see the relationships between the numbers in the problem in order to choose the proper operations.

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• Developing Students’ Number Sense: The development of number sense is a critical part of a student’s learning in the mathematics classroom. The ability to reason about numbers and their relationships allows students the opportunity to think instead of just following a rote set of procedures. The standard algorithms for computation may provide students with a quick answer, but they do not allow for development of student thinking and reasoning. The standard algorithms should not be abandoned completely, but should be used as one of many ways of approaching a computation problem. It is, however, very important that students have the opportunity to develop their number sense through alternative computation strategies before learning the standard algorithm in order to prevent students from having a limited view of number relationships.

• Creating an Environment of Student Engagement: The most effective math classrooms are places in which students have chances to interact with their teacher, their classmates, and the math content. Students should be given plenty of opportunities to explore and investigate new math concepts through higher-order, rigorous, and hands-on activities. Cooperative learning opportunities are critical in order for students to talk through what they are learning. The goal should be for the student to work harder than the teacher and for the student to do more of the talking.

• Higher Level Questioning: The key to developing student thinking is in the types of questions teachers ask their students. Teachers should strive to ask questions from the top three levels of Bloom’s Taxonomy to probe student thinking.

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NISD Math Focus

Developing Problem Solving through a 4-Square Model Approach • The 4-square problem solving model should be used to help guide students through the problem solving process. It is important that

students complete step 2 (pictorial modeling) before attempting to solve the problem abstractly (with computation). When students create a visual model for the problem they are better able to recognize the appropriate operation(s) for solving the problem.

Dragon Problem Solving 1. What is the problem asking?

2. This is how I see the problem. (pictorial/ strip diagram)

3. This is how I solve the problem. (computation)

4. I know my work is correct because... (justify)

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NISD Math Focus

Developing Problem Solving through Pictorial Modeling/ Strip Diagrams Part-Part-Whole: Addition and Subtraction

Whole Unknown Alan had some marbles. He lost 12 of them. Then he had 32 left. How many did he have to begin with?

? 12 32

Part Unknown Steven had 122 peanuts. He ate 71 of them. How many peanuts are left?

122 71 ?

Another Part Unknown Some adults and 12 children are on a bus. There are 31 people on the bus. How many adults are on the bus?

31 ? 12

Comparison: Addition and Subtraction

How Much More Unknown Alex has 47 toy cars. Keisha has 12 toy cars. How many more cars does Alex have than Keisha?

47 12 ?

Smaller Quantity Unknown Fran spent $84 which was $26 more than Alice spent. How much did Alice spend?

84 ? 26

Larger Quantity Unknown Barney has 23 old coins. Steve has 16 more old coins than Barney. How many old coins does Steve have?

? 23 16

Equal Parts: Multiplication and Division

Whole Unknown Kim has 4 photo albums. Each album has 85 pictures. How many photos are in her 4 albums?

? 85 85 85 85

Amount for Each Part Unknown Pam put the same number of apples in each of 4 bags. She ended up with 52 apples in bags. How many apples did she put in each bag?

52 ? ? ? ?

Number of Equal Parts Unknown Fred bought some books. Each book cost $16. He spent $48 on books. How many books did he buy?

48 16

Comparison: Multiplication and Division

Larger Quantity Unknown Raul has 17 markers. Linda has 3 times as many. How many markers does Linda have?

? 17 17 17 17

Smaller Quantity Unknown Maria has 24 cards. She has 3 times as many cards as Jamal has. How many cards does Jamal have?

24 ? ? ? ?

Number of Times as Many Unknown Ann’s teacher is 39 years old. Ann is 13 years old. Ann’s teacher is how many times as old as Ann?

39 13 13

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NISD Math Focus

Developing Number Sense through Number Talks

What is a Number Talk? A Number Talk is a short, ongoing daily routine that provides students with meaningful ongoing practice with computation. A Number Talk is a powerful tool for helping students develop computational fluency because the expectation is that they will use number relationships and the structures of numbers to add, subtract, multiply and divide.

Number Talks should be structured as short sessions alongside (but not necessarily directly related to) the ongoing math curriculum. It is important to keep Number Talks short, as they are not intended to replace current curriculum or take up the majority of the time spent on mathematics. In fact, teachers need to spend only 5 to 15 minutes on Number Talks. Number Talks are most effective when done every day.

A Rationale for Number Talks

http://www.mathsolutions.com/documents/9781935099116_ch1.pdf

Number Talks 6-Weeks Focus

Operation Strategies 1st 6-Weeks Addition Breaking Each Number Into its Place Value (Decomposing Each Number), Making Landmark

or Friendly Numbers, Doubles/ Near-Doubles, Making Tens, Compensation, Adding Up in Chunks (Decomposing One Number)

2nd 6-Weeks Subtraction Adding Up, Removal or Counting Back, Place Value and Negative Numbers, Keeping a Constant Difference, Adjusting One Number to Create and Easier Problem

3rd 6-Weeks Multiplication Repeated Addition or Skip Counting, Making Landmark or Friendly Numbers, Partial Products, Doubling and Halving, Breaking Factors into Smaller Factors

4th 6-Weeks Multiplication Repeated Addition or Skip Counting, Making Landmark or Friendly Numbers, Partial Products, Doubling and Halving, Breaking Factors into Smaller Factors

5th 6-Weeks Division Repeated Subtraction or Sharing/ Dealing Out, Multiplying Up, Partial Quotients, Proportional Reasoning

6th 6-Weeks Division Repeated Subtraction or Sharing/ Dealing Out, Multiplying Up, Partial Quotients, Proportional Reasoning

* Highlighted strategies must be introduced. Other strategies may be used with teacher discretion.

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Number Talks

Addition Strategy Examples Strategy Example(s)

Breaking Each Number Into its Place Value (Decomposing Each Number)

116 + 118 (100 + 10 + 6) + (100 + 10 + 8) 100 + 100 = 200 10 + 10 = 20 6 + 8 = 14 200 + 20 + 14 = 234

Making Landmark or Friendly Numbers

116 + 118 +2 116 + 120 = 236 236 – 2 = 234

Doubles/ Near-Doubles 116 + 118 -2 -3 115 + 115 = 230 230 + 4 = 234

Making Tens 116 + 118 (110 + 4 + 2) + (110 + 8) 110 + 110 + (2 + 8) + 4 110 + 110 + 10 + 4 230 + 4 = 234

116 + 118 (110 + 6) + (110 + 4 + 4) 110 + 110 + (6 + 4) + 4 110 + 110 + 10 + 4 = 234

Compensation 116 + 118 - 2 +2 114 + 120 = 234

116 + 118 +4 -4 120 + 114 = 234

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Adding Up in Chunks (Decomposing One Number)

116 + 118 116 + (100 + 10 + 4 + 4) 116 + 100 = 216 216 + 10 = 226 226 + 4 = 230 230 + 4 = 234

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Number Talks

Subtraction Strategy Examples Strategy Example(s)

Adding Up 123 – 59 59 + 1 = 60 60 + 40 = 100 100 + 23 = 123 1 + 40 + 23 = 64

Removal or Counting Back

123 – 59 123 – (10+10+10+10+10+3+6)

123- 59 123 – (20+30+3+6) 123 -20 = 103 103 – 30 = 73 73 – 3 = 70 70 – 6 = 64

123 – 59 123 – (50 + 9) 123 – 50 = 73 73 – 9 = 64

Place Value and Negative Numbers

123 – 59

100 20 3 - 50 9

100 -30 -6

100 – 30 – 6 100 – 30 = 70 70 – 6 = 64

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Keeping a Constant Difference

123 – 59 123 + 1 = 124 - 59 + 1 = - 60 64

300 – 78 300 – 1 = 299 - 78 – 1 = - 77 222

Adjusting One Number to Create and Easier Problem

123 – 59 123 123 -59 + 1 = - 60 63 + 1 = 64

123 – 59 123 123 - 59 – 6 = - 53 70 – 6 = 64

123 – 59 123 + 6 = 129 - 59 -59 70 – 6 = 64

123 – 59 123 – 4 = 119 -59 -59 60 + 4 = 64

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Number Talks

Multiplication Strategy Examples Strategy Example(s)

Repeated Addition or Skip Counting

6 x 15 15 + 15 + 15 + 15 + 15 + 15 15 + 15 = 30 30 + 15 = 45 45 + 15 = 60 60 + 15 = 75 75 + 15 = 90

6 x 15 15 + 15 + 15 + 15 + 15 + 15 2 x 15 = 30 2 x 15 = 30 90 2 x 15 = 30

6 x 15 15 + 15 + 15 + 15 + 15 + 15 6 x 10 = 60 6 x 5 = 30 90

Making Landmark or Friendly Numbers

9 x 15 9 groups of 15 + 1 (group of 15) 10 x 15 = 150 150 – 15 = 135

Partial Products 12 x 9 (10 + 2) x 9 10 x 9 = 90 2 x 9 = 18 90 + 18 = 108

12 x 9 (6 + 6) x 9 6 x 9 = 54 6 x 9 = 54 54 + 54 = 108

12 x 9 12 x (5 + 4) 12 x 5 = 60 12 x 4 = 48 60 + 48 = 108

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Doubling and Halving

Breaking Factors into Smaller Factors

8 x 25 (4 x 25) + (4 x 25) 100 + 100 = 200

8 x 25 (2 x 4) x 25 2 x (4 x 25) 2 x 100 = 200

8 x 25 8 x (5 x 5) (8 x 5) x 5 40 x 5 = 200

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Number Talks

Division Strategy Examples Strategy Example(s)

Sharing/ Dealing Out 30 ÷ 5

1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1

5 x 6 = 30

30 ÷ 5

2 2 2 2 2 2 2 2 2 2 2 2 2 2 2

5 x 6 = 30 First Deal: 5 x 2 = 10

Second Deal: 5 x 2 = 10 Third Deal: 5 x 2 = 10

Multiplying Up 72 ÷ 8 8 x 5 = 40 8 x 4 = 32 8 x 9 = 72 72 ÷ 8 = 9 72 ÷ 8

8 x 5 = 40

8 x 4 = 32

76 ÷ 4 4 x 10 = 40 4 x 5 = 20 4 x 4 = 16 4 x 19 = 76 76 ÷ 4 = 19

4 x 10 = 40

4 x 5 = 20 4 x 4 = 16

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Partial Quotients 56 ÷ 4 40 ÷ 4 = 10 16 ÷ 4 = 4 56 ÷ 4 = 14

88 ÷ 4 40 ÷ 4 = 10 40 ÷ 4 = 10 8 ÷ 4 = 2 88 ÷ 4 = 22

Proportional Reasoning 88 ÷ 4 88 ÷ 4 = 44 ÷ 2 = 22 ÷ 1 = 22

88 44 22 ---- = ---- = ----- = 22 4 2 1

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Year at a Glance

First Semester Second Semester 1st 6-Weeks 4th 6-Weeks

• Bundle #1- Representing and Comparing Positive Rational Numbers (7 days)

• Bundle #2- Whole Number Operations (23 days)

• Bundle #6 (cont.)- Geometry and Measurement (9 days)

• Bundle #7- Data Analysis (10 days)

• Bundle #8- Personal Financial Literacy (5 days)

• Bundle #9- STAAR Review and Testing (4 days) 2nd 6-Weeks 5th 6-Weeks

• Bundle #3- All Operations with Decimals (16 days)

• Bundle #4- All Operations with Fractions (13 days)

• Bundle #9 (cont.)- STAAR Review and Testing (17 days)

• Bundle #10- Extended Learning (16 days)

3rd 6-Weeks 6th 6-Weeks • Bundle #4 (cont.)- All Operations with Fractions (5 days)

• Bundle #5- Graphing on Coordinate Planes (15 days)

• Bundle #6- Geometry and Measurement (9 days)

• Bundle #10 (cont.)- Extended Learning (11 days)

• Bundle #13- Continued Extended Learning (18 days)

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TEKS Spiraling Checklist

*The following checklist is intended to assist with planning for spiraled review of previously taught material. The lavender shaded boxes indicate the 6-weeks in which the TEKS are introduced/taught. For each 6-weeks following initial instruction, there are six boxes in which to indicate when the TEKS were reviewed. For documentation purposes, the date the TEKS were reviewed can be written in the provided boxes. TEKS can be reviewed through center activities, problem of the day, entry/exit tickets, mini-lessons, etc.

1st 6-Weeks 2nd 6-Weeks 3rd 6-Weeks 4th 6-Weeks 5th 6-Weeks 6th 6-Weeks 5.2A

5.2B

5.2C

5.3A

5.3B

5.3C

5.3D

5.3E

5.3F

5.3G

5.3H

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5.3I

5.3J

5.3K

5.3L

5.4A

5.4B

5.4C

5.4D

5.4E

5.4F

5.4G

5.4H

5.5A

5.6A

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5.6B

5.7A

5.8A

5.8B

5.8C

5.9A

5.9B

5.9C

5.10A

5.10B

5.10C

5.10D

5.10E

5.10F

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Mathematical Process Standards • Process standards MUST be integrated within EACH bundle to ensure the success of students.

