Unit 1
6 in
Jay found the volume of the prism pictured to the right by
multiplying 5 × 6 and then adding: 30 + 30 + 30 = 90. He says
the volume is 90 cubic inches.
5 in
3 in
Step A: Maria says he did it wrong. He should have multiplied
the bottom first (3 × 5) and then multiplied by the height. Explain
to Maria why Jay’s method works and is equivalent to her method.
Possible explanation:
Jay thought about this problem as layers. He calculated the area of the side (5 x 6) and
imagined there were three of them, so he added them up. 3 x 5 x 6= 90 as well.
Step B: Use Jay’s method to find the volume of this right rectangular prism.
5 ft
5 x 3 = 15
2 ft
3 ft
Wicomico County Board of Education
15 + 15 = 30 cubic feet
2014-2015
Grade 5
2
Unit 1
The following structure is composed of two right rectangular prisms
that each measure 12 inches by 10 inches by 5 inches, and one right
rectangular prism that measures 10 inches by 8 inches by 36 inches.
What is the total volume of the structure? Explain your thinking.
Wicomico County Board of Education
2014-2015
Grade 5
3
Unit 1
Juliet wants to know if the chicken broth in this beaker will fit into this
rectangular food storage container. Explain how you would figure it out
without pouring the contents in. If it will fit, how much more broth could
the storage container hold? If it will not fit, how much broth would be left
over? (Remember 1 cm3 = 1 mL.)
4L
15 cm
2L
7 cm
20 cm
Wicomico County Board of Education
2014-2015
Grade 5
4
Unit 1
An aquarium is in the shape of a rectangular prism. The base of the
aquarium has an area of 40 square inches. When 160 cubic inches of
water is poured into the aquarium, the water level is 2 inches from the
top of the container.
Step A: What is the volume of the aquarium? Show your work.
240
200
160
120
0 80
40
Step B: Marty said the dimension of the aquarium must be 8 x 5 x 6 in.
However, Jackson said the dimension of the aquarium were 10 x 4 x 6 in
Who is correct? Explain how you know.
Both Marty and Jackson could be correct.
Marty: 8 x 5 = 40 sq in. This would be the area of the base, and
40 x 6 = 240 cubic inches would be the volume.
Jackson: 10 x 4 = 40 sq in. This would be the area of the base, and
40 x 6 = 240 cubic inches would be the volume.
Wicomico County Board of Education
2014-2015
Grade 5
5
Unit 1
Cari is the lead architect for the city’s new aquarium. All of the
tanks in the aquarium will be rectangular prisms where the side
lengths are whole numbers.
 Cari knows that a certain species of fish needs at least 240 cubic feet
of water in their tank. Create 3 separate tanks that hold exactly 240
cubic feet of water.
 In the back of the aquarium, Cari realizes that the ceiling is only 10
feet high. She needs to create a tank that can hold exactly 100 cubic
feet of water. Name one way that she could build a tank that is not
taller than 10 feet.
There are many possible solutions to this task. Here are several possible answers:
Length
Width
Height
8 ft
6 ft
5 ft
10 ft
8 ft
3 ft
10 ft
6 ft
4 ft
12 ft
5 ft
4 ft
15 ft
4 ft
4 ft
The teacher should prompt students to think about what their tank would look like in
real-life. A tank that is 40 feet long, 3 feet wide and 2 feet high would be very long,
skinny and short. This tank would not be good for bigger fish, but it could be useful to
watch something small that does not need a lot of space to move, such as a turtle.
There are many possible solutions to this task. Here are several possible answers:
Length
5 ft
2 ft
10 ft
25 ft
25ft
Wicomico County Board of Education
Width
5ft
10 ft
1 ft
2 ft
4 ft
Height
4 ft
5 ft
10 ft
2 ft
1 ft
2014-2015
Grade 5
6
Unit 1
A local home improvement store sold mulch. The store had
a large container to hold the mulch that was 20 yards long, 12 yards
wide and 8 yards tall. The store manager filled the container with
bags that each contained 12 cubic yards of mulch. How many bags
of mulch will it take to fill the container? ________ bags
8 yards
12 yards
20 yards
The volume of the box is 12 x 8 x 20 = 1,920 cubic yards.
1,920 ÷ 12 = 160. So 160 bags of mulch will fill the container.
Wicomico County Board of Education
2014-2015
Grade 5
7
Unit 1
Using the digits 2,8,6,3 figure out the arrangement of the
digits which gives you the greatest and least product.
Greatest Product:
x
8
2
6
3
82
X 63
6
160
180
4800
5146
5,166
38
X 26
48
180
160
600
988
Least Product:
x
3
8
2
6
988
Wicomico County Board of Education
2014-2015
Grade 5
8
Unit 1
Draw a visual model to solve this problem:
Jasmine was helping her mother color eggs for a party. She carefully
placed 41 eggs into egg cartons. Each carton held 12 eggs. How many
cartons would she need for the eggs? ____________
4 cartons
Find all the numbers that divide 100 with a remainder of 4.
What is the largest two-digit divisor that can divide 100 and still
have a remainder of 4?
Use base ten blocks to model this problem. Students should realize they are finding
numbers which divide into 96 evenly. However not all of the factors of 96 will work for
this problem. For example 96 is divisible by 2, but when dividing 100 by 2 will not result
in a remainder of 4.
6, 8, 12, 16, 24, 32, 48, 96 all divide 100 and have a remainder 4. 96 is the largest.
