← 3-1 · Find fractions between two on the number line · Compare Fractions and Decimals by Structure

Find fractions between two on the number line · 10 practice problems

3.NF.A.23.NF.A.3

Generated variants — 10

Freshly produced from the archetype’s parameters — problem, figure, and solution derived together.

Variant 1 answer: 1/2 and 5/12 and 3/4

Mark each given fraction on the number line, then write every one of them that lies between $\dfrac{1}{3}$ and $\dfrac{5}{6}$ .

$\dfrac{1}{2} \qquad \dfrac{11}{12} \qquad \dfrac{5}{12} \qquad \dfrac{3}{4}$

(figure) A number line from 0 to 1 is divided into 12 equal parts, with 0 labeled at the left end and 1 at the right end. Mark the given fractions $\dfrac{1}{2}$ , $\dfrac{11}{12}$ , $\dfrac{5}{12}$ , $\dfrac{3}{4}$ on this number line.

Show solution

Understand

On a number line from 0 to 1 divided into 12 equal parts, mark the fractions 1/2, 11/12, 5/12, 3/4, then list every one of them that lies strictly between 1/3 and 5/6.

Givens

The number line from 0 to 1 is split into 12 equal parts, so each tick is 1/12
The fractions to place are 1/2, 11/12, 5/12, 3/4
The target interval is between 1/3 and 5/6

Unknowns

Which of the fractions fall between 1/3 and 5/6

Constraints

'Between' means greater than 1/3 and less than 5/6
Comparisons are easiest when all fractions share the denominator 12

Plan

#2 Make a Systematic List · also uses: #1 Draw a Diagram

Rewriting every fraction over the common denominator 12 (matching the number-line ticks) lets us list each one's position and read off which lie between the two endpoints. The number line is the diagram that anchors the comparison.

Execute

#1 Draw a Diagram 3.NF.A.3

Since the line has 12 equal parts, write the boundaries over 12: 1/3 is 4/12, and 5/6 is 10/12. So we want fractions between 4/12 and 10/12.

\dfrac{5}{6} = \dfrac{10}{12}

Matching the tick spacing makes every position a count of parts.

#2 Make a Systematic List 3.NF.A.3

Rewrite each fraction with denominator 12: 1/2 = 6/12, 11/12 = 11/12, 5/12 = 5/12, 3/4 = 9/12.

\dfrac{1}{2}=\dfrac{6}{12},\;\; \dfrac{3}{4}=\dfrac{9}{12}

Equivalent fractions over 12 line up exactly with the marked ticks.

#2 Make a Systematic List 3.NF.A.2

Compare each in /12 to the range (4/12, 10/12): 6/12 (=1/2) yes, 11/12 (=11/12) too big, 5/12 (=5/12) yes, 9/12 (=3/4) yes. So 1/2 and 5/12 and 3/4 lie between.

4/12 < \text{kept} < 10/12

On the number line, a fraction is between two others exactly when its tick sits between their ticks.

Answer: 1/2 and 5/12 and 3/4

Review

In /12 the target window is 4/12 through 10/12; the kept fractions sit inside it and the others sit outside, and the marks on the number line agree.

Draw the diagram (tool 1) and physically mark all ticks: place the two endpoints, then read which of the marked points land between them, giving the same answer.

Standards · min grade 3

3.NF.A.2 Understand a fraction as a number on the number line — Placing each fraction at its tick and reading which lie between the endpoints
3.NF.A.3 Explain equivalence of fractions and compare fractions by reasoning — Rewriting the fractions as equivalent /12 to compare

💡 Put every fraction over the same 12 ticks, and you can see at a glance which ones sit in between!

Variant 2 answer: 1/2 and 1/3

Mark each given fraction on the number line, then write every one of them that lies between $\dfrac{1}{6}$ and $\dfrac{2}{3}$ .

$\dfrac{1}{2} \qquad \dfrac{5}{6} \qquad \dfrac{1}{3} \qquad \dfrac{1}{6}$

(figure) A number line from 0 to 1 is divided into 6 equal parts, with 0 labeled at the left end and 1 at the right end. Mark the given fractions $\dfrac{1}{2}$ , $\dfrac{5}{6}$ , $\dfrac{1}{3}$ , $\dfrac{1}{6}$ on this number line.

