Unit 2 Plan - Grade 4

Title	Takeaways	Assessment
Lesson 1 Representations of Fractions (Part 1)	—	What Do the Diagrams Show? (1 problem) Each whole diagram represents 1. What fraction does each shaded part represent? Explain or show how you could use this diagram to represent sixths. Show Solution $\frac{1}{6}$ , $\frac{1}{10}$ , and $\frac{1}{5}$ . Sample response: Split each half into 3 equal parts so there will be a total of 6 parts. Each part is a sixth.
Lesson 2 Representations of Fractions (Part 2)	—	What Do the Diagrams Show? (1 problem) Use a blank diagram to create a representation for each fraction. Both blank diagrams represent the same quantity. $\frac{5}{8}$ $\frac{9}{8}$ Show Solution Sample response: $\frac{5}{8}$ $\frac{9}{8}$
Lesson 3 Same Denominator or Numerator	—	Sizing Up Fractions (1 problem) In each pair of fractions, which one is greater? Explain or show your reasoning. $\frac{7}{8}$ or $\frac{10}{8}$ $\frac{4}{10}$ or $\frac{4}{5}$ Show Solution $\frac{10}{8}$ is greater, because the two fractions have the same fractional parts (eighths). There are more eighths in $\frac{10}{8}$ than in $\frac{7}{8}$ . $\frac{4}{5}$ is greater because 1 fifth is greater than 1 tenth, so 4 fifths are greater than 4 tenths.
Lesson 4 Same Size, Related Sizes	—	Where on the Number Line? (1 problem) Locate and label each fraction on one of the number lines. Show your reasoning. $\frac{3}{6}$ $\frac{2}{10}$ $\frac{6}{8}$ $\frac{4}{12}$ Show Solution Sample response:
Lesson 5 Fractions on Number Lines	—	Two of the Same (1 problem) Show $\frac{5}{6}$ on the number line. Be sure to include labels. Then explain or show that the fraction $\frac{10}{12}$ is equivalent to $\frac{5}{6}$ . Show Solution Sample response: Each third can be partitioned into 2 sixths, and each sixth into 2 twelfths. There are 10 twelfths in 5 sixths.
Lesson 6 Relate Fractions to Benchmarks	—	Greater than or Less than . . .? (1 problem) For each question, explain or show your reasoning. Use a number line if it is helpful. Is $\frac{6}{10}$ greater or less than $\frac{1}{2}$ ? Is $\frac{11}{12}$ greater or less than 1? Show Solution Greater than $\frac{1}{2}$ . Sample response: I know that $\frac{5}{10}$ is equivalent to $\frac{1}{2}$ and $\frac{6}{10}$ is greater than $\frac{5}{10}$ . Less than 1. Sample response: I know that $\frac{12}{12}$ is 1 and $\frac{11}{12}$ is less than $\frac{12}{12}$ .
Section A Check Section A Checkpoint	Problem 1 Label the point on each number line with a fraction it represents. Show Solution $\frac78$ . There are 8 equal parts in 1, and the point is on the 7th tick mark from 0 to show 7 parts. $\frac85$ . There are 5 equal parts in each whole, and the point is on the 8th tick mark from 0 to show 8 parts. Problem 2 Is $\frac{7}{12}$ greater than or less than $\frac{1}{2}$ ? Explain your reasoning. Use the number line if it is helpful. Show Solution $\frac{7}{12}$ is greater than $\frac{1}{2}$ . Sample responses: $\frac{7}{12}$ is the 7th tick mark from 0 on this number line, and it’s more than halfway to 1. Half of 12 is 6 and $\frac{1}{2}$ is $\frac{6}{12}$ , so $\frac{7}{12}$ is greater. Problem 3 Explain why $\frac{4}{12}$ is equivalent to $\frac{1}{3}$ . Use the number line if it is helpful. Show Solution Sample response: If I divide each third into 4 equal pieces, each of those pieces is 1 twelfth, and $\frac{1}{3}$ is on the 4th tick mark from 0, showing it is equivalent to 4 twelfths.