Wicomico County Board of Education
2014-2015
Grade 5
9
Unit 1
Draw a visual model to solve this problem.
The art teacher has 384 paintbrushes for his classroom. He is organizing
his closet and plans to buy plastic boxes for the paintbrushes. He places
16 paintbrushes in each box. How many boxes will he need? _________
The art teacher will need 24 plastic boxes.
Wicomico County Board of Education
2014-2015
Grade 5
10
Unit 1
Generate and solve another division problem with the same quotient and
remainder as the two problems below. Explain your strategy for creating
the new problem.
3
1 7 6 3
5 1
1 2
3
4 2 1 3 8
1 2 6
1 2
The student should generate a division problem with a quotient of 3 and remainder of 12 and uses a
sound strategy (e.g., writes a checking equation ____ = 3 x_____+ 12). For example:
Wicomico County Board of Education
2014-2015
Grade 5
11
Unit 2
A baker uses 5.5 lb of flour daily.
Step A: How many ounces of flour will he use in two weeks? Use words,
numbers, and pictures to explain your thinking. (1 lb = 16 oz)
Step B: The baker’s recipe for a loaf of bread calls for 12 oz of flour. If he
uses all of his flour to make loaves of bread, how many full loaves can he
bake in two weeks?
12
1232
100
-1200
102 R 8
The baker can bake 102
full loaves in two weeks.
32
-24
2
8
Wicomico County Board of Education
2014-2015
Grade 5
12
Unit 2
Step C: The baker sends all his bread to one store.
If he can pack up to 15 loaves of bread
in a box for shipping, what is the minimum number of boxes required to ship all the loaves
baked in two weeks. Explain your reasoning.
15 102
6
The baker will need 7 boxes to ship all the bread.
The last box won’t be full. It will have 12 loaves in
it.
-90
12
Step D: The baker pays $0.80 per pound for sugar and $1.25 per pound for
butter. Write an expression that shows how much the baker will spend if he
buys 6 pounds of butter and 20 pounds of sugar.
(6 x $1.25) + (20 x $0.80)
Step E: Chocolate sprinkles cost 1/10 as much per pound as sugar. Find
the baker’s total cost for 100 pounds of chocolate sprinkles. Explain the
number of zeros and the placement of the decimal in your answer using a
place value chart.
Wicomico County Board of Education
2014-2015
Grade 5
13
Unit 2
Model the number 8.88 on the place value chart.
Step A: Use words, numbers, and your model to explain why each of the
digits has a different value. Be sure to use “ten times as much” and “one
tenth of” in your explanation.
Step B: Multiply 8.88 x 104. Explain the shift of the digits, the change in
the value of each digit, and the number of zeroes in the product.
Wicomico County Board of Education
2014-2015
Grade 5
14
Unit 2
Step C: Divide the product from (b) by 102. Explain the shift of the digits
and how the value of each digit changed.
Unit 2
Rainfall collected in a rain gauge was found to be 2.3 cm when
rounded to the nearest tenth of a centimeter.
Step A: Circle all the measurements below that could be the actual
measurement of the rainfall.
2.251 cm
2.349 cm
2.352 cm
2.295 cm
Step B: Convert the rounded measurement to meters. Write an equation
to show your work.
2.3 cm ÷ 102 = 0.023m or 2.3 ÷100 = 0.023
Wicomico County Board of Education
2014-2015
Grade 5
15
Unit 2
Express the missing divisor using a power of 10. Explain your reasoning
using a place value model.
a. 5.2 ÷ _____ = 0.052
b. 7,650 ÷ _______ = 7.65
The student should express the divisors as powers of 10 either as multiples of 10 or
exponents and shows correct reasoning on a place value chart for both Part (a) and
Part (b).
a. 100 or 102 or both
Wicomico County Board of Education
b. 1000 or 103 or both
2014-2015
Grade 5
16
Unit 2
Show three equivalent forms for the number seventy-two hundredths.
Possible solutions:
0.72
72/100
720/1000
70/100 + 2/100
7/10 + 2/100
(7 x .10) + (2 x .01)
Write this number in standard form
0.267
2 x (1/10) + 6 x (1/100) + 7 x (1/1000)
Write two more equivalent forms for this number.
267/1000
2/10 + 6/100 + 7/1000
0.2 + 0.06 + 0.007
Wicomico County Board of Education
2014-2015
Grade 5
17
Unit 2
The table below shows the times each grade took to
complete a Math Olympics Task. The 2nd grade
completed their task in a time longer than the 4th grade but
less than the 1st grade. Write a possible time in the table
in which the second graders could have completed the
Math Olympics Task.
Grade
Minutes
1
8.406
2
8.401,8.402, 8.403,
8.404, and 8.405
3
8.045
4
8.4
Use a number line model to show how you know you are correct.
1st Grade
8.406
8.4
8.405
8.410
4th Grade
Acceptable answers
Wicomico County Board of Education
2014-2015
Grade 5
18
Unit 2
The Wilson Family owns four dogs. These are the
weights in pounds of the four dogs:
51.6
51.06
51.066
51.549
Write the decimals in order from least to greatest.
51.06 ,
51.066,
51.549,
51.6
Use what you know about decimals to explain how you ordered the
decimal numbers.
Answers may vary. Students may include a number line or a representation of the
numbers using base ten blocks, or a place value table to prove their answer.
