Unit 3 Plan - Grade 4

Lesson 1

Equal Groups of Unit Fractions

Sandwiches on Plates

Lin has 9 plates. She puts $\frac14$ of a sandwich on each plate.

Which expression represents the sandwiches in this situation?
1. $9 \times 4$
2. $9 \times \frac{1}{4}$
3. $4 \times 9$
4. $4 \times \frac {1}{9}$
How many sandwiches did Lin put on plates? Explain or show your reasoning.

Show Solution

B
$\frac{9}{4}$ sandwiches or $2\frac{1}{4}$ sandwiches. Sample responses:
- A diagram showing 9 groups of $\frac{1}{4}$ .
- I counted by $\frac{1}{4}$ nine times.
- I know 4 groups of $\frac{1}{4}$ sandwiches is 1 whole sandwich, so 8 groups of $\frac{1}{4}$ sandwiches make 2 whole sandwiches. Adding $\frac{1}{4}$ sandwich makes $2\frac{1}{4}$ .

Lesson 2

Representations of Equal Groups of Fractions

Equal Groups of Fractions

Write a multiplication expression to represent the shaded parts of the diagram. Then find its value. Explain or show your reasoning.

Show Solution

$6 \times \frac{1}{12}$ . Its value is $\frac{6}{12}$ . Sample response:

There are 6 equal groups of $\frac{1}{12}$ .
I counted by $\frac{1}{12}$ six times.
If all the shaded parts are moved to a single rectangle that represents 1 whole, they would take up 6 parts, which represent $\frac{6}{12}$ .

Lesson 3

Patterns in Multiplication

Fraction Multiplication

Complete each equation to make it true. Show your thinking, using words or diagrams.
1. $5 \times \frac{1}{8} = \underline{\hspace{.5in}}$
2. $\underline{\hspace{0.5in}} \times \frac{1}{3} = \frac{7}{3}$
Write each fraction as the product of a whole number and a unit fraction.
1. $\frac{8}{9} = \underline{\hspace{.5in}} \times \underline{\hspace{.5in}}$
2. $\frac{6}{5} = \underline{\hspace{.5in}} \times \underline{\hspace{.5in}}$

Show Solution

1. $5 \times \frac{1}{8} = \frac{5}{8}$ , five groups of 1 eighth make 5 eighths.
2. $7 \times \frac{1}{3} = \frac{7}{3}$ , because I drew a diagram showing 7 groups of $\frac{1}{3}$ .
1. $8 \times \frac{1}{9}$
2. $6 \times \frac{1}{5}$

Lesson 4

Equal Groups of Non-unit Fractions

What’s the Value?

Find the value of each expression. Explain or show your reasoning. Use a diagram if it is helpful.

$6 \times \frac{2}{5}$
$5 \times \frac{3}{10}$

Show Solution

$\frac{12}{5}$ . Sample response: 6 groups of 2 fifths make& 12 fifths.
$\frac{15}{10}$ . Sample response: 5 groups of $\frac{3}{10}$ make $\frac{15}{10}$ .

Lesson 5

Equivalent Multiplication Expressions

Expressions for Fractions

Kiran says that the expressions $2 \times \frac{6}{8}$ and $3 \times 4 \times \frac{1}{8}$ both represent the same fraction. Do you agree? Explain or show your reasoning.
Write two new expressions that have the same value as $12 \times \frac{1}{9}$ . You can use a diagram if it is helpful.

Show Solution

Agree. Sample response: $2 \times \frac{6}{8}$ is $\frac{12}{8}$ or 12 groups of $\frac{1}{8}$ , and $3 \times 4 \times \frac{1}{8}$ is $12 \times \frac{1}{8}$ , which is also 12 groups of $\frac{1}{8}$ .
Sample responses: $4 \times \frac{3}{9}$ , $6 \times \frac{2}{9}$ , $2 \times 3 \times \frac{2}{9}$

Lesson 6

Problems with Equal Groups of Fractions

The Same or Not the Same?

Tyler bought 5 cartons of milk. Each carton contains $\frac{3}{4}$ liter. How many liters of milk did Tyler buy? Explain or show your reasoning.
Han bought 3 cartons of chocolate milk. Each carton contains $\frac{5}{8}$ liter. Did Han buy the same amount of milk as Tyler? Explain or show your reasoning.