Lesson 7 Equivalent Fractions	—	Two Equivalent Fractions (1 problem) Name two fractions that are equivalent to $\frac{5}{3}$ . Explain or show your reasoning. Show Solution Sample responses: $\frac{10}{6}$ . I drew a tape diagram to show 5 thirds. Then I partitioned each 1 third into 2 equal parts, so in 5 thirds there are 10 parts. Each part is 1 sixth, so in 5 thirds there are 10 sixths. $\frac{20}{12}$ . I drew a tape diagram to show 5 thirds. Then I partitioned each 1 third into 2 equal parts to make sixths and then each 1 sixth into 2 equal parts again to make 12 parts. Each part is 1 twelfth, so in 10 sixths there are 20 twelfths.
Lesson 8 Equivalent Fractions on the Number Line	—	In Search of Equivalence (1 problem) Name a fraction that is equivalent to $\frac{3}{4}$ . Explain or show your reasoning. Use a number line, if it helps. Show Solution Sample responses: $\frac{6}{8}$ , $\frac{9}{12}$ , or equivalent. If using a number line, students may partition it into fourths, and further partition each fourth into 2 parts to get eighths or into 3 parts to get twelfths.
Lesson 9 Explain Equivalence	—	To Be or Not to Be (Equivalent) (1 problem) Explain or show why this statement is true: $\frac{5}{4}$ is equivalent to $\frac{15}{12}$ . Use a number line, if it helps. Diego wrote $\frac{11}{5}$ and $\frac{55}{10}$ as equivalent fractions. Are those fractions equivalent? Explain or show how you know. Use a number line, if it helps. Show Solution Sample responses: Students may use number lines to show their reasoning. When I split each fourth into 3 equal parts, I can see that $3 \times 5 = 15$ . Splitting each fourth into 3 equal parts makes twelfths, and 5 fourths is the same size as 15 twelfths. No, 1 fifth can be partitioned into 2 parts to get tenths, so 11 fifths has $11 \times 2$ or 22 tenths, not 55 tenths.
Lesson 10 Use Multiples to Find Equivalent Fractions	—	Fractions of the Same Size (1 problem) Find two fractions that are equivalent to $\frac{3}{8}$ . Explain or show your reasoning. Decide if each of the following fractions are equivalent to $\frac{9}{4}$ . $\frac{10}{8}$ $\frac{16}{10}$ $\frac{18}{8}$ $\frac{27}{12}$ Show Solution Sample response: $\frac{6}{16}$ and $\frac{9}{24}$ . $\frac{3 \ \times \ 2}{8 \ \times \ 2} = \frac{6}{16}$ and $\frac{3 \ \times \ 3}{8 \ \times \ 3} = \frac{9}{24}$ . No No Yes Yes
Lesson 11 Use Factors to Find Equivalent Fractions	—	Find Three or More (1 problem) Name at least 3 fractions that are equivalent to $\frac{20}{100}$ . Explain or show your reasoning. Show Solution Sample responses: $\frac{2}{10}$ , $\frac{4}{20}$ , $\frac{10}{50}$ , $\frac{40}{200}$ $\frac{20 \ \div \ 2}{100 \ \div \ 2}=\frac{10}{50}$ , $\frac{20 \ \div \ 5}{100 \ \div \ 5}=\frac{4}{20}$ , $\frac{20 \ \div \ 10}{100 \ \div \ 10}=\frac{2}{10}$ , $\frac{20 \ \times \ 2}{100 \ \times \ 2}=\frac{40}{200}$
Section B Check Section B Checkpoint	Problem 1 List two fractions that are equivalent to $\frac{3}{4}$ . Explain or show your reasoning. Show Solution Sample response: $\frac68$ and $\frac{9}{12}$ . Each $\frac14$ is the same size as two $\frac18$ s. So, three $\frac14$ s is the same size as six $\frac18$ s. Each $\frac14$ is the same size as three $\frac{1}{12}$ s. So, three $\frac14$ s is the same size as nine $\frac{1}{12}$ s. Problem 2 List two equivalent fractions that the point represents. Explain your reasoning. Show Solution Sample responses: $\frac16$ and $\frac{2}{12}$ . There are 6 equal parts on the number line and the point is the first tick mark, so that’s $\frac16$ . The point is the second tick mark when I divide each $\frac16$ into 2 equal parts. Those parts are $\frac{1}{12}$ s because there are 12 in 1 whole. So, the point is also $\frac{2}{12}$ . Problem 3 To show that $\frac{7}{12}$ is equivalent to $\frac{14}{24}$ , Kiran wrote: $\frac{2 \ \times \ 7}{2 \ \times \ 12} = \frac{14}{24}$ . Do you agree with Kiran? Explain your reasoning. Show Solution Yes, I agree with Kiran. Sample response: $\frac{7}{12}$ is 7 equal parts when there are 12 of those parts in 1 whole. If I partition each of those 12 parts into 2 smaller parts, then I get $2 \times 7$ of the smaller parts with $2 \times 12$ of them in the whole. That’s $\frac{2 \ \times \ 7}{2 \ \times \ 12}$ .

