teaching multiplication (and division) conceptually

Teaching Multiplication(and Division) Conceptually

Professional Learning Targets…• I can describe what it means and what it looks like to teach

multiplication (and division) conceptually.• I can describe how standards progress across grade levels,

giving details for the grade span in which I teach.

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Agenda• Warming up with Multiplication and Division

• Number Strings• Quick Images

• Number Talks with Multiplication and Division• Big Ideas of Multiplication and Division • Types of Multiplication and Division Problems• Multiplication/Division Games

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A warm–up mental number string• 100 x 13• 2 x 13• 102 x 13• 99 x 13• 14 x 99• 199 x 34

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A warm–up mental number string• 100 x 13• 2 x 13• 102 x 13• 99 x 13• 14 x 99• 199 x 34

• What strategy does this string support?

• What big ideas underlie this strategy?

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Pictures for early multiplication

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Small Group Discussions• What strategies

would you expect to see?

• How would you represent them?

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How many apples? How many lemons?

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How many tiles in each patio?

• The furniture obscures some of the tiles possibly providing a constraint to counting by ones and supporting the development of the distributive property

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Here’s one that I found.

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Prior Understandings—Grades K-2

• Counting numbers in a set (K)• Counting by tens (K)• Understanding the numbers 10, 20, 30, 40, …, 90

refer to one, two, three, four, …, nine tens (1)• Counting by fives (2)

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Prior Understandings• 2.G.2. Partition a rectangle into rows and columns of same-

size squares and count to find the total number of them.

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So What Else About Multiplication?

Thinking that multiplication should always be interpreted as repeated addition.X 14

Let’s Review

Multiplication can be interpreted in a variety of familiar ways.

3 x 5 = 15

array

3’

5’

repeated addition area model

5 + 5 + 5

3 + 3 + 3 + 3 + 3 3

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Array Cards

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Scaling

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Core Lesson

“Gary’s flashlight shines three times farther than mine!”

My Flashlight

5 feet =

3 times farther than 5 feet

15 feet

Gary’s Flashlight

153 x 5

Core Lesson

When can multiplication be interpreted as scaling?

When it represents the relationship between the size of the product and the factors.

3 x 5 = 15

15 is 3 times > 5

15 is 5 times > 3

So What About Division?

How many of our students understand dividing a number by 3 is the same as multiplying the number by 1/3?

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To begin thinking about division, solve this problem using a strategy other than the conventional division algorithm. You may use symbols, diagrams, words, etc. Be prepared to show your strategy

169 ÷ 14 =

Hedges, Huinker and Steinmeyer. Unpacking Division to Build Teachers’ Mathematical Knowledge, Teaching Children Mathematics, November 2004, p. 4-8.

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Forgiveness Method

2112 252

- 120 10 132 - 120 10 12 - 12 1

0 21

Issic Leung, Departing from the Traditional Long Division Algorithm: An Experimental Study. Hong Kong Institute of Education, 2006.

Change it UP!!!!

1. Deal each player five cards. The remaining cards are placed face down on the center of the table.

2. Player one places a card face up on the table reads the division problem and provides the quotient. The next player must place a card with the same quotient on the first card. If the player cannot match, he/she may place a “Math Wizard” card on top and then a card with a different quotient.

3. If the player in unable to make either move, he/she must draw from the deck until a match is made.

4. The first player to use all of his/her cards is the winner.

Lies my teacher told me…

Division is about “fair sharing”.

35 ÷ 8 =

The Remainder

Can be discarded.The remainder can “force the answer to

the next highest whole number.The answer is rounded to the nearest

whole number for an approximate result.

1. Landon bought 80 piece bag of bubble gum to share with his five person soccer team. How many pieces did each player receive?

2. Brittany is making 7 foot jump ropes for the school team. She has a 25 foot piece of rope. How many can she make?

3. The ferry can hold 8 cars. How many trips will it need to make to carry 25 cars across the river?

Near Facts…

Find the largest factor without going over the target number

Partial Quotients 18 R 25

26 493 - 260 10 233 - 130 5 103 - 78 3

25 18

The Remainder Game

1. To begin the game, both players place their token on START.

2. Player one spins the spinner and divides the number beneath his/her marker by the number on the spinner. If there is a remainder, he/she is allowed to move his/her token as many spaces as the remainder indicates. If the division does not result in a remainder, he/she must leave his/her marker where it is.

3. The play alternates between the two players (a new spin must occur each time) until some reaches HOME.

Lies my teacher told me…Any number divided by zero is zero!

6 ÷ 0 =

How many times can 0 be subtracted from 6?

How many 0 equal groups are there in six?

