Mrs. James left a tray of cookies on the counter early one morning. Larry walked by before lunch and decided to take 1/3 of the cookies on the tray. Later that afternoon Barry came in and ate 1/4 of the remaining cookies. After supper Terry saw the tray of cookies and ate 1/2 of the cookies remaining at that time.
The next morning Mrs. James found the tray with only 6 cookies left. How many cookies were on the tray when Mrs. James first left it on the counter?
Georgia Formative Assessment tasks

Mrs. James found the tray with only 6 cookies left.
6
Terry ate 1/2 of the cookies remaining.
6 Terry’s remaining
6
Barry ate 1/4 of the remaining cookies.
12 remaining
Larry took 1/3 of the cookies.
Barry
16 starting
Larry
4
6
4 4 4

There are 338 players on a soccer team. 186 are girls and the rest are boys. How many boys are on the soccer team?
186
338
?
Misha has 34 dollars. How many dollars does she have to earn to have 47 dollars?

Visual representations for multiplication

Visual representations translate to symbolic (CRA)

Estimate:
1.4 x 1.3 is somewhere between 1 and 4
Distributive Property:
1.4
x 1.3
0.12
0.3
0.4
1
1.82

Knowing the reasons behind a procedure is as important as being fluent with it.

Reason?
2
13 13 x
8 x
8
104 24
+80
104
8
8∙3 + 8∙10 = 8(3+10) a∙b + a∙c = a(b+c)
10 3

Reason?
1
3
×
1
4
=
1
12

What problem does this illustrate?

Division by Partitioning
354 photos to share among 3 children

Work with manipulatives also translates to procedures
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.

Problem-solving is enhanced by knowing the structures of word problems.
Part-whole where a part is unknown
There are 23 players on a soccer team. 18 are girls and the rest are boys. How many boys are on the soccer team?
18 ?
23
Joining (adding to) where the change is unknown
Misha has 34 dollars. How many dollars does she have to earn to have 47 dollars?

Equal groups
Each person on a relay race team runs 5/8 of a mile. There are 4 people on the team. How long is the total race?
5
8
+
5
8
+
5
8
+
5
8
=
4×5
8
Multiplicative comparison
The tree is 40 times taller than the person. If the person is 5 feet tall, how tall is the tree?

Measurement division
Our class baked 225 cookies for a bake sale. We want to put them in bags with 6 in each bag. How many bags can we make?
225 – 60 = 165 10 bags
165 – 60 = 105 10 bags
105 – 60 = 55 10 bags
45 – 30 = 15 5 bags
15 – 12 = 3 2 bags
37 bags with 3 cookies left over
Try this with 251 ÷ 8. What’s a corresponding word problem?

A recipe that makes 20 cookies needs ¾ cup of sugar. How much sugar is needed to make 100 cookies?
Measurement division
How many groups of 20 are in 100?
(repeated subtraction, or
20 x what = 100?)
Equal groups
How much is 5 groups of 3 fourths?
(Skip count on a number line)

If a serving size for Cheerios is 3/4 cup, how many servings are in a box that has 12 1/2 cups?
While division by a fraction is in the 6 th grade curriculum, this example shows how important it is to understand the underlying structure of word problems.

T-shirts with the school logo cost $8 wholesale. The Pep
Club has saved $496. How many t-shirts can they buy for their fund-raiser?

Number Talks
32 x 15

14
4
64
-38
26

Do these multiplication problems as quickly as you can using the partial product method.
25 x 164 562 x 36 285 x 102
Practice makes perfect: Frequent cumulative review is the most underused method for ensuring long-term retention.