Multiplication and Division: The Inside Story

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

http://elementary-math-resources.wiki.inghamisd.org/home

Three sessionsToday: Multiplication and DivisionMar. 27: Fractions and DecimalsApr. 23: Geometric Shapes and

Volume

TodayHow children learnMultiplication and division

problem-solvingMultiplication and division

combinationsMulti-digit multiplication and

divisionConnections with area and

perimeter

How many ways?With a partner, make as many rectangles as you can using all the tiles at your table. Record each rectangle as a multiplication sentence.

6 = 2 x 3

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 drawing that show jumps of 2-3-4.

Key Strategies1. Use visual representations

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.

Key Strategies2. Teach the underlying structures

of word problems

Three blended goals of math education:

Conceptual understandingFluencyProblem solving

Related problem typesRate Price CombinationThe Friendly Old Ice Cream Shop has 3 types of ice cream cones. They also have 4 flavors of ice cream. How many different combinations of an ice cream flavor and cone type can you get at the Friendly Old Ice Cream Shop?

Here’s how I would do it:

sugar conechocolatemint choc. chiprocky roadvanilla swirl

4

safety conechocolatemint choc. chiprocky roadvanilla swirl

4

waffle conechocolatemint choc. chiprocky roadvanilla swirl

4

4 x 3 = 12

Key Strategies3. Explain your thinking carefully

when showing proceduresRequire students to explain their thinking also.

If one lamp costs $45, how much do 12 lamps cost? Explain how you figured this out.

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 into 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

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, then DrillExtensions

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 (practice)4) Drill in game situations (fluency)5) Use in division situations

(generalization)

IISD Fluency PacketResources for helping those

students who still need work on combinations.

Fluency – Practice and Drill“Practice” refers to lessons that are problem-based and that encourage students to develop flexible and useful strategies that are personally meaningful.“Drill” is repetitive non-problem-based activity to help children become facile with strategies they know already in order to internalize (remember) the fact combinations.

From Van de Walle, Elementary and Middle School Mathematics: Teaching Developmentally

4. Practice, practice, practice

And use cumulative review often.

Daily Double!!!

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 Factsfrom ORIGO Mathematics

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 many rectangles…?What does this represent?

Watch Associative Property 12 x 15

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

Division by Partitioning354 photos to share among 3 children

Work with manipulatives also translates to procedures354 ÷ 3(300 + 50 + 4) ÷ 3 = 100 + 10 + 1 r 21

100 + 10 + 1 + 7

Try this with 251 ÷ 8. Partition base 10 blocks, then write a corresponding algorithm.

What about remainders?The remainder is simply left over and not taken into account (ignored)It takes 3 eggs to make a cake. How many cakes can you make with 17 eggs? The remainder means an extra is needed20 people are going to a movie. 6 people can ride in each car. How many cars are needed to get all 20 people to the movie?

The remainder is the answer to the problemMs. Baker has 17 cupcakes. She wants to share them equally among her 3 children so that no one gets more than anyone else. If she gives each child as many cupcakes as possible, how many cupcakes will be left over for Ms. Baker to eat? The answer includes a fractional part9 cookies are being shared equally among 4 people. How much does each person get?

A Remainder of One