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MULTIPLICATION THE ALGEBRA WAY ii © 2006 AIMS Education Foundation

The Distributive Property ...................................... 1

Part One—Building a Model for Multiplication ..... 6Multiplying with Tens .............................................. 7Building Rectangles .............................................. 10Picturing Rectangles ............................................. 19Writing Rectangles ................................................ 22

Part Two—Developing Understandings and Skills ....................................................... 27Display Multiplication ........................................... 28Expanding the View .............................................. 33Horizontal Multiplication ....................................... 37Picturing Multiplication ......................................... 42Interpretations ...................................................... 49

Part Three—Introducing Quadratic Expressions and Equations in Base Ten ............................. 56Filling Frames ...................................................... 57Models of Square Numbers ................................... 65Constructions Plus ................................................ 70

Part Four—Decimal Numbers and the Distributive Property ...................................... 74From Tens to Tenths ............................................. 75More From Tens to Tenths .................................... 80From Tens to Tenths—Again ................................ 81More From Tens to Tenths—Again ........................ 82Filling Frames—From Tens to Tenths ................... 83Multiplication of Fractions .................................. 84Fraction Squares .................................................. 85Fraction Factors ................................................... 87Product Pictures ................................................... 89Fraction Frames ................................................... 90The Distributive Property and the Multiplication of Mixed Numbers ................... 91Expanding on Fractions ........................................ 93

Part Five—Transitioning to Other Number Bases ................................................ 97Style Tiles ............................................................. 98Filling Par 3, 4, and 5 Frames ............................. 107Par 3, 4, and 5 Constructions ............................. 118Mystery Numbers ............................................... 127

Part Six—Moving Into Algebra .......................... 134Filling Quadrilaterals .......................................... 135Building Quadrilaterals ....................................... 143Modeling Quadrilaterals ...................................... 150Patterns of Special Squares ................................ 155Parts of Quadralaterals ....................................... 158Factors from Quadralaterals ............................... 163Factor Practice ................................................... 170Factoring Special Squares .................................. 175

Part Seven—Dealing With Integers ................... 178Combinations ..................................................... 179Positives and Negatives ...................................... 185Quadratic Equation Snapshots ........................... 191Dealing With Negatives ...................................... 195Moving Into Four Quadrants ............................... 201

MULTIPLICATION THE ALGEBRA WAY iii © 2006 AIMS Education Foundation

NCTM Standards* Addressed in Multiplication the Algebra Way

Number and OperationsUnderstand numbers, ways of representing numbers, relationships among numbers, and number systems:• Understand the place-value structure of the base-ten number system and be able to

rep re sent and compare whole numbers and decimals• Explore numbers less than 0 by extending the number line and through similar

ap pli ca tions• Develop meaning for integers and represent and compare quantities with them• Understand meanings of operations and how they relate one to another• Understand the effects of multiplying and dividing whole numbers• Use the associative and commutative properties of addition and mul ti pli ca tion and the

distributive property of multiplication over addition to simplify computations with in te gers, fractions, and decimals

• Compute fl uently and make reasonable estimates• Develop and analyze algorithms for computing with fractions, decimals, and integers and

develop fl uency in their use

AlgebraRepresent and analyze mathematical situations and structures using algebraic symbols:• Represent the idea of a variable as an unknown quantity using a letter or a symbol• Express mathematical relationships using equations• Use mathematical models to represent and understand quantitative re la tion ships• Model problem situations with objects and use representations such as graphs, tables, and

equations to draw conclusions

GeometryUse visualization, spatial reasoning, and geometric modeling to solve problems:• Use geometric models to solve problems in other areas of mathematics such as number

and measurement

Communication• Use the language of mathematics to express mathematical ideas precisely

Connections• Recognize and use connections among mathematical ideas• Understand how mathematical ideas interconnect and build on one

another to produce a coherent whole

Problem Solving• Build new mathematical knowledge through problem solving• Monitor and refl ect on the process of mathematical problem solving

Representation• Select, apply, and translate among mathematical representations to

solve problems• Use mathematical models to represent and understand quantitative

relationships

* Reprinted with permission from Principles and Standards for School Mathematics, 2000 by the Na tion al Council of Teachers of Mathematics. All rights reserved.