Show solution

Understand

On a number line from 0 to 1 divided into 6 equal parts, mark the fractions 1/2, 5/6, 1/3, 1/6, then list every one of them that lies strictly between 1/6 and 2/3.

Givens

The number line from 0 to 1 is split into 6 equal parts, so each tick is 1/6
The fractions to place are 1/2, 5/6, 1/3, 1/6
The target interval is between 1/6 and 2/3

Unknowns

Which of the fractions fall between 1/6 and 2/3

Constraints

'Between' means greater than 1/6 and less than 2/3
Comparisons are easiest when all fractions share the denominator 6

Plan

#2 Make a Systematic List · also uses: #1 Draw a Diagram

Rewriting every fraction over the common denominator 6 (matching the number-line ticks) lets us list each one's position and read off which lie between the two endpoints. The number line is the diagram that anchors the comparison.

Execute

#1 Draw a Diagram 3.NF.A.3

Since the line has 6 equal parts, write the boundaries over 6: 1/6 is 1/6, and 2/3 is 4/6. So we want fractions between 1/6 and 4/6.

\dfrac{2}{3} = \dfrac{4}{6}

Matching the tick spacing makes every position a count of parts.

#2 Make a Systematic List 3.NF.A.3

Rewrite each fraction with denominator 6: 1/2 = 3/6, 5/6 = 5/6, 1/3 = 2/6, 1/6 = 1/6.

\dfrac{1}{2}=\dfrac{3}{6},\;\; \dfrac{1}{3}=\dfrac{2}{6}

Equivalent fractions over 6 line up exactly with the marked ticks.

#2 Make a Systematic List 3.NF.A.2

Compare each in /6 to the range (1/6, 4/6): 3/6 (=1/2) yes, 5/6 (=5/6) too big, 2/6 (=1/3) yes, 1/6 (=1/6) too small. So 1/2 and 1/3 lie between.

1/6 < \text{kept} < 4/6

On the number line, a fraction is between two others exactly when its tick sits between their ticks.

Answer: 1/2 and 1/3

Review

In /6 the target window is 1/6 through 4/6; the kept fractions sit inside it and the others sit outside, and the marks on the number line agree.

Draw the diagram (tool 1) and physically mark all ticks: place the two endpoints, then read which of the marked points land between them, giving the same answer.

Standards · min grade 3

3.NF.A.2 Understand a fraction as a number on the number line — Placing each fraction at its tick and reading which lie between the endpoints
3.NF.A.3 Explain equivalence of fractions and compare fractions by reasoning — Rewriting the fractions as equivalent /6 to compare

💡 Put every fraction over the same 6 ticks, and you can see at a glance which ones sit in between!

Variant 3 answer: 1/2 and 2/5 and 3/10

Mark each given fraction on the number line, then write every one of them that lies between $\dfrac{1}{5}$ and $\dfrac{7}{10}$ .

$\dfrac{1}{2} \qquad \dfrac{2}{5} \qquad \dfrac{3}{10} \qquad \dfrac{9}{10}$

(figure) A number line from 0 to 1 is divided into 10 equal parts, with 0 labeled at the left end and 1 at the right end. Mark the given fractions $\dfrac{1}{2}$ , $\dfrac{2}{5}$ , $\dfrac{3}{10}$ , $\dfrac{9}{10}$ on this number line.

Show solution

Understand

On a number line from 0 to 1 divided into 10 equal parts, mark the fractions 1/2, 2/5, 3/10, 9/10, then list every one of them that lies strictly between 1/5 and 7/10.

Givens

The number line from 0 to 1 is split into 10 equal parts, so each tick is 1/10
The fractions to place are 1/2, 2/5, 3/10, 9/10
The target interval is between 1/5 and 7/10

Unknowns

Which of the fractions fall between 1/5 and 7/10

Constraints

'Between' means greater than 1/5 and less than 7/10
Comparisons are easiest when all fractions share the denominator 10

Plan

#2 Make a Systematic List · also uses: #1 Draw a Diagram

Rewriting every fraction over the common denominator 10 (matching the number-line ticks) lets us list each one's position and read off which lie between the two endpoints. The number line is the diagram that anchors the comparison.

Execute

#1 Draw a Diagram 3.NF.A.3

Since the line has 10 equal parts, write the boundaries over 10: 1/5 is 2/10, and 7/10 is 7/10. So we want fractions between 2/10 and 7/10.

