ReseaRch—Best PRactices Putting Research into Practice · ReseaRch—Best PRactices Putting...

Research & Math BackgroundContents Planning

Dr. Karen C. Fuson, Math Expressions Author

ReseaRch—Best PRactices

Putting Research into Practice

From Our Curriculum Research Project: Multiplication and Division

A core concept that students will learn is that multiplication and division are inverse operations. The students, with the teacher’s help, will learn to use and understand the language to describe underlying concepts and situations of multiplication and division, including repeated equal groups and arrays.

Students will learn multiplications and divisions for each number by looking for patterns that become the basis for count-bys for that number.

Students will learn how to use products they know to find products they don’t know or don’t recall. Students study division almost as soon as they learn multiplication. Studying these together makes the process faster because each division is just finding an unknown factor. Through their daily in-class work and goal-setting, students build fluency with multiplication and division.

1t | UNIT 1 | Overview

UN

IT 1

Re

se

aR

ch

From Current Research: Multiplication

Children learn skip-count lists for different multipliers (e.g., they count 4, 8, 12, 16, 20, … to multiply by four). They then count on and count down these lists using their fingers to keep track of different products. They invent thinking strategies in which they derive related products from products they know.

As with addition and subtraction, children invent many of the procedures they use for multiplication. They find patterns and use skip counting (e.g., multiplying 4 × 3 by counting “3, 6, 9, 12”). Finding and using patterns and other thinking strategies greatly simplifies the task of learning multiplication tables. Moreover, finding and describing patterns are a hallmark of mathematics. Thus, treating multiplication learning as pattern finding both simplifies the task and uses a core mathematical idea.

National Research Council. “Developing Proficiency with Whole Numbers.”

Adding It Up: Helping Students Learn Mathematics. Washington, D.C.: National Academy Press, 2001. pages 191–192.

Beckman, Sybilla. Mathematics for Elementary Teachers with Activity Manual, Addison Wesley, third edition, 2010.

Lemaire, P., and R.S. Siegler, “Four aspects of strategic change: Contributions to children’s learning of multiplication.” Journal of Experimental Psychology: General, 124, (1995): 83–97.

Mulligan, J., and M. Mitchelmore, “Young children’s intuitive models of multiplication and division.” Journal for Research in Mathematics Education, 28 (1997): 309–330.

Steffe, L. “Children’s multiplying schemes.” The Development of Multiplicative Reasoning in the Learning of Mathematics. Eds. G. Harel and J. Confrey, Albany: State University of New York Press, 1994. 3–39.

Other Useful References: Multiplication

UNIT 1 | Overview | 1U

ACTIVITY 3

ACTIVITY 4


Getting Ready to Teach Unit 1Using the Common Core Standards for Mathematical PracticeThe Common Core State Standards for Mathematical Content indicate what concepts, skills, and problem solving students should learn. The Common Core State Standards for Mathematical Practice indicate how students should demonstrate understanding. These Mathematical Practices are embedded directly into the Student and Teacher Editions for each unit in Math Expressions. As you use the teaching suggestions, you will automatically implement a teaching style that encourages students to demonstrate a thorough understanding of concepts, skills, and problems. In this program, Math Talk suggestions are a vehicle used to encourage discussion that supports all eight Mathematical Practices. See examples in Mathematical Practice 6.

Mathematical Practice 1Make sense of problems and persevere in solving them.

Students analyze and make conjectures about how to solve a problem. They plan, monitor, and check their solutions. They determine if their answers are reasonable and can justify their reasoning.

TeaCher ediTion: examples from Unit 1

MP.1 Make Sense of Problems Analyze the Problem Direct students to Problem 4 on Student Book page 34. Read aloud the problem. Then write this equation on the board. 5 × 10 = Discuss with students what each part of the equation represents.

→ What does the 5 represent? The 5 represents the number of packs of trading cards Zoe bought.

→ What does the 10 represent? The 10 represents the number of trading cards in each pack.

→ What does the box represents? The box represents the total number of trading cards Zoe bought.

→ Explain to students that the box represents the unknown number.

Lesson 7

MP.1 Make Sense of Problems Check Answers Ask students to solve Problems 7–12 on Student Book page 48. Encourage students to use the 5s shortcut method to check their answers when finding products for multipliers greater than 5.

Lesson 10

Mathematical Practice 1 is integrated into Unit 1 in the following ways:

Make Sense of Problems Analyze the Problem Check Answers

1V | UNIT 1 | Overview

ACTIVITY 1

ACTIVITY 3

UN

IT 1

MA

TH

BA

CK

GR

OU

ND

Mathematical Practice 2Reason abstractly and quantitatively.

Students make sense of quantities and their relationships in problem situations. They can connect diagrams and equations for a given situation. Quantitative reasoning entails attending to the meaning of quantities. In this unit, this involves connecting symbols, diagrams, and words to basic multiplications and divisions to build fluency.

TeACHeR eDITION: examples from Unit 1

MP.2 Reason Abstractly and Quantitatively Connect Symbols and Words As a class, continue to write multiplication equations up to 50 = 10 × 5. Have students, in unison, read each equation after writing it: “15 equals 3 times 5,” “20 equals 4 times 5,” and so on. After all the equations are written, the board should appear as shown below.

Lesson 1

MP.2 Reason Abstractly and Quantitatively Connect Diagrams and Equations Remind students that, in the last lesson, they found that 3 groups of 5 have the same total as 5 groups of 3. Tell them that arrays can help them see why this is true. Draw and label a picture of 3 groups of 5

3 × 5 = 15

Next, rearrange the groups to form rows. Make the drawing on the right next to the first drawing.