MULTIPLICATION THE ALGEBRA WAY v © 2006 AIMS Education Foundation

Topic Multiplication—area model

Key QuestionHow can you determine how many base-ten cubes it would take to cover your rectangle?

Learning GoalsStudents will:• learn to model multiplication by con struct ing

an array,• learn to use powers of ten to count large ar rays, and• recognize patterns of powers of tens within ar rays.

Guiding DocumentsProject 2061 Benchmarks• Add, subtract, multiply, and divide whole num bers

mentally, on paper, and with a cal cu la tor.• Express numbers like 100, 1000, and 1,000,000 as

powers of 10.

NCTM Standards 2000*• Understand the place-value structure of the base-ten

nu mer a tion system and be able to rep re sent and compare whole numbers and dec i mals

• Understand the effects of multiplying and dividing numbers

• Develop and analyze al go rithms for computing with fractions, decimals, and integers and develop fl uency in their use

• Use the associative and com mu ta tive properties of addition and mul ti pli ca tion and the dis trib u tive property of multiplication over ad di tion to simplify com pu ta tions with integers, fractions, and decimals

• Use mathematical models to represent and understand quan ti ta tive relationships

• Model problem sit u a tions with objects and use rep re sen ta tions such as graphs, tables, and equa tions to draw conclusions

• Select, apply, and translate among math e mat i cal rep re sen ta tions to solve problems

MathMultiplicationPlace valueEstimation

Integrated ProcessesObservingComparing and contrastingGeneralizing

MaterialsBase-ten blocksButcher paper, cm grid paper, or sample rect an gles

Background Information This investigation has stu dents model the mul ti pli ca tion process in an area ar ray. Prior experience will al low some students to move more quickly than oth ers. Students just be ing in tro duced to mul ti pli ca tion will con tin ue with the con crete models for an ex tend ed time. Older students will of ten move directly to the visual record and quickly to a numeric al go rithm once the model helps them make sense of prior skills. In the area model, the two factors give the di men sions of the rectangle, and the number of squares with in the rect an gle—the area—gives the product. If a rect an gle is made from base-ten ma te ri als, the squares can be count ed rapidly since a fl at covers 100 squares, a stick 10 squares, and a cube one square. It is ev i dent that to count the num ber of squares quick ly, the rect an gle should be covered with the fewest pieces. This makes it necessary to start with the big gest piece possible. When one fac tor uses the tens place, sticks will be in volved. When both fac tors have tens in them, fl ats can be uti lized. By be ing con sis tent with where rect an gles are be gun, the pat terns become easier to rec og nize and communicate. Students may choose a number of ways to record their ex pe ri ence and may switch from one to another as they become more fa mil iar with the process. Young students ini tial ly like to make a record on a rectangle that is identical to the sample they receive. They quick ly move on to either recording it on a reduced grid; mak ing a box, line, and dot sketch; or de vel op ing a rect an gle marked into re gions.

By building and representationally re cord ing a num ber of rectangles, students will dis cov er some very

10

10 8

2

10 x 10 = 100 8 x 10 = 80

10 x 2 = 20 8 x 2 =16 1002080

+ 16216

18x 12216

MULTIPLICATION THE ALGEBRA WAY 10 © 2006 AIMS Education Foundation

2

10

10 3

10 x 10 =100

10 x 2 = 20

3 x

10 =

30

3 x 2 = 6

use ful patterns. Vi su al ly they will see that fl ats form a rect an gle in the lower left corner. The sticks form rect an gles in two regions—a rectangle of sticks placed hor i zon tal ly in the upper left corner, and a rect an gle of sticks placed ver ti cal ly in the lower right corner. The cubes form a rect an gle in the upper right corner. As num bers are add ed to these representations, nu mer ic pat terns can be iden ti fi ed. The fl ats in the low er left cor ner are tens by tens products. The two regions of sticks are ones by tens products. The cubes in the upper right corner are the products of the ones place.

This visual approach provides a very powerful way for many students to make sense of mul ti pli ca tion pro cess es. Some visual learners begin build ing the rect an gles in their minds and only record the products of each region (the partial products) to sum. Other stu dents become so confi dent of the patterns that they can make box, line, and dot sketch es or a rectangle with regions much faster than fi nding the solution with tra di tion al al go rithms.