What does six divided into equal groups of 0 look like?

What number times 0 gives you 6?

Division Vocabulary

Quotient

Divisor Dividend

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THE PARTITIVE PROBLEM

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Partitive

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Example 1: Write a word problem to represent this model of division?

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THE MEASUREMENT PROBLEM

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Measurement

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Example 2: Write a word problem to represent this model of division?

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Two basic types of problems in divisionPartitive (Sharing): You have a group of objects and you share them equally. How many will each get?

Example: You have 15 lightning bugs to share equally in three jars. How many will you put in each jar?

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Two basic types of problems in divisionMeasurement: You have a group of objects and you remove subgroups of a certain size repeatedly. The basic question is—how many subgroups can you remove?Example: You have 15 lightning bugs and you put three in each jar. How many jars will you need?

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Measurement Model

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Ratio

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Grade 3 Introduction• In Grade 3, instructional time should focus on four critical areas: (1)

developing understanding of multiplication and division and strategies for multiplication and division within 100; …

• Students develop an understanding of the meanings of multiplication and division of whole numbers through activities and problems involving equal-sized groups, arrays, and area models;

• multiplication is finding an unknown product, and division is finding an unknown factor in these situations.

• For equal-sized group situations, division can require finding the unknown number of groups or the unknown group size.

• Students use properties of operations to calculate products of whole numbers, using increasingly sophisticated strategies based on these properties to solve multiplication and division problems involving single-digit factors.

• By comparing a variety of solution strategies, students learn the relationship between multiplication and division.

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Commutative Property• It is not intuitively obvious that 3 x 8 = 8 x 3. A picture of 3

sets of 8 objects cannot immediately be seen as 8 piles of 3 objects. Eight hops of 3 land at 24, but it is not clear that 3 hops of 8 will land at 24.

• The array, however, can be quite powerful in illustrating the commutative property.

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Distributive Property & Area Models

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3

5 + 2

15 6+

3 x 7 =3 x (5 + 2) = (3 x 5) + (3 x 2)= 15 + 6 = 21

3 x 7 =__

Grade 3•3.MD.7. Relate area to the operations of multiplication and addition.

• Find the area of a rectangle with whole-number side lengths by tiling it, and show that the area is the same as would be found by multiplying the side lengths.

• Multiply side lengths to find areas of rectangles with whole-number side lengths in the context of solving real world and mathematical problems, and represent whole-number products as rectangular areas in mathematical reasoning.

• Use tiling to show in a concrete case that the area of a rectangle with whole-number side lengths a and b + c is the sum of a × b and a × c. Use area models to represent the distributive property in mathematical reasoning.

• Recognize area as additive. Find areas of rectilinear figures by decomposing them into non-overlapping rectangles and adding the areas of the non-overlapping parts, applying this technique to solve real world problems.

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Grade 3Represent and solve problems involving multiplication and division

(Glossary-Table 2)

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Not until 4th GradeNot until 4th Grade

Connections to other 3rd Grade Standards

3.NBT.3: Multiply one-digit whole numbers by multiples of 10 in the range 10–90 (e.g., 9 × 80, 5 × 60) using strategies based on place value and properties of operations.

9 x 80: 80 is ten 8’s. So, if I know that 8x9 is 72, then I have ten 72’s. That equals 720.

Or..80 is 8 tens. So, if 10 x 9 = 90, then I know I have 8 of those (90s). 90 + 90 + 90 + 90 + 90 + 90 + 90 + 90 = (800-80) = 720

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Sample Activity: Finding Factors (Elementary and Middle School Mathematics: Teaching Developmentally by Van de Walle, Karp, Bay-Williams)

1. With a partner, choose one of the following numbers:12, 18, 24, 30, 36, or 48

2. Use equal sets, arrays, or number lines to find as many multiplication expressions as possible to represent your number.

3. For each multiplication expression, write its equivalent addition expression showing your groupings.

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Sample Activity: Finding Factors from Elementary and Middle School Mathematics: Teaching Developmentally by Van de Walle, Karp, Bay-Williams

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Grade 4A focus on teaching multiplication (and division) conceptually

Grade 4 Introduction• In Grade 4, instructional time should focus on three critical areas: (1)

developing understanding and fluency with multi-digit multiplication, and developing understanding of dividing to find quotients involving multi-digit dividends

• They apply their understanding of models for multiplication (equal-sized groups, arrays, area models), place value, and properties of operations, in particular the distributive property, as they develop, discuss, and use efficient, accurate, and generalizable methods to compute products of multi-digit whole numbers.

• Depending on the numbers and the context, they select and accurately apply appropriate methods to estimate or mentally calculate products.