5.1A 5.1B 5.1C 5.1D 5.1E 5.1F 5.1G apply mathematics to problems arising in everyday life, society, and the workplace

use a problem-solving model that incorporates analyzing given information, formulating a plan or strategy, determining a solution, justifying the solution, and evaluating the problem-solving process and the reasonableness of the solution

select tools, including real objects, manipulatives, paper and pencil, and technology as appropriate, and techniques, including mental math, estimation, and number sense as appropriate, to solve problems

communicate mathematical ideas, reasoning, and their implications using multiple representations, including symbols, diagrams, graphs, and language as appropriate

create and use representations to organize, record, and communicate mathematical ideas

analyze mathematical relationships to connect and communicate mathematical ideas

display, explain, and justify mathematical ideas and arguments using precise mathematical language in written or oral communication

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Math Instructional Resources

Resource Print/Online Description EnVision Both Textbook Adoption https://www.pearsontexas.com/#/ Motivation Math Both Supplemental Curriculum https://www.mentoringminds.com/customer/account/login/ Engaging Mathematics Print Collection of Mini-Lessons for All TEKS from Region IV http://www.region4store.com/catalog.aspx?catid=1171582 Reasoning Mind (STAAR Readiness) Online Online STAAR Supplemental Curriculum http://www.rmcity.org/ Think Through Math Online Online Supplemental Curriculum https://lms.thinkthroughmath.com/users/sign_in\ Thinking Blocks Online Online Problem Solving Practice with Strip Diagrams http://www.mathplayground.com/thinkingblocks.html 5th Grade Math Games Print Collection of Engaging and Low-Prep Math Games for Skill

Practice http://maccss.ncdpi.wikispaces.net/file/view/5thgrade_GAMES_8.22.14.pdf/519548106/5thgrade_GAMES_8.22.14.pdf Epic! For Educators Online Search for Literature Connections for Math Content https://www.getepic.com/educators Number Talks (Sherry Parrish) Print Develop Number Sense Through a Daily Number Talk Routine TEKSing Toward STAAR Print STAAR Based Supplemental Curriculum Lessons for Learning (North Carolina) Print Collection of Engaging and Rigorous Math Lessons http://maccss.ncdpi.wikispaces.net/file/view/5thgrade%20Lessons%204Learning%2022616.pdf/575957189/5thgrade%20Lessons%204Learning%2022616.pdf Math Learning Center (Bridges) Print Collection of Engaging and Rigorous Math Lessons http://catalog.mathlearningcenter.org/catalog/supplemental-materials-elementary/lessons-activities-grade-5-free NCTM Illuminations Online Search for Engaging and Rigorous Math Lessons by Grade and

Topic http://illuminations.nctm.org/ Math Coach’s Corner Online Math Blog from a Master Texas Math Teacher, Coach, and

Consultant http://www.mathcoachscorner.com/ Promethean Planet Online Tools and Lessons for Interactive Whiteboard http://www.prometheanplanet.com/en-us/ Interactive Math Glossary Online TEA Interactive Math Glossary http://www.texasgateway.org/resource/interactive-math-glossary?field_resource_keywords_tid=math%20teks&sort_by=title&sort_order=ASC&items_per_page=5

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TEKS Information for Teachers TEA STAAR Resources Online TEA Information Regarding STAAR Math http://tea.texas.gov/student.assessment/staar/math/ TEA Math Resources Online TEA Supporting Information for Math TEKS http://tea.texas.gov/Curriculum_and_Instructional_Programs/Subject_Areas/Mathematics/Resources_for_the_Revised_Mathematics_TEKS/ Lead4Ward Resources Online Math TEKS Instructional Resources and Supporting

Information http://lead4ward.com/resources/ TEKS Resource System Online Math TEKS Instructional Resources and Supporting

Information http://www.teksresourcesystem.net/module/profile/Account/LogOn

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Course: Grade 5 Math Bundle 1: Representing and Comparing Positive Rational Numbers

Dates: August 22nd – August 30th (7 days)

TEKS 5.2A: represent the value of the digit in decimals through the thousandths using expanded notation and numerals 5.2B: compare and order two decimals to thousandths and represent comparisons using the symbols >, <, or =

ELPS Learning Strategies 1E: internalize new basic and academic language by using and reusing it in meaningful ways in speaking and writing activities that build concept and language attainment Listening 2C: learn new language structures, expressions, and basic and academic vocabulary heard during classroom instruction and interactions 2D: monitor understanding of spoken language during classroom instruction and interactions and seek clarification as needed 2E: use visual, contextual, and linguistic support to enhance and confirm understanding of increasingly complex and elaborated spoken language Reading 4E: read linguistically accommodated content area material with a decreasing need for linguistic accommodations as more English is learned

Vocabulary

Unit Vocabulary Billions Equal to (=) Hundredths Numerals Tens Compare Expanded notation Least to greatest Ones Tenths Comparison symbol Greater than (>) Less than (<) Order Thousands Decimals Greatest to least Millions Place value Thousandths Digit Hundreds

Cognitive Complexity Verbs: represent, use, compare, order Academic Vocabulary by Standard: 5.2A: compare, order, billions, millions, thousands, hundreds, tens, ones, tenths, hundredths, thousandths,, decimals, digit, expanded notation, numerals, place value 5.2B: comparison symbol, equal to (=), greater than (>), less than (<), greatest to least, least to greatest, tenths, hundredths, thousandths

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Suggested Math Manipulatives Base 10 Blocks 10 x 10 Grids Place Value Charts Place Value Dice Place Value Disks Number Lines Cons/Money

Bundle 1: Vertical Alignment

4.2B represent the value of the digit in whole numbers through 1,000,000,000 and decimals to the hundredths using expanded notation and numerals 4.2A interpret the value of each place-value position as 10 times the position to the right and as one-tenth of the value of the place to its left 4.2E represent decimals, including tenths and hundredths, using concrete and visual models and money

5.2A: represent the value of the digit in decimals through the thousandths using expanded notation and numerals

4.2C compare and order whole numbers to 1,000,000,000 and represent comparisons using the symbols >,<,= 4.2F compare and order decimals using concrete and visual models to the hundredths

5.2B: compare and order two decimals to thousandths and represent comparisons using the symbols >, <, or =

6.2D order a set of rational numbers arising from mathematical and real-world contexts

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Bundle 1: Teacher Notes

TEKS/Student Expectations

Instructional Implications Distractor Factors Supporting Readiness Standards

TEA Supporting Information

5.2A: represent the value of the digit in decimals through the thousandths using expanded notation and numerals

In conjunction with 4.2B, students have been representing decimal values through the hundredths and this standard extends that knowledge to the thousandths. Students should relate how to represent the value of each digit through expanded notation (i.e. 1.357 = 1 + 0.3 + 0.05 + 0.007). In alignment with 4.2A, students should understand how digits moving to the left on a place value chart become ten-times larger and digits moving to the right on a place value chart become one-tenth the size.

Focusing on the value of each digit in decimal representations will be critical for students to be able to compare and order decimal values through the thousandths. 5.2B: compare and order two decimals to thousandths and represent comparisons using the symbols >, <, or =

Represent includes using place value to read and write using numerals and expanded notation. The expanded notation for 3.94 can be represented as 3.94 = (3 x 1) + (9 x 0.1) + (4 x 0.01) or 3.94 = (3 x 1) + (9 x 1/10) + (4 x 1/100). The conversion between expanded notation, verbal representation, and numerals builds on the fourth grade skill. Expanded notation is written following the order of place value.

5.2B: compare and order two decimals to thousandths and represent comparisons using the symbols >, <, or =

In conjunction with 5.2A, as students compare the value of decimals they need to be able to relate their understanding of place value (i.e. 2.42 is less than 2.7 because the digit 4 in 2.42 means there are four tenths which is a value of 0.40; however, the digit 7 in 2.7 means there are seven tenths which is a value of 0.70). Students will compare two decimal values using the correct academic vocabulary (i.e. 2.42 is less than 2.7). It is important for students to recognize the inverse comparison statement as well (i.e. 2.7 is greater than 2.42). Instruction should connect the comparative language to the symbols (>, <, =).

* Students that rely on a trick to determine directionality (i.e. the alligator’s mouth eats the bigger number) may not be able to read comparison symbols correctly. * Students may compare the number of digits instead of applying their understanding of place value to determine the value of decimals (i.e. 0.451 is greater than 0.98 because it has more digits). * Students may not understand that 0.7 is equivalent to 0.70. * Students may not view the comparison statement 5.246<5.43

Specificity regarding notation has been included with the inclusion of the symbols >, <, or =. A set of decimals can be compared in pairs in the process of ordering decimals. Comparing, ordering, and representing comparisons of decimals in this grade level bridges the SEs in grade 4 and 6.

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It is critical that students understand how to correctly read each of the symbols without using a trick to remember directionality of the symbols (i.e. the alligator’s mouth eats the bigger number). Encourage students to write and articulate two comparison statements during activities (i.e. 2.42 < 2.7 and 2.7 > 2.42). The standard also has students ordering decimals from least to greatest or greatest to least. The use of number lines will allow students to order more efficiently. Number increase from left to right on a number line which can be associated with ordering from least to greatest; numbers decrease from right to left on a number line which can be associated with ordering from greatest to least.

as the same as 5.43> 5.246. * Students need to understand the context of problems to order decimals correctly (i.e. when ordering time from fastest to slowest, students may want to order from greatest to least instead of least to greatest).

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Course: Grade 5 Math Bundle 2: Whole Number Operations Dates: August 31st- September 30th (23 days) TEKS

5.3A: estimate to determine solutions to mathematical and real-world problems involving addition, subtraction, multiplication, or division 5.3B: multiply with fluency a three-digit number by a two-digit number using the standard algorithm 5.3C: solve with proficiency for quotients of up to a four-digit dividend by a two-digit divisor using strategies and the standard algorithm 5.3K: add and subtract positive rational numbers fluently (whole numbers) 5.4A: identify prime and composite numbers 5.4B: represent and solve multi-step problems involving the four operations with whole numbers using equations with a letter standing for the unknown quantity (Whole numbers only. Other positive rational numbers added through spiraled instruction.) 5.4E: describe the meaning of parentheses and brackets in a numeric expression 5.4F: simplify numerical expressions that do not involve exponents, including up to two levels of grouping (Whole numbers only. Other positive rational numbers added through spiraled instruction.)

ELPS Learning Strategies 1A: use prior knowledge and experiences to understand meanings in English 1D: speak using learning strategies such as requesting assistance, employing non-verbal cues, and using synonyms and circumlocution (conveying ideas by defining or describing when exact English words are not known) Listening 2C: learn new language structures, expressions, and basic and academic vocabulary heard during classroom instruction and interactions Speaking 3D: speak using grade-level content area vocabulary in context to internalize new English words and build academic language proficiency Reading 4D: use pre-reading supports such as graphic organizers, illustrations, and pre-taught topic-related vocabulary and other pre-reading activities to enhance comprehension of written text 4F: use visual and contextual support and support from peers and teachers to read grade-appropriate content area text, enhance and confirm understanding, and develop vocabulary, grasp of language structures, and background knowledge needed to comprehend increasingly challenging language

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Vocabulary

Unit Vocabulary Add Divisor Grouping symbols Place value Round Brackets Equation Multiplication Prime number Simplify Compatible number Equivalent Number patterns Product Subtract Composite number Estimate Numeric expressions Quotient Sum Difference Factor Order of operations Rational number Variable Dividend Factor pairs Parentheses Reasonableness

Cognitive Complexity Verbs: estimate, multiply, use, solve, add, subtract, identify, describe, represent, simplify Academic Vocabulary by Standard: 5.3A: compatible numbers, estimate, reasonableness, round 5.3B: factor, place value, product 5.3C: dividend, divisor, place value, quotient 5.3K: add, difference, rational number, subtract, sum 5.4A: composite number, prime number, factors, factor pairs, number patterns 5.4B: addition, sum, division, quotient, equation, multiplication, product, subtraction, difference, variable (letter standing for unknown quantity) 5.4E: brackets, numeric expressions, parentheses, grouping symbols, order of operations 5.4F: equivalent, numerical expression, simplify, grouping symbols, order of operations

Suggested Math Manipulatives

Base 10 Blocks Money Color Tiles

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4.4A add and subtract whole numbers and decimals to the hundredths place using the standard algorithm

5.3A: estimate to determine solutions to mathematical and real-world problems involving addition, subtraction, multiplication, or division

4.4B determine products of a number and 10 or 100 using properties of operations and place value understandings 4.4C represent the product of 2 two-digit numbers using arrays, area models, or equations, including perfect squares through 15 by 15 4.4D use strategies and algorithms, including the standard algorithm, to multiply up to a four-digit number by a one-digit number and to multiply a two-digit number by a two-digit number. Strategies may include mental math, partial products, and the commutative, associative, and distributive properties 4.4H solve with fluency one- and two-step problems involving multiplication and division, including interpreting remainders

5.3B: multiply with fluency a three-digit number by a two-digit number using the standard algorithm

6.3E multiply and divide positive rational numbers fluently

4.4E represent the quotient of up to a four-digit whole number divided by a one-digit whole number using arrays, area models, or equations 4.4F use strategies and algorithms, including the standard algorithm, to divide up to a four-digit dividend by a one-digit divisor 4.4H solve with fluency one-and two-step problems involving multiplication and division, including interpreting remainders

5.3C: solve with proficiency for quotients of up to a four-digit dividend by a two-digit divisor using strategies and the standard algorithm



5.3K: add and subtract positive rational numbers fluently (whole numbers)

5.4A: identify prime and composite numbers 4.5A represent multi-step problems involving the four operations with whole numbers using strip diagrams and equations with a letter standing for the unknown quantity

5.4B: represent and solve multi-step problems involving the four operations with whole numbers using equations with a letter standing for the unknown quantity

6.7B distinguish between expressions and equations verbally, numerically, and algebraically 6.10A model and solve one-variable, one-step equations and inequalities that represent problems, including geometric concepts 6.10B determine if the given value(s) make(s) one-variable, one-step equations or inequalities true

5.4E: describe the meaning of parentheses and brackets in a numeric expression

6.7A generate equivalent numerical expressions using order of operations, including whole number exponents and prime factorization

5.4F: simplify numerical expressions that do not involve exponents, including up to two levels of grouping

6.7A generate equivalent numerical expressions using order of operations, including whole number exponents and prime factorization

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Instruction should model the use of estimations to all operational problems prior to solving for the exact answer. Estimations will become even more critical as students begin working with decimal and fraction problems in order to determine reasonableness to various solutions. Rounding (i.e. 4.5 x 1.25 ≈ 5 x 1 = 5) and compatible numbers (i.e. 4.5 x 1.25 ≈ 4 x 1.25 = 6) are two such examples of estimation. In adherence to the standard, the intent of rounding is to estimate a solution; therefore, employing a specific rounding rule is not necessary. It is important for students to determine if their estimates will yield an over-estimated amount or under-estimated amount (i.e. 3.75 x 6.7 could yield 4 x 7 = 28. This would be an over-estimated product as both factors were rounded up to the next whole number).

As students are asked to solve problems using all four operations involving whole numbers, fractions, and decimals, it will be important for them to estimate solutions prior to solving. The use of rounding and/or compatible numbers will also allow students to evaluate the reasonableness of solutions. 5.3E: solve for products of decimals to the hundredths, including situations involving money, using strategies based on place-value understandings, properties of operations, and the relationship to the multiplication of whole numbers 5.3G: solve for quotients of decimals to the hundredths, up to four-digit dividends and two-digit whole number divisors, using strategies and algorithms, including the standard algorithm 5.3K: add and subtract positive rational numbers fluently 5.4B: represent and solve multi-step problems involving the four operations with whole numbers using equations with a letter standing for the unknown quantity

The word “problems” has been clarified with “mathematical and real-world problems.” Strategies and methods may include front-end estimation (one keeps the first digit of the number and changes all remaining digits to zero), compatible numbers (with values that lend themselves to mental calculations), rounding up or down, and/or compensations (one adjusts estimates to draw closer to an exact calculation). This SE includes estimation with whole numbers, fractions, and decimals.