Lesson 12 Ways to Compare Fractions	—	Pick the Greater Fraction (1 problem) In each pair of fractions, which fraction is greater? Explain or show your reasoning. $\frac{5}{12}$ and $\frac{5}{8}$ $\frac{11}{10}$ and $\frac{18}{100}$ $\frac{6}{10}$ and $\frac{7}{12}$ Show Solution $\frac{5}{8}$ , Sample response: 1 eighth is greater than 1 twelfth, so 5 eighths is greater than 5 twelfths. $\frac{11}{10}$ , Sample response: $\frac{11}{10}$ is greater than 1, and $\frac{18}{100}$ is less than 1. $\frac{6}{10}$ , Sample response: $\frac{7}{12}$ is $\frac{1}{12}$ more than $\frac{1}{2}$ , and $\frac{6}{10}$ is $\frac{1}{10}$ more than $\frac{1}{2}$ . I know $\frac{1}{10}$ is greater than $\frac{1}{12}$ , so $\frac{6}{10}$ is greater.
Lesson 13 Use Equivalent Fractions to Compare	—	Make It True (1 problem) Compare each pair of fractions. Use the symbols $>$ , $<$ , or $=$ to make each statement true. Explain or show your reasoning. $\frac{15}{8} \underline{\hspace{1.05cm}} \frac{7}{4}$ $\frac{2}{5} \underline{\hspace{1.05cm}} \frac{30}{100}$ Show Solution $\frac{15}{8}> \frac{7}{4}$ , Sample response: $\frac{7}{4}$ is equivalent to $\frac{14}{8}$ , so it is less than $\frac{15}{8}$ . $\frac{2}{5}> \frac{30}{100}$ , Sample response: $\frac{2}{5}$ is equivalent to $\frac{40}{100}$ , so it is greater than $\frac{30}{100}$ .
Lesson 14 Fraction Comparison Problems	—	Who Ran the Farthest? (1 problem) Jada, Kiran, and Lin tried to run as far as possible before they had to stop and rest. Jada ran $\frac{3}{4}$ mile. Kiran ran $\frac{7}{12}$ mile. Lin ran $\frac{4}{6}$ mile. Who ran the farthest before stopping? Explain or show your reasoning. Show Solution Jada ran the farthest. Sample responses: Comparing $\frac{7}{12}$ and $\frac{4}{6}$ : $\frac{4}{6}$ is equivalent to $\frac{8}{12}$ which is greater than $\frac{7}{12}$ , so Lin ran farther than Kiran. Comparing $\frac{8}{12}$ and $\frac{3}{4}$ : $\frac{3 \ \times \ 3}{4 \ \times \ 3}=\frac{9}{12}$ , so $\frac{3}{4}$ is greater than $\frac{8}{12}$ . That means Jada ran farther than both Lin and Kiran.
Lesson 15 Use Common Denominators to Compare	—	Which Is Greater? (1 problem) In each pair of fractions, which fraction is greater? Explain or show your reasoning. $\frac{3}{10}$ or $\frac{2}{6}$ $\frac{99}{100}$ or $\frac{9}{10}$ Show Solution $\frac{2}{6}$ is greater. Sample response: $\frac{3 \ \times \ 6}{10 \ \times \ 6}=\frac{18}{60}$ , $\frac{2 \ \times \ 10}{6 \ \times \ 10}=\frac{20}{60}$ and $\frac{18}{60}$ $<$ $\frac{20}{60}$ . $\frac{99}{100}$ is greater. Sample response: $\frac{99}{100}$ is $\frac{1}{100}$ less than 1, and $\frac{9}{10}$ is $\frac{1}{10}$ less than 1. That means that $\frac{99}{100}$ is closer to 1, so it is greater.