Show Solution

$\frac{15}{4}$ liters. Sample response: $5 \times \frac{3}{4} = \frac{15}{4}$
No, Han bought less milk than Tyler did. Sample response: $3 \times \frac{5}{8} = \frac{15}{8}$ , and $\frac{15}{8}$ is less than $\frac{15}{4}$ because an eighth is less than a fourth.

Section A Check

Section A Checkpoint

Problem 1

Select all diagrams that show $4 \times \frac{1}{3}$ .

A.

3 diagrams of equal length. 4 equal parts. 1 part shaded. Total length, 1.

B.

4 Diagrams of equal length. 3 equal parts. 1 part shaded. Total length, 1.

C.

3 diagrams of equal length. 3 equal parts, 1 part shaded. Total length, 1.

D.

Diagram. 3 equal parts each labeled 1 fourth.

E.

Diagram. 4 equal parts each labeled 1 third.

Show Solution

B, E

Problem 2

Select all expressions that are equivalent to $\frac{8}{5}$ .

A.

5 \times \frac{1}{8}

B.

8 \times \frac{1}{5}

C.

2 \times \frac{4}{5}

D.

4 \times \frac{2}{5}

E.

2 \times \frac{6}{5}

Show Solution

B, C, D

Problem 3

Draw a diagram showing $4 \times \frac{2}{3}$ .
Use the diagram to calculate $4 \times \frac{2}{3}$ .

Show Solution

Sample response:
$\frac{8}{3}$ . Sample response: There are $4 \times 2$ or 8 shaded parts and each part is $\frac{1}{3}$ of a full rectangle.

Lesson 7

Fractions as Sums

Make a Sum of $\frac{7}{4}$

Find three different ways to use fourths to make a sum of $\frac{7}{4}$ .

Write an equation for each.

Show Solution

Sample responses:

$\frac{1}{4} + \frac{2}{4} + \frac{4}{4} = \frac{7}{4}$
$\frac{6}{4} + \frac{1}{4} = \frac{7}{4}$
$\frac{5}{4} + \frac{2}{4} = \frac{7}{4}$

Lesson 8

Addition of Fractions

Lucky Thirteen-Tenths

On each number line, draw two “jumps” to show how to use tenths to make a sum of $\frac{13}{10}$ .
1. Represent each combination of jumps as an equation.
2. Write $\frac{13}{10}$ as a sum of a whole number and a fraction.
Find the value of $\frac{8}{5} + \frac{6}{5}$ . Use the number line if you find it helpful.

Show Solution

Sample response:
1. $\frac{10}{10} + \frac{3}{10} = \frac{13}{10}$ and $\frac{5}{10} + \frac{8}{10} = \frac{13}{10}$
2. $1 + \frac{3}{10} = \frac{13}{10}$
Number line. 21 evenly spaced tick marks. First tick mark, 0. Eleventh, 1. Twenty first, 2. Arrow, labeled ten tenths, from first tick mark to eleventh tick mark. Arrow, labeled three tenths, from eleventh tick mark to point at fourteenth tick mark, labeled thirteen tenths.

Number line. 21 evenly spaced tick marks. First tick mark, 0. Eleventh, 1. Twenty first, 2. Arrow, labeled five tenths, from first tick mark to sixth tick mark. Arrow, labeled eight tenths, from sixth tick mark to point at fourteenth tick mark, labeled thirteen tenths.
$\frac{14}{5}$ or $2\frac{4}{5}$

Lesson 9

Differences of Fractions

Differences of Fifths

Use a number line to represent each difference and find its value.

$\frac{12}{5} - \frac{4}{5}$
$2\frac{1}{5} - \frac{7}{5}$

Show Solution

$\frac{12}{5} - \frac{4}{5} = \frac{8}{5}$ or $\frac{12}{5} - \frac{4}{5} = 1\frac{3}{5}$ . Sample responses:
$2\frac{1}{5} - \frac{7}{5} = \frac{4}{5}$ . Sample responses:

Lesson 10

The Numbers in Subtraction

Two Differences

Find the value of each difference. Show your reasoning.