3 × 5 = 15 3 × 5 = 15

Lesson 3


Connect Symbols and WordsReason Abstractly and

Quantitatively

Reason Quantitatively Connect Diagrams and Equations

UNIT 1 | Overview | 1W

ACTIVITY 2ACTIVITY 3


Mathematical Practice 3Construct viable arguments and critique the reasoning of others.

Students use stated assumptions, definitions, and previously established results in constructing arguments. They are able to analyze situations and can recognize and use counterexamples. They justify their conclusions, communicate them to others, and respond to the arguments of others.

Students are also able to distinguish correct logic or reasoning from that which is flawed, and—if there is a flaw in an argument—explain what it is. Students can listen to or read the arguments of others, decide whether they make sense, and ask useful questions to clarify or improve the arguments.

MATH TALK is a conversation tool by which students formulate ideas and analyze responses and engage in discourse. See also MP.6 Attend to Precision.

TeaCher ediTion: examples from Unit 1

MP.3, MP.6 Construct Viable arguments/Critique reasoning of others Puzzled Penguin Ask a volunteer to read the Puzzled Penguin problem on Student Book page 25. Give students an opportunity to complete Problem 5 and write an answer to Puzzled Penguin. Remind students they need to describe Puzzled Penguin’s error and explain why it is wrong. In this exercise, Puzzled Penguin used an incorrect factor in the related multiplication.

Lesson 4

MP.3, MP.6 Construct Viable arguments/Critique reasoning of others Puzzled Penguin After students read the letter from Puzzled Penguin, give them time to write a response. Then allow several students to share ideas about what Puzzled Penguin did wrong. Encourage them to make drawings on the board if it helps them explain.See the Math Talk in Action in the side column for a sample classroom dialogue.

Lesson 12


Puzzled Penguin Critique the Reasoning of Others Justify Conclusions

1X | UNIT 1 | Overview

DO NOT EDIT--Changes must be made through “File info” CorrectionKey=B

ACTIVITY 2

ACTIVITY 2

UN

IT 1

MA

TH

BA

CK

GR

OU

ND

Mathematical Practice 4Model with mathematics.

Students can apply the mathematics they know to solve problems that arise in everyday life. This might be as simple as writing an equation to solve a problem. Students might draw diagrams to lead them to a solution for a problem.

Students apply what they know and are comfortable making assumptions and approximations to simplify a complicated situation. They are able to identify important quantities in a practical situation and represent their relationships using such tools as diagrams, tables, graphs, and formulas.


MP.4 Model with Mathematics Draw a Diagram Encourage students to make very simple drawings. Make sure various drawings and solution methods are presented. Below are two possible drawings. The second is a form of the Equal Shares drawings students made in Lesson 2. If no one makes drawings like these, you may want to suggest them yourself.

6 × 2 = 12 6 × 2 = 12

2 2 2 2 2 2

Lesson 3

MP.4 Model with Mathematics Write an Equation Circulate as students work and observe the types of drawings they make and equations they write. Encourage them to make drawings that don’t show each individual item, such as Equal Shares drawings. Make sure a variety of drawings and solution methods are presented.

Lesson 9


Draw a DiagramMathBoard

Model with MathematicsWrite an Equation

Draw an Array

UNIT 1 | Overview | 1Y

ACTIVITY 2ACTIVITY 3


Mathematical Practice 5Use appropriate tools strategically.

Students consider the available tools and models when solving mathematical problems. Students make sound decisions about when each of these tools might be helpful. These tools might include paper and pencil, a straightedge, a ruler, or the MathBoard. They recognize both the insight to be gained from using the tool and the tool’s limitations. When making mathematical models, they are able to identify quantities in a practical situation and represent relationships using modeling tools such as diagrams, grid paper, tables, graphs, and equations

Modeling numbers in problems and in computations is a central focus in Math Expressions lessons. Students learn and develop models to solve numerical problems and to model problem situations. Students continually use both kinds of modeling throughout the program.

Teacher ediTion: examples from Unit 1

MP.5 Use appropriate Tools Class MathBoard Model the steps of this activity on the Class MathBoard as students follow along on their MathBoards. The completed board is reproduced on Student Activity Book page 33. You can use it to facilitate a summary discussion of the patterns in 10s multiplications. Students can draw line segments separating sequential groups of 10, up to 100, on the Number Path and write the totals so far next to each group. As they work, have students say in unison, “1 group of 10 is 10,” “2 groups of 10 are 20,” and so on.

Lesson 7

Explore Multiplication as Area PA IRS

MP.5 Use appropriate Tools Square Tiles Distribute Inch Grid Paper (TRB M8) for Student Pairs to cut out and use as tiles. Hold up one of the tiles. Tell students this small square tile is called 1 square unit. Since it measures 1 inch on a side it is also called 1 square inch. This unit square can be used to measure sizes of other squares and rectangles.Ask Student Pairs to arrange three tiles in a line with no gaps or overlaps on their MathBoards to form a rectangle.

Lesson 11


MathBoardClass MathBoardClass Multiplication TableSquare Tiles

Fingers5s ChartSignature Sheet

Study SheetsCheck SheetsFast Array Drawings

1Z | UNIT 1 | Overview

ACTIVITY 1

ACTIVITY 1

ACTIVITY 2

ACTIVITY 2

ACTIVITY 2

ACTIVITY 2

UN

IT 1

MA

TH

BA

CK

GR

OU

ND

Mathematical Practice 6Attend to precision.

