Transcript Multiplication and Division - elementary-math

Our Purpose and Agenda

Important outcomes for 4th grade
Multi-digit multiplication and division
Concept of decimal numbers
Basic fraction concepts

Important outcomes for 5th grade
Multi-digit multiplication and division
Operations with decimal numbers
Advanced fraction concepts
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.
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.
Multiplication and Division
Problem types
Equal size groups
5 groups with 4 in each group, the total is 5 x 4
4
4
4
4
4
5 groups of 4
I have 9 boxes with 13 books in each box.
How many books do I have?
See handout and poster
Division is a “missing factor” problem.
20 cookies in 5 bags, how many in each
bag?
20 cookies, 4 in each bag, how many bags?
20 = 5 x __
20 = __ x 4
5 groups of ___
___ groups of 4
Partitive division
Measurement division
Division answers the questions
1) how many in a group, or 2) how many groups
Related problem types
Rate
 Price
 Combination

The 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?
A robin eats 24 worms every day for 15 days.
How many worms did the robin eat?
How might children solve these?
1.
2.
3.
4.
5.
6.
Chad had 5 bags of candy with 7 pieces of candy in each
bag. How many pieces of candy did Chad have?
At Sally’s birthday party there were 15 children and 3
blankets. If the same number of children sat on each
blanket, how many children sat on each blanket?
Lee collects stamps. He has 45 stamps. If he sticks 9
stamps on each page, how many pages would he fill with
stamps?
Mr. Wong has 4 children. He wants his children to share 12
marbles so that they each get the same amount. How
many marbles should Mr. Wong give each child?
19 children are going to the circus. 5 children can ride in
each car. How many cars are needed so that all the
children can go to the circus?
Maria has 17 fish. If 3 fish can be put into one fish bowl,
how many bowls does she need to hold her fish?
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.
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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).
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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.
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?
12 x 15 = 20 x ___
Number Talks:5th grade 12 x 15
32 x 15 Then look at CCSS

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.

5.NBT.5 Fluently multiply multi-digit whole
numbers using the standard algorithm.
8 x 7 = (8 x 5) + (8 x 2)
This is the distributive property
Now you can multiply larger 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.
Rectangle Multiplication
Factors
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4.OA.4 Find all factor pairs for a whole number in the range 1-100.
Recognize that a whole number is a multiple of each of its factors.
Determine whether a given whole number in the range 1-100 is a
multiple of a given one-digit number. Determine whether a given
whole number in the range 1-100 is prime or composite.

Start with a number like 20 or 24.
Give each pair of students 20 (or 24) tiles.
Ask them to make as many different rectangles as they
can out of the tiles (using all of them).
Record each arrangement as a product.
Try this by drawing, using the number 36.

The Factor Game
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:
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.
Learning Progression
Why do we line up the decimal points?
2.1
x4.7
0.07
1.4
0.4
8
9.87
What does this represent?
Think about division…
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.

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.

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.
Getting to fluency C-R-A
Division by Partitioning – C or R
(objects or pictures)
354 photos to share among 3 children
Work with manipulatives also translates to
procedures – Abstract (symbols)
354 ÷ 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.
Partial quotient method
6 )234
-120
114
-60
54
-30
24
-24
0
20
10
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.
5
4
39
This type of division is also
called repeated subtraction.
You try it
24)8280
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.
Keep in mind that 8280 = 8000 + 200 + 80 + 0 or
8200 + 80 or
82 hundreds + 8 tens
Now the standard algorithm
24)8280
What’s hard about this for students?
Why use it?
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?
A Remainder of One
Which is
answer.
3
more,
4
2
10
3
7
of a pizza or ? Explain your
1
2
Is closer to 0 or ? How can you prove your
answer?
7
8
1
2
3
5
1
2
Is closer to or 1? How can you show this?
1
2
Is less than or greater than ? What is your
reasoning?
3
4
Compare and
2
. Which
3
Therefore which is
Which is
5
larger,
6
Which is
3
larger,
5
is closer to 1? Why?
3
larger,
4
2
3
or ?
4
5
or ? Explain your answer.
2
3
or ? Explain your answer.
Order from smallest to largest this set of
3 1 5 3 14
fractions: , , , ,
Explain how you
4 10 12 5 15
figured this out.