In conjunction with 5.3A, students should estimate the product of a multi-digit factor problem before solving for the exact answer. As students begin to use the standard algorithm to determine

Students will need to become fluid with multi-digit multiplication as they will be asked to solve problems using all four operations and extend their

The introductory paragraph (a)(3) communicated the following: “Students in grade 5 are expected to

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the product of multi-digit factors, it is essential that they communicate the correct place value understanding (i.e. when completing the steps of a multiplication problem such as 345 x 26, instead of referring to a step as 6 x 4 is 24, so record the 4 and carry the 2, students should know that this step represent 6 x 40, which yields 240).

understanding of multiplying whole numbers to that of decimals. 5.3E: solve for products of decimals to the hundredths, including situations involving money, using strategies based on place-value understandings, properties of operations, and the relationship to the multiplication of whole numbers 5.4B: represent and solve multi-step problems involving the four operations with whole numbers using equations with a letter standing for the unknown quantity

perform their work without the use of calculators.” When paired with 5.1A, the expectation is that students solve problems. Specificity has been provided with the inclusion of the phrase “using the standard algorithm.” Work with the standard algorithm builds on the work from grade 4 with mental math, partial products, and the commutative, associative and distributive properties for 4.4D. The phrase “with fluency” is included. “Procedural fluency refers to knowledge of procedures, knowledge of when and how to use them appropriately, and skill in performing them flexibly, accurately, and efficiently” (National Research Council, 2001, p. 121).

5.3C: solve with proficiency for quotients of up to a four-digit dividend by a two-digit divisor using strategies and the standard algorithm

In accordance to the standard, students will move from the concrete understanding of division to more abstract. Students should apply their understanding of number patterns (i.e. 25 x 100 = 2,500; 25 x 10 = 250; 25 x 1 = 25), flexibility with numbers (i.e. if 25 x 100 = 2,500 then 25 x 50 = 1,250; if 25 x 10 = 250 then 25 x 5 = 125; if 25 x 1 = 25 then 25 x 2 = 50), and place value (i.e. 425 = 400 + 20 + 5) to develop the standard algorithm (i.e. 425 ÷ 25 = x). 25 x 10 = 250

Students will need to become fluid with multi-digit division as they will be asked to solve problems using all four operations and extend their understanding of division of whole number to division of decimals. 5.4B: represent and solve multi-step problems involving the four operations with whole numbers using equations with a letter

The introductory paragraph (a)(3) communicated the following: “Students in grade 5 are expected to perform their work without the use of calculators.” When paired with 5.1A, the expectation is that students solve problems. Specificity has been

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25 x 5 = 125 25 x 2 = 50 -so- 25 x 17 = 425 and 425 ÷ 25 = 17

standing for the unknown quantity 5.3G: solve for quotients of decimals to the hundredths, up to four-digit dividends and two-digit whole number divisors, using strategies and algorithms, including the standard algorithm

provided with the inclusion of the phrase “using strategies and the standard algorithm.” The application of strategies and the standard algorithm includes four-digit dividends. Students are expected to solve with proficiency. Procedural fluency and conceptual understanding weave together to develop mathematical proficiency along with strategic competence, adaptive reasoning, and productive disposition (National Research Council, 2001). “Procedural fluency refers to knowledge of procedures, knowledge of when and how to use them appropriately, and skill in performing them flexibly, accurately, and efficiently” (National Research Council, 2001, p. 121).

5.3K: add and subtract positive rational numbers fluently (whole numbers and decimals)

Rational numbers include whole numbers, fractions, and decimals. In adherence to the standard, operations with rational numbers are limited to those that are positive. Students should be able to apply all of their experiences with positive rational numbers and apply them to addition/ subtraction fluidly. Instruction should include whole numbers through the billions, decimals through the thousandths, and all representations of fractions (i.e. improper, mixed numbers, like

* Students may apply the use of “key words” to select addition or subtraction instead of understanding the context of the problem. * Students may line up the decimal point incorrectly (or not at all) when adding or subtracting decimal numbers. * Students may forget to use a common denominator when

The addition and subtraction of positive rational numbers includes the addition and subtraction of whole numbers and decimals. This SE includes the addition and subtraction of fractions, including improper fractions and mixed numbers.

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denominators, unlike denominators). In conjunction with 4.2G, students have related decimals to fractions and future instruction may include the addition/ subtraction of a mixture of rational number representations (i.e. 4.75 + 2 ½ = x).

adding or subtracting fractions or mixed numbers.

Rational numbers include decimals and fractions. Within a problem situation, students may be asked to add 0.2 and 2/5. When paired with 5.1A, students are expected to solve problems. The intent of this SE is not a sole focus on computation. Within the TEKS, fluency with fraction and decimal addition and subtraction occurs in grade 5. The word “fluently” has been added. “Procedural fluency refers to knowledge of procedures, knowledge of when and how to use them appropriately, and skill in performing them flexibly, accurately, and efficiently” (National Research Council, 2001, p. 121). This SE builds to the grade 7 skills on operations with rational numbers.

5.4A: identify prime and composite numbers

Students need to identify prime numbers as those that only have two factors; one and itself (i.e. 13 is prime because the only factors for 13 are 1 and 13, 1 x 13 = 13). A composite number has more than two factors (i.e. 18 is composite because 1 x 18 = 18; 2 x 9 = 18; 3 x 6 = 18). The number one is neither prime nor composite. Instruction should model multiple representations of composite numbers as this will support future needs of finding a common denominator and/

Identifying prime and composite numbers will support students as they will have to determine a common denominator in order to add and subtract fractions with unlike denominators, and identify a common factor in order to represent sums/differences in simplified form. 5.3K: add and subtract positive

When paired with 5.1C, students may use real (concrete) objects to identify prime and composite numbers. When paired with 5.1D, students may use pictorial models including arrays as a representation to identify prime and composite numbers.

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or common factor. Listing factors in a t-chart may help students to ensure that all factors have been identified.

Factors of 24 1 24 2 12 3 8 4 6

rational numbers fluently

When paired with 5.1F, students may analyze factor pairs to connect the concepts of prime and composite numbers to defining characteristics of factors using patterns in factor pairs. The concept of prime factorization appears in grade 6 and then reappears in the Algebra I TEKS.

5.4B: represent and solve multi-step problems involving the four operations with whole numbers using equations with a letter standing for the unknown quantity

In adherence to the standard, instruction should include multi-step problems involving a mixture of operations. The focus is not just on the students being able to solve the problems, but also represent them using the appropriate equation (i.e. Maricella buys 75 tickets at a carnival. She keeps 35 tickets for herself and gives the remaining tickets to her 2 sisters who will share them equally. How many tickets does each sister get?

75 tickets (whole)

Maricella (part)

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Sister (part)

t

Sister (part)

t (75 – 35) ÷ 2 = t Encourage students to write more than one equation (i.e. 75 = 35 ÷ 2t; (75 – 35) ÷ 2 = t). The representing and solving of multi-step problems lays a strong foundation for future algebraic reasoning.

* Students may not recognize equivalent equations (i.e. 75 = 35 ÷ 2t is the same as (75 – 35) ÷ 2 = t). * Students may use “key words” to determine the operation instead of understanding the context of the problem. * Students may not understand the variety of representations for multiplication (i.e. 3 x 4; 3 ⋅ 4; 3(4); 3t) and division (i.e. 20/4; 20 ÷ 4). * Students may think that they have to use a letter that stands for the unknown quantity (i.e. the letter “t” has to be used since we are looking for “tickets” when any variable would be appropriate). * Students may not appropriately apply the use of parentheses and brackets when representing the equation of a multiple step problem. * Students may not understand that the variable/ letter has value.

“Multi-step problems involving the four operations with whole numbers included meaningful problem situations. When paired with 5.1D, students may use diagrams, such as strip diagrams, to represent the known and unknown quantities in a multi-step problem. The letter standing for the unknown quantity may be any part of the equation.

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5.4E: describe the meaning of parentheses and brackets in a numeric expression

In adherence to the standard, students should understand the intentional use of parentheses and brackets. Instruction should apply to a real-world problem for students to better relate (i.e. two adults and three children go to the movies; cost of adult tickets are $7.50; cost of a child’s ticket is $4.50; how much will it cost for the family to go to the movies? (2 x $7.50) + (3 x 4.50) = p; we put parentheses around (2 x 7.50) and (3 x 4.50) to show how we would determine the cost of the adult and children before adding the two costs together). This standard is setting the foundational understanding for order of operations. Students should recognize how each of the following would yield a different result due to the use of parentheses and brackets: 6 x 2 + 3 6 x (2 + 3) 6 x [(2 + 3) – 1]

This SE supports 5.4F. Students are expected to describe which operations are included within a set of parentheses and/or brackets and that operations within these parentheses and/or brackets should be completed to simplify to a value for use in the remainder of a numeric expression. For example, 3 + (2 x 6) is 3 + 12 not 5 x 6. Students are expected to describe the relationship between numbers and operations separated by parentheses and brackets. For example, 4(14 + 5) is 4 times as large as (14 + 5). When paired with 5.4F, the expectation is that students describe the meaning of parentheses and brackets up to two levels of grouping.

5.4F: simplify numerical expressions that do not involve exponents, including up to two levels of grouping

In conjunction with 5.4E, this standard is setting the foundational understanding for order of operations. Instruction should begin with contextual understanding of order of operations as it applies in the real world (i.e. Two adults and three children go to the movies. The cost of an adult ticket is $7.50 and the cost of a child’s ticket is $4.50. How much will it cost for the family to go to the movies? (2 x 7.50) + (3 x 4.50) = p; determine the cost of the two adults first;

* Students may not relate fractions to division (i.e. 33/11 = 33 ÷ 11) * Students may not recognize the different representations of multiplication [i.e. 3 x 4; 3 ⋅ 4; 3(4)]. * Students may just work problems from left to right (i.e. 5 + 10 x 4 ≠ 15 x 4 = 60) instead

An example of two levels of grouping is (3 + 7) ÷ (5 – 3). Students are expected to use the order of operations to simplify numerical expressions. Because fluency with addition and subtraction of positive rational numbers

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the cost of the three children second; and then add the expenses to yield the total cost). Instruction should model how the work found within brackets/ parentheses is simplified first. Moving from left to right, simplify all multiplication/ division problems; moving from left to right simplify all addition/ subtraction problems. 6 x (2 + 3) ÷ 10 6 x (5) ÷ 10 30 ÷ 10 3 Two levels of grouping mean students need to understand that fractions also represent a division problem.

of applying order of operations (i.e. 5 + 10 x 4 = 5 + 40 = 45). * Students may want to work addition problems first then subtraction (i.e. 15 – 6 + 3 ≠ 15 – 9 = 6) instead of which operation comes first in reading from left to right (i.e. 15 – 6 + 3 = 9 + 3 = 12). * Students may want to work multiplication problems first then division (i.e. 24 ÷ 3 x 2 ≠ 24 ÷ 6 = 4) instead of which operation comes first when reading from left to right (i.e. 24 ÷ 3 x 2 = 8 x 2 = 16). * Students may not understand that “simplifying” an expression does not change the value. Each step in order of operations yields an equivalent expression.

is expected within the TEKS, expressions may include fractional values when adding and subtracting. Exponents are included in grade 6.

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Course: Grade 5 Math Bundle 3: All Operations with Decimals Dates: October 4th- October 25th (16 days) TEKS

5.2C: round decimals to tenths or hundredths 5.3A: estimate to determine solutions to mathematical and real-world problems involving addition, subtraction, multiplication, or division 5.3D: represent multiplication of decimals with products to the hundredths using objects and pictorial models, including area models 5.3E: solve for products of decimals to the hundredths, including situations involving money, using strategies based on place-value understandings, properties of operations, and the relationship to the multiplication of whole numbers 5.3F: represent quotients of decimals to the hundredths, up to four-digit dividends and two-digit whole number divisors, using objects and pictorial models, including area models 5.3G: solve for quotients of decimals to the hundredths, up to four-digit dividends and two-digit whole number divisors, using strategies and algorithms, including the standard algorithm 5.3K: add and subtract positive rational numbers fluently (whole numbers and decimals)

ELPS Learning Strategies 1A: use prior knowledge and experiences to understand meanings in English 1D: speak using learning strategies such as requesting assistance, employing non-verbal cues, and using synonyms and circumlocution (conveying ideas by defining or describing when exact English words are not known) Listening 2C: learn new language structures, expressions, and basic and academic vocabulary heard during classroom instruction and interactions 2D: monitor understanding of spoken language during classroom instruction and interactions and seek clarification as needed 2I: demonstrate listening comprehension of increasingly complex spoken English by following directions, retelling or summarizing spoken messages, responding to questions and requests, collaborating with peers, and taking notes commensurate with content and grade-level needs Speaking 3B: expand and internalize initial English vocabulary by learning and using high-frequency English words necessary for identifying and describing people, places, and objects, by retelling simple stories and basic information represented or supported by pictures, and by learning and using routine language needed for classroom communication

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3D: speak using grade-level content area vocabulary in context to internalize new English words and build academic language proficiency 3E: share information in cooperative learning interactions 3F: ask and give information ranging from using a very limited bank of high-frequency, high-need, concrete vocabulary, including key words and expressions needed for basic communication in academic and social contexts, to using abstract and content-based vocabulary during extended speaking assignments 3G: express opinions, ideas, and feelings ranging from communicating single words and short phrases to participating in extended discussions on a variety of social and grade-appropriate academic topics 3H: narrate, describe, and explain with increasing specificity and detail as more English is acquired Reading 4C: develop basic sight vocabulary, derive meaning of environmental print, and comprehend English vocabulary and language structures used routinely in written classroom materials 4E: read linguistically accommodated content area material with a decreasing need for linguistic accommodations as more English is learned 4F: use visual and contextual support and support from peers and teachers to read grade-appropriate content area text, enhance and confirm understanding, and develop vocabulary, grasp of language structures, and background knowledge needed to comprehend increasingly challenging language 4G: demonstrate comprehension of increasingly complex English by participating in shared reading, retelling or summarizing material, responding to questions, and taking notes commensurate with content area and grade level needs