$2 - \frac{5}{6}$
$4 - \frac{11}{6}$

Show Solution

$\frac{7}{6}$ or $1\frac{1}{6}$ . Sample response:
- $2 - \frac{5}{6} = \frac{12}{6} - \frac{5}{6} = \frac{7}{6}$
- $2 - \frac{5}{6} = \left(1 + \frac{6}{6}\right) - \frac{5}{6} = 1 + \left(\frac{6}{6} - \frac{5}{6}\right) = 1 + \frac{1}{6} = 1\frac{1}{6}$
$\frac{13}{6}$ or $2\frac{1}{6}$ . Sample response:
- $4 - \frac{11}{6} = \frac{24}{6} - \frac{11}{6} = \frac{13}{6}$
- $4 - \frac{11}{6} = \left(2 + \frac{12}{6}\right) - \frac{11}{6} = 2 + \left(\frac{12}{6} - \frac{11}{6} \right) = 2 + \frac{1}{6} = 2\frac{1}{6}$
- $\frac{11}{6}$ is $\frac{1}{6}$ away from 2. I subtracted 2 from 4, and then add $\frac{1}{6}$ back to get $2 \frac{1}{6}$

Lesson 11

Subtract Fractions Flexibly

A Shorter Strip, Please

Lin has a strip of paper that is $7\frac{1}{4}$ inches long and needs to be trimmed by $2\frac{3}{4}$ inches. What is the length of the paper strip after it is trimmed? Explain or show your reasoning.

Show Solution

$4\frac{2}{4}$ inches. Sample reasoning:

$7\frac{1}{4}$ is $6 + 1 + \frac{1}{4}$ , which is $6 + \frac{4}{4} + \frac{1}{4}$ or $6 + \frac{5}{4}$ . I subtracted 2 wholes from 6 wholes, which gives 4 wholes, and then subtracted $\frac{3}{4}$ from $\frac{5}{4}$ , which gives $\frac{2}{4}$ .
I know 3 is $\frac{1}{4}$ more than $2\frac{3}{4}$ . I subtracted 3 from $7 \frac{1}{4}$ to get $4\frac{1}{4}$ , and the I added $\frac{1}{4}$ back because I subtracted $\frac{1}{4}$ more than needed earlier.

Lesson 12

Sums and Differences of Fractions

How Would You Find the Difference?

Consider the expression $\frac {13}{5} - 1\frac{2}{5}$ .

What would be your first step for finding the value of the expression?
Find the value of the expression. Show your reasoning.

Show Solution

Sample responses:
- I would decompose $\frac{13}{5}$ into a whole number and a fraction and write it as a mixed number.
- I would write $1\frac{2}{5}$ as a fraction without a whole number.
$1\frac{1}{5}$ or $\frac{6}{5}$ . Sample response:
- $\frac{13}{5} = \frac{10}{5} + \frac{3}{5} = 2+\frac{3}{5} = 2\frac{3}{5}$ and $2\frac{3}{5} - 1\frac{2}{5} = 1\frac{1}{5}$
- $1\frac{2}{5} = \frac{5}{5} + \frac{2}{5} = \frac{7}{5}$ and $\frac{13}{5}-\frac{7}{5} = \frac{6}{5}$

Lesson 13

Fractional Measurements on Line Plots

Jada’s Pencil Data

Jada measured the lengths of her pencils and displayed her data on a line plot.

The last three pencils in her collection are not yet plotted. Their lengths are: $3\frac{1}{4}$ , $4\frac{3}{8}$ , and $5\frac{1}{4}$ . Plot them on the line plot.
What is the difference in the lengths of the shortest and the longest pencils in her collection? Show your reasoning.

Show Solution

$3\frac{7}{8}$ inches. Sample response: $5\frac{6}{8} - 1\frac{7}{8} = 4 \frac{14}{8} -1\frac{7}{8} = 3\frac{7}{8}$

Lesson 14

Problems about Fractional Measurement Data

Fourth-grade Height Data

The students in a fourth-grade class keep track of their heights all year long. The line plot shows the number of inches each student in the class has grown this year.

Dot plot titled Growth in One Year from 1 to 4 by 1’s. Hash marks by eighths. Horizontal axis, length in inches. Beginning at 7 eighths, the numbers of X’s above each eighth increment are 1, 0, 0, 3, 0, 4, 0, 0, 0, 2, 0, 0, 2, 0, 0, 0, 0, 0, 1.

