Transcript Multiplication and Division: The Inside Story - elementary

Multiplication and Division:
The Inside Story
A behind-the-scenes look at the most
powerful operations
Three sessions

Today: Multiplication and Division

Dec. 3: Fractions and Decimals

Jan. 28: Geometric Shapes and Volume
Today
How children learn

Multiplication and division problemsolving

Multiplication and division combinations

Multi-digit multiplication and division

Connections with area and perimeter
The first way we teach children to
think about multiplication:
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x 5
x 10
x 15
x 20
Skip-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
x
x
x
x
x
x
x
x
x
x
x
x 4
x 8
x 12
x 16
x 20
Make up 3 examples with jumps of 2-3-4.
The second way we teach children
to think about multiplication:
4
4
4
4
4
Equal groups. This is a generalization of equalsize 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.
The third way we teach children
to think about multiplication:

My dog can run 5 times as fast as your rabbit.
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Your rabbit can jump 3 times as far as my dog.
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My dog eats 10 times more food than your rabbit.
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Your rabbit is 1/4 the height of my dog (or my
dog is 4 times taller than your rabbit).
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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 types
Rate
 Price
 Combination
 See the handout

Why is it important to recognize
types of multiplication problems?
The fixed costs of manufacturing basketballs in a factory
are $1,400.00 per day. The variable costs are $5.25 per
basketball. Which of the following expressions can be
used to model the cost of manufacturing b basketballs
in one day?
A. $1,405.25b
B. $5.25b − $1,400.00
C. $1,400.00b + $5.25
D. $1,400.00 − $5.25b
E. $1,400.00 + $5.25b
Number Talk
What 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 DVD

Number 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
Factors
The word “factor” is an academic
vocabulary term that is essential to
understanding multiplication.
6x1=6
3x2=6
Which 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 Game
Common Core Collaboration Cards
 With 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
fluent
Acquisition – Fluency – Generalization
Concepts, strategies, procedures
Practice, practice, practice
Extensions
This learning progression is true for single
digit “math facts” and for fluency with
multi-digit procedures.
Math facts, if not already known
Math fact strategy:
1) Only work on unknown combinations
2) 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 Packet

Resources for helping those students who
still need work on combinations.
The Product Game

Good practice for children who don’t
have all their combinations from memory
yet.

A combination game from PhET
Research Recommendation
Interventions 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-A
Concrete-Representational-Abstract
Concrete: Multiply 16 x 12 using base 10
blocks.
Procedures… The C-R-A
Concrete-Representational-Abstract
Representational:
National Library of Virtual Manipulatives nlvm.usu.edu
Procedures… The C-R-A
Concrete-Representational-Abstract
Abstract:
Learning Progression
Problem-solving with area and
perimeter
Table for 22: Real-World
Geometry Problem
How is division tied to
multiplication?
List several ways the two are connected…
Two types of division
Partitive (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 method
6 )234
-120
114
-60
54
-30
24
-24
0
20
10
5
4
39
Find whole-number
quotients and
remainders with up
to four-digit
dividends and onedigit 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
3
24)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
34
24)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?
24
24
24
24
12
345
24)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
345
24)8280
72
1080
96
120
120
0
8280 = 8000 + 200 + 80 + 0 or
= 7200 + 960 + 120
= 24 x 300 + 24 x 40 + 24 x 5
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
needed
20 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
problem
Ms. 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 part
9 cookies are being shared equally among 4
people. How much does each person get?