Vocabulary

Unit Vocabulary Add Dividend Mixed number Properties of operations Round Area model Divisor Number line Quotient Subtract Compatible numbers Estimate Place value Rational number Sum Consecutive multiples Factor Product Reasonableness Tenths Decimals Hundredths

Cognitive Complexity Verbs: round, estimate, determine, represent, use, solve, add, subtract Academic Vocabulary by Standard: 5.2C: consecutive multiples, number line, place value, round, tenths, hundredths 5.3A: compatible numbers, estimate, reasonableness, round 5.3D: area model, factor, hundredths, place value, product, tenths 5.3E: factors, products, tenths, hundredths, decimals, place value, properties of operations 5.3F: dividend, divisor, place value, quotient 5.3G: dividend, divisor, hundredths, quotient, tenths 5.3K: add, decimal, difference, mixed number, rational number, subtract, sum

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Base 10 Blocks Graph Paper 10 x 10 Grids


4.2D round whole numbers to a given place value through the hundred thousands place 4.4G round to the nearest 10, 100, or 1,000 or use compatible numbers to estimate solutions involving whole numbers

5.2C: round decimals to tenths or hundredths

4.4G round to the nearest 10, 100, or 1,000 or use compatible numbers to estimate solutions involving whole numbers


4.4B determine products of a number and 10 or 100 using properties of operations and place value understandings 4.4C represent the product of 2 two-digit numbers using arrays, area models, or equations, including perfect squares through 15 by 15 4.4D use strategies and algorithms, including the standard algorithm, to multiply up to a four-digit number by a one-digit number and to multiply a two-digit number by a two-digit number. Strategies may include mental math, partial products, and the commutative, associative, and distributive properties 4.4H solve with fluency one- and two-step problems involving multiplication and division, including interpreting remainders



4.4H solve with fluency one- and two-step problems involving multiplication and division, including interpreting remainders 4.4C represent the product of 2 two-digit numbers using arrays, area models, or equations, including perfect squares through 15 by 15

5.3D: represent multiplication of decimals with products to the hundredths using objects and pictorial models, including area models


4.4H solve with fluency one- and two-step problems involving multiplication and division, including interpreting remainders

5.3E: solve for products of decimals to the hundredths, including situations involving money, using strategies based on place-value understandings, properties of


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4.4C represent the product of 2 two-digit numbers using arrays, area models, or equations, including perfect squares through 15 by 15 4.4D use strategies and algorithms, including the standard algorithm, to multiply up to a four-digit number by a one-digit number and to multiply a two-digit number by a two-digit number. Strategies may include mental math, partial products, and the commutative, associative, and distributive properties

operations, and the relationship to the multiplication of whole numbers

4.4E represent the quotient of up to a four-digit whole number divided by a one-digit whole number using arrays, area models, or equations 4.4H solve with fluency one-and two-step problems involving multiplication and division, including interpreting remainders

5.3F: represent quotients of decimals to the hundredths, up to four-digit dividends and two-digit whole number divisors, using objects and pictorial models, including area models


4.4E represent the quotient of up to a four-digit whole number divided by a one-digit whole number using arrays, area models, or equations 4.4F use strategies and algorithms, including the standard algorithm, to divide up to a four-digit dividend by a one-digit divisor 4.4H solve with fluency one-and two-step problems involving multiplication and division, including interpreting remainders

5.3G: solve for quotients of decimals to the hundredths, up to four-digit dividends and two-digit whole number divisors, using strategies and algorithms, including the standard algorithm




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5.2C: round decimals to tenths or hundredths

Instruction should begin with skip counting by tenths and hundredths for students to understand consecutive multiples. Students should represent these benchmark values through the use of an open number line. As students locate a given number on an open number line, you will begin to assess their understanding of place value (i.e. students place the number 1.23 between 1.2 and 1.3), the relative position of numbers (i.e. the number 1.2 would be indicated first and the number 1.3 would be indicated second on the open number line), and magnitude of numbers (i.e. students physically place the number 1.23 closer to 1.2 than 1.3). Students will apply this understanding to the rounding of whole numbers to the nearest tenth and hundredth. Relating money to decimals may also support the understanding of rounding (i.e. Is $1.23 closer to $1.20 or $1.30?).

As students will be asked to solve problems using all four operations, it will be important to estimate solutions prior to solving. Rounding is one way to estimate values in order to evaluate solutions for reasonableness. 5.3A estimate to determine solutions to mathematical and real-world problems involving addition, subtraction, multiplication, or division

This SE mentions rounding and provides specificity of rounding with decimals. Because the work with decimals in the TEKS extends to the thousandths place, students are expected to round decimals to the tenths or hundredths.


Instruction should model the use of estimations to all operational problems prior to solving for the exact answer. Estimations will become even more critical as students begin working with decimal and fraction problems in order to determine reasonableness to various solutions. Rounding (i.e. 4.5 x 1.25 ≈ 5 x 1 = 5) and compatible numbers (i.e. 4.5 x 1.25 ≈ 4 x 1.25 = 6) are two such examples of estimation. In adherence to the standard, the intent of rounding is to estimate a solution; therefore, employing a specific rounding rule is not necessary. It is important for students to determine if their estimates

As students are asked to solve problems using all four operations involving whole numbers, fractions, and decimals, it will be important for them to estimate solutions prior to solving. The use of rounding and/or compatible numbers will also allow students to evaluate the reasonableness of solutions. 5.3E: solve for products of decimals to the hundredths, including situations involving money, using strategies based on place-value understandings,

The word “problems” has been clarified with “mathematical and real-world problems.” Strategies and methods may include front-end estimation (one keeps the first digit of the number and changes all remaining digits to zero), compatible numbers (with values that lend themselves to mental calculations), rounding up or down, and/or compensations (one adjusts

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will yield an over-estimated amount or under-estimated amount (i.e. 3.75 x 6.7 could yield 4 x 7 = 28. This would be an over-estimated product as both factors were rounded up to the next whole number).

properties of operations, and the relationship to the multiplication of whole numbers 5.3G: solve for quotients of decimals to the hundredths, up to four-digit dividends and two-digit whole number divisors, using strategies and algorithms, including the standard algorithm 5.3K: add and subtract positive rational numbers fluently 5.4B: represent and solve multi-step problems involving the four operations with whole numbers using equations with a letter standing for the unknown quantity

estimates to draw closer to an exact calculation). This SE includes estimation with whole numbers, fractions, and decimals.

5.3D: represent multiplication of decimals with products to the hundredths using objects and pictorial models, including area models

In conjunction with 4.2E, students will extend their understanding of how to represent decimals using concrete and visual models (i.e. if a hundred flat represents one whole, then a ten rod represents one-tenth and a unit cube represents one-hundredth) to multiplication (i.e. 0.3 x 0.4 = x). Instruction should model the use of the term “of” when multiplying decimals (i.e. 0.3 x 0.4 = x; what is three-tenths “of” four-tenths). An area model is one example of how to represent the multiplication of decimals. Here, if the hundred flat represents one whole, then an area model is one example of how to represent the multiplication of decimals. See models below:

Using concrete objects and pictorial models to represent the multiplication of decimals will support students in developing the concrete understanding of the concept before moving to the abstract of developing a strategy and/or algorithm. 5.3E: solve for products of decimals to the hundredths, including situations involving money, using strategies based on place-value understandings, properties of operations, and the relationship to the multiplication of whole numbers

The limitation of products to the hundredths constrains the values with which students are asked to work in grade 5. When paired with 5.1A, the expectation is that students solve problems. The intent of this SE is not a sole focus on the computation. This SE builds to the sixth grade skill of multiplying and dividing rational numbers fluently.

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Let one of the factors represent the length of the whole: 0.3

Let the other factor represent the width of the whole: 0.4

Shade the length (factor) and width (factor) on the same whole; the overlapped area represents the product: 0.3 x 0.4 = 0.12

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5.3E: solve for products of decimals to the hundredths, including situations involving money, using strategies based on place-value understandings, properties of operations, and the relationship to the multiplication of whole numbers

In conjunction with 5.3D, the intent of this standard is to build flexibility in multiplying decimals (4 x $1.35 = x). Beginning instruction with the use of money will allow students to better relate their understanding of the concept (i.e. 4 x $1.35 = x; representing a one dollar bill, three dimes, and five pennies and asking students to represent four times that amount would yield four dollar bills, 12 dimes, and 20 pennies; $4.00 + $1.20 + $0.20 = $5.40). Instruction can move to the use of place value understanding and properties of operations to solve for products of decimals (i.e. 4 x 1.35 = 4 x (1 + 0.30 + 0.05) = (4 x 1) + (4 x 0.30) + (4 x 0.05) = 4 + 1.20 + 0.20 = 5.40). In alignment with 5.3A, it is essential for students to estimate solutions to decimal multiplication problems before solving the problems (i.e. 4 x 1.35 ≈ 4 x 1 = 4). This will allow students to continue to use whole number multiplication strategies and apply reasonableness when determining the product of decimals (i.e. When computing a product such as 1.6 x 8.2, a student could estimate the answer to be close to 2 x 8 = 16. Then, when multiplying the related whole numbers (16 x 82) to get an answer of 1312, the student could reason that neither 1.312, 131.2 or 1312 would be reasonable. Thus, 1.6 x 8.2 = 13.12). In accordance with the standard, products are limited to hundredths.

* Students may think that the multiplying of two numbers always yields a larger product. * When applying the standard algorithm, students may want to align place values like in addition/ subtraction of decimals.

Place-value understandings would include estimating factors and products to determine reasonable placement of the decimal in the product. Properties of operations would include thinking with the distributive property. For example, if students are asked to determine the total price of 12 books if each book is $4.50, students might think of this as $4.50(10 + 2) and determine that $4.50(10) is $45 and that doubling $4.50 yields $9. $45 and $9 would yield a solution of $54. The intention is for students to develop flexible thinking with numbers using properties of operations. When paired with 5.1A, the expectation is that students solve problems. The intent of this SE is not a sole focus on the computation. Within the TEKS, fluency with decimal multiplication occurs in grade 6.

5.3F: represent quotients of decimals to the hundredths, up to four-digit dividends and two-digit whole number divisors, using objects and pictorial models,

In conjunction with 4.2E, students will extend their understanding of how to represent decimals using concrete and visual models (i.e. if a hundred flat represent one whole,

Using concrete objects and pictorial models to represent the division of decimals will support students in developing the concrete understanding of the

The limitation of quotients to the hundredths constrains the values with which students are asked to work in grade 5.

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including area models then a ten rod represents one-tenth and a unit cube represents one-hundredth) to division (i.e. 0.12 ÷ 0.4 = x). Instruction should model how division with decimals is very much like division of whole numbers (i.e. 12 ÷ 4 = x; how many equal groups of 4 divide into 12? Or 4 x ___=12; 0.12 ÷ 0.4 = x; how many equal groups of four-tenths divide into twelve-hundredths? Or 0.4 x = 0.12). An area model is one example of how to represent the division of decimals (i.e. if a hundreds flat represents one whole, then a square unit represents a hundredth; the length of 0.3 and width of 0.4 yield an area of 0.12).

0.12 ÷ 0.4 = 0.3

concept before moving to the abstract of developing a strategy and/or algorithm. 5.3G: solve for quotients of decimals to the hundredths, up to four-digit dividends and two-digit whole number divisors, using strategies and algorithms, including the standard algorithm

When paired with 5.1A, the expectation is that students solve problems. The intent of this SE is not a sole focus on the computation. Within the TEKS, fluency with decimal division occurs in grade 6.

5.3G: solve for quotients of decimals to the hundredths, up to four-digit dividends and two-digit whole number divisors, using strategies and algorithms, including the standard algorithm

In conjunction with 5.3F, the intent of this standard is to build flexibility in dividing decimals. Instruction is limited to four-digit dividends and two-digit whole number divisors (i.e. 4.6 ÷ 2 = x). Beginning instruction with concrete objects will allow students to develop their understanding of the concept and build flexibility with numbers (i.e. if a hundreds flat represents one whole, then 4.6 would be represented with four hundred flats and six ten rods in each group; 4.6 ÷ 2 = 2.3). In alignment with 5.3A, it will be essential for students to estimate (rounding and compatible numbers) quotients before solving

* Students may think that dividing two numbers always yields a smaller quotient. * When applying the standard algorithm, students may not articulate the correct place value understanding (i.e. 384 ÷ 3 = x; “three goes into three one time” instead of “there are 100 groups of three in 300”). * Students may revert to past experience when using base-ten blocks to represent whole numbers and become confused when using

Strategies may include mental math, place value, partial products, and properties of operations. The intention is for students to develop flexible thinking with numbers. When paired with 5.1A, the expectation is that students solve problems. The intent of this SE is not a sole focus on the computation. Within the TEKS, fluency with decimal division occurs

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division problems. This will allow students to continue to use whole number division strategies and apply reasonableness when determining the quotient of decimals (i.e. 38.4 ÷ 3 ≈ 39 ÷ 3 = 13 if the answer should be around 13, then .128, 1.28, and 128 would be unreasonable; hence 38.4 ÷ 3 = 12.8). This understanding can also be applied when using the standard algorithm for division.

the manipulatives to represent decimal values.

in grade 6. These division skills support development of computational skills needed for students’ work with division and rates in grade 6 science.


Rational numbers include whole numbers, fractions, and decimals. In adherence to the standard, operations with rational numbers are limited to those that are positive. Students should be able to apply all of their experiences with positive rational numbers and apply them to addition/ subtraction fluidly. Instruction should include whole numbers through the billions, decimals through the thousandths, and all representations of fractions (i.e. improper, mixed numbers, like denominators, unlike denominators). In conjunction with 4.2G, students have related decimals to fractions and future instruction may include the addition/ subtraction of a mixture of rational number representations (i.e. 4.75 + 2 ½ = x).

* Students may apply the use of “key words” to select addition or subtraction instead of understanding the context of the problem. * Students may line up the decimal point incorrectly (or not at all) when adding or subtracting decimal numbers. * Students may forget to use a common denominator when adding or subtracting fractions or mixed numbers.

The addition and subtraction of positive rational numbers includes the addition and subtraction of whole numbers and decimals. This SE includes the addition and subtraction of fractions, including improper fractions and mixed numbers. Rational numbers include decimals and fractions. Within a problem situation, students may be asked to add 0.2 and 2/5. When paired with 5.1A, students are expected to solve problems. The intent of this SE is not a sole focus on computation. Within the TEKS, fluency with fraction and decimal addition and subtraction occurs in grade 5. The word “fluently” has been added. “Procedural fluency refers to knowledge of procedures, knowledge of when and how to use them

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appropriately, and skill in performing them flexibly, accurately, and efficiently” (National Research Council, 2001, p. 121). This SE builds to the grade 7 skills on operations with rational numbers.