Wicomico County Board of Education
2014-2015
Grade 5
19
Unit 2
Bob is making a snack tray for a party. He bought two
packages of cheese, ½ pound of Swiss and 3/5 of a pound
of Cheddar. Bob also bought 0.8 pound of ham, 0.75 pound
of turkey, and 0.375 pound of salami. Write the foods in order
from least to greatest weight.
Students may convert all of the weights to equivalent fractions and or decimals in
order to compare the foods.
Food
Fraction
Swiss
½
Equivalent
Fraction
500/1000
Cheddar
3/5
600/1000
.600
Ham
8/10
800/1000
.800
Turkey
75/100
750/1000
.750
Salami
375/1000
375/1000
.375
Decimal
.500
Use this number line to show how you would round 14.375 to
the nearest tenth.
14.375
14.3
14.350
14.370
14.4
14.375 is between 14.3 and 14. Since 14.375 is closer to 14.4than 14.3, it would
round to 14.4.
Wicomico County Board of Education
2014-2015
Grade 5
20
Units 2 & 3
A race is limited to 1,000 cyclists divided into 8 equal groups.
Each group will start the race at a different time. The first group
starts at 8:00 AM, the second group starts at 8:15 AM, the third
group starts at 8:30 AM, and so on. Write and model a decimal
number for the fraction of the 1,000 cyclists that is represented by
each group. What time will the last group start the race?
1000 ÷ 8 = 125. Each group consists of 125 cyclists. This can be represented
as a fraction and decimal; 125/1,000 and 0.125.
The last group of cyclists will begin the race at 9:45.
Wicomico County Board of Education
2014-2015
Grade 5
21
Unit 3
Carolina’s Banana Pudding Recipe
Ingredients





2 cups sour cream
5 cups whipped cream
3 ¾ cups vanilla pudding mix
4 cups milk
8 bananas
Carolina is making her special banana pudding recipe. She is looking for
her measuring cup, but can only find the "1/4 cup container."
Step A: How many times must she fill the "1/4 cup container" to put in
the correct amount of sour cream? Draw a picture to illustrate your
solution, and write an equation that represents the situation.
Step B: Carolina does not remember in what order she added the
ingredients but the last ingredient added required 16 quarter cups. What
was the last ingredient Carolina added to the pudding? Draw a picture to
illustrate your solution, and write an equation that represents the
situation.
The purpose of this task is to provide students with a concrete situation they can model by
dividing a whole number by a unit fraction. For students who are just beginning to think about the
meaning of division by a unit fraction (or students who have never cooked), the teacher can bring
in a 1/4 cup measuring cup so that students can act it out. Students may need a great deal of
practice seeing the connection between the visual representation and the more abstract
equations. This task provides an excellent opportunity for teachers to emphasize the relationship
between multiplication and division, and if students do not automatically see the connection, the
teacher should draw their attention to it.
The approaches described in the solution can be used to make the connection between dividing
by a whole number and multiplying by its reciprocal. If students are just beginning to understand
the meaning of dividing by a unit fraction, such a conversation should be postponed. However, if
they feel comfortable solving these kinds of problems with pictures, this task provides a perfect
opportunity to see that dividing a whole number by a unit fraction is the same as multiplying by
the reciprocal of the unit fraction. This, in turn, prepares them for the more general result that
dividing by any fraction is the same as multiplying by its reciprocal, which students will see in 6th
grade.
Wicomico County Board of Education
2014-2015
Grade 5
22
1. Carolina would need to fill the "1/4 cup container" with sour cream 8 times
because (as we can see in the picture below), there are 4 "1/4 cups" in 1 cup, and
so it takes 2×4=8 "1/4 cups" to make 2 cups:
This is a "how many groups?" division problem because it asks "How many '1/4
cups' are in 2 cups?" There are two correct equations:
2÷1/4= ?
or equivalently:
?×1/4=2
We can verify that 8 is the correct solution by noting that 8×1/4 equals 2.
2.
We can think of this problem several ways. First, we can ask, "How many cups
did I start with if I ended up with 16 '1/4 cups'?" We could write an equation for
this:
? ÷14=16
We could also say, "I have 16 '1/4 cups', and I want to know how many cups this
is, so I can multiply 16 and 1/4 to find the number of cups I started with." This can
be represented by the following equation:
16×1/4= ?
Notice that these two equations are equivalent and can both be interpreted in
terms of the following picture:
Here is yet another approach: If we note that there are 4 "1/4 cups" in 1 cup, we
can also think of this as a "how many groups?" division problem. We know that
there are a total of 16 "1/4 cups," and that there are 4 "1/4 cups" in a cup (a
group), and we want to know how many cups (or how many groups) this makes.
Here the equation is:
16÷4=?
Since 16÷4=4, she must have milk as the last ingredient.
Wicomico County Board of Education
2014-2015
Grade 5
23
Unit 3
Draw a visual model to solve this problem.
Darrien took a Science test on Friday. Three-eighths of the questions
were multiple choice and one-fourth of the questions were true-false.
What fractional part of the total number of questions were either multiple
choice or true-false questions?
Four Possible Models that show 3/8 + ¼ = 5/8:
Set Model
Two Hexagons make the whole
An Area Model
Wicomico County Board of Education
Cuisenaire Rods
2014-2015
Grade 5
24
Unit 3
Emilio rides 7/8 mile from his house to his friend Jake’s house.
Together they ride 1/4 mile to school. How far does Emilio ride to
school that day? Use a visual model to prove your answer is correct.