Students try to communicate precisely to others. They try to use clear definitions in discussion with others and in their own reasoning. They state the meaning of the symbols they choose. They are careful about specifying units of measure to clarify the correspondence with quantities in a problem. They calculate accurately and efficiently, express numerical answers with a degree of precision appropriate for the problem context. Students give carefully formulated explanations to each other.


MP.6 Attend to Precision As students look at the top of Student Activity Book page 6, point out and discuss the meaning of the important vocabulary words in this lesson: equation, multiplication, factor, and product. You may want to add these new vocabulary words to your chart paper vocabulary list. The three symbols for multiplication are also shown. Have students work individually to complete Exercises 1–12.

Lesson 1

MP.6 Attend to Precision Describe a Method Next, discuss how students have been using the Strategy Cards to study.

• As you have used the cards, have you discovered any good strategies for finding and learning your multiplications and divisions?

Allow several students to share their ideas. If they do not mention the two points below, bring them up yourself.

Lesson 13

MATH TALK Ask students to describe what they see down the column. Make sure the following points are discussed:

→ The column shows the 5s multiplications from 1 • 5 = 5 to 10 • 5 = 50.

→ In all the multiplications in the column, the 5 is the second factor.

→ The large, bold numbers are the products, which are also the “5s count-bys” (the numbers we say when we count by 5).

Lesson 6

MATH TALKin ACTION

Let’s think about other patterns you found in the 3s count-bys and equations. Who would like to share another pattern they found?

Jose: I think I see a pattern in the products. The tens digits are 0, 0, 0, 1, 1, 1, 2, 2, 2 and then a 3. They look like they go in order.

Shayna: I see that too, and I see another pattern. If you add 3 + 27, you get 30. If you add 6 + 24, you also get 30.

Larry: And if you add 9 + 21 or 12 + 18, that’s 30 also!

Lesson 10


Attend to PrecisionExplain a Method

Puzzled PenguinDescribe a Method

Explain a Representation

78319.U4.L13.02.TG

9

3 27

3 9

27

9 3

27 9

18 27

3 6 9

12 15

18 21 24 27

UNIT 1 | Overview | 1AA

ACTIVITY 4

ACTIVITY 2


Mathematical Practice 7Look for structure.

Students analyze problems to discern a pattern or structure. They draw conclusions about the structure they have identified.

Teacher ediTion: examples from Unit 1

MP.7 Look for Structure Identify Relationships Ask students to look at the top of Student Book page 52. Discuss how the multipliers of 3 are related. They should see that the multiplier in the equation for the large rectangle is the total of the multipliers in the equations for the small rectangles. Also be sure students understand how to write the multiplication and addition of the smaller rectangles as one equation as shown in the example of the Distributive property. Then have students complete Exercises 7–11 on Student Book page 52. Discuss the answers.

Lesson 11

MP.7 Look for Structure Identify Relationships Each multiplication card can be paired with a division card that has the same count-by lists and the same Fast Array drawing. Here is one such pair:

9

3 27

3 9

27

9 3

279

1827

369

1215

18212427

39

93 369

1215

18212427

91827

9

3 27

27 27

Be sure to mention this last point. Talk about the relationships in a fast array. Make sure students understand that for any factor-factor-product combination, they can write two multiplication equations and two division equations.

Lesson 13


Identify Relationships Look for Structure Use Structure

1BB | UNIT 1 | Overview

DO NOT EDIT--Changes must be made through “File info” CorrectionKey=B

ACTIVITY 4

ACTIVITY 2

Class Activity

(Yellow Notebook Paper)

(Yellow Notebook Paper)

1-19

© H

ough

ton

Miff

lin H

arco

urt

Pub

lishi

ng C

ompa

ny •

Im

age

Cre

dits

: (G

irls

play

grou

nd)

©D

igita

l Vis

ion/

Get

ty I

mag

es; (

Ups

ide

dow

n) ©

Pho

toA

lto/G

etty

Im

ages

; (R

ecyc

ling)

©Le

land

Bob

be/

Dig

ital V

isio

n/G

etty

Im

ages

; (T

ire s

win

g) ©

Com

stoc

k/G

etty

Im

ages

; (C

hild

ren

runn

ing)

©O

JO I

mag

es/G

etty

Im

ages

; (C

hild

ren

smili

ng)

©G

olde

n P

ixel

s LL

C/A

lam

y Im

ages

; (R

aisi

ng h

ands

) ©

Sto

ckby

te/A

lam

y Im

ages

; (B

low

ing

bubb

les)

©S

ami S

arki

s/G

etty

Im

ages

; (Z

ooke

eper

) ©

Imag

e S

ourc

e/G

etty

Im

ages

; (P

ushi

ng b

ox)

©P

eter

Mul

ler/

Cul

tura

/Get

ty I

mag

es

► PATH toFLUENCY Math and Hobbies

A hobby is something you do for fun. Owen’s hobby is photography. He took pictures on a field trip and displayed them on a poster.

Solve.

1. How many photos did Owen display on the poster? Explain the different strategies you can use to find the answer. Write an equation for each.

