multiplication and division: the inside story

Multiplication and Division: The Inside Story

A behind-the-scenes look at the most powerful operations

Three sessionsToday: Multiplication and DivisionDec. 3: Fractions and DecimalsJan. 28: Geometric Shapes and

Volume

TodayHow children learnMultiplication and division

problem-solvingMultiplication and division

combinationsMulti-digit multiplication and

divisionConnections with area and

perimeter

The first way we teach children to think about multiplication:

x x x x x 5x x x x x 1

0x x x x x 1

5x x x

 x x 2

0Skip-counting of rows in an array.An example is 4 rows of 5 chairs lined up in a room.

Is 5 rows of 4 the same number?

x x x x 4x x x x 8x x x x 1

2x x x x 1

6x x x x 2

0Make up 3 examples with jumps of 2-3-4.

The second way we teach children to think about multiplication:

Equal groups. This is a generalization of equal-size rows of objects in an array.An example is 5 bags with 4 cookies in each bag. Make up 3 more examples using 10-11-12.

4 4 4 4 4

The third way we teach children to think about multiplication:My dog can run 5 times as fast as your rabbit.Your rabbit can jump 3 times as far as my dog.My dog eats 10 times more food than your

rabbit.Your rabbit is 1/4 the height of my dog (or my

dog is 4 times taller than your rabbit).Your rabbit is twice as old as my dog.My dog can bark 100 times louder than your

rabbit!Multiplicative comparison.Make up 3 more that involve everyday things.

Related problem typesRate Price CombinationSee the handout

Why is it important to recognize types of multiplication problems?The fixed costs of manufacturing basketballs in a factoryare $1,400.00 per day. The variable costs are $5.25 perbasketball. Which of the following expressions can beused to model the cost of manufacturing b basketballs in one day?A. $1,405.25b

B. $5.25b − $1,400.00C. $1,400.00b + $5.25D. $1,400.00 − $5.25b

E. $1,400.00 + $5.25b

Number TalkWhat number do you think will go in the blank to make the equation true? Try to solve this by reasoning, without doing the calculations.4 x 9 = 12 x ___How did you think about this?

The most powerful way of thinking about multiplication:

This is powerful because it connects multiplication to the area of a rectangle.8 x 7 = 56 8 in. x 7 in. = 56 sq. in.

The most powerful way of thinking about multiplication:

Plus, it gives us insight in the process of multiplication, and new ways to compute:8 x 7 = (8 x 5) + (8 x 2) This is the distributive property (3.MD.7)

The most powerful way of thinking about multiplication:

Now you can multiply bigger numbers in your head. Try 56 x 5. Try 8 x 23.

Find a way to multiply 38 x 6 by representing 38 as a subtraction.Try 3,426 x 5 by decomposing into thousands, hundreds, tens and ones.

Number Talks book and DVDNumber Talks:

Helping Children Build Mental Math and Computation Strategies, Grades K-5, by Sherry Parrish (DVD)

Watch Array Discussion

How many rectangles…?How many different rectangles can you make from your bag of squares? Write a multiplication sentence to go with each rectangle.

Watch Associative Property 12 x 15

FactorsThe word “factor” is an academic vocabulary term that is essential to understanding multiplication.6 x 1 = 63 x 2 = 6Which are the factors and which are the products in your rectangles?Watch 16 x 35

Rectangle multiplication What does this visual representation tell you about multiplication? (knees to knees, eyes to eyes)

http://nlvm.usu.edu

The Factor GameCommon Core Collaboration CardsWith your team member, see if

you can figure out a strategy for winning.

Also linked from our Elementary Math Resources wiki: Go to inghamisd.org, then click on Wiki Spaces.

How to help a child become fluentAcquisition – Fluency – Generalization Concepts, strategies, procedures

Practice, practice, practiceExtensions

This learning progression is true for single digit “math facts” and for fluency with multi-digit procedures.

Math facts, if not already knownMath fact strategy:1) Only work on unknown combinations2) Ensure knowledge of meaning of

multiplication (acquisition)3) Learn strategies through repeated

problem-solving (acquisition)4) Practice in game situations (fluency)5) Use in division situations

(generalization)

IISD Fluency PacketResources for helping those

students who still need work on combinations.

The Product GameGood practice for children who

don’t have all their combinations from memory yet.

A combination game from PhET

Research RecommendationInterventions at all grade levels should devote about 10 minutes in each session to building fluent retrieval of basic arithmetic facts.Provide about 10 minutes per session of

instruction to build quick retrieval of basic arithmetic facts. Consider using technology, flash cards, and other materials for extensive practice to facilitate automatic retrieval.

For students in kindergarten through grade 2, explicitly teach strategies for efficient counting to improve the retrieval of mathematics facts.

Teach students in grades 2-8 how to use knowledge of properties, such as commutative, associative, and distributive laws, to derive facts in their heads.

Box and Books of Facts

Procedures… The C-R-AConcrete-Representational-AbstractConcrete: Multiply 16 x 12 using base 10 blocks.

Procedures… The C-R-AConcrete-Representational-AbstractRepresentational:

National Library of Virtual Manipulatives nlvm.usu.edu

Procedures… The C-R-AConcrete-Representational-AbstractAbstract:

Learning Progression

Problem-solving with area and perimeterTable for 22: Real-World Geometry Problem

How is division tied to multiplication?List several ways the two are connected…

Two types of divisionPartitive (fair shares)We want to share 12 cookies equally among 4 kids. How many cookies does each kid get? How would you solve this with a picture?The number of groups is known; the number in each group is unknown.

Measurement (repeated subtraction)For our bake sale, we have 12 cookies and want to make bags with 2 cookies in each bag. How many bags can we make?How would you solve this with a picture?The number in each group is known; the number of groups is unknown.

Partial quotient method6 )234 -120 20 114 -60 10 54 -30 5 24 -24 4 0 39

Find whole-number quotients and remainders with up to four-digit dividends and one-digit divisors, using strategies based on place value. 4.NBT.6

This type of division is called repeated subtraction

You try it

24)8280

Now the standard algorithm

Keep in mind that 8280 = 8000 + 200 + 80 + 0 or 8200 + 80 or 82 hundreds + 8 tens

324)8280 72 10

The standard algorithm: 1) How many equal groups of 24

can be made from 82? 3 groups, with 10 left over. 82 what? 10 what? Why do we put the 3 there?

24 24 24 10

3424)8280 72 1080 96 12

The standard algorithm: 1) How many equal groups of 24

can be made from 82? 3 groups, with 10 left over. 82 what? Why do we put the 3 there?

2) How many equal groups of 24 can be made from 108? 4 groups, with 12 left over.108 what?Why do we put the 4 there?

1224 24 24 24

34524)8280 72 1080 96 120 120 0

The standard algorithm: 1) How many equal groups of 24

can be made from 82? 3 groups, with 10 left over. 82 what? 10 what?Why do we put the 3 there?

2) How many equal groups of 24 can be made from 108? 4 groups, with 12 left over.108 what? 12 what?Why do we put the 4 there?

3) How many equal groups of 24 can be made from 120? 5 groups, with 0 left over.120 what? Why do we put the 5 there?

24 24 24 24 24

34524)8280 72 1080 96 120 120 0

8280 = 8000 + 200 + 80 + 0 or

= 7200 + 960 + 120= 24x300 + 24x40 + 24x5

What about remainders?

Algorithms