7/8 + 1/4 = 7/8 + 2/8 = 9/8 miles or 1 1/8 miles (Although students may write their
answer as a mixed number, it is not expected or required.)
Four Possible Models that show 7/8 + 1/4 = 9/8:
Set Model
Two Hexagons make the whole
7/8
¼ = 2/8
An Area Model
Wicomico County Board of Education
Cuisenaire Rods
2014-2015
Grade 5
25
Unit 3
The difference between two mixed numbers with unlike denominators
is 3 5/8 . What might the two mixed numbers be? Show as many
different solutions as you can. Explain your strategy for finding
solutions.
Answers may vary. One possible solution is 5 3/4 - 2 1/8 = 3 5/8.
Students should use manipulatives such as pattern blocks, fraction circles or squares,
or Cuisenaire rods to model this problem.
Wicomico County Board of Education
2014-2015
Grade 5
26
Unit 3
3
A cornbread recipe calls for cup of milk. Rachel uses the
4
5
last cup of milk in one carton. She opens another carton and
8
pours the remaining amount needed. How much milk does Rachel
use from the newly opened carton?
Four Possible Models that show 3/4 – 5/8 = 1/8:
Set Model
Two Hexagons make the whole
3/4
3/4 = 6/8 take
away 5/8 and
1/8 is left
Begin with 3/4 then turn it into an
equivalent fraction of 6/8 then take
away 5/8.
An Area Model
Cuisenaire Rods
The white rods
represent eighths.
The red rods
represent fourths
The brown rod represents the whole.
Wicomico County Board of Education
2014-2015
Grade 5
27
Unit 3
Mom bakes a chocolate flavored cake, a vanilla flavored cake, and a
strawberry flavored cake. Each cake is the same size. Angela eats
four-eights of the chocolate cake, two-eighths of the vanilla cake and
two-fourths of the strawberry cake. David eats one-fourth of the
chocolate cake, three-eighths of the vanilla cake, and one-half of the
strawberry cake. Who eats the most cake? How much of each cake is
left? Show all of your mathematical thinking.
A
A
A
A
A
A
D
D
D
D
Chocolate
D
Vanilla
A
A
A
A
D
D
D
D
Strawberry
Angela ate 10/8 and David ate 9/8. Angela ate more cake than David.
The chocolate cake has 2/8 left and the vanilla cake has 3/8 left. There is no
strawberry cake left.
Wicomico County Board of Education
2014-2015
Grade 5
28
Unit 3
If the whole bar is 3 units long, what is the length of the shaded part of
the bar? Write a multiplication equation for the diagram, and then solve.
0
Circle the expression(s) that are equal to
3
5
× 6. Explain why the others
are not equal using words, pictures, or numbers.
a. 3 × (6 ÷5)
b. 3 ÷ (5 × 6) = 3 ÷ 30 = 0.1
c. (3 × 6) ÷5
d. 3 ×
6
5
Wicomico County Board of Education
2014-2015
Grade 5
29
Unit 3
Jackson claims that multiplication always makes a number bigger.
He gave the following examples:
 If I take 6, and I multiply it by 4, I get 24, which is bigger than 6.
1
2
1
 If I take , and I multiply it by 2 (whole number), I get , or which
4
4
1
2
is bigger than .
4
Jackson’s reasoning is incorrect. Give an example that proves he is wrong,
and explain his mistake using pictures, words, or numbers.
Draw a visual model to solve this problem.
Helen gave her son Matt two hot dogs for dinner. She cuts each hot
dog into sixths. Matt eats all but half of the last piece. How much of his
dinner is left?
Each hot dog is cut into sixths, however both of the hot dogs make the whole dinner. Therefore the
pieces of hot dog represent twelfths. If Matt eats all but ½ of a twelfth, then he left 1/24 of his dinner.
Use two pattern blocks to represent the hot dogs. Each pattern
block can be divided into sixths with the green triangle. The whole
becomes twelfths. Matt eats all of the pieces except half of the last
twelfth. Half of the last twelfth would be represented by the purple
triangle. Since it takes 24 purple triangles to make the whole, Matt
left 1/24 of his dinner.
Half of the last piece would be a purple triangle. The purple triangle represents
1/24 of the whole which is two hexagons.
Wicomico County Board of Education
2014-2015
Grade 5
30
Unit 3
Draw a visual model to solve and represent this problem.
Sophie’s gave Mike 1/2 of a candy bar. Mike ate 1/3 of it. How
much did he eat? 1/6 of the whole candy bar
Draw a visual model to solve and represent this problem.
3
The Smith’s garden covers of an acre. One third of the garden is
4
vegetables, and the rest is flowers. What part of the acre is planted
in flowers? 6/12 of the acre is planted in flowers
1/3 vegetables
Wicomico County Board of Education
2014-2015
Grade 5
31
Unit 3
Draw a visual model to solve and represent this problem.
Kim is at the movies with her friend. The movie lasts 2 hours long.
3
Kim’s friend decided into the movie that she wanted some popcorn.
4
How long did they watch the movie, before getting popcorn? _______
Each hexagon represents an hour of the movie. Since the movie lasts 2 hours, the
whole is represented by both hexagons. A red trapezoid represents a fourth of the
whole. Three trapezoids represent 3/4. This 3/4 would be an hour and a half into the
movie.
Another possible model using Cuisenaire rods:
3/4 of the whole
1 hour
1 hour
whole
Wicomico County Board of Education
2014-2015
Grade 5
32
Unit 3
Draw a visual model to solve and represent this problem.