Lesson 16 Compare and Order Fractions	—	All in Order (1 problem) Put these fractions in order, from least to greatest. Show your reasoning. $\frac{5}{12}$ $\frac{8}{6}$ $\frac{4}{10}$ $\frac{7}{5}$ Show Solution $\frac{4}{10}$ , $\frac{5}{12}$ , $\frac{8}{6}$ , $\frac{7}{5}$ . Sample response: $\frac{4}{10}$ and $\frac{5}{12}$ are less than 1. $\frac{8}{6}$ and $\frac{7}{5}$ are greater than 1. Comparing $\frac{4}{10}$ and $\frac{5}{12}$ : $\frac{4 \ \times \ 6}{10 \ \times \ 6}=\frac{24}{60}$ and $\frac{5 \ \times \ 5}{12 \ \times \ 5}=\frac{25}{60}$ , so $\frac{5}{12}$ is greater. Comparing $\frac{8}{6}$ and $\frac{7}{5}$ : $\frac{8 \ \times \ 5}{6 \ \times \ 5}=\frac{40}{30}$ and $\frac{7 \ \times \ 6}{5 \ \times \ 6}=\frac{42}{30}$ . Or: $\frac{8}{6}$ is $\frac{2}{6}$ more than 1, while $\frac{7}{5}$ is $\frac{2}{5}$ more than 1. Since $\frac{2}{5}$ is greater than $\frac{2}{6}$ , $\frac75$ is greater than $\frac86$ .
Lesson 17 Paper Clip Games	—	No cool-down
Section C Check Section C Checkpoint	Problem 1 Use a >, <, or = symbol to make each statement true. Explain your reasoning. $\frac{4}{12} \, \underline{\phantom{\frac{1}{1}\hspace{1.05cm}}} \, \frac{1}{3}$ $\frac{53}{100} \, \underline{\phantom{\frac{1}{1}\hspace{1.05cm}}} \, \frac{5}{12}$ $\frac{265}{100} \, \underline{\phantom{\frac{1}{1}\hspace{1.05cm}}} \, \frac{28}{10}$ $\frac{13}{8} \, \underline{\phantom{\frac{1}{1}\hspace{1.05cm}}} \, \frac{7}{5}$ Show Solution Sample responses: $\frac{4}{12} \, = \frac{1}{3}$ , because $\frac{1}{3}$ is the same as $\frac{1 \times 4}{3 \times 4}$ , which is $\frac{4}{12}$ . $\frac{53}{100} > \frac{5}{12}$ , because $\frac{53}{100}$ is more than $\frac{1}{2}$ , and $\frac{5}{12}$ is less than $\frac{1}{2}$ . $\frac{265}{100} < \frac{28}{10}$ , because $\frac{28}{10}$ is $\frac{280}{100}$ . $\frac{13}{8} > \frac{7}{5}$ , because if I take away 1 from each then I have $\frac{5}{8}$ and $\frac{2}{5}$ . I know $\frac{5}{8}$ is greater than $\frac{1}{2}$ , and $\frac{2}{5}$ is less than $\frac{1}{2}$ . Problem 2 Clare walked $\frac{4}{5}$ of the way around a lake. Tyler walked $\frac{7}{12}$ of the way around a pond. Explain why you don’t have enough information to determine who walked farther. Show Solution Sample response: I know that $\frac{4}{5}$ is greater than $\frac{7}{12}$ because $\frac{4}{5}$ is close to 1 (or is $\frac{1}{5}$ less than 1) and $\frac{7}{12}$ is just a little over $\frac{1}{2}$ (or is $\frac{1}{12}$ more than $\frac{6 }{12}$ , which is $\frac{1}{2}$ ). But I don’t know how far it is all the way around the lake or the pond. If the pond Tyler is walking around has a much longer distance than the lake Clare is walking around, then Tyler could be walking farther.
Unit 2 Assessment End-of-Unit Assessment