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Course: Grade 5 Math Bundle 4: All Operations with Fractions Dates: October 26th- November 18th (18 days)

TEKS 5.3A: estimate to determine solutions to mathematical and real-world problems involving addition, subtraction, multiplication, or division 5.3H: represent and solve addition and subtraction of fractions with unequal denominators referring to the same whole using objects and pictorial models and properties of operations 5.3I: represent and solve multiplication of a whole number and a fraction that refers to the same whole using objects and pictorial models, including area models 5.3J: represent division of a unit fraction by a whole number and the division of a whole number by a unit fraction such as 1/3 ÷ 7 and 7 ÷ 1/3 using objects and pictorial models, including area models 5.3K: add and subtract positive rational numbers fluently (fractions and mixed numbers) 5.3L: divide whole numbers by unit fractions and unit fractions by whole numbers 5.4A: identify prime and composite numbers (previously taught in bundle 2)

ELPS Learning Strategies 1A: use prior knowledge and experiences to understand meanings in English Listening 2C: learn new language structures, expressions, and basic and academic vocabulary heard during classroom instruction and interactions Speaking 3B: expand and internalize initial English vocabulary by learning and using high-frequency English words necessary for identifying and describing people, places, and objects, by retelling simple stories and basic information represented or supported by pictures, and by learning and using routine language needed for classroom communication 3H: narrate, describe, and explain with increasing specificity and detail as more English is acquired Reading 4C: develop basic sight vocabulary, derive meaning of environmental print, and comprehend English vocabulary and language structures used routinely in written classroom materials 4F: use visual and contextual support and support from peers and teachers to read grade-appropriate content area text, enhance and confirm understanding, and develop vocabulary, grasp of language structures, and background knowledge needed to comprehend increasingly challenging language

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Vocabulary

Unit Vocabulary Add Division Fractions Prime number Round Area model Divisor Improper fraction Product Subtract Compatible numbers Equal parts Mixed number Quotient Sum Composite number Estimate Number patterns Rational number Unit fraction Denominator Factor Numerator Reasonableness Whole Dividend Factor pairs

Cognitive Complexity Verbs: estimate, represent, solve, use, add, subtract, divide, identify Academic Vocabulary by Standard: 5.3A: compatible numbers, estimate, reasonableness, round 5.3H: area model, denominator, equal parts, fractions, numerator, whole 5.3I: area model, factor, fractions, product 5.3J: area model, dividend, division, divisor, quotient, unit fraction 5.3K: add, denominator, difference, fraction, mixed number, improper fraction, numerator, rational number, subtract, sum 5.3L: dividend, divisor, quotient, unit fraction, whole number 5.4A: composite number, prime number, factors, factor pairs, number patterns


Fraction Bars Fraction Circles Cuisenaire Rods Geoboards

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4.4A add and subtract whole numbers and decimals to the hundredths place using the standard algorithm 4.4G round to the nearest 10, 100, or 1,000 or use compatible numbers to estimate solutions involving whole numbers


4.3E represent and solve addition and subtraction of fractions with equal denominators using objects and pictorial models that build to the number line and properties of operations

5.3H: represent and solve addition and subtraction of fractions with unequal denominators referring to the same whole using objects and pictorial models and properties of operations

5.3I: represent and solve multiplication of a whole number and a fraction that refers to the same whole using objects and pictorial models, including area models


5.3J: represent division of a unit fraction by a whole number and the division of a whole number by a unit fraction such as 1/3 ÷ 7 and 7 ÷ 1/3 using objects and pictorial models, including area models


4.3E represent and solve addition and subtraction of fractions with equal denominators using objects and pictorial models that build to the number line and properties of operations

5.3K: add and subtract positive rational numbers fluently (fractions and mixed numbers)

5.3L: divide whole numbers by unit fractions and unit fractions by whole numbers



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Instruction should model the use of estimations to all operational problems prior to solving for the exact answer. Estimations will become even more critical as students begin working with decimal and fraction problems in order to determine reasonableness to various solutions. Rounding (i.e. 4.5 x 1.25 ≈ 5 x 1 = 5) and compatible numbers (i.e. 4.5 x 1.25 ≈ 4 x 1.25 = 6) are two such examples of estimation. In adherence to the standard, the intent of rounding is to estimate a solution; therefore, employing a specific rounding rule is not necessary. It is important for students to determine if their estimates will yield an over-estimated amount or under-estimated amount (i.e. 3.75 x 6.7 could yield 4 x 7 = 28. This would be an over-estimated product as both factors were rounded up to the next whole number).

As students are asked to solve problems using all four operations involving whole numbers, fractions, and decimals, it will be important for them to estimate solutions prior to solving. The use of rounding and/or compatible numbers will also allow students to evaluate the reasonableness of solutions. 5.3E: solve for products of decimals to the hundredths, including situations involving money, using strategies based on place-value understandings, properties of operations, and the relationship to the multiplication of whole numbers 5.3G: solve for quotients of decimals to the hundredths, up to four-digit dividends and two-digit whole number divisors, using strategies and algorithms, including the standard algorithm 5.3K: add and subtract positive rational numbers fluently 5.4B: represent and solve multi-step problems involving the four operations with whole numbers using equations with a letter standing for the unknown quantity

The word “problems” has been clarified with “mathematical and real-world problems.” Strategies and methods may include front-end estimation (one keeps the first digit of the number and changes all remaining digits to zero), compatible numbers (with values that lend themselves to mental calculations), rounding up or down, and/or compensations (one adjusts estimates to draw closer to an exact calculation). This SE includes estimation with whole numbers, fractions, and decimals.

5.3H: represent and solve addition and subtraction of fractions with unequal denominators referring to the same whole using objects and pictorial models and properties of operations

In conjunction with 4.3E, students will extend their understanding of how to add and subtract fractions with like denominators (i.e. 2/3 + 5/3 = 7/3) to adding and subtracting fractions with unlike denominators (i.e. 2/3 + ¼ = x). An area model is one example of how to represent the addition/ subtraction of

Using concrete objects and pictorial models to represent the addition and subtraction of fractions with unlike denominators will support students in developing the concrete understanding of the concept before moving to the

When paired with 5.1A, the expectation is that students solve problems. The intent of this SE is not a sole focus on the computation. Specificity addresses the

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fractions with unlike denominators. Represent one of the addends as the length of the whole: 2/3

Represent the other addend as the width of the whole: 1/4

The representation of the two addends on the same whole yield a common denominator: 3 x 4 = 12

Determine an equivalent fraction for the first addend: 2/3 = 8/12

Determine an equivalent fraction for the second addend: 1/4 = 3/12

Combine the two addends on the same whole to determine the sum: 2/3 + 1/4 = 8/12 + 3/12 = 11/12

Instruction should include models of mixed numbers (i.e. 2 ¼ + 1 2/3 = x). Subtraction of fractions with unlike

abstract of developing a strategy and/ or algorithm. 5.3K: add and subtract positive rational numbers fluently

accuracy of the mathematics. The equivalent values should be used to describe the same whole. Within the TEKS, fluency with fraction and decimal addition and subtraction occurs in grade 5. “Procedural fluency refers to knowledge of procedures, knowledge of when and how to use them appropriately, and skill in performing them flexibly, accurately, and efficiently” (National Research Council, 2001, p. 121). Pictorial models may include strip diagrams. This SE builds to the grade 7 skills on operations with rational numbers.

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denominators can also be modeled using the area model, but the difference would be represented by taking area unit away. In conjunction with 4.3G, students could use a number line to model the addition/ subtraction of fractions with unlike denominators.

5.3I: represent and solve multiplication of a whole number and a fraction that refers to the same whole using objects and pictorial models, including area models

In adherence to the standard, instruction is limited to multiplication of a whole number and fraction (i.e. 2/5 x 3 = x), not a fraction times a fraction (2/5 x 1/3 = x). Instruction should associate how multiplication is repeated addition. The use of fraction circles, rectangles, number lines, etc. will support the concrete understanding of repeated addition (i.e. 2/5 x 3 = 2/5 + 2/5 + 2/5 = 6/5).

Instruction should also model the use of the term “of” when multiplying fractions (i.e. ¾ x 2 = x; what is three-fourths “of” two wholes?).

¾ x 2 = ¾ + ¾ = 6/4 = 1 2/4 = 1 1/2

Using concrete objects and pictorial models to represent the multiplication of fractions and whole numbers will support students in developing the concrete understanding of the concept before moving to the abstract of developing a strategy and/ or algorithm. Understanding the use of the area model for multiplication of fractions will support its use in the division of fractions as well. 5.3L: divide whole numbers by unit fractions and unit fractions by whole numbers

When paired with 5.1A, the expectation is that students solve problems. The intent of this SE is not a sole focus on the computation. Specificity addresses the accuracy of the mathematics. The equivalent values should be used to describe the same whole. Within the TEKS, fluency with fraction multiplication occurs in grade 6.

5.3J: represent division of a unit fraction by a whole number and the division of a whole number by a unit fraction such as 1/3 ÷ 7 and 7 ÷ 1/3 using objects and pictorial models, including area models

In adherence to the standard, instruction surrounding the division of fractions is limited to unit fractions and whole numbers (i.e. ¼ ÷ 5 = x or 5 ÷ ¼ = x). Instruction will not include a non-unit fraction and whole number (i.e. ¾ ÷ 5 = x) or the division of two fractions (i.e. 4/5 ÷ ¾ = x). Instruction should model how division of fractions is like division of whole numbers (i.e. 8 ÷ 2 = x; how many

Using concrete objects and pictorial models to represent the division of fractions by whole numbers and whole numbers by fractions will support students in developing the concrete understanding of the concept before moving to the abstract of developing a strategy and/ or algorithm.

When paired with 5.1A, the expectation is that students solve problems. The intent of this SE is not a sole focus on the computation. Within the TEKS, fluency with fraction division occurs in grade 6.

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equal groups of two divide into 8? Or 8 ÷ ½ = x; how many equal groups of halves divide into 8? If a circle represents one whole, then 8 ÷ ½ = 16 because 16 halves will divide into 8 whole circles).

(i.e. ½ divided by 4 = x; divide half of a whole into four equal parts, how many would be in each group? If a rectangle represents one whole and the shaded portion represents half of a whole, then ½ ÷ 4 = 1/8 because a half divided into four parts would yield eighths and each group would have one-eighth).


A unit fraction is a fraction with a numerator of 1. Students first see unit fractions in grade 3.

5.3K: add and subtract positive rational numbers fluently

Rational numbers include whole numbers, fractions, and decimals. In adherence to the standard, operations with rational numbers are limited to those that are positive. Students should be able to apply all of their experiences with positive rational numbers and apply them to addition/ subtraction fluidly. Instruction should include whole numbers through the billions, decimals through the thousandths, and all representations of fractions (i.e. improper, mixed numbers, like denominators, unlike denominators). In conjunction with 4.2G, students have related decimals to fractions and future instruction may include the addition/ subtraction of a mixture of rational number representations (i.e. 4.75 + 2 ½ = x).

* Students may apply the use of “key words” to select addition or subtraction instead of understanding the context of the problem. * Students may line up the decimal point incorrectly (or not at all) when adding or subtracting decimal numbers. * Students may forget to use a common denominator when adding or subtracting fractions or mixed numbers.

The addition and subtraction of positive rational numbers includes the addition and subtraction of whole numbers and decimals. This SE includes the addition and subtraction of fractions, including improper fractions and mixed numbers. Rational numbers include decimals and fractions. Within a problem situation, students may be asked to add 0.2 and 2/5.

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When paired with 5.1A, students are expected to solve problems. The intent of this SE is not a sole focus on computation. Within the TEKS, fluency with fraction and decimal addition and subtraction occurs in grade 5. The word “fluently” has been added. “Procedural fluency refers to knowledge of procedures, knowledge of when and how to use them appropriately, and skill in performing them flexibly, accurately, and efficiently” (National Research Council, 2001, p. 121). This SE builds to the grade 7 skills on operations with rational numbers.


In conjunction with 5.3J, the intent of this standard is to build the concrete understanding of division of fractions. In adherence to the standard, examples are limited to whole numbers divided by a unit fraction (i.e. 4 ÷ ¼ = x) or a unit fraction divided by a whole number (i.e. ¼ ÷ 4 = x). A unit fraction is one part of a whole that is divided into equal parts; the numerator is 1. Students would not be expected to divide a non-unit fraction by a whole number (i.e. ¾ ÷ 5 = x). A non-unit fraction has a numerator other than 1. The standard also excludes the division of two fractions (i.e. 4/5 ÷ ¾ = x). This foundation will prepare students in better understanding how dividing by a fraction and multiplying by its reciprocal yield equivalent answers (i.e. 3 ÷ ¼ = 12 and 3 x 4 = 12). Instruction should model how division of

* Students may confuse the dividend from the divisor (i.e. 4 ÷ ¼ = x; “how many fours divide into ¼” instead of “how many fourths divide into four wholes?”).

When paired with 5.1A, students are expected to solve problems. The intent of this SE is not a sole focus on computation. Within the TEKS, fluency with fraction division occurs in grade 6.

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fractions is like division of whole numbers (i.e. 18 ÷ 3 = x; how many equal groups of three divide into 18? Or 18 ÷ 1/3 = x; how many times can one-third go into 18? Or 1/3 ÷ 8 = x; given one-third of a whole, divide it into eight equal parts). See 5.3J for examples.


Students need to identify prime numbers as those that only have two factors; one and itself (i.e. 13 is prime because the only factors for 13 are 1 and 13, 1 x 13 = 13). A composite number has more than two factors (i.e. 18 is composite because 1 x 18 = 18; 2 x 9 = 18; 3 x 6 = 18). The number one is neither prime nor composite. Instruction should model multiple representations of composite numbers as this will support future needs of finding a common denominator and/ or common factor. Listing factors in a t-chart may help students to ensure that all factors have been identified.