4
4
5
5
At lunch Gloria ate of a bag of potato chips, Suzie ate of a bag
1
of potato chips, and John ate of what Suzie ate. How many bags
2
of potato chips did they eat in all? _________
The first step would be to calculate the amount of chips John ate.
John ate1/2 of 4/5:
1/2 x 4/5 = 4/10
Then add the amount of potato chips eaten by Gloria, Suzie, and John.
4/5 + 4/5 + 4/10
8/10 + 8/10 + 4/10= 20/10
Two whole bags of potato chips were eaten.
Wicomico County Board of Education
2014-2015
Grade 5
33
Unit 3
Draw a visual model to solve this problem.
3
Megan has yard of green felt to use on two bulletin boards.
4
She wants to use the same amount of felt on each board. How
much felt will she use on each bulletin board?
1 yard of green felt
This task focuses on dividing a fraction by a whole number ¾ ÷ 2 or the inverse
which would be multiplying a fraction by a fraction ½ of ¾.
Students may begin by using manipulatives such as paper or pattern blocks to model
the problem. Students should then transfer their model to a representation.
Each bulletin board will be covered with 3/8 yard of felt.
Wicomico County Board of Education
2014-2015
Grade 5
34
Unit 3
Kulani is painting his room. He needs 1/3 of a gallon to paint the whole
room. What fraction of a gallon will he need for each of his 4 walls if he
uses the same amount of paint on each? Explain your work and draw a
picture to support your reasoning.
In order to find out what fraction of a gallon of paint Kulani will use on each wall, we
need to divide the amount of paint that he has, 1/3 of a gallon, by 4.
We know that the fraction 1/3 represents 1 piece where 3 equal pieces make a whole
gallon. We can draw a rectangle that represents a whole gallon and shade 1/3 of it to
represent the amount of paint he has. If we divide each of the 3 equal pieces in the
picture into 4 smaller, equal-sized pieces, then the whole gallon is now divided into
4×3=12 equal pieces. This means that one of these new, smaller equal pieces
represents the fraction 1/12:
Note that by dividing every third into 4 equal pieces, we divided the 1/3 that represents
Kulani's paint into 4 equal pieces, and in particular each of these pieces is 1/12 of the whole
gallon. This means that 1/3÷4=1/12. Based on our understanding of the relationship between
multiplication and division, we can also see that 4× 1/12= 1/3.
So Kulani will need to use 1/12 of a gallon of paint on each wall.
Wicomico County Board of Education
2014-2015
Grade 5
35
Unit 3
Salad Dressing Recipe




1/3 cup olive oil
1/6 cup balsamic vinegar
a pinch of herbs
a pinch of salt
Makes 6 servings
1. How much salad dressing will this recipe make? Write an equation to
represent your thinking. Assume that the herbs and salt do not change
the amount of dressing.
2. If this recipe makes 6 servings, how much dressing would there be in
one serving? Write a number sentence to represent your thinking.
The purpose of this task is to have students add fractions with unlike denominators and
divide a unit fraction by a whole number. This accessible real-life context provides
students with an opportunity to apply their understanding of addition as joining two
separate quantities. Additionally, the context presents a "how many groups" division
problem where a unit fraction should be divided into 6 equal groups.
1. The total amount of dressing can be found by adding the oil and vinegar (since
the other two ingredients won't change the amount). Here is a picture that
represents the amount of oil:
and here is a picture that represents the amount of vinegar:
We can show the sum like this:
Wicomico County Board of Education
2014-2015
Grade 5
36
Unit 3
After a class potluck, Emily has three equally sized apple pies left and
she wants to divide them into eight equal portions to give to eight
students who want to take some pie home.
 Draw a picture showing how Emily might divide the pies into eight
equal portions. Explain how your picture shows eight equal portions.
 What fraction of a pie will each of the eight students get?
 Explain how the answer to (b) is related the division problem 3÷8.
The purpose of this task is to help students see the connection between a÷b and ab in a particular
concrete example. The relationship between the division problem 3÷8 and the fraction 3/8 is actually
very subtle. 3÷8 is the number you multiply 8 by to get 3. 3/8 is the number you get by taking 3 copies
of the unit fraction 1/8.
So 3÷8 is defined in terms of multiplication, and 3/8 is defined in terms of unit fractions. It is not
obvious that these two numbers are the same, so students need opportunities to see that they will
necessarily always be the same. Note that if b people share a pies equally, then each person will get
a/b of a pie by the same kind of reasoning shown in the solution below.
Below is a picture of how Emily might divide the three apple pies into eight equal portions. Here each
color (yellow, orange, red, green, blue, dark purple, light purple, and light blue) represents one
portion. So each portion consists of three pieces of pie and each piece of pie represents 18 of a full
pie:
Because these pies are all the same size and they are all apple pies, Emily does not need to give
each student one piece of each of the three pies: two or three pieces of the same pie could go to one
student. This picture, however, shows clearly that the pies have been divided into eight equal
portions. If multiple pieces of a particular pie were to go to the same student, it would be necessary to
analyze the picture more closely and count how many slices of pie each student received to check
that it has been divided evenly. As the picture shows, each portion consists of three slices of apple
pie. Since these slices represent 1/8 of a pie, this means that each student gets 3/8 of an apple pie.