\dfrac{7}{10}

Matching the tick spacing makes every position a count of parts.

#2 Make a Systematic List 3.NF.A.3

Rewrite each fraction with denominator 10: 1/2 = 5/10, 2/5 = 4/10, 3/10 = 3/10, 9/10 = 9/10.

\dfrac{1}{2}=\dfrac{5}{10},\;\; \dfrac{2}{5}=\dfrac{4}{10}

Equivalent fractions over 10 line up exactly with the marked ticks.

#2 Make a Systematic List 3.NF.A.2

Compare each in /10 to the range (2/10, 7/10): 5/10 (=1/2) yes, 4/10 (=2/5) yes, 3/10 (=3/10) yes, 9/10 (=9/10) too big. So 1/2 and 2/5 and 3/10 lie between.

2/10 < \text{kept} < 7/10

On the number line, a fraction is between two others exactly when its tick sits between their ticks.

Answer: 1/2 and 2/5 and 3/10

Review

In /10 the target window is 2/10 through 7/10; the kept fractions sit inside it and the others sit outside, and the marks on the number line agree.

Draw the diagram (tool 1) and physically mark all ticks: place the two endpoints, then read which of the marked points land between them, giving the same answer.

Standards · min grade 3

3.NF.A.2 Understand a fraction as a number on the number line — Placing each fraction at its tick and reading which lie between the endpoints
3.NF.A.3 Explain equivalence of fractions and compare fractions by reasoning — Rewriting the fractions as equivalent /10 to compare

💡 Put every fraction over the same 10 ticks, and you can see at a glance which ones sit in between!

Variant 4 answer: 5/12 and 1/2 and 7/12

Mark each given fraction on the number line, then write every one of them that lies between $\dfrac{1}{4}$ and $\dfrac{2}{3}$ .

$\dfrac{5}{12} \qquad \dfrac{1}{2} \qquad \dfrac{7}{12} \qquad \dfrac{11}{12}$

(figure) A number line from 0 to 1 is divided into 12 equal parts, with 0 labeled at the left end and 1 at the right end. Mark the given fractions $\dfrac{5}{12}$ , $\dfrac{1}{2}$ , $\dfrac{7}{12}$ , $\dfrac{11}{12}$ on this number line.

Show solution

Understand

On a number line from 0 to 1 divided into 12 equal parts, mark the fractions 5/12, 1/2, 7/12, 11/12, then list every one of them that lies strictly between 1/4 and 2/3.

Givens

The number line from 0 to 1 is split into 12 equal parts, so each tick is 1/12
The fractions to place are 5/12, 1/2, 7/12, 11/12
The target interval is between 1/4 and 2/3

Unknowns

Which of the fractions fall between 1/4 and 2/3

Constraints

'Between' means greater than 1/4 and less than 2/3
Comparisons are easiest when all fractions share the denominator 12

Plan

#2 Make a Systematic List · also uses: #1 Draw a Diagram

Execute

#1 Draw a Diagram 3.NF.A.3

Since the line has 12 equal parts, write the boundaries over 12: 1/4 is 3/12, and 2/3 is 8/12. So we want fractions between 3/12 and 8/12.

\dfrac{2}{3} = \dfrac{8}{12}

Matching the tick spacing makes every position a count of parts.

#2 Make a Systematic List 3.NF.A.3

Rewrite each fraction with denominator 12: 5/12 = 5/12, 1/2 = 6/12, 7/12 = 7/12, 11/12 = 11/12.

\dfrac{1}{2}=\dfrac{6}{12}

Equivalent fractions over 12 line up exactly with the marked ticks.

#2 Make a Systematic List 3.NF.A.2

Compare each in /12 to the range (3/12, 8/12): 5/12 (=5/12) yes, 6/12 (=1/2) yes, 7/12 (=7/12) yes, 11/12 (=11/12) too big. So 5/12 and 1/2 and 7/12 lie between.

3/12 < \text{kept} < 8/12

On the number line, a fraction is between two others exactly when its tick sits between their ticks.

Answer: 5/12 and 1/2 and 7/12

Review

In /12 the target window is 3/12 through 8/12; the kept fractions sit inside it and the others sit outside, and the marks on the number line agree.

Draw the diagram (tool 1) and physically mark all ticks: place the two endpoints, then read which of the marked points land between them, giving the same answer.

Standards · min grade 