2. What other ways could Owen have arranged the photos in an array on the poster?

Name Date

24 photos; Possible answer: Count by 4s: 4, 8, 12,

2 rows of 12, 12 rows of 2, 1 row of 24, 24 rows

16, 20, 24; Addition: 4 + 4 + 4 + 4 + 4 + 4 = 24;

of 1, 3 rows of 8, 8 rows of 3, 6 rows of 4

Multiply: 6 × 4 = 24; Multiply and add: 6 × (2 + 2) =

(6 × 2) + (6 × 2) = 12 + 12 = 24

UNIT 1 LESSON 19 Focus on Mathematical Practices 85

3_MNLESE824468_U01L19.indd 85 14/02/12 11:04 AM

Class Activity

HobbiesDancing

Photography

Games

Reading

Each stands for 2 third graders.

GamesEight third graders said

games.

DancingFour third graders

said dancing.

PhotographyEight more than dancing

said photography.

ReadingSix less than photography said

reading.

1-19

© H

oughton Mifflin H

arcourt Publishing C

ompany • Im

age Credits: (G

irl dancing) ©Jupiterim

ages/Getty Im

ages; (Cam

era) ©P

hotoDisc/G

etty Images (V

ideo game) ©

Jose Luis Pelaez Inc/G

etty Images

► PATH toFLUENCY What is Your Hobby?

Carina asked some third graders, “What is your hobby?” The answers are shown under the photos.

3. Use the information above to complete the chart below.

What is Your Hobby?

4. Use the chart to complete the pictograph below.

5. How many third graders answered Carina’s question?

Hobby Number of Students

Dancing

Photography

Games

Reading

Name Date

30

4

12

8

6

86 UNIT 1 LESSON 19 Focus on Mathematical Practices

3_MNLESE824468_U01L19.indd 86 16/02/12 3:33 PM

UN

IT 1

MA

TH

BA

CK

GR

OU

ND

Mathematical Practice 8Look for and express regularity in repeated reasoning.

Students use repeated reasoning as they analyze patterns, relationships, and calculations to generalize methods, rules, and shortcuts. As they work to solve a problem, students maintain oversight of the process, while attending to the details. They continually evaluate the reasonableness of their intermediate results.


MP.8 Use Repeated Reasoning Ask what patterns students see in the count-bys and equations. Two common patterns are:

→ The sums of the digits of the count-bys follow the pattern 3, 6, 9, 3, 6, 9, . . . .

→ The products follow the pattern odd, even, odd, even, . . . .

For additional patterns students may find, see the Math Talk in Action in the side column for a sample classroom discussion.

Lesson 10

MP.8 Use Repeated Reasoning Generalize Have the students discuss the multiplication properties and division rules for 1 and 0. Ask for volunteers to come to the board and draw a picture to show each rule. Then draw the pictures and write the expressions shown below on the classroom board. Use them to review the Associative Property by asking questions like the ones on the previous page.

78319.U4.L14.08.TG 78319.U4.L14.09.TG

Lesson 15

Mathematical Practice 8 is integrated into Unit 1 in the following ways:Use Repeated Reasoning Generalize

Focus on Mathematical Practices Unit 1 includes a special lesson that involves solving real world problems and incorporates all 8 Mathematical Practices. In this lesson students describe strategies for multiplying and make a pictograph.

STUDeNT eDITION: LeSSON 19, PAGeS 85–86


Math Expressions VOCABULARY

•EqualSharesdrawing•count-bys•FastArraydrawing•EqualGroupsdrawing

See the Teacher Glossary.

Getting Ready to Teach Unit 1Learning Path in the Common Core StandardsInthisunitandUnit2,studentsparticipateintestingandgoaldirectionpracticeinschoolandathome.Avarietyofpracticesheets,checksheetsandroutineshelpstudentslearnthebasicmultiplicationsanddivisionsandhelpstudentskeeptrackoftheirprogress.Thisisalsoanimportantopportunityforstudentstobecomeself-directedandorganized.

Studentsalsolearnhowtousedifferentstrategiesformultiplyinganddividing,howmultiplicationanddivisionarerelated,andhowtousemathdrawingsandequationstorepresentandsolvewordproblems.

Visualmodelsandrealworldsituationsareusedthroughouttheunittoillustrateimportantmultiplicationanddivisionconcepts.

Help Students Avoid Common ErrorsMath Expressionsgivesstudentsopportunitiestoanalyzeandcorrecterrors,explainingwhythereasoningwasflawed.

InthisunitweusePuzzledPenguintoshowtypicalerrorsthatstudentsmake.StudentsenjoyteachingPuzzledPenguinthecorrectway,whythiswayiscorrectandwhyPuzzledPenguinmadetheerror.CommonerrorsarepresentedinPuzzledPenguinfeaturesinthefollowinglessons:

→ Lesson 4:Usesanincorrectfactorintherelatedmultiplicationtosolveadivision

→ Lesson 7: Writesawordproblemthatcannotbesolvedusing40÷10

→ Lesson 12:Combinesmultiplicationequationsbyaddingbothfactors

InadditiontoPuzzledPenguin,thereareothersuggestionslistedintheTeacherEditiontohelpyouwatchforsituationsthatmayleadtocommonerrors.AsapartoftheUnitTestTeacherEditionpages,youwillfindacommonerrorandprescriptionlistedforeachtestitem.

1DD|UNIT1|Overview

UN

IT 1

MA

TH

BA

CK

GR

OU

ND

Basic Multiplications and Divisions

Lessons

1 2 3 4 5 6 7 8 9 10

11 12 13 14 15 16 17 18

Path to Fluency The basic facts in Unit 1 are introduced and practiced from easiest to hardest: 5s, 2s, 9s, 10s, 3s, 4s, 1s, and 0s. The harder facts 6s, 7s, and 8s are introduced and practiced in Unit 2.