Lesson 1

Representations of Fractions (Part 1)

—

What Do the Diagrams Show? (1 problem)

Each whole diagram represents 1.

What fraction does each shaded part represent?
Explain or show how you could use this diagram to represent sixths.

Show Solution

$\frac{1}{6}$ , $\frac{1}{10}$ , and $\frac{1}{5}$ .
Sample response: Split each half into 3 equal parts so there will be a total of 6 parts. Each part is a sixth.

Lesson 2

Representations of Fractions (Part 2)

—

What Do the Diagrams Show? (1 problem)

Use a blank diagram to create a representation for each fraction. Both blank diagrams represent the same quantity.

$\frac{5}{8}$
$\frac{9}{8}$

Show Solution

Sample response:

$\frac{5}{8}$
$\frac{9}{8}$

Lesson 3

Same Denominator or Numerator

—

Sizing Up Fractions (1 problem)

In each pair of fractions, which one is greater? Explain or show your reasoning.

$\frac{7}{8}$ or $\frac{10}{8}$
$\frac{4}{10}$ or $\frac{4}{5}$

Show Solution

$\frac{10}{8}$ is greater, because the two fractions have the same fractional parts (eighths). There are more eighths in $\frac{10}{8}$ than in $\frac{7}{8}$ .
$\frac{4}{5}$ is greater because 1 fifth is greater than 1 tenth, so 4 fifths are greater than 4 tenths.

Lesson 4

Same Size, Related Sizes

—

Where on the Number Line? (1 problem)

Locate and label each fraction on one of the number lines. Show your reasoning.

$\frac{3}{6}$

$\frac{2}{10}$

$\frac{6}{8}$

$\frac{4}{12}$

Number line. From 0 to 1. 4 evenly spaced tick marks. First tick mark, 0. Last tick mark, 1.

Number line from 0 to 1. 5 evenly spaced tick marks.

number line. 6 evenly spaced tick marks. first tick mark, 0. last tick mark, 1.

Number line. 7 evenly spaced tick marks. First tick mark, 0. Last tick mark, 1.

Show Solution

Sample response:

Number line from 0 to 1. Evenly spaced by sixths. Point at 3 sixths.

Lesson 5

Fractions on Number Lines

—

Two of the Same (1 problem)

Show $\frac{5}{6}$ on the number line. Be sure to include labels. Then explain or show that the fraction $\frac{10}{12}$ is equivalent to $\frac{5}{6}$ .

number line. 4 evenly spaced tick marks. first tick mark, 0. fourth tick mark, 1.

Show Solution

Sample response:

Each third can be partitioned into 2 sixths, and each sixth into 2 twelfths. There are 10 twelfths in 5 sixths.

Lesson 6

Relate Fractions to Benchmarks

—

Greater than or Less than . . .? (1 problem)

For each question, explain or show your reasoning. Use a number line if it is helpful.

Is $\frac{6}{10}$ greater or less than $\frac{1}{2}$ ?
Is $\frac{11}{12}$ greater or less than 1?

Show Solution

Greater than $\frac{1}{2}$ . Sample response: I know that $\frac{5}{10}$ is equivalent to $\frac{1}{2}$ and $\frac{6}{10}$ is greater than $\frac{5}{10}$ .
Less than 1. Sample response: I know that $\frac{12}{12}$ is 1 and $\frac{11}{12}$ is less than $\frac{12}{12}$ .

Section A Check

Section A Checkpoint

Problem 1

Label the point on each number line with a fraction it represents.

Show Solution

$\frac78$ . There are 8 equal parts in 1, and the point is on the 7th tick mark from 0 to show 7 parts.
$\frac85$ . There are 5 equal parts in each whole, and the point is on the 8th tick mark from 0 to show 8 parts.

Problem 2

Is $\frac{7}{12}$ greater than or less than $\frac{1}{2}$ ? Explain your reasoning. Use the number line if it is helpful.

Number Line. From 0 to 1, evenly spaced by twelfths.

Show Solution

$\frac{7}{12}$ is greater than $\frac{1}{2}$ . Sample responses: $\frac{7}{12}$ is the 7th tick mark from 0 on this number line, and it’s more than halfway to 1. Half of 12 is 6 and $\frac{1}{2}$ is $\frac{6}{12}$ , so $\frac{7}{12}$ is greater.

Problem 3

Explain why $\frac{4}{12}$ is equivalent to $\frac{1}{3}$ . Use the number line if it is helpful.

Number Line. From 0 to 1, evenly spaced by thirds.

Show Solution

Sample response: If I divide each third into 4 equal pieces, each of those pieces is 1 twelfth, and $\frac{1}{3}$ is on the 4th tick mark from 0, showing it is equivalent to 4 twelfths.

Number Line. From 0 to 1, evenly spaced by twelfths. One third is labeled.

Lesson 7

Equivalent Fractions

—

Two Equivalent Fractions (1 problem)

Name two fractions that are equivalent to $\frac{5}{3}$ . Explain or show your reasoning.