Management1. Initially students will want to trace their solutions onto

their sample rectangles. Depending on the level of the students, they may choose to make sim ple box, stick, and dot sketch es for their records.

2. The sample rectangles provided are limited by the print space on a page. Larger rectangles can be made by tracing rectangles onto butcher paper, centimeter grid paper that is available in a roll, or by piecing stan dard sheets together. These can be laminated for re peat ed use.

Procedure1. Give students outlines of rectangles (those pro vid ed

or teach er generated) and discuss strat e gies for quick ly de ter min ing how many unit cubes would be needed to cover the rectangle.

2. Provide the students with base-ten materials and direct them to cover their rectangles. Inform them that in order to make comparisons between groups, they should cover the rectangles with the ma te ri als using the following guidelines:• Fill from the lower left corner where the x is in

the circle.• Use the fewest number of pieces by fi lling with

fl ats fi rst, sticks second, and cubes last.

3. Have each student make an individual record of the group’s solution on the sample rectangle sheets or with a sketch.

4. Working with their group, direct the students to determine a way to count the squares quickly us ing their record or the model. Have them record the num ber of cubes that would be needed to cov er their rect an gle.

5. When the students have completed a number of rectangles, discuss as a class:• their counting methods and• patterns of arrangement of fl ats, sticks, and cubes.

6. Direct the students to return to their record and add the following numeric information:• Record the dimensions of the rect an gle with tens

and ones by the respective pieces of the bottom and left side.

• Within each region, record the di men sions and prod uct of that region.

• Within a blank margin, record and fi nd the sum of all the regions’ products.

• Within a blank margin, record the num ber of cubes and the rectangle’s di men sions as a mul ti pli ca tion problem.

7. While the students refer to their records, dis cuss what visual and numeric patterns they observe emerging.

Connecting Learning1. What strategies did you use to count up the num ber

of cubes rapidly? [fl ats by 100, sticks by 10, cubes by one]

2. What patterns did you notice in how the types of pieces were placed in each rectangle? [Flats in a rect an gle in the lower left corner, a rectangle of sticks placed horizontally in the upper left corner, a rect an gle of sticks placed vertically in the lower right cor ner, cubes in a rect an gle in the upper right cor ner.]

3. What number patterns do you notice in the record of your solution? [tens by tens problem in low er left corner, ones by tens problem in upper left, tens by ones in lower right corner, and ones by ones problem in upper right corner]

4. How can you use your patterns to help you count the number of cubes in a rectangle if you are only told how long each side is?

ExtensionHave students outline rectangles or generate the di men sions of a rectangle on their own. Have them ap ply their patterns and determine the amount of cubes required to cover the rectangle, and then have them check it by building and recording the model.

* Reprinted with permission from Principles and Stan dards for School Mathematics, 2000 by the National Council of Teachers of Mathematics. All rights reserved.


Solutions

1003090

+ 27247

3 x 10 = 30 3 x 9 = 27

10 x 10 = 100 10 x 9 = 90

10 9

10

+

3

+

5 x 10 = 50 5 x 5= 25

10 x 10 = 100 10 x 5= 50

5

+

10

10 5+

1005050

+ 25225

2 x 10 = 20

10 x 10 = 100 10 x 8 = 80

2 x 8 = 162

10

+

10 + 8

1002080

+ 16216

5 x 20 = 100

10 x 20 = 200

5 x 1 = 5

10 x 1 = 10

5

+20

10

+

1

20010010

+ 5315

10 x 20 = 200

4 x 20 = 80 4 x 3= 12

10 x 3= 30

4

+

10

20 + 3

2008030

+ 12322

1 x 20 = 20

10 x 20 = 200

1 x 3 = 3

10 x 3= 30

1

+

10

20 + 3

2002030

+ 3253

1.

2.

3.

4.

5.

6.

15 x 15 = 225

18 x 12 = 216

21 x 15 = 315

23 x 14 = 322

23 x 11 = 253

19 x 13 = 247


x

1.


2.

x


x3.


x4.


x

5.


x

6.