If 3 pies are divided into 8 equal portions, then 8 of these portions makes 3 pies, a fact that is clearly
illustrated in part (a). We can write this in symbols if we use a question mark to represent the amount
of pie in one portion:
8×?=3
When we know a factor and the product, we can find the other factor by dividing:
3÷8=?
So one person's portion is whatever we get when we divide 3 by 8. In part (b), we saw that one
portion is 3/8. So that means that
3÷8=3/8.
Wicomico County Board of Education
2014-2015
Grade 5
37
Unit 4
The following equations involve different quantities and use different
operations, yet produce the same result. Use a place value mat and
words to explain why this is true.
4.13 x 103 = 4130
Wicomico County Board of Education
413,000 ÷ 102 = 4130
2014-2015
Grade 5
38
Unit 4
Joaquin hiked 0.56 mile in the morning and 1.45 miles in the
afternoon. Joaquin wants to record in his journal how many miles
he hiked altogether and how much farther he hiked in the afternoon.
What distances should he put in his journal?
Use base ten blocks to represent this addition and subtraction problem:
+
=
Joaquin hiked 2.01 miles altogether.
_
=
Joaquin hiked 0.89 miles farther in the afternoon
Wicomico County Board of Education
2014-2015
Grade 5
39
Unit 4
The cost of making money varies from coin to coin:
Pennies 0.8 cents each
Nickels 2.9 cents each
Dimes 1.7 cents each
Quarters 3.7 cents each
Half dollars 7.8 cents each
What is the cost of making $5.00 worth of each coin?
It may be helpful to suggest students make a table to organize their information.
Students need to determine how many of each coin it takes to equal $5.00. Then they
multiply that number by the cost.
Coin
Penny
Nickel
Dime
Quarter
Half-dollar
Number of
Coins Needed
to Make $5.00
500
100
50
20
10
Wicomico County Board of Education
Cost to Make
One
Total Cost
0.8
2.9
1.7
3.7
7.8
$4.00
$2.90
$0.85
$0.74
$0.78
2014-2015
Grade 5
40
Unit 4
The table shows four people who earn the typical amount for their
education level.
Name
Level of Education
Weekly Income
Miley
Drop out
High School
Graduate
2-Year College
Graduate
4-Year College
Graduate
$440.50
Niko
Taylor
Pinky
$650.35
$771.25
$1,099.20
How much money will Pinky earn in a month? About how long will Miley
have to work to earn the same amount?
There are four weeks in a month. Pinky makes $1099.20 in a week so she will make
4×1099.20=4×1100−4×0.80=4400−3.20=4396.80
Pinky will make $4396.80 in a month. We can divide this by how much Miley makes in
one week to find out how many weeks she will have to work.
4396.8÷440.5≈4400÷440=10
Miley will have to work about 10 weeks, or two and a half months, to earn the same
amount that Pinky will make in one month.
Wicomico County Board of Education
2014-2015
Grade 5
41
Unit 4
Kevin works in a factory. He has to make 3.2 car parts in an hour to
make the number of parts needed in a 6.5 hour workday. Use an area
model to show how many parts Kevin should make in one day.
6 x 3 = 18
15 x .10 = 1.5
10 x .01 = .10
12 x .10 = 1.2
18.00
1.20
1.50
0.10
Partial Products
20.80
Students will use mathematical tools and model with mathematics as they use base ten
blocks to make the area model for 3.2 x 6.5. They attend to precision as they transfer
their model to the representation and determine the total of the partial products shown
in their models.
Wicomico County Board of Education
2014-2015
Grade 5
42
Unit 4
Sarah says that 26 ÷ 8 equals 14 ÷ 4 because both are “3 R2.” Explain
her mistake using decimal division.
Wicomico County Board of Education
2014-2015
Grade 5
43
Unit 5
Mr. Schmidt used 10 buckets to collect rainfall in various locations on his
property. The following line plot shows the amount of rain collected in
each bucket in gallons. Write an expression that includes multiplication
to show how to find the total amount of water collected in gallons. Then
solve your expression.
Rainfall Collected
Gallons
Wicomico County Board of Education
2014-2015
x = 1 bucket
Grade 5
44
Unit 5
The students in Raul’s class were growing grass seedlings in
different conditions for a science project. He noticed that Pablo’s
seedlings were 11/2 times a tall as his own seedlings. He also saw
that Celina’s seedlings were 3/4 as tall as his own. Which of the
seedlings shown below must belong to which student? Explain your
reasoning.
Since Pablo’s seedlings are 11/2 times a tall as Raul’s, Pablo’s seedlings must be taller
than Raul’s. Pablo’s seedlings looks like Raul’s seedlings scaled by a factor larger than
1.
Since Celina’s seedlings are 3/4 as tall as Raul’s, Celina’s seedlings must be shorter
than Raul’s. Celina’s seedlings looks like Raul’s seedlings scaled by a factor smaller
than 1.
Celina’s seedlings are shorter than Raul’s and Pablo’s are taller than Raul’s. Thus, the
seedlings in pot A belong to Celina, the seedlings in pot B to Raul, and the seedlings in
pot C belong to Pablo.
Wicomico County Board of Education
2014-2015
Grade 5
45
Unit 5
Mrs. Williams uses the following recipe for crispy rice treats. She
2
decides to make of the recipe.
3
2 cups melted butter
24 oz marshmallows
13 cups rice crispy cereal
Step A: How much of each ingredient will she need? Write an expression
which includes multiplication. Solve by multiplying.
Step B; How many fluid ounces of butter will she use?