Show Solution

Sample responses:

$\frac{10}{6}$ . I drew a tape diagram to show 5 thirds. Then I partitioned each 1 third into 2 equal parts, so in 5 thirds there are 10 parts. Each part is 1 sixth, so in 5 thirds there are 10 sixths.
$\frac{20}{12}$ . I drew a tape diagram to show 5 thirds. Then I partitioned each 1 third into 2 equal parts to make sixths and then each 1 sixth into 2 equal parts again to make 12 parts. Each part is 1 twelfth, so in 10 sixths there are 20 twelfths.

Lesson 8

Equivalent Fractions on the Number Line

—

In Search of Equivalence (1 problem)

Name a fraction that is equivalent to $\frac{3}{4}$ . Explain or show your reasoning. Use a number line, if it helps.

number line. two evenly spaced tick marks. first tick mark, 0. second tick mark, not labeled.

Show Solution

Sample responses:

$\frac{6}{8}$ , $\frac{9}{12}$ , or equivalent. If using a number line, students may partition it into fourths, and further partition each fourth into 2 parts to get eighths or into 3 parts to get twelfths.

Lesson 9

Explain Equivalence

—

To Be or Not to Be (Equivalent) (1 problem)

Explain or show why this statement is true: $\frac{5}{4}$ is equivalent to $\frac{15}{12}$ . Use a number line, if it helps.
Diego wrote $\frac{11}{5}$ and $\frac{55}{10}$ as equivalent fractions. Are those fractions equivalent? Explain or show how you know. Use a number line, if it helps.

Show Solution

Sample responses: Students may use number lines to show their reasoning.

When I split each fourth into 3 equal parts, I can see that $3 \times 5 = 15$ . Splitting each fourth into 3 equal parts makes twelfths, and 5 fourths is the same size as 15 twelfths.
No, 1 fifth can be partitioned into 2 parts to get tenths, so 11 fifths has $11 \times 2$ or 22 tenths, not 55 tenths.

Lesson 10

Use Multiples to Find Equivalent Fractions

—

Fractions of the Same Size (1 problem)

Find two fractions that are equivalent to $\frac{3}{8}$ . Explain or show your reasoning.
Decide if each of the following fractions are equivalent to $\frac{9}{4}$ .
1. $\frac{10}{8}$
2. $\frac{16}{10}$
3. $\frac{18}{8}$
4. $\frac{27}{12}$

Show Solution

Sample response: $\frac{6}{16}$ and $\frac{9}{24}$ . $\frac{3 \ \times \ 2}{8 \ \times \ 2} = \frac{6}{16}$ and $\frac{3 \ \times \ 3}{8 \ \times \ 3} = \frac{9}{24}$ .
1. No
2. No
3. Yes
4. Yes

Lesson 11

Use Factors to Find Equivalent Fractions

—

Find Three or More (1 problem)

Name at least 3 fractions that are equivalent to $\frac{20}{100}$ . Explain or show your reasoning.

Show Solution

Sample responses: $\frac{2}{10}$ , $\frac{4}{20}$ , $\frac{10}{50}$ , $\frac{40}{200}$

$\frac{20 \ \div \ 2}{100 \ \div \ 2}=\frac{10}{50}$ , $\frac{20 \ \div \ 5}{100 \ \div \ 5}=\frac{4}{20}$ , $\frac{20 \ \div \ 10}{100 \ \div \ 10}=\frac{2}{10}$ , $\frac{20 \ \times \ 2}{100 \ \times \ 2}=\frac{40}{200}$

Section B Check

Section B Checkpoint

Problem 1

List two fractions that are equivalent to $\frac{3}{4}$ . Explain or show your reasoning.

Show Solution

Sample response: $\frac68$ and $\frac{9}{12}$ . Each $\frac14$ is the same size as two $\frac18$ s. So, three $\frac14$ s is the same size as six $\frac18$ s. Each $\frac14$ is the same size as three $\frac{1}{12}$ s. So, three $\frac14$ s is the same size as nine $\frac{1}{12}$ s.

Problem 2

List two equivalent fractions that the point represents. Explain your reasoning.

Number line. Scale 0 to 1. 7 evenly spaced tick marks. First tick mark, 0. Point at second tick mark, no label. Last tick mark, 1.