Factors of 24 1 24 2 12 3 8 4 6

Identifying prime and composite numbers will support students as they will have to determine a common denominator in order to add and subtract fractions with unlike denominators, and identify a common factor in order to represent sums/differences in simplified form. 5.3K: add and subtract positive rational numbers fluently

When paired with 5.1C, students may use real (concrete) objects to identify prime and composite numbers. When paired with 5.1D, students may use pictorial models including arrays as a representation to identify prime and composite numbers. When paired with 5.1F, students may analyze factor pairs to connect the concepts of prime and composite numbers to defining characteristics of factors using patterns in factor pairs. The concept of prime factorization appears in grade 6 and then reappears in the Algebra I TEKS.

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Course: Grade 5 Math Bundle 5: Graphing on Coordinate Planes Dates: November 28th-December 16th (15 days) TEKS

5.4C: generate a numerical pattern when given a rule in the form y = ax or y = x + a and graph 5.4D: recognize the difference between additive and multiplicative numerical patterns given in a table or graph 5.8A: describe the key attributes of the coordinate plane, including perpendicular number lines (axes) where the intersection (origin) of the two lines coincides with zero on each number line and the given point (0, 0); the x-coordinate, the first number in an ordered pair, indicates movement parallel to the x-axis starting at the origin; and the y-coordinate, the second number, indicates movement parallel to the y-axis starting at the origin 5.8B: describe the process for graphing ordered pairs of numbers in the first quadrant of the coordinate plane 5.8C: graph in the first quadrant of the coordinate plane ordered pairs of numbers arising from mathematical and real-world problems, including those generated by number patterns or found in an input-output table

ELPS Learning Strategies 1A: use prior knowledge and experiences to understand meanings in English 1B: monitor oral and written language production and employ self-corrective techniques or other resources 1F: use accessible language and learn new and essential language in the process Listening 2C: learn new language structures, expressions, and basic and academic vocabulary heard during classroom instruction and interactions 2I: demonstrate listening comprehension of increasingly complex spoken English by following directions, retelling or summarizing spoken messages, responding to questions and requests, collaborating with peers, and taking notes commensurate with content and grade-level needs Speaking 3D: speak using grade-level content area vocabulary in context to internalize new English words and build academic language proficiency 3G: express opinions, ideas, and feelings ranging from communicating single words and short phrases to participating in extended discussions on a variety of social and grade-appropriate academic topics Reading 4C: develop basic sight vocabulary, derive meaning of environmental print, and comprehend English vocabulary and language structures used routinely in written classroom materials 4F: use visual and contextual support and support from peers and teachers to read grade-appropriate content area text, enhance and confirm understanding, and develop vocabulary, grasp of language structures, and background knowledge needed to comprehend increasingly challenging language

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Vocabulary

Unit Vocabulary Additive pattern Graph Number pattern Perpendicular x-axis Attributes Input/ output table Ordered pairs Quadrant x-coordinate Axis/axes Multiplicative pattern Origin (0,0) Rule y-axis Coordinate plane Number line Parallel movement Table y-coordinate Coordinates

Cognitive Complexity Verbs: generate, graph, recognize, describe Academic Vocabulary by Standard: 5.4C: additive pattern, graph, input/output table, multiplicative pattern, rule, x-axis, y-axis 5.4D: additive pattern, graph, multiplicative pattern, table 5.8A: attributes, axis/ axes, coordinate plane, coordinates, number line, ordered pairs, origin (0,0), parallel movement, perpendicular, x-axis, x-coordinate, y-axis, y-coordinate 5.8B: attributes, axis/ axes, coordinate plane, coordinates, graph, number line, ordered pairs, origin (0,0), parallel movement, perpendicular, quadrant, x-axis, x-coordinate, y-axis, y-coordinate 5.8C: coordinate place, graph, input/output table, number pattern, ordered pairs, quadrant, x-axis, y-axis


Balance Scales Graph Paper

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4.5B represent problems using an input-output table and numerical expressions to generate a number pattern that follows a given rule representing the relationship of the values in the resulting sequence and their position in the sequence

5.4C: generate a numerical pattern when given a rule in the form y = ax or y = x + a and graph

6.4A compare two rules verbally, numerically, graphically, and symbolically in the form of y = ax or y = x + a in order to differentiate between additive and multiplicative relationships

4.5B represent problems using an input-output table and numerical expressions to generate a number pattern that follows a given rule representing the relationship of the values in the resulting sequence and their position in the sequence

5.4D: recognize the difference between additive and multiplicative numerical patterns given in a table or graph

6.4A compare two rules verbally, numerically, graphically, and symbolically in the form of y = ax or y = x + a in order to differentiate between additive and multiplicative relationships

5.8A: describe the key attributes of the coordinate plane, including perpendicular number lines (axes) where the intersection (origin) of the two lines coincides with zero on each number line and the given point (0, 0); the x-coordinate, the first number in an ordered pair, indicates movement parallel to the x-axis starting at the origin; and the y-coordinate, the second number, indicates movement parallel to the y-axis starting at the origin

6.11A graph points in all four quadrants using ordered pairs of rational numbers

5.8B: describe the process for graphing ordered pairs of numbers in the first quadrant of the coordinate plane


5.8C: graph in the first quadrant of the coordinate plane ordered pairs of numbers arising from mathematical and real-world problems, including those generated by number patterns or found in an input-output table


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TEKS/Student Expectations Instructional Implications Distractor Factors Supporting Readiness Standards


5.4C: generate a numerical pattern when given a rule in the form y = ax or y = x + a and graph

In adherence to the standard, instruction should provide additive (i.e. y = x + 2) and multiplicative (i.e. y = 2x) rules and students will generate a number pattern (i.e. list of ordered pairs, input/output table, etc.). Given y = 2x (0,0); (1,2); (2,4); (3,6); (4,8); (5,10)

x Y 0 0 1 2 2 4 3 6 4 8 5 10

y = 2x

In support of 5.8C, students will graph the number patterns developed. Graphing of coordinates is limited to those found in the first quadrant. When generating a graph, students will informally discover the difference between discrete (data not represented between the scaled intervals) and continuous data (data values represented between scaled intervals). In conjunction with 5.4D, it will be important for students to include the zero value for x in both the table and the graph in order to identify differences between an additive and a multiplicative pattern.

* Students may not recognize y = 3x as a multiplication problem. * Students may not begin the number pattern at zero. * Students may think the order in plotting coordinate points is not important [i.e. (3, 4) is the same location as (4, 3)].

This SE provides clarity for which types of patterns align to grade 5 expectations. This SE refers to patterns that are bivariate and does not directly refer to sequences.

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5.4D: recognize the difference between additive and multiplicative numerical patterns given in a table or graph

In conjunction with 5.4C and 5.8C, as students begin graphing coordinates from tables of data, instruction will focus on the similarities and differences between an additive and multiplicative pattern. Additive patterns in a table yield an addition relationship between the input and output data on a table and include the coordinate (0, a) due to the additive property of zero (i.e. a + 0 = a). Multiplicative patterns in a table yield a multiplicative relationship between the input and output data on a table and include the coordinate (0, 0) due to the multiplicative property of zero (i.e. a ⋅ 0 = 0). When graphing additive and multiplicative patterns, both will yield a linear representation but additive will start the graph at (0, a) and multiplicative will start the graph at (0, 0).

Recognizing the difference between an additive and multiplicative pattern sets the foundational understanding for proportionality as additive patterns yield non-proportional situations and multiplicative patterns yield proportional situations. 5.4C: generate a numerical pattern when given a rule in the form y = ax or y = x + a and graph

The SE specifies the relationship between paired sets of data. The focus is on additive versus multiplicative patterns. The relationships are of form y = x + a (additive) or y = ax (multiplicative) where x is the input value and y is the output value. This SE is related to 5.4C where students generate a pattern given an additive or multiplicative rule and its graph. This SE is related to 5.8C where students graph the relationships found in input/output tables. The graphical representation has been added. However, graphing in the first quadrant will be the relevant skill for these relationships. While new to algebraic reasoning, this skill is not new to grade 5. This skill builds to the introduction of simple lines (y = kx and y = x + b) in grade 6.

5.8A: describe the key attributes of the coordinate plane, including perpendicular number lines (axes) where the intersection (origin) of the two lines coincides with zero on each number line and the given point (0,

In adherence to the standard, instruction should focus on the key attributes of a coordinate grid. The x-axis is represented by the horizontal number line; the y-axis is represented by the vertical number line. Coordinate

Understanding the proper movement along the coordinate grid in addition to communicating those actions appropriately will be the foundation to interpreting graphed mathematical and real-world data.

The SE adds specificity for what students are expected to understand regarding the structure of the coordinate plane.

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0); the x-coordinate, the first number in an ordered pair, indicates movement parallel to the x-axis starting at the origin; and the y-coordinate, the second number, indicates movement parallel to the y-axis starting at the origin

planes may include scaled axes. The point at which the two axes intersect to form a perpendicular line is identified as the origin (0, 0). The origin is the starting point for the graphing of all ordered pairs. Instruction should relate ordered pairs to the coordinate plane. The first number is referred to as the x-coordinate which will be located by moving parallel to the x-axis. The second number is referred to as the y-coordinate which will be located by moving parallel to the y-axis.


Students are expected to graph ordered pairs only in the first quadrant.

5.8B: describe the process for graphing ordered pairs of numbers in the first quadrant of the coordinate plane

In adherence to the standard, instruction should focus on appropriately communicating the actions of locating a given point on a coordinate plane (i.e. relating the x-coordinate to the parallel movement along the x-axis; relating the y-coordinate to the parallel movement along the y-axis; movement beginning at the origin). In conjunction with 5.8C, data points can be in the form of ordered pairs [i.e. (3, 4)] or those generated from an input/ output table.

Understanding the proper movement along the coordinate grid in addition to communicating those actions appropriately is the foundation for interpreting graphed mathematical and real-world data. 5.8C: graph in the first quadrant of the coordinate plane ordered pairs of numbers arising from mathematical and real-world problems, including those generated by number patterns or found in an input-output table

This SE makes the process of graphing explicit.


In conjunction with 5.8A/B, graphing coordinates are limited to those in the first quadrant (positive numbers only). In adherence to the standard, coordinates are generated from mathematical (i.e. 12 inches = 1 foot) and real-world (i.e. 3 wheels = 1 tricycle) number patterns. These patterns can be represented through generated number patterns (i.e. 3, 6, 9, etc.) or within an input-output table, such as the one below.

Number of Tricycles

Number of Wheels

0 0 1 3 2 6 3 9 4 12 5 15

* Students may confuse the x- and y-axis. * Students may not understand how a coordinate communicates information. * Students may not locate coordinates correctly given graphs of intervals other than one. * Students may think the order in plotting a coordinate point is not important [i.e. (1, 3), is the same location as (3, 1)].

The SE condenses “locate and name points on a coordinate plane using ordered pairs of whole numbers” to “graph ordered pairs.” Students may be expected to graph points with fractional values because of work in grade 4 on the number line with 4.3G: Represent fractions and decimals to the tenths or hundredths as distances from zero on a number line. The fractional values may be between grid lines or represented by grid lines.

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Instruction should ensure that all graphs are titled, x- and y-axis are labeled correctly, and intervals of each axis are identified. Students should be able to relate mathematical and real-world problems as they are represented on a graph (i.e. The coordinate, (1, 3) represents one tricycle yielding three wheels and the coordinate (2, 6) represents two tricycles yielding six wheels, etc.).

The graphing in this SE is related to 5.4C and 5.4D which is the reason for including “real-world problems” and “including those generated by number patterns or found in an input-output table.”

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Course: Grade 5 Math Bundle 6: Geometry and Measurement Dates: January 2nd- January 27th (18 days) TEKS

5.4G: use concrete objects and pictorial models to develop the formulas for the volume of a rectangular prism, including the special form for a cube (V = l x w x h, V = s x s x s, and V = Bh) 5.4H: represent and solve problems related to perimeter and/or area and related to volume 5.5A: classify two-dimensional figures in a hierarchy of sets and subsets using graphic organizers based on their attributes and properties 5.6A: recognize a cube with side length of one unit as a unit cube having one cubic unit of volume and the volume of a three-dimensional figure as the number of unit cubes (n cubic units) needed to fill it with no gaps or overlaps if possible 5.6B: determine the volume of a rectangular prism with whole number side lengths in problems related to the number of layers times the number of unit cubes in the area of the base 5.7A: solve problems by calculating conversions within a measurement system, customary or metric.

ELPS Learning Strategies 1A: use prior knowledge and experiences to understand meanings in English 1F: use accessible language and learn new and essential language in the process Listening 2C: learn new language structures, expressions, and basic and academic vocabulary heard during classroom instruction and interactions 2D: monitor understanding of spoken language during classroom instruction and interactions and seek clarification as needed 2E: use visual, contextual, and linguistic support to enhance and confirm understanding of increasingly complex and elaborated spoken language Speaking 3D: speak using grade-level content area vocabulary in context to internalize new English words and build academic language proficiency 3E: share information in cooperative learning interactions 3F: ask and give information ranging from using a very limited bank of high-frequency, high-need, concrete vocabulary, including key words and expressions needed for basic communication in academic and social contexts, to using abstract and content-based vocabulary during extended speaking assignments Reading 4F: use visual and contextual support and support from peers and teachers to read grade-appropriate content area text, enhance and confirm understanding, and develop vocabulary, grasp of language structures, and background knowledge needed to comprehend increasingly challenging language 4G: demonstrate comprehension of increasingly complex English by participating in shared reading, retelling or summarizing material, responding to questions, and taking notes commensurate with content area and grade level needs

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Vocabulary

Unit Vocabulary Angles Equation Metric Polygon Three-dimensional Area Expression Number of layers Quadrilateral Trapezoid Area of the base Figure Number of sides Rectangle Triangle Attribute Formula Octagon Rectangular prism Two-dimensional Base Geometric property Parallel lines Rhombus Unit cube Circle Height Parallelogram Shape Vertex Conversions Hexagon Pentagon Side length Vertices Cube Length Perimeter Sides Volume Cubic units Measurement system Perpendicular lines Square Width Customary

Cognitive Complexity Verbs: use, represent, solve, classify, use, recognize, determine, solve Academic Vocabulary by Standard: 5.4G: formula, volume, rectangular prism, cube, length, width, height, side length 5.4H: area, area of the base, base, expression, equation, length, width, height, perimeter, volume 5.5A: angles, attribute, figure, geometric property, parallel lines, perpendicular lines, polygon, quadrilateral, parallelogram, rectangle, rhombus, square, trapezoid, triangle, circle, hexagon, pentagon, octagon, shape, sides, two-dimensional, vertex, vertices 5.6A: cubic units, side length, three-dimensional, unit cube, volume 5.6B: area of the base, number of layers, number of sides, rectangular prism, unit cubes, volume 5.7A: conversions, customary, measurement system, metric


Geometric Solids (Cubes and Rectangular Prisms)

Rulers Pattern Blocks Power Polygons

Interlocking Cubes Color Tiles Graph Paper

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4.5C use models to determine the formulas for the perimeter of a rectangle (l + w +l + w or 2l + 2w), including the special form for perimeter of a square (4s) and the area of a rectangle (l x w)

5.4G: use concrete objects and pictorial models to develop the formulas for the volume of a rectangular prism, including the special form for a cube (V = l x w x h, V = s x s x s, and V = Bh)

6.8B model area formulas for parallelograms, trapezoids, and triangles by decomposing and rearranging parts of these shapes

4.5D solve problems related to perimeter and area of rectangles where dimensions are whole numbers

5.4H: represent and solve problems related to perimeter and/or area and related to volume

6.8C write equations that represent problems related to the area of rectangles, parallelograms, trapezoids, and triangles and volume of right rectangular prisms where dimensions are positive rational numbers 6.8D determine solutions for problems involving the area of rectangles, parallelograms, trapezoids, and triangles and volume of right rectangular prisms where dimensions are positive rational numbers

3.6A classify and sort two- and three-dimensional figures, including cones, cylinders, spheres, triangular and rectangular prisms, and cubes, based on attributes using formal geometric language 4.6D classify two-dimensional figures based on the presence or absence of parallel or perpendicular lines or the presence or absence of angles of a specified size

5.5A: classify two-dimensional figures in a hierarchy of sets and subsets using graphic organizers based on their attributes and properties

5.6A: recognize a cube with side length of one unit as a unit cube having one cubic unit of volume and the volume of a three-dimensional figure as the number of unit cubes (n cubic units) needed to fill it with no gaps or overlaps if possible

3.6C determine the area of rectangles with whole number side lengths in problems using multiplication related to the number of rows times the number of unit squares in each row

5.6B: determine the volume of a rectangular prism with whole number side lengths in problems related to the number of layers times the number of unit cubes in the area of the base

4.8A identify relative sizes of measurement units within the customary and metric systems 4.8B convert measurements within the same measurement system, customary or metric, from a smaller unit into a larger unit or a larger unit into a smaller unit when given other equivalent measures represented in a table 4.8C solve problems that deal with measurements of length, intervals of time, liquid volumes, mass, and money using addition, subtraction, multiplication, or division as appropriate

5.7A: solve problems by calculating conversions within a measurement system, customary or metric.

6.4H convert units within a measurement system, including the use of proportions and unit rates

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TEKS/Student Expectations Instructional Implications Distractor Factors Supporting Readiness Standards


5.4G: use concrete objects and pictorial models to develop the formulas for the volume of a rectangular prism, including the special form for a cube (V = l x w x h, V = s x s x s, and V = Bh)

As students are physically layering the base of a rectangular prism with unit cubes, it is essential that instruction relate the actions to determining the area of the base (A = l x w). As students begin filling the rectangular prism with unit cubes, instruction should relate to how the height of the rectangular prism will determine the number of layers or the number of times the area of the base will be repeated (V = l x w x h). In conjunction with 5.4E, students begin applying the use of parentheses to show how the first step was determining the area of the base [V= (l x w) x h]. As students become comfortable with this representation, instruction can move to replacing the (l x w) with the capital B as a means of communicating the area of the base 9V = Bh). In adherence to the standard, instruction should include the study of cubes as a special type of rectangular prism in which all sides are of equal length. As students experience filling cubes, they will discover how the length, width, and height are all the same length and its formula can be represented as V = s x s x s. In accordance to the state standards, the use of exponents does not get introduced until grade 6 (see 6.7A).

Physically layering/ filling a rectangular prism and associating those actions to the formula will move instruction from the concrete to the abstract. 5.4H: represent and solve problems related to perimeter and/or area and related to volume

The SE specifies the use of concrete objects and pictorial models. Volume is specifically tied to rectangular prisms, and three formulas have been stated within the SE.


This standard merges the understanding of perimeter, area, and volume. Instruction should not only include problems involving the three concepts in isolation (i.e. given a 4 ft. x 6 ft. rectangle, what is the area? What is the perimeter?), but also extend problems to include merging the different concepts into one question (i.e. if the perimeter of a square is 32 cm, what is the area? If a rectangular prism has a height of 12 inches and the area of its rectangular measures 15 square inches, what is the volume? Be sure to vary the given information within the problems (i.e. given the length, width, and height, determine the

* Students may confuse the concepts of perimeter, area, and volume. * When determining perimeter of irregular shapes, students may only add the number of sides given and disregard side lengths that are not labeled. * When determining the perimeter/ area of a square or volume of a cube, students may forget that the side lengths must

Students are expected to determine the perimeter of polygons, area of rectangles and composite figures formed by rectangles, and volume of rectangular prisms. “Use appropriate…. formulas” is stated more appropriately as “represent and solve.” Students may still be expected to measure lengths to

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volume; given one length, the height and the volume of a rectangular prism, determine the length of the missing side; given the area of the base and the volume of a rectangular prism, determine the height; given the volume of a cube is 216 cubic meters, determine the dimensions of the base). Students are not only to solve problems, but represent them in equations (i.e. given the perimeter is 45 yards and a side length of 15 yards; 45 = 2(15) + 2w). In conjunction with 5.7A, problems should include conversions within the same measurement system (i.e. given a 4 ft. by 6 ft. rectangle, what is the perimeter in inches?). Area and perimeter problems are not limited to rectangular shapes; therefore instruction should include composite structures.

be equal. * Students may not correctly label the units of measure (i.e. perimeter in units; area in square units; and volume in cubic units). * Students may not relate how the formula for area is a component of the formula for volume.

determine perimeter, area, and volume if the problem requires it. Because fluency with addition and subtraction of positive rational numbers is expected within the TEKS, lengths may reflect fractional measures with perimeter.

5.5A: classify two-dimensional figures in a hierarchy of sets and subsets using graphic organizers based on their attributes and properties

In adherence to the standard, students need to be able to not only describe the attributes and properties of a two-dimensional figure (i.e. A square is a closed figure that has four vertices, four sides of equal length, two sets of parallel lines, two sets of perpendicular lines, and four right angles) but relate that knowledge to other two-dimensional shapes in order to classify sets and/or subsets (i.e. A square is also classified as a polygon because it is a closed figure with at least three straight sides. It is also a quadrilateral because it is a polygon with four sides. It is also a rectangle because it is a quadrilateral with two pair of congruent, parallel sides and four right

* Students may rely only on the look of a geometric figure to determine classification instead of comparing its attributes/ properties. * Students may not understand the inverse relationship between terms (i.e. all squares are rectangles, but not all rectangles are squares).

This SE focuses on classification by attributes and properties. An attribute is a characteristic or component of a geometric figure. The attributes of a square include side lengths and angle measures. The attributes combine to form the properties of a square: 4 right angles, 4 congruent sides, 2 sets of parallel sides.

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angles. It is also a parallelogram because it has four sides with two pair of parallel congruent sides; it is also a rhombus because it is a quadrilateral with all four sides congruent in length). Encourage students to identify all of the appropriate geometric terms for any given shape. According to the standard, students need to make use of a graphic organizer to display their understanding of the relationships.

The SE clarifies the purpose of identifying essential attributes: classification within a hierarchy of sets and subsets. For example, all rectangles have the property that opposite sides are parallel; therefor, every rectangle is a parallelogram. The SE specifies the use of graphic organizers as a classification tool. See graphic below:

5.6A: recognize a cube with side length of one unit as a unit cube having one cubic unit of volume and the volume of a three-dimensional figure as the number of unit cubes (n cubic units) needed to fill it with no gaps or overlaps if possible

As students tend to confuse the concepts of perimeter, area, and volume, it is essential that they physically layer/ fill three-dimensional figures with unit cubes. Just like rulers are used to measure length, clocks are used to measure time, and thermometers are used to measure temperature, unit cubes are used to determine the volume. Recognizing the geometric properties of a unit cube (i.e. it has three dimensions: length, width, and height) will allow students to understand how volume is identified in cubic units. Instruction should provide a variety of three-dimensional figures for students to fill to experience how gaps will not yield an accurate volume for a given figure.

This analysis of geometric attributes of a unit cube supports the understanding of how volume is measured in cubic units. Physically filling a three-dimensional figure with unit cubes with no gaps or overlaps will provide the concrete development of volume. 5.4H: represent and solve problems related to perimeter and/or area and related to volume

The SE specifies how to use concrete objects and pictorial models and develops formulas as described in 5.4G.

5.6B: determine the volume of a rectangular prism with whole number side lengths in problems related to the number of layers times the number of unit cubes in the area of the base

As students are physically layering the base of a rectangular prism with unit cubes, it is essential that instruction relate those actions to determining the area of the base. Before students fill the rectangular prism with unit cubes, instruction should relate how the height of the rectangular prism will

Relating the area of the base to determine the volume of a rectangular prism will support students’ understanding of the relationship between the two concepts.

The SE specifies how to use concrete objects and pictorial models and develops formulas as described in 5.4G. This SE supports V=Bh.

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determine the number of layers or the number of times the area of the base will be repeated.


5.7A: solve problems by calculating conversions within a measurement system, customary or metric.

In grade 4 (see 4.8B) students used a table to convert measurement units. Multiplicative patterns were identified so students begin to recognize the rules for conversions (i.e. to convert from kilometers to meters multiply by 1,000).

Kilometers Process Meters 1 1 x 1,000 1,000 2 2 x 1,000 2,000 3 3 x 1,000 3,000 4 4 x 1,000 4,000

21 21 x 1,000 21,000 Students would have experience converting both metric and customary units for length, liquid volume, and mass. Grade 5 instruction now moves to the application of these discovered rules of conversions.

Conversions within the customary or metric system may be necessary when asked to solve problems involving perimeter, area, and volume. 5.4H: represent and solve problems related to perimeter and/or area and related to volume

Specificity provides the purpose for performing conversions. The conversions should serve the purpose of solving a problem. Performing conversions will still be required to solve problems. These conversions may include decimal values with metric units or fractional values with customary units that align to the Number and Operations strand. Conversions may be multistep within a measurement system such as yards to feet to inches.

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Course: Grade 5 Math Bundle 7: Data Analysis

Dates: January 30th- February 10th (10 days)

TEKS 5.9A: represent categorical data with bar graphs or frequency tables and numerical data, including data sets of measurements in fractions or decimals, with dot plots or stem-and-leaf plots 5.9B: represent discrete paired data on a scatterplot 5.9C: solve one- and two-step problems using data from a frequency table, dot plot, bar graph, stem-and-leaf plot, or scatterplot

ELPS Learning Strategies 1B: monitor oral and written language production and employ self-corrective techniques or other resources Listening 2C: learn new language structures, expressions, and basic and academic vocabulary heard during classroom instruction and interactions 2D: monitor understanding of spoken language during classroom instruction and interactions and seek clarification as needed 2I: demonstrate listening comprehension of increasingly complex spoken English by following directions, retelling or summarizing spoken messages, responding to questions and requests, collaborating with peers, and taking notes commensurate with content and grade-level needs Speaking 3F: ask and give information ranging from using a very limited bank of high-frequency, high-need, concrete vocabulary, including key words and expressions needed for basic communication in academic and social contexts, to using abstract and content-based vocabulary during extended speaking assignments 3G: express opinions, ideas, and feelings ranging from communicating single words and short phrases to participating in extended discussions on a variety of social and grade-appropriate academic topics Reading 4C: develop basic sight vocabulary, derive meaning of environmental print, and comprehend English vocabulary and language structures used routinely in written classroom materials 4G: demonstrate comprehension of increasingly complex English by participating in shared reading, retelling or summarizing material, responding to questions, and taking notes commensurate with content area and grade level needs

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Vocabulary

Unit Vocabulary Bar graph Decimal Fraction Numerical data Stem-and-leaf plot Categorical data Discrete paired data Frequency table Scatterplot Whole number Data Dot plot Graph

Cognitive Complexity Verbs: represent, solve, use Academic Vocabulary by Standard: 5.9A: bar graph, categorical data, decimal, dot plot, fraction, frequency table, numerical data, stem-and-leaf plot, whole number 5.9B: discrete paired data, scatterplot 5.9C: bar graph, data, numerical data, categorical data, dot plot, frequency table, graph, scatterplot, stem-and-leaf plot


Chart Paper Graph Paper Rulers

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4.9A represent data on a frequency table, dot plot, or stem-and-leaf plot marked with whole numbers and fractions

5.9A: represent categorical data with bar graphs or frequency tables and numerical data, including data sets of measurements in fractions or decimals, with dot plots or stem-and-leaf plots

6.12A represent numeric data graphically, including dot plots, stem-and-leaf plots, histograms, and box plots

5.9B: represent discrete paired data on a scatterplot

4.9B solve one- and two-step problems using data in whole number, decimal, and fraction form in a frequency table, dot plot, or stem-and-leaf plot

5.9C: solve one- and two-step problems using data from a frequency table, dot plot, bar graph, stem-and-leaf plot, or scatterplot

6.13A interpret numeric data summarized in dot plots, stem-and-leaf plots, histograms, and box plots

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5.9A: represent categorical data with bar graphs or frequency tables and numerical data, including data sets of measurements in fractions or decimals, with dot plots or stem-and-leaf plots

According to the TEKS, students need to collect, organize, and display their own data. Personalizing such activities will allow students to make more sense of the data and summarize more appropriately. Instruction needs to include the collection of categorical data (i.e. “what is your favorite animal?”) and numeric data (i.e. “how tall are you?”). In accordance with the standard, categorical data should be represented on a bar graph or frequency table; whereas, numeric data should be represented on a dot plot or stem-and-leaf plot. In adherence to the standard, numeric data needs to extend to include the use of fractional and decimal representations. Students will need more support with the development of stem-and leaf plots. This type of graph will distribute the data by separating one place value from the other or the parts from the whole when working with fractions or decimals (i.e. 55.2, 55.9, 56.5, 56.4, etc.).

Stem Leaf 55 2 9 56 1 4 4 5 57 0 0

56 l 4 = 56.4 The larger place value or the whole of a fraction/ decimal value is identified as the stem and the smaller place value or the part of the whole is called the leaf. Instruction should include representing the same set of data on different types of graphs to compare.

