Proposed Unit 5
Slides 63- 73
These have been reformatted--feedback?
Structures of Addition
How many altogether?
Join and Part-Part Whole
– There is something, and you get more of it?
– There are two kinds, how many all together?
?
Start Unknown
– Some are given away, some are left, how many were there to
start?
?
What did
I start with?
Taken
Left
Compare--total unknown
- I know one amount and I have some amount more than
that--how many do I have?
?
How many do I have?
Addition types Adapted from Carpenter, Fennema, Franke, Levi and Empson, 1999, p. 12 in Adding it Up, NRC 2001.
Structure
3 types of subtraction
Ask yourself if a problem is a subtraction problem—
Does it fit one of these three types?:
You’ve got some amount
and “take away” from it:
What’s left?
?
– The Classic “Take away”
(how many left?)
?
– Comparison
(difference between? who has more?)
You compare to see:
Who has more or less?
?
– Deficit/Missing amount
(what’s missing?)
You need some more to get
where you want to be:
What is the missing amount?
Addition
Start Unknown
Julie had a bunch of fruit. She
gave away 30 oranges and she
still has 50 pieces of fruit left.
How many pieces of fruit did she
have to start with?
?
Subtraction
Classic “take-away”
Julie had 80 pieces of fruit. She
gave away 30 oranges. How
many pieces of fruit did she
have left?
?
30
50
30
?
left + gave away =start
?
80
start - gave away = left
start – left = gave away
gave away + left = start
addend + addend = Sum(Total)
Sum(Total) – addend = addend
minuend – subtrahend = difference
Addition
Join or Part/Part -Whole
Julie had 50 apples and then
bought 30 oranges. How many
pieces of fruit does she have
now?
Subtraction
Deficit/ Missing Amount
Julie wanted to collect 80
pieces of fruit for a food drive.
She already has 50 apples. How
many more pieces of fruit does
she need?
?
?
?
30
50
50
?
part + other part = whole
addend + addend = sum
80
whole – part = other part
whole – part accounted for
= part needed
whole – part = difference
minuend – subtrahend = difference
Addition
Compare: Total unknown
Julie had 30 oranges and some
apples. She had 20 more
apples than oranges. How
many apples does she have?
Subtraction
Compare: difference unknown
Julie had 50 apples and 30
oranges. Does she have more
apples or oranges? How many
more?
?
50
?
20
30
30
?
20
30
?
Amount of one set + the difference between
two sets
= amount of second set
Addend + addend
= sum total (of unknown set)
30
50
Amount in one set – amount of an other set
= difference between sets
sum total (needed) – amount of one set
(have)
= difference
STRUCTURE:
3 Types of Multiplication: 4 x 3
Repeated Addition
Array/row-column
Counting Principle
STRUCTURE:
10  2
3 Types of Division:
Measurement/Repeated Subtraction
“How many 2s can I get out of 10?”
2
2
10
?
2
2 2
If I have 10 cups of
beans and I give out 2
cup portions, how many
servings will that
provide?
Partitive/Unitizing/Fair Shares
“How many would 1 person get?
Or “What would that mean in relation
to 1?”
?
10
?
If 2 people find $10 how
much will each person get ?
Product/Factor
“If I have an area of 10 and one side is 2, how long is the other side?”
10
?
2
Multiplication
Repeated Addition
Division
Repeated
Subtraction/Measurement
Julie had 4 baskets with 5
pieces of fruit in each basket.
How many pieces of fruit does
she have?
5 + 5 +
Julie has 20 pieces of fruit. She
wants to eat 5 pieces of fruit a
day. How many days can she
eat her fruit?
5 + 5
-5
- 5
-5
-5
20
0
0 5 10 15 20
0
5 10 15 20
How many is 4 5s?
How many 5s can you get out of twenty?
# of Groups * Objects in group = Total
objects
Total ÷ portions = servings
Product ÷ factor = factor
Factor * Factor = Product
Multiplication
Array/Row-Column (Area/Side
Length)
Julie has a rectangular surface she wants
to cover with square unit tiles. The
length of one side is 5 units long and the
length of the other side is 4 units long.
How many pieces of tile does Julie need?
Division
Product/Factor (Area/Side
Length)
Julie has a rectangular surface that is
20 square units. The length of one
side is 5 units long. What is the
length of the other side?
1
1
2
3
4
2
3
4
5
5
1
?
2
3
4
linear side ∙ linear side = area of rectangle
row ∙ column = total
factor ∙ factor = product of area
area of rectangle ÷ linear side =
other linear side
Total ÷ column = row
Total ÷ row = column
Product ÷ factor = factor
Multiplication
Fundamental Counting Principle
Julie packed 4 pair of jeans and 5 shirts for her trip. How many different unique outfits can
she make?
S1 S2 S3 S4 S5
J1
S1 S2 S3 S4 S5
J2
S1 S2 S3 S4 S5
S1 S2 S3 S4 S5
J3
J4
Total outfits?
This is also an excellent model for probability:
Julie has four dice in different colors: blue, red, green and white. If she picks one die at
random and then rolls it, what are the chances that she would have rolled a blue 5?
1 2 3 4
5 6
blue
1 2 3 4 5 6
red
green
1 2 3 4 5 6
white
1 2 3 4
5 6
Division
Partitive/Unitizing/Fair Shares
Julie is packing her suitcase for a trip. She is planning her outfits for the trip and will
wear one shirt and one pair of jeans each day. She brought 5 shirts. How many pairs
of jeans must she bring if she needs 20 unique outfits?
S1 S2 S3 S4 S5
S1 S2 S3 S4 S5 S1 S2 S3 S4 S5
5 outfits 10 outfits
15 outfit
S1 S2 S3 S4 S5
20 outfits
This model is the way students first learn division, through ‘fair shares’?
How many will each one person get?
It is also the structure for a Unit Rate: 20 per every 4, how many per 1?
D1 D5 D9 D13 D17
n1
D2 D5 D10 D14 D18
n2
D3 D7 D11 D15 D19
n3
D4 D8 D12 D16 D20
n4
Julie has 20 dollars and wants to give out the money equally among her four nieces. She
passes the dollars out to them one at a time. How much will each niece get?