Show Solution

Sample responses: $\frac16$ and $\frac{2}{12}$ . There are 6 equal parts on the number line and the point is the first tick mark, so that’s $\frac16$ .

The point is the second tick mark when I divide each $\frac16$ into 2 equal parts. Those parts are $\frac{1}{12}$ s because there are 12 in 1 whole. So, the point is also $\frac{2}{12}$ .

Problem 3

To show that $\frac{7}{12}$ is equivalent to $\frac{14}{24}$ , Kiran wrote: $\frac{2 \ \times \ 7}{2 \ \times \ 12} = \frac{14}{24}$ . Do you agree with Kiran? Explain your reasoning.

Show Solution

Yes, I agree with Kiran. Sample response: $\frac{7}{12}$ is 7 equal parts when there are 12 of those parts in 1 whole. If I partition each of those 12 parts into 2 smaller parts, then I get $2 \times 7$ of the smaller parts with $2 \times 12$ of them in the whole. That’s $\frac{2 \ \times \ 7}{2 \ \times \ 12}$ .

Lesson 12

Ways to Compare Fractions

—

Pick the Greater Fraction (1 problem)

In each pair of fractions, which fraction is greater? Explain or show your reasoning.

$\frac{5}{12}$ and $\frac{5}{8}$
$\frac{11}{10}$ and $\frac{18}{100}$
$\frac{6}{10}$ and $\frac{7}{12}$

Show Solution

$\frac{5}{8}$ , Sample response: 1 eighth is greater than 1 twelfth, so 5 eighths is greater than 5 twelfths.
$\frac{11}{10}$ , Sample response: $\frac{11}{10}$ is greater than 1, and $\frac{18}{100}$ is less than 1.
$\frac{6}{10}$ , Sample response: $\frac{7}{12}$ is $\frac{1}{12}$ more than $\frac{1}{2}$ , and $\frac{6}{10}$ is $\frac{1}{10}$ more than $\frac{1}{2}$ . I know $\frac{1}{10}$ is greater than $\frac{1}{12}$ , so $\frac{6}{10}$ is greater.

Lesson 13

Use Equivalent Fractions to Compare

—

Make It True (1 problem)

Compare each pair of fractions. Use the symbols $>$ , $<$ , or $=$ to make each statement true. Explain or show your reasoning.

$\frac{15}{8} \underline{\hspace{1.05cm}} \frac{7}{4}$
$\frac{2}{5} \underline{\hspace{1.05cm}} \frac{30}{100}$

Show Solution

$\frac{15}{8}> \frac{7}{4}$ , Sample response: $\frac{7}{4}$ is equivalent to $\frac{14}{8}$ , so it is less than $\frac{15}{8}$ .
$\frac{2}{5}> \frac{30}{100}$ , Sample response: $\frac{2}{5}$ is equivalent to $\frac{40}{100}$ , so it is greater than $\frac{30}{100}$ .

Lesson 14

Fraction Comparison Problems

—

Who Ran the Farthest? (1 problem)

Jada, Kiran, and Lin tried to run as far as possible before they had to stop and rest.

Jada ran $\frac{3}{4}$ mile.
Kiran ran $\frac{7}{12}$ mile.
Lin ran $\frac{4}{6}$ mile.

Who ran the farthest before stopping? Explain or show your reasoning.

Show Solution

Jada ran the farthest. Sample responses:

Comparing $\frac{7}{12}$ and $\frac{4}{6}$ : $\frac{4}{6}$ is equivalent to $\frac{8}{12}$ which is greater than $\frac{7}{12}$ , so Lin ran farther than Kiran.
Comparing $\frac{8}{12}$ and $\frac{3}{4}$ : $\frac{3 \ \times \ 3}{4 \ \times \ 3}=\frac{9}{12}$ , so $\frac{3}{4}$ is greater than $\frac{8}{12}$ . That means Jada ran farther than both Lin and Kiran.

Lesson 15

Use Common Denominators to Compare

—

Which Is Greater? (1 problem)

In each pair of fractions, which fraction is greater? Explain or show your reasoning.