Wicomico County Board of Education
2014-2015
Grade 5
46
Unit 5
Step C: When the crispy rice treats have cooled, Mrs. Williams cuts
them into 30 equal pieces. She gives two-fifths of the treats to
her son and takes the rest to school. How many treats will
Mrs. Williams take to school? Use any method to solve.
Luke had a calculator that will only display numbers less than or equal
to 999,999,999. Without solving these problems, which of the following
products will his calculator display? Explain.
1.
2.
3.
4.
792 × 999,999,999
½ × 999,999,999
15/4 × 999,999,999
0.67 × 999,999,999
It is much easier to answer the question if you can think of multiplying a
number by a factor as scaling the number. Since multiplying a positive number by
a factor greater than 1 always results in a larger number, the first and third
products will be too large to display.
Since multiplying a positive number by a factor less than 1 (but greater than zero)
always results in a smaller number, the second and fourth products will be
displayed on his calculator.
Wicomico County Board of Education
2014-2015
Grade 5
47
Unit 5
Draw a visual model to solve this problem.
A parking space for a school bus measures 15 3/4 yards long
and 4 1/2 yards wide. What is the area of the parking space?
4 ½ yards
To find the area of
the parking space
add the partial
products:
15 ¾ yards
60+ 7 ½ + 3 + 3/8 =
70 7/8 sq yards
15 x 4= 60 sq yards
15 x ½ = 7 ½ sq yards
½ x ¾ = 3/8 sq yards
yards
¾ x 4 = 3 sq yards
Wicomico County Board of Education
2014-2015
Grade 5
48
Unit 6
Read the statements. Circle “True” or “False.” Explain your choice
for each using words and/or pictures.
a. All parallelograms are quadrilaterals.
True
False
True
False
A parallelogram is a kind of quadrilateral with 2 sets of parallel sides.
b. All squares are rhombuses.
A rhombus has 4 equal sides, and so does a square. But not all rhombuses are squares.
c. Squares are rhombuses, but not rectangles.
True
False
Squares are both because they also have 4 right angles.
d. The opposite angles in a parallelogram have the same measure.
True
False
It’s because the sides are parallel to each other and they have to add to 360, so both pairs have to
be the same.
e. Because the angles in a rectangle are 90, it is not a parallelogram.
The sides are still parallel pairs, so it’s a parallelogram.
True
False
f. The sum of the angle measures of any trapezoid is greater than the
sum of the angle measures of any parallelogram.
The angle measures of any quadrilateral, including trapezoids and
parallelograms is always 360.
True
False
g. The following figure is a parallelogram.
True
False
115
60
The angles would have to add up to 180, but they only make 175. This would mean all the angles
together would make 350 degrees, which is impossible. They must total 360 degrees.
Wicomico County Board of Education
2014-2015
Grade 5
49
Unit 6
How many DIFFERENT quadrilaterals can be made by
joining the dots on these circles? Classify these
quadrilaterals.
Students should have an opportunity to make quadrilaterals and other shapes with
toothpicks or geoboards prior to completing this activity. Clarify that drawing a
congruent shape but using different dots does not count as a different shape.
Students may look for and make use of structure by beginning with the same dot
and joining different dots to make new quadrilaterals.
Wicomico County Board of Education
2014-2015
Grade 5
50
Unit 7
Using the order of operations and parentheses, write at least three
expressions which equal 27.
Possible solutions:
(2 + 7) x 3 = 27
(2 x 10) + ( 3 + 4) = 27
(5 x 3) + (4 x 3) = 27
Evaluate this equation. Is it correct or incorrect?
How do you know?
3 x (18 – 3) + (6 + 4) ÷ 2 = 50
3 x (15) + (10) ÷ 2
45 + 5 = 50
Wicomico County Board of Education
2014-2015
Grade 5
51
Unit 7
Write an expression using grouping symbols to find how many
sides 6 triangles, 4 hexagons, and 5 quadrilaterals have altogether.
(6 x 3) + ( 4 x 6) + ( 5 x 4) = X
18 + 24 + 20 = 62 sides
What is the value of the x?
3 x (8 + 1) - 12 ÷ 3 = x
x =
23
If the parentheses in the expression above were removed, explain
how the value of the expression would change. Use words, numbers,
and/or symbols in your explanation.
If the parentheses are removed, the value of the expression changes to 21
Write an expression which equals 50. You must use all
four operations at least once and at least one set of parentheses
in your expression.
Possible answer: (5 x 10) + (20 ÷ 4) – 5 = 50
Wicomico County Board of Education
2014-2015
Grade 5
52
Unit 8
Complete the table for the rule multiply by 2 then add 2 for the
values of 𝑥 from 0 to 4. Then use the coordinate plane to answer the
questions.
𝒙
𝒚
(𝒙, 𝒚)
0
2
(0,2)
1
4
(1,4)
2
6
(2,6)
3
8
(3,8)
4
10
(4,10)
e
a. Which line shows the rule in the table? Line a
b. Give the coordinates for the intersection of lines 𝓫 and 𝓬. (2,6)
c. Draw a line on the graph such that any point on the line has a 𝑦-coordinate of 2. Label
your line as 𝒆.
d. Which coordinate is 2 for any point on line 𝓬? The x-coordinate
e. Kim and Lacy want to draw a line on the coordinate plane that is parallel to line 𝓪. Kim
uses the rule, multiply by 4 and add 2 to generate her 𝑦-coordinates. Lacy uses the rule
multiply by 2 and add 4 to generate her y-coordinates. Which girl’s line will be parallel to
line 𝓪? Without graphing the lines, explain how you know.