When first learning the facts, students use count bys with diagrams to give the multiplication meaning and to connect symbols with words and equations. Next they look for patterns, rules, or strategies to make learning that fact easier. Students practice and check fluency of the facts using a routine with a variety of tools such as Study Sheets, Check Sheets and Strategy Cards.

Students should work with 1 fact at a time: studying, practicing, and checking recall. Then students should combine checking previously learned facts with new facts. Multiplication and division are taught together to make the process meaningful and faster. Each division is just finding an unknown factor. Some students may not learn all the basic multiplications and divisions this year.

Practice Materials and Routines for Learning the Basic Multiplications and Divisions

Lessons

1 2 3 4 5 6 7 8 9 10

11 12 13 14 15 16 17 18

Study Plans Each day students will fill out a study plan at the top of a homework page, indicating which basic multiplications and divisions he or she will study that evening at home. At first it contains just the count-bys, multiplications, and divisions for the new number introduced. Later it will be the new number and any count-bys, multiplications, or divisions they do not recall when tested by their partner during the Check Up.

When a student has finished practicing/studying, the Homework Helper should sign the study plan.

3-4 Name Date1–1

Study Plan

Homework Helper

5s count bys5s multiplications

from the ProgreSSionS For the Common Core State StandardS on oPerationS and algebraiC thinking

the meaning of multiplication

Students focus on understanding

the meaning and properties of

multiplication and division and on

finding products of single-digit

multiplying and related quotients.

These skills and understandings

are crucial; students will rely on

them for years to come as they

learn to multiply and divide with

multidigit whole numbers and to

add, subtract, multiply and divide

with fractions and with decimals in

later grades.

UNIT 1 | Overview | 1EE


Practice Charts In Lessons 1, 5, 7, 8, 10, 12, and 15, each time a new number is introduced, a student’s Homework page will include a practice chart. See Teacher Edition page 8 for an explanation of how to practice the count-bys, multiplications, and divisions by covering the answers with a pencil and sliding it.

1 5 5

2 5 10

3 5 15

4 5 20

5 5 25

6 5 30

7 5 35

8 5 40

9 5 45

10 5 50

9 5 45

5 5 25

2 5 10

7 5 35

4 5 20

6 5 30

10 5 50

8 5 40

1 5 5

3 5 15

In Order Mixed Up

5s

Study Sheets Students use both a class and home study sheet, which includes 3 or 4 practice charts on one page. This sheet can be used to practice all the count-bys, multiplications, and divisions or to practice just the ones a student doesn’t know.

p U d e x i M s y b - t n u o C p U d e x i M

s 5

2 5 0 1

9 5 5 4

1 5 5

5 5 5 2

7 5 5 3

3 5 5 1

0 1 5 0 5

6 5 0 3

4 5 0 2

8 5 0 4

0 1 5 2

5 3 5 7

0 5 5 0 1

5 5 1

0 2 5 4

5 1 5 3

0 3 5 6

0 4 5 8

5 2 5 5

5 4 5 9

1 5 5

2 5 0 1

3 5 5 1

4 5 0 2

5 5 5 2

6 5 0 3

7 5 5 3

8 5 0 4

9 5 5 4

0 1 5 0 5

p U d e x i M s y b - t n u o C p U d e x i M

s 2

7 2 4 1

1 2 2

3 2 6

5 2 0 1

6 2 2 1

8 2 6 1

2 2 4

0 1 2 0 2

4 2 8

9 2 8 1

0 2 2 0 1

2 2 1

6 2 3

6 1 2 8

2 1 2 6

4 2 2

0 1 2 5

8 2 4

4 1 2 7

8 1 2 9

1 2 2

2 2 4

3 2 6

4 2 8

5 2 0 1

6 2 2 1

7 2 4 1

8 2 6 1

9 2 8 1

0 1 2 0 2

A t e e h S y d u t S e m o H

A routine is built into this program so each day at school and at home students practice count-bys, multiplications, and divisions and are tested when ready. When a student is ready for a Check Up on a number, a student’s partner or Homework Helper tests the student marking any missed exercises lightly with a pencil. If a student gets all the answers in a column correct, the partner or Homework Helper signs the Signature Sheet or the Home Signature Sheet.

Name Date 1–3

Count-Bys Homework Helper

Multiplications Homework Helper

Divisions Homework Helper

0

1

Home Signature Sheet

from the PRogReSSionS foR the CoMMon CoRe State StandaRdS on oPeRationS and algeBRaiC thinking

Building fluency Fluency may

be reached by becoming fluent

for each number (e.g. the 2s, the

5s, etc) and then extending the

fluency to several, then all numbers

mixed together. Organizing

practice so that it focuses most

heavily on understood but not

yet fluent products and unknown

factors can speed learning.

1FF | UNIT 1 | Overview

DO NOT EDIT--Changes must be made through "File info" CorrectionKey=B

UN

IT 1

MA

TH

BA

CK

GR

OU

ND

Signature Sheet When a student gets all the answers in a column on the Study Sheet correct, the partner or Homework Helper signs the appropriate column on the Signature Sheet. When signatures are in all the columns, the student turns the Signature Sheet in to the teacher so there is a record that the multiplications and divisions have been mastered.

Check Sheet In Lessons 5, 8, 9, 12, and 17, when a student has signatures for a fact they use an answer strip to complete a Check Sheet for that fact and place it in their Fluency Progress Folder. Both the class and home check sheets include columns of 20 multiplications and divisions in mixed order.