$\frac{3}{10}$ or $\frac{2}{6}$
$\frac{99}{100}$ or $\frac{9}{10}$

Show Solution

$\frac{2}{6}$ is greater. Sample response: $\frac{3 \ \times \ 6}{10 \ \times \ 6}=\frac{18}{60}$ , $\frac{2 \ \times \ 10}{6 \ \times \ 10}=\frac{20}{60}$ and $\frac{18}{60}$ $<$ $\frac{20}{60}$ .
$\frac{99}{100}$ is greater. Sample response: $\frac{99}{100}$ is $\frac{1}{100}$ less than 1, and $\frac{9}{10}$ is $\frac{1}{10}$ less than 1. That means that $\frac{99}{100}$ is closer to 1, so it is greater.

Lesson 16

Compare and Order Fractions

—

All in Order (1 problem)

Put these fractions in order, from least to greatest. Show your reasoning.

$\frac{5}{12}$

$\frac{8}{6}$

$\frac{4}{10}$

$\frac{7}{5}$

Show Solution

$\frac{4}{10}$ , $\frac{5}{12}$ , $\frac{8}{6}$ , $\frac{7}{5}$ . Sample response:

$\frac{4}{10}$ and $\frac{5}{12}$ are less than 1. $\frac{8}{6}$ and $\frac{7}{5}$ are greater than 1.
Comparing $\frac{4}{10}$ and $\frac{5}{12}$ : $\frac{4 \ \times \ 6}{10 \ \times \ 6}=\frac{24}{60}$ and $\frac{5 \ \times \ 5}{12 \ \times \ 5}=\frac{25}{60}$ , so $\frac{5}{12}$ is greater.
Comparing $\frac{8}{6}$ and $\frac{7}{5}$ : $\frac{8 \ \times \ 5}{6 \ \times \ 5}=\frac{40}{30}$ and $\frac{7 \ \times \ 6}{5 \ \times \ 6}=\frac{42}{30}$ . Or: $\frac{8}{6}$ is $\frac{2}{6}$ more than 1, while $\frac{7}{5}$ is $\frac{2}{5}$ more than 1. Since $\frac{2}{5}$ is greater than $\frac{2}{6}$ , $\frac75$ is greater than $\frac86$ .

Lesson 17

Paper Clip Games

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No cool-down

Section C Check

Section C Checkpoint

Problem 1

Use a >, <, or = symbol to make each statement true. Explain your reasoning.

$\frac{4}{12} \, \underline{\phantom{\frac{1}{1}\hspace{1.05cm}}} \, \frac{1}{3}$
$\frac{53}{100} \, \underline{\phantom{\frac{1}{1}\hspace{1.05cm}}} \, \frac{5}{12}$
$\frac{265}{100} \, \underline{\phantom{\frac{1}{1}\hspace{1.05cm}}} \, \frac{28}{10}$
$\frac{13}{8} \, \underline{\phantom{\frac{1}{1}\hspace{1.05cm}}} \, \frac{7}{5}$

Show Solution

Sample responses:

$\frac{4}{12} \, = \frac{1}{3}$ , because $\frac{1}{3}$ is the same as $\frac{1 \times 4}{3 \times 4}$ , which is $\frac{4}{12}$ .
$\frac{53}{100} > \frac{5}{12}$ , because $\frac{53}{100}$ is more than $\frac{1}{2}$ , and $\frac{5}{12}$ is less than $\frac{1}{2}$ .
$\frac{265}{100} < \frac{28}{10}$ , because $\frac{28}{10}$ is $\frac{280}{100}$ .
$\frac{13}{8} > \frac{7}{5}$ , because if I take away 1 from each then I have $\frac{5}{8}$ and $\frac{2}{5}$ . I know $\frac{5}{8}$ is greater than $\frac{1}{2}$ , and $\frac{2}{5}$ is less than $\frac{1}{2}$ .

Problem 2

Clare walked $\frac{4}{5}$ of the way around a lake. Tyler walked $\frac{7}{12}$ of the way around a pond. Explain why you don’t have enough information to determine who walked farther.

Show Solution

Sample response: I know that $\frac{4}{5}$ is greater than $\frac{7}{12}$ because $\frac{4}{5}$ is close to 1 (or is $\frac{1}{5}$ less than 1) and $\frac{7}{12}$ is just a little over $\frac{1}{2}$ (or is $\frac{1}{12}$ more than $\frac{6 }{12}$ , which is $\frac{1}{2}$ ). But I don’t know how far it is all the way around the lake or the pond. If the pond Tyler is walking around has a much longer distance than the lake Clare is walking around, then Tyler could be walking farther.

Unit 2 Fraction Equivalence And Comparison — Unit Plan