Lacy’s rule will make a line parallel to line a. The rule for a is multiply x by 2, add 2. The
rule for Lacy’s line is multiply x-coordinate by 2 and add 4. Lacy’s line is parallel
because the steepness of the line is the same. The multiplication part of the rule is the
same. The adding part of the rule will make the y-coordinates two more than those in
line a.
Wicomico County Board of Education
2014-2015
Grade 5
53
Unit 8
An airplane is descending into an airport. When its altitude is 5 miles, it is 275 miles from the
airport. When its altitude is 4 miles, it is 200 miles from the airport. At 3 miles, it is 125 miles
from the airport.
5 mi
The plane’s altitude will be 2 miles
4 mi
Step B: For the plane to land at the airport, the
altitude will need to be 0 and the distance from
the airport will need to be 0. Should the pilot
continue this pattern? Why or why not?
Altitude
Step A: If the pilot follows the same pattern, what will
the plane’s altitude be at 50 miles from the airport?
3 mi
2 mi
1 mi
50
The pilot should not continue this pattern. If the pilot continues
this course the plane will be past the airport when it reaches 0
altitude.
Wicomico County Board of Education
2014-2015
100
150
200
Miles from Airport
Grade 5
54
250
Unit 9
2
In the diagram, the length of S is the length of T. If S has an area of
3
368 cm2, find the perimeter of the figure.
S
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16 cm
T
2014-2015
Grade 5
55
Unit 10
Mrs. Hinoj had 75 feet of ribbon. If each of the 18 students in her class
gets an equal length of ribbon, how long will each piece be? Write
your answer.
 Using only feet,
 Using a whole number of feet and a whole number of inches,
and
 Using only inches.
Each student will get a piece of ribbon that is 4 1/6 feet long, which is also 4 feet 2
inches or 50 inches. In this problem the whole set consists of the ribbon and the
number of subsets is 18, namely the number of students in the class. The student
must know that there are 12 inches in one foot to complete the problem.
Wicomico County Board of Education
2014-2015
Grade 5
56
Unit 10
James M. Bennett is replacing the existing grass soccer field
with a new type of grass. They are looking for the best buy for
their money.
Below are price quotes from various local nurseries:
6’ x 2’ roll $1.50
6’ x 4’ roll $3.50
6’ x 5’ roll $4.00
8’ x 2’ roll $2.00
The field dimensions are 120ft x 160 ft
How many rolls of sod will be needed for each size?
What will be the total cost of the sod for each size?
Which is the best buy?
If the truck which delivers the sod holds a maximum of 1500 rolls, and
charges $100 for each trip, which is the best buy?
(Optional) Make a scale diagram of how the sod will be laid down on
the field.
Roll Size
6’ x 2’
6’ x 4’
6’ x 5’
8’ x 2’
Number of
Rolls
Needed
1600
800
640
1200
Cost Per
Roll
Total Cost
$1.50
$3.50
$4.00
$2.00
$2400.00
$2800.00
$2560.00
$2400.00
The 6’ x 2 and 8’ x 2’ sizes will be the best buy, however once the delivery charge is
added the 8’ x 2’ size would cost the least because it would take two deliveries for
the 6’ x 2’ size which would add an additional $200 to the cost. The 8’ x2’ size
would require only one delivery which would be an additional $100 in cost.
Wicomico County Board of Education
2014-2015
Grade 5
57
Unit 10
Five fifth grade classes are planning n end-of-the-year celebration.
There are a total of 120 students invited to the celebration. The
teachers have decided to rent tables and chairs from the local Rent-ACenter. They charge $15 per square table and $8 per chair. The
square tables can seat 2 people on each side. Write an expression for
all the ways you could arrange the tables to seat 120 people. Use
pictures and charts for your solution. Find the largest number of tables
as well as the smallest number of tables that could be used to seat 120
people.Calculate the cost of the different seating arrangements.
15 of these
6 of these
10 of these
Number Cost Number Cost
of
of
of
of
Tables Tables Chairs Chairs
Total
Cost
15
$225
120
$960
$1185
20
$300
120
$960
$1260
24
$360
120
$960
$1320
15 x 8 = 120
10 x 12 = 120
6 x 20 = 120
Wicomico County Board of Education
2014-2015
Grade 5
58
Unit 10
Miguel and Jacqui built towers out of craft sticks. Miguel’s tower had
a 4-inch square base. Jacqui’s tower had a 6-inch square base. If
Miguel’s tower had a volume of 128 cubic inches, and Jacqui’s had a
volume of 288 cubic inches, whose tower was taller? Explain your
reasoning.
Wicomico County Board of Education
2014-2015
Grade 5
59
Unit 10
Jeanne makes hair bows to sell at the craft fair. Each bow requires 1.5
yards of ribbon.
Step A: At the fabric store, ribbon is sold by the foot. If Jeanne wants
to make 84 bows, how many feet of ribbon must she buy? Show all
your work.
Step B: If the ribbon costs 10¢ per foot, what is the total cost of the
ribbon in dollars? Explain your reasoning, including how you
decided where to place the decimal.
Step C: A manufacturer is making 1,000 times as many bows as
Jeanne to sell in stores nationwide. Write an expression using
exponents to show how many yards of ribbon the manufacturer will
need. Do not calculate the total.
Wicomico County Board of Education
2014-2015
Grade 5
60