How many students grew more than $1\frac{3}{8}$ inches? Explain your reasoning.
What is the difference between the greatest amount of growth and the least amount of growth, in inches?

Show Solution

Nine students grew more than $1\frac{3}{8}$ inches. Sample response: $1\frac{3}{8}$ is located between $1\frac{1}{4}$ and $1\frac{2}{4}$ , and there are 9 points to the right of $1\frac{3}{8}$ .
$2\frac{2}{8}$ inches. Sample response: $3\frac{1}{8} - \frac{7}{8} = 2\frac {9}{8} - \frac{7}{8} =2\frac{2}{8}$ .

Section B Check

Section B Checkpoint

Problem 1

Select all expressions that are equivalent to $\frac{5}{8}$ .

A.

\frac{1}{8} + \frac{1}{8} + \frac{1}{8} + \frac{1}{8} + \frac{1}{8}

B.

\frac{3}{8} + \frac{2}{8} + \frac{2}{8}

C.

\frac{2}{8} + \frac{2}{8} + \frac{1}{8}

D.

\frac{4}{8} + \frac{1}{8}

E.

\frac{3}{5} + \frac{2}{3}

Show Solution

A, C, D

Problem 2

Find the value of each expression. Explain or show your reasoning. Use the number line if it is helpful.

$3\frac{2}{5} + \frac{4}{5}$
$2 - \frac{1}{8}$

Show Solution

$\frac{21}{5}$ . Sample response: Each whole is $\frac{5}{5}$ , so there are $\frac{15}{5}$ in 3 and $\frac{17}{5}$ in $3 \frac{2}{5}$ . Then $\frac{4}{5}$ more makes $\frac{21}{5}$ .
$\frac{15}{8}$ . Sample response: Each whole is $\frac{8}{8}$ , so 2 is equivalent to $\frac{16}{8}$ , and subtracting $\frac{1}{8}$ gives $\frac{15}{8}$ .

Problem 3

The line plot shows the weights of some puppies at a pet store.

What is the difference between the weights of the heaviest puppy and the lightest puppy? Explain or show your reasoning.
How much did the 5 heaviest puppies weigh all together? Explain or show your reasoning.

Show Solution

$4\frac {1}{8}$ pounds. Sample response: $4\frac {6}{8} - \frac {5}{8} = 4\frac {1}{8}$
$18 \frac {4}{8}$ pounds. Sample response: I first added the whole numbers of pounds which was $4 + 3 + 3+ 3 + 3$ or 16. Then I added the eighths which was $\frac {6}{8} + \frac{7}{8} + \frac{7}{8} = \frac{20}{8}$ , which is the same as $2\frac{4}{8}$ . $16 + 2\frac {4}{8} = 18\frac {4}{8}$

Lesson 15

An Assortment of Fractions

Which Stack Is Taller?

Which stack of foam blocks is taller:

Two $\frac {1}{3}$ -foot blocks and one $\frac {1}{6}$ -foot block, or
One $\frac {1}{2}$ -foot block and two $\frac {1}{6}$ -foot blocks?

Explain or show your reasoning.

Show Solution

They are the same height. Sample response: First stack:

2 \times \frac{1}{3} = \frac{2}{3}

, which is equivalent to

\frac{4}{6}

. Adding another

\frac{1}{6}

makes

\frac{5}{6}

. Second stack:

\frac{1}{2}

is equivalent to

\frac{3}{6}

. Adding another

\frac{2}{6}

makes

\frac{5}{6}

foot.

Lesson 16

Add Tenths and Hundredths Together

Some Sums

Find the value of each sum. Show your reasoning. Use number lines if you find them helpful.

$\frac{1}{10} + \frac{50}{100}$
$\frac{20}{100} + \frac{4}{10}$
$\frac{6}{10} + \frac {3}{100}$
$\frac{18}{100} + \frac{7}{10}$

Show Solution

$\frac{6}{10}$ or $\frac{60}{100}$
$\frac{6}{10}$ or $\frac{60}{100}$
$\frac{63}{100}$
$\frac{88}{100}$

Lesson 17

Sums of Tenths and Hundredths

Missing Fractions

Each equation is missing a fraction in tenths or hundredths. Find the fraction that makes each equation true.