• They develop fluency with efficient procedures for multiplying whole numbers; understand and explain why the procedures work based on place value and properties of operations; and use them to solve problems.

• Students apply their understanding of models for division, place value, properties of operations, and the relationship of division to multiplication as they develop, discuss, and use efficient, accurate, and generalizable procedures to find quotients involving multi-digit dividends.

• They select and accurately apply appropriate methods to estimate and mentally calculate quotients, and interpret remainders based upon the context.

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Selected Standards…

• 4.NBT.5.• Multiply a whole number of up to four digits by a one-

digit whole number, and multiply two two-digit numbers, using strategies based on place value and the properties of operations. • Illustrate and explain the calculation by using equations,

rectangular arrays, and/or area models.

• (Area models for this standard are directly linked to the understanding of partitioning a rectangle into equal parts and 3.MD.7c)

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Area Models

25 x 38=

57

20 +

530 + 8

150 40

600 160

750 + 200 = 950

950

Partitioning Strategies for Multiplication• 27 x 4 27 x 4• 4 x 20 = 80 27 + 27 + 27 + 27• 4 x 7 = 28 108• 54 54• 108• -----------------------------------• 267 x 7• 7 x 200 = 1400 1820 • 7 x 60 = 420 1876• 7 x 8 = 56

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Selected Standards…

• 4.NBT.6. • Find whole-number quotients and remainders with up to

four-digit dividends and one-digit divisors, using strategies based on place value, the properties of operations, and/or the relationship between multiplication and division. • Illustrate and explain the calculation by using equations,

rectangular arrays, and/or area models.

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Selected Standards…• 4.OA.3.• Solve multistep word problems posed

with whole numbers and having whole-number answers using the four operations, including problems in which remainders must be interpreted. • Represent these problems using equations

with a letter standing for the unknown quantity.

• Assess the reasonableness of answers using mental computation and estimation strategies including rounding.

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Grade 5A focus on teaching multiplication (and division) conceptually

Grade 5 Introduction• (2) extending division to 2-digit divisors, integrating decimal fractions

into the place value system and developing understanding of operations with decimals to hundredths, and developing fluency with whole number and decimal operations;

• Students develop understanding of why division procedures work based on the meaning of base-ten numerals and properties of operations.

• They finalize fluency with multi-digit addition, subtraction, multiplication, and division.

• They apply their understandings of models for decimals, decimal notation, and properties of operations to add and subtract decimals to hundredths. They develop fluency in these computations, and make reasonable estimates of their results. Students use the relationship between decimals and fractions, as well as the relationship between finite decimals and whole numbers (i.e., a finite decimal multiplied by an appropriate power of 10 is a whole number), to understand and explain why the procedures for multiplying and dividing finite decimals make sense. They compute products and quotients of decimals to hundredths efficiently and accurately. 64

Grade 5 Selected Standards

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

•Fluently multiply multi-digit whole numbers using the standard algorithm.

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Gr. 5 Selected Standards

5.NF.5Interpret multiplication as scaling (resizing), by:•a. Comparing the size of a product to the size of one factor on the basis of the size of the other factor, without performing the indicated multiplication.•b. Explaining why multiplying a given number by a fraction greater than 1 results in a product greater than the given number (recognizing multiplication by whole numbers greater than 1 as a familiar case); explaining why multiplying a given number by a fraction less than 1 results in a product smaller than the given number; and relating the principle of fraction equivalence a/b =(n×a)/(n×b) to the effect of multiplying a/b by 1.

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Scaling

•Recognize that 3 x (25,421 + 376) is 3 times larger than

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Gr. 5 Selected Standards

• 5.NBT.6. • Find whole-number quotients of whole numbers

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

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There must be a clear connection

between multiplication and division

using manipulatives, before this

understanding takes place.

Zero and Identity Properties

• Rules with no reasons?• No…ask students to reason.• Ex: How many grams of fat are there in 7

servings of celery? Celery has 0 grams of fat.• Ex: Note that on a number line, 5 hops of 0 land

at 0. Also, 0 hops of 20 also stays at 0.

• Arrays with factors of 1 are also worth investigation to determine the identity property 69

Distributive Property Activity

• Slice it Up…• Each pair, please use the grid paper to make a

rectangle that has a total area greater than 10 square units.

• Make a slice through the rectangle and write an equation that matches using the lengths and widths of the smaller rectangles created.

• Continue this process until you have found all the ways to “slice it up.”

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14 x 25: An Area Model

*Sketch is not drawn to scale. 71

20 + 5

80 20

200 50

Algebra 1: Multiplying Binomials

*Sketch is not drawn to scale. 72

x + 5

4x 20

x2 5x