Name Date

UNIT 4 LESSON 5 Multiply and Divide with 2 221

Check Sheet 1: 5s and 2s

5sMultiplications

2 5 10

5 • 6 30

5 * 9 45

4 5 20

5 • 7 35

10 * 5 50

1 5 5

5 • 3 15

8 * 5 40

5 5 25

5 • 8 40

7 * 5 35

5 4 20

6 • 5 30

5 * 1 5

5 10 50

9 • 5 45

5 * 2 10

3 5 15

5 • 5 25

5sDivisions

30 / 5 6

5 5 1

15 / 5 3

50 5 10

20 / 5 4

10 5 2

35 / 5 7

40 5 8

25 / 5 5

45 / 5 9

20 5 4

15 / 5 3

30 5 6

25 / 5 5

10 5 2

45 / 5 9

35 5 7

50 5 10

40 / 5 8

5 5 1

2sDivisions

8 / 2 4

18 2 9

2 / 2 1

16 2 8

4 / 2 2

20 2 10

10 / 2 5

12 2 6

6 / 2 3

14 / 2 7

4 2 2

2 / 2 1

8 2 4

6 / 2 3

20 2 10

14 / 2 7

10 2 5

16 2 8

12 / 2 6

18 2 9

2sMultiplications

4 2 8

2 • 8 16

1 * 2 2

6 2 12

2 • 9 18

2 * 2 4

3 2 6

2 • 5 10

10 * 2 20

2 7 14

2 • 10 20

9 * 2 18

2 6 12

8 • 2 16

2 * 3 6

2 2 4

1 • 2 2

2 * 4 8

5 2 10

7 • 2 14

Check SheetAnswer Strip

Date:

Name Date

UNIT 4 LESSON 5 Multiply and Divide with 2 221

Check Sheet 1: 5s and 2s

5sMultiplications

2 5 10

5 • 6 30

5 * 9 45

4 5 20

5 • 7 35

10 * 5 50

1 5 5

5 • 3 15

8 * 5 40

5 5 25

5 • 8 40

7 * 5 35

5 4 20

6 • 5 30

5 * 1 5

5 10 50

9 • 5 45

5 * 2 10

3 5 15

5 • 5 25

5sDivisions

30 / 5 6

5 5 1

15 / 5 3

50 5 10

20 / 5 4

10 5 2

35 / 5 7

40 5 8

25 / 5 5

45 / 5 9

20 5 4

15 / 5 3

30 5 6

25 / 5 5

10 5 2

45 / 5 9

35 5 7

50 5 10

40 / 5 8

5 5 1

2sDivisions

8 / 2 4

18 2 9

2 / 2 1

16 2 8

4 / 2 2

20 2 10

10 / 2 5

12 2 6

6 / 2 3

14 / 2 7

4 2 2

2 / 2 1

8 2 4

6 / 2 3

20 2 10

14 / 2 7

10 2 5

16 2 8

12 / 2 6

18 2 9

2sMultiplications

4 2 8

2 • 8 16

1 * 2 2

6 2 12

2 • 9 18

2 * 2 4

3 2 6

2 • 5 10

10 * 2 20

2 7 14

2 • 10 20

9 * 2 18

2 6 12

8 • 2 16

2 * 3 6

2 2 4

1 • 2 2

2 * 4 8

5 2 10

7 • 2 14

Targets In Lesson 6, students begin using their Targets with a Multiplication Table to practice multiplications and divisions they have studied so far and to see inverse and commutative relationships. By covering the Target circle, students can check on whether they know the product for two factors. By covering one end of the Target, they can check on a related division.

Students can take one Target home and use it with the Multiplication Table on the inside back cover of the Homework and Remembering book or with a copy of TRB M12. They can use the Targets with a multiplication table or scrambled multiplication table throughout the year to maintain fluency with basic multiplications and divisions.

×

10

10

0

10

20

30

40

50

60

70

80

90

100

0 1 2 3 4 5 6 7 8 9

0 0 0 0 0 0 0 0 0 0 0

1 0 1 2 3 4 5 6 7 8 9

2 0 2 4 6 8 10 12 14 16 18

3 0 3 6 9 12 15 18 21 24 27

4 0 4 8 12 16 20 24 28 32 36

5 0 5 10 15 20 25 30 35 40 45

6 0 6 12 18 24 30 36 42 48 54

7 0 7 14 21 28 35 42 49 56 63

8 0 8 16 24 32 40 48 56 64 72

9 0 9 18 27 36 45 54 63 72 81

0 10 20 30 40 50 60 70 80 90

Signature Sheet

Count-BysPartner

MultiplicationsPartner

DivisionsPartner

Multiplications Check Sheets

Divisions Check Sheets

5s 1: 1:

2s 1: 1:

10s 2: 2:

9s 2: 2:

from THe progreSSionS for THe Common Core STaTe STandardS on operaTionS and algebraiC THinking

The relationship between

multiplication and division

The extensive work relating

multiplication and division

means that division can be

solved by thinking of the related

multiplication. Multiplication and

division can be learned at the same

time and can reinforce each other.

UNIT 1 | Overview | 1GG


Strategy Cards In Lesson 11, students are introduced to the Strategy Cards. Students can use the cards to practice multiplications and divisions. They should sort their cards into three piles: those with answers they know quickly, those with answers they know slowly, and those with answers they don’t know yet. As they practice, they may be able to move some of the cards into other piles.

There is a home set of Strategy Cards on Homework and Remembering pages 41–66. As part of their homework, students should cut out the cards and use them to study.