$\frac{26}{100} + \frac{8}{10} = \underline{\hspace{.5in}}$
$\frac{7}{10} + \underline{\hspace{.5in}} = \frac{92}{100}$
$\underline{\hspace{.5in}} + \frac{8}{100} = \frac{128}{100}$
$\frac{12}{100} + \frac{12}{10} = \underline{\hspace{.5in}}$

Show Solution

$\frac{106}{100}$ or $1\frac{6}{100}$
$\frac{22}{100}$
$\frac{12}{10}$ or $\frac{120}{100}$
$\frac{132}{100}$ or $1\frac{32}{100}$

Lesson 18

A Lot of Fractions to Add

U.S. Coins

The table shows the thicknesses of U.S. coins, in centimeters.

coin	thickness (cm)
penny	$\frac{15}{100}$
nickel	$\frac{2}{10}$
dime	$\frac{14}{100}$
quarter	$\frac{18}{100}$
half dollar	$\frac{22}{100}$
dollar	$\frac{2}{10}$

Find the combined thickness of:

a penny, a nickel, a quarter
a dollar, a half dollar, a quarter, and a dime

Show Solution

$\frac{53}{100}$ cm. Sample response: $\frac{15}{100} + \frac{2}{10} + \frac{18}{100} = \frac{33}{100} + \frac{20}{100} = \frac{53}{100}$
$\frac{74}{100}$ cm. Sample response: $\frac{2}{10} + \frac{22}{100} + \frac{18}{100} + \frac{14}{100} = \frac{20}{100} + \frac{54}{100} = \frac{74}{100}$

Lesson 19

Flexible with Fractions

Han’s Design

Han is using small sticky notes to make an H shape to decorate a notebook that is 6 inches wide and 9 inches tall. His design is shown here.

Sticky notes shaping the capital letter H. 5 sticky notes each set horizontally to form the left and right columns. 2 sticky note set set horizontally in between the two columns on the third row.

The longer side of the sticky note is $\frac{15}{8}$ inches. The shorter side is $\frac{11}{8}$ inches.

Is the notebook tall enough for his design? Show your reasoning.

Show Solution

Yes. Sample response: The H shape is $5 \times \frac{11}{8}$ or $\frac{55}{8}$ inches tall. The notebook is $9 \times \frac{8}{8}$ or $\frac{72}{8}$ inches tall.

Lesson 20

Sticky Notes

No cool-down

Section C Check

Section C Checkpoint

Problem 1

Select all expressions that are equivalent to $\frac{53}{100}$ .

A.

\frac{3}{10} + \frac{5}{100}

B.

\frac{50}{100} + \frac{3}{10}

C.

\frac{5}{10} + \frac{3}{100}

D.

\frac{1}{10} + \frac{4}{10} + \frac{3}{100}

E.

\frac{31}{100} + \frac{12}{100} + \frac{1}{10}

Show Solution

C, D, E

Problem 2

Find the value of each expression. Explain or show your reasoning.

$\frac{19}{100} + \frac{26}{100} + \frac{1}{100}$
$\frac{4}{10} + \frac{3}{10} + \frac{18}{100}$

Show Solution

$\frac{46}{100}$ . Sample response: I first added $\frac{19}{100}$ and $\frac{1}{100}$ to get $\frac{20}{100}$ , and then added $\frac{26}{100}$ .
$\frac{88}{100}$ . Sample response: I first added $\frac{4}{10}$ and $\frac{3}{10}$ to get $\frac{7}{10}$ , which is the same as $\frac{70}{100}$ , which I put together with $\frac{18}{100}$ .

Problem 3

If we combine each person's times for the two races, who finished in less time? Explain or show your reasoning.

	Lin	Tyler
first race	$6\frac{5}{10}$ minutes	$6\frac{72}{100}$ minutes
second race	$6\frac{41}{100}$ minutes	$6\frac{26}{100}$ minutes

Show Solution

Lin finished in less time. Her two times add to $12 \frac{91}{100}$ minutes and Tyler’s times add to $12 \frac{98}{100}$ minutes.

Unit 3 Extending Operations To Fractions — Unit Plan