This supporting standard provides an opportunity for students to collect, organize, display, and interpret data using a bar graph, frequency table, dot plot, and stem-and-leaf plot. This concrete experience will allow them to better understand the data represented in order to solve one- and two-step problems. 5.9C: solve one- and two-step problems using data from a frequency table, dot plot, bar graph, stem-and-leaf plot, or scatterplot

Data may be in the form of whole numbers, decimals, and/or fractions. A bar graph may be used to represent frequencies of an item, a category, a number, or a range of numbers. For example, the horizontal axis is labeled with the item, category, number or range of numbers being counted. The vertical axis is labeled with appropriate frequencies. The length of the bar represents the frequency of the items, categories, numbers, or range of numbers being counted. This SE limits the representation of data using a bar graph to categorical data. Students begin work with bar graphs in grade 1. A dot plot may be used to represent frequencies. A number line may be used for counts related to numbers. A line labeled with categories may be used as well if the context requires. Dots are recorded vertically above the number line to indicate frequencies. Dots may represent one count or multiple counts if so noted. Students begin work with dot plots in grade 3.

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A stem-and-leaf plot organizes data in numerical order according to place value. The stem represents the place values preceding the last digit. The leaves represent the last digits. The leaves provide the frequency counts for the range of numbers included in that row of the stem-and-leaf plot.

Students begin work with stem-and-leaf plots in grade 4.

5.9B: represent discrete paired data on a scatterplot

Discrete paired data compares two numeric data types to determine if there is a relationship between the two (i.e. the number of hours studied vs. the grade earned on a test). According to the TEKS, students need to collect, organize, and display their own data. Personalizing such activities will allow students to make more sense of the data and summarize more appropriately. Outcomes of the collection, representation, and interpretation of such data may or may not reflect a relationship between the two variables.

This supporting standard provides an opportunity for students to collect, organize, display, and interpret data using a scatterplot. This concrete experience will allow them to better understand the data represented in order to solve one- and two-step problems. 5.9C: solve one- and two-step problems using data from a frequency table, dot plot, bar graph, stem-and-leaf plot, or scatterplot

A scatterplot graphs two sets of data as corresponding numbers represented as ordered pairs. A scatterplot is comparable to a Quadrant I graph with paired data as ordered pairs.

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5.9C: solve one- and two-step problems using data from a frequency table, dot plot, bar graph, stem-and-leaf plot, or scatterplot

Instruction should vary the context of the problems being asked of the students (i.e. joining, separating, comparing). In conjunction with 5.3K/E/G/L. problems involving whole number/ decimal/ and fractional data should be included when interpreting graphed data. In alignment with 5.9A, as students have graphed the same data set on different types of graphs, students could then solve the same problem using different representations to model their flexibility in moving among the various types of graphs.

* In using the stem-and-leaf plot, students may have difficulty determining what values represent the stem and leaf. * When using data on different types of graphs, students may interpret the data as different because of the difference in the visual representations. * When using data on different frequency tables and/ or bar graphs, students may interpret the data as different because of the difference in the visual representations.

A frequency table shows how often an item, a number, or a range of numbers occurs. Tallies and counts are used to record frequencies. Students begin work with frequency tables in grade 3. Students begin work with dot plots in grade 3. Students begin work with bar graphs in grade 1. Students begin work with stem-and- leaf plots in grade 4. Students begin work with scatterplots in grade 5.

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Course: Grade 5 Math Bundle 8: Personal Financial Literacy

Dates: February 13th- February 17th (5 days)

TEKS 5.10A: define income tax, payroll tax, sales tax, and property tax 5.10B: explain the difference between gross income and net income 5.10C: identify the advantages and disadvantages of different methods of payment, including check, credit card, debit card, and electronic payments 5.10D: develop a system for keeping and using financial records 5.10E: describe actions that might be taken to balance a budget when expenses exceed income 5.10F: balance a simple budget

ELPS Learning Strategies 1A: use prior knowledge and experiences to understand meanings in English Listening 2C: learn new language structures, expressions, and basic and academic vocabulary heard during classroom instruction and interactions 2I: demonstrate listening comprehension of increasingly complex spoken English by following directions, retelling or summarizing spoken messages, responding to questions and requests, collaborating with peers, and taking notes commensurate with content and grade-level needs Speaking 3C: speak using a variety of grammatical structures, sentence lengths, sentence types, and connecting words with increasing accuracy and ease as more English is acquired Reading 4C: develop basic sight vocabulary, derive meaning of environmental print, and comprehend English vocabulary and language structures used routinely in written classroom materials 4E: read linguistically accommodated content area material with a decreasing need for linguistic accommodations as more English is learned 4F: use visual and contextual support and support from peers and teachers to read grade-appropriate content area text, enhance and confirm understanding, and develop vocabulary, grasp of language structures, and background knowledge needed to comprehend increasingly challenging language

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Vocabulary

Unit Vocabulary Balanced budget Debit card Gross income Net income Property tax Budget Electronic payment Income Payment methods Sales tax Check Expenses Income tax Payroll tax System Credit card Financial records

Cognitive Complexity Verbs: define, explain, identify, develop, describe, balance Academic Vocabulary by Standard: 5.10A: income tax, payroll tax, property tax, sales tax 5.10B: gross income, net income 5.10C: check, credit card, debit card, electronic payment, payment methods 5.10D: financial records, system 5.10E: budget, expenses, income 5.10F: balanced budget

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4.10A distinguish between fixed and variable expenses 5.10A: define income tax, payroll tax, sales tax, and property tax

5.10B: explain the difference between gross income and net income

6.14H compare the annual salary of several occupations requiring various levels of postsecondary education or vocational training and calculate the effects of the different annual salaries on lifetime income

4.10E describe the basic purpose of financial institutions, including keeping money safe, borrowing money, and lending

5.10C: identify the advantages and disadvantages of different methods of payment, including check, credit card, debit card, and electronic payments

6.14B distinguish between debit cards and credit cards

4.10D describe how to allocate a weekly allowance among spending; saving, including for college; and sharing

5.10D: develop a system for keeping and using financial records

6.14A compare the features and costs of a checking account and a debit card offered by different local financial institutions 6.14C balance a check register that includes deposits, withdrawals, and transfers

5.10E: describe actions that might be taken to balance a budget when expenses exceed income

5.10F: balance a simple budget

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5.10A: define income tax, payroll tax, sales tax, and property tax

In adherence to the standard, instruction should address the different types of taxes. Instruction should explain how once a year individuals/ businesses must file an income tax return to determine whether they owe any taxes or are eligible for a tax refund. Income tax is a key source of funds that the government uses to fund activities and serve the public. Payroll tax is money withheld by employers from the income earned by employees. Payroll taxes fund such programs as Social Security, healthcare, and unemployment compensation. Sales tax is money collected when goods and services are rendered. Each state sets its own sales tax and it is collected accordingly. Property tax is money paid by individuals/ businesses that own land/ property. The tax is usually based on the value of the land/property. The money raised from this tax is mainly used for road repairs, building of local schools, etc.

Defining income tax, payroll tax, sales tax, and property tax will support one’s ability to manage their financial resources more effectively for a lifetime of financial security.

Income tax is paid by the employee. Payroll tax is paid by the employer. Sales tax is based upon the cost of items purchased. Property tax is based upon real property held and is paid directly by the owner.

5.10B: explain the difference between gross income and net income

Gross income is the amount of money earned before taxes; insurance costs, etc. are taken out. Net income is the amount of money earned after all taxes, insurance, etc. are taken out.

Understanding the difference between gross and net income will support one’s ability to manage their financial resources more effectively for a lifetime of financial security.

This SE builds on the concept of profit in 4.10B.

5.10C: identify the advantages and disadvantages of different methods of payment, including check, credit card, debit card, and electronic payments

In adherence to the standard, instruction should include discussions on the advantages and disadvantages of different methods of payment. (See table below)

Identifying the advantages and disadvantages of different methods of payment will support one’s ability to manage their financial resources more effectively for a lifetime of financial security.

This SE builds to 6.14B.

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5.10D: develop a system for keeping and using financial records

Organizing a system for securing important financial records should be considered (i.e. storing all checks, bank statements, payroll stubs, income tax returns in a safe; scanning all important financial records for electronic access).

Developing a system for keeping and using financial records will support one’s ability to manage their financial resources more effectively for a lifetime of financial security.

This SE builds to 6.14C.

5.10E: describe actions that might be taken to balance a budget when expenses exceed income

Instruction should include discussion on managing one’s budget appropriately. Students should brainstorm ideas of what can be done when expenses exceed income.

Describing a plan that might be taken to balance a budget when expenses exceed income will support one’s ability to manage their financial resources more effectively for a lifetime of financial security.


5.10F: balance a simple budget In conjunction with 5.3K, instruction can merge the balancing of a simple budget with addition and subtraction of whole numbers and decimals (i.e. Margarita has a monthly net pay of $1,544.56. She has the following expenses to pay:

Expense Cost Rent $450.00 Utilities $124.56 Food $275.00 Car Payment $250.00 Car Insurance $74.50 Gas $75.00

Margarita wants to trade her car for a newer model. Can she afford a more expensive car? Why or why not?

Balancing a simple budget will support one’s ability to manage their financial resources more effectively for a lifetime of financial security.


Payment Type Advantages Disadvantages Check Do not have to carry cash Cost of checks Credit Card Do not have to carry cash

Establishing good credit Interest charges if total amount charged is not paid off at the end of each month

Debit Card Do not have to carry cash Money automatically deducted from your account

Service fee

Electronic Payments Do not have to mail payment Save on the cost of stamps

Usage fee Errors in entering amount which can create additional fees

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Course: Grade 5 Math Bundle 9: STAAR Review and Testing

Dates: February 20th-March 28th (22 days)

TEKS -Review previously taught skills/TEKS -Address STAAR questions STEMS at varied DOK levels

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Course: Grade 5 Math Bundle 10: Extended Learning

Dates: March 29th- May 8th (26 days)

TEKS Continued Problem Solving 5.4B: represent and solve multi-step problems involving the four operations with whole numbers using equations with a letter standing for the unknown quantity 5.4H: represent and solve problems related to perimeter and/or area and related to volume 5.9C: solve one- and two-step problems using data from a frequency table, dot plot, bar graph, stem-and-leaf plot, or scatterplot Continued Number Sense 5.3A: estimate to determine solutions to mathematical and real-world problems involving addition, subtraction, multiplication, or division 5.3B: multiply with fluency a three-digit number by a two-digit number using the standard algorithm 5.3C: solve with proficiency for quotients of up to a four-digit dividend by a two-digit divisor using strategies and the standard algorithm 5.3D: represent multiplication of decimals with products to the hundredths using objects and pictorial models, including area models 5.3E: solve for products of decimals to the hundredths, including situations involving money, using strategies based on place-value understandings, properties of operations, and the relationship to the multiplication of whole numbers 5.3F: represent quotients of decimals to the hundredths, up to four-digit dividends and two-digit whole number divisors, using objects and pictorial models, including area models 5.3G: solve for quotients of decimals to the hundredths, up to four-digit dividends and two-digit whole number divisors, using strategies and algorithms, including the standard algorithm 5.3H: represent and solve addition and subtraction of fractions with unequal denominators referring to the same whole using objects and pictorial models and properties of operations 5.3I: represent and solve multiplication of a whole number and a fraction that refers to the same whole using objects and pictorial models, including area models

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5.3J: represent division of a unit fraction by a whole number and the division of a whole number by a unit fraction such as 1/3 ÷ 7 and 7 ÷ 1/3 using objects and pictorial models, including area models 5.3K: add and subtract positive rational numbers fluently 5.3L: divide whole numbers by unit fractions and unit fractions by whole numbers Continued Review of 5th Grade Readiness Standards 5.2B: compare and order two decimals to thousandths and represent comparisons using the symbols >, <, or = 5.4C: generate a numerical pattern when given a rule in the form y = ax or y = x + a and graph 5.4F: simplify numerical expressions that do not involve exponents, including up to two levels of grouping 5.5A: classify two-dimensional figures in a hierarchy of sets and subsets using graphic organizers based on their attributes and properties 5.8C: graph in the first quadrant of the coordinate plane ordered pairs of numbers arising from mathematical and real-world problems, including those generated by number patterns or found in an input-output table Project Based Learning -Introduce project-based learning activities with strong integration of TEKS that need extended exposure for mastery.

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Course: Grade 5 Math Bundle 11: Continued Extended Learning

Dates: May 9th – June 1st (17 days)

TEKS Continued Problem Solving 5.4B: represent and solve multi-step problems involving the four operations with whole numbers using equations with a letter standing for the unknown quantity 5.4H: represent and solve problems related to perimeter and/or area and related to volume 5.9C: solve one- and two-step problems using data from a frequency table, dot plot, bar graph, stem-and-leaf plot, or scatterplot Continued Number Sense 5.3A: estimate to determine solutions to mathematical and real-world problems involving addition, subtraction, multiplication, or division 5.3B: multiply with fluency a three-digit number by a two-digit number using the standard algorithm 5.3C: solve with proficiency for quotients of up to a four-digit dividend by a two-digit divisor using strategies and the standard algorithm 5.3D: represent multiplication of decimals with products to the hundredths using objects and pictorial models, including area models 5.3E: solve for products of decimals to the hundredths, including situations involving money, using strategies based on place-value understandings, properties of operations, and the relationship to the multiplication of whole numbers 5.3F: represent quotients of decimals to the hundredths, up to four-digit dividends and two-digit whole number divisors, using objects and pictorial models, including area models 5.3G: solve for quotients of decimals to the hundredths, up to four-digit dividends and two-digit whole number divisors, using strategies and algorithms, including the standard algorithm 5.3H: represent and solve addition and subtraction of fractions with unequal denominators referring to the same whole using objects and pictorial models and properties of operations 5.3I: represent and solve multiplication of a whole number and a fraction that refers to the same whole using objects and pictorial models, including area models

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5.3J: represent division of a unit fraction by a whole number and the division of a whole number by a unit fraction such as 1/3 ÷ 7 and 7 ÷ 1/3 using objects and pictorial models, including area models 5.3K: add and subtract positive rational numbers fluently 5.3L: divide whole numbers by unit fractions and unit fractions by whole numbers Project Based Learning -Introduce project-based learning activities with strong integration of TEKS that need extended exposure for mastery.

grade 5 mathematics curriculum document...

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