Sample Multiplication Card Sample Division Card

78319.U4.L10.01.TG

3 9 9 3

9

3 27

3 9

27

9 3

27 9

18 27

3 6 9

12 15

18 21 24 27

78319.U4.L10.02.TG

3 27

27 3 3

9 27 9

3 27 3 6 9

12 15

18 21 24 27

9 18 27

9

3 27

from the PRogReSSionS foR the CoMMon CoRe State StanDaRDS on oPeRationS anD algeBRaiC thinking

Studying facts Facts should not

be instilled divorced from their

meanings, but rather the outcome

of a carefully designed learning

process that heavily involves the

interplay of practice and reasoning.

1HH | UNIT 1 | Overview

UN

IT 1

MA

TH

BA

CK

GR

OU

ND

Dashes When students have successfully completed all Check Sheets, they complete a Dash and place it in their Fluency Progress folder. The Dashes are 20 multiplications or divisions in mixed order. Dashes with the same facts in a different order are included so students can time themselves and try to improve their speed on the next Dash with the same facts. For example, 9A, 9B, 9C, and 9D have the same facts but are in a different order. A dash record sheet is included for students to record their speed and accuracy.

from the progressions for the Common Core state stanDarDs on operations anD algebraiC thinking

Checking fluency Fluency in Grade 3

involves a mixture of just knowing

some answers from knowing other

answers, knowing some answers

from pattern (e.g., "multiplying

1 yields the same number"), and

knowing some answers from the

use of strategies. It is important to

push sensitively and encouragingly

toward fluency of the designated

numbers in each lesson, recognizing

that fluency will be a mixture of

these kinds of thinking which may

differ across students.

Dash 9A 2s, 5s, 9s, 10s

Multiplications

Dash 10A 2s, 5s, 9s, 10s

Divisions

Dash 11A 0s, 1s, 3s, 4s

Multiplications

Dash 12A 1s, 3s, 4s Divisions

a. 9× 9 = a. 30 / 5 = a. 0 × 4 = a. 10 / 1 =

b. 4 * 5 = b. 18 ÷ 2 = b. 4 * 9 = b. 40 ÷ 4 =

c. 10 • 3 = c. 40 / 5 = c. 3 • 8 = c. 12 / 3 =

d. 3 × 9 = d. 6 ÷ 2 = d. 3 × 0 = d. 6 ÷ 3 =

Dash 9B 2s, 5s, 9s, 10s

Multiplications

Dash 10B 2s, 5s, 9s, 10s

Divisions

Dash 11B 0s, 1s, 3s, 4s

Multiplications

Dash 12B 1s, 3s, 4s Divisions

a. 6× 2 = a. 18 / 2 = a. 7×1 = a. 2 / 1 =

b. 9 • 4 = b. 25 ÷ 5 = b. 0 • 6 = b. 28 ÷ 4 =

c. 8 * 5 = c. 70 / 10 = c. 4 * 4 = c. 3 / 3 =

d. 1×10 = d. 54 ÷ 9 = d. 7×3 = d. 1 ÷ 1 =

e. 2 • 7 = e. 50 / 5 = e. 3 • 1 = e. 40 / 4 =

UNIT 1 | Overview | 1II


from the PRogRessions foR the CoMMon CoRe state standaRds on oPeRations and algeBRaiC thinking

Patterns and strategies Mastering

this material, and reaching fluency

in single digit multiplications

and related divisions with

understanding, may be quite time

consuming because there are no

general strategies for multiplying

or dividing all single-digit numbers

as there are for addition and

subtraction. Instead, there are

many patterns and strategies

dependent upon specific numbers.

So it is imperative that extra time

and support be provided if needed.

Fluently Multiply and Divide within 100

Lessons

6 9 14 18

fluency lessons These lessons are included to reinforce facts learned, to give more time to students who need it to study and practice a fact, and to review strategies.

Independent activities are also included in these lessons.

independent activities Students who do not need extra teaching or practice for fluency after completing a Check Sheet may choose from one of the activity options shown below.

→ go ahead

student Pairs go ahead to the next fact, using the same plan as for 2s and 5s: study, partner check, complete Check Sheet for that fact.

→ go for speed

student Pairs go for speed. They can use an answer strip from the back of the Student Activity Book or Activity Workbook, or Check Sheet Answer Strips (TRB M7) to complete a Check Sheet again for one of the facts and record the time it takes to complete it. Then complete it again using another answer strip and compare it with the first time.

→ invent a game or Play a game

student Pairs can invent a game to learn or practice multiplications and divisions. Students can also play the games Solve the Stack and High Card Wins introduced in Lesson 13.

→ Write a Word Problem

student Pairs can write word problems that can be solved using a multiplication or division they know. Then exchange to solve.

→ invent Rhymes or songs

student Pairs write rhymes or songs that will help everyone remember the hardest multiplication and division facts and lead the class to practice them.

1JJ | UNIT 1 | Overview

UN

IT 1

MA

TH

BA

CK

GR

OU

ND

from the Progressions for the Common Core state standards on oPerations and algebraiC thinking

equal groups In Equal Groups, the

roles of the factors differ. One factor

is the number of objects in a group

(like any quantity in addition and

subtraction situations), and the other

is a multiplier that indicates the

number of groups. So, for example,

4 groups of 3 objects is arranged

differently than 3 groups of 4

objects. Thus there are two kinds

of division situations depending on

which factor is the unknown (the

number of objects in each group or

the number of groups).

Strategies for Multiplying and Dividing

Lessons

1 3 5 7 8 10 11

12 15

identify and explain Patterns in arithmetic Students circle sequential groups of a given number (such as 4) on their Number Path and write the sequential totals. The totals show the multiplication products. Students analyze patterns they see in the count-bys for each number.

78319.U4.L12.03.TG Patterns for 0s and 1s. Students use patterns to make multiplication with 0s and 1s easy. Any number times 0 is 0. Division is not possible with 0. Any number multiplied by 1 is the original number. Any number divided by 1 is the number being divided.

Use the relationship between multiplication and division Students use their circled sequential groups on the Number Path and their knowledge of multiplication to write the related division equations. By studying the two operations together, students see that doing division is the same as finding an unknown factor in a multiplication situation.

Use drawings Students use Equal Shares, Equal Groups, and Fast Array drawings to represent known and unknown factors and products in a conceptual format and to write equations and solve problems.

78319.U4.L12.03.TG

UNIT 1 | Overview | 1KK


Use Properties of Multiplication The array model for multiplication leads students to understand the Commutative Property of Multiplication. For example, 3 rows of 5 objects results in the same number of objects as 5 rows of 3 objects. This helps students derive new facts from facts they already know.

3 × 5 = 15 5 × 3 = 15

The Associative and Distributive Properties help students build fluency with multiplication by using facts they know to find unknown products.

The Associative Property allows students to change the grouping of factors presented.

(4 × 2) × 3 = 4 × (2 × 3)

8 × 3 = 4 × 6

(4 × 2) × 3 4 × (2 × 3)

Multiplication and Area The Distributive Property allows students to break apart facts they don’t know into known facts by relating area.

7 × 3 =(5 + 2) × 3 = (5 × 3) + (2 × 3)

Add the areas of the two smaller rectangles.

5 × 3 = 15 square units

2 × 3 = 6 square units

= 21 square units

3

3

2

5

7

from the PRogRessions foR the CoMMon CoRe stAte stAndARds on oPeRAtions And AlgeBRAiC thinking

Arrays In the array situations, the

roles of the factors do not differ.

One factor tells the number of

rows in the array, and the other

factor tells the number of columns

in the situation. But rows and

columns depend on the orientation

of the array. If an array is rotated,

the rows become columns and

the columns become rows. This is

useful for seeing the Commutative

Property for Multiplication in

rectangular arrays and areas.

from the PRogRessions foR the CoMMon CoRe stAte stAndARds on oPeRAtions And AlgeBRAiC thinking

Multiplication and Area Area

problems where regions are

partitioned by unit squares are

foundational for Grade 3 standards

because area is used as a model

for single-digit multiplication

and division strategies such as

decomposing to find the sum of

two known facts.

78319.U4.L14.07.TG 78319.U4.L14.06.TG

1LL | UNIT 1 | Overview

UN

IT 1

MA

TH

BA

CK

GR

OU

ND

Represent and Solve Problems Involving Multiplication and Division

Lessons

2 3 4 9

In Math Expressions a research-based problem solving approach that focuses on problem types is used.

• Interpret the problem• Represent the situation• Solve the problem • Check that the answer makes sense

Students using Math Expressions are taught a variety of ways to represent word problems. Some are conceptual in nature (making math drawings), while others are symbolic (writing equations). Students move from using math drawings to solving problems symbolically with equations. The following are math drawings students use to represent multiplication and division word problems in this unit.

bags of lemons

Equal Groups Drawing

4 × 6 = 24 6 × 4 = 24 4 × 6 = 244 × 6 = 24 24 ÷ 4 = 6

24

Equal Shares Drawing Array Drawing Area Model

6 6 6 64

6

4 ×

Situation and Solution Equations Students are introduced to situation and solution equations in Lesson 5. Students may represent a word problem with a situation equation. A situation equation shows the order of the information. Then they may rewrite the situation equation as a solution equation. A solution equation shows the operation that can be used to solve the problem.

Situation: Rhonda divided 8 crayons equally between her twin brothers. How many crayons did each boy get?

You might write 8 ÷ 2 = as a situation equation, but you would actually think 2 × = 8 to find the answer. That is your solution equation.

Focus on Mathematical Practices

Lesson

19

The Standards for Mathematical Practice are included in every lesson of this unit. However, there is an additional lesson that focuses on all eight Mathematical Practices. In this lesson, students describe strategies for multiplying and make a pictograph.

from thE ProgrESSionS for thE Common CorE StatE StandardS on oPErationS and algEbraiC thinking

relating Equal groups and array

Situations Array situations can be

seen as Equal Group situations if

each row or column is considered

as a group. Relating Equal Group

situations to Arrays, and indicating

rows or columns within arrays, can

help students see that a corner

object in an array (or a corner

square in an area model) is not

double counted: at a given time, it

is counted as part of a row or as a

part of a column but not both.

rows and Columns Row and

column language can be difficult.

The Array problems are of the

simplest form in which a row is a

group and Equal Groups language

is used (“with 6 apples in each

row”). Such problems are a good

transition between the Equal

Groups and array situations and

can support the generalization

of the Commutative Property.

Problems in terms of “rows” and

“columns,” e.g., “The apples in the

grocery window are in 3 rows and

6 columns,” are difficult because

of the distinction between the

number of things in a row and the

number of rows.

UNIT 1 | Overview | 1MM

ReseaRch—Best PRactices Putting Research into Practice · ReseaRch—Best PRactices Putting...

Documents

Transcript of ReseaRch—Best PRactices Putting Research into Practice · ReseaRch—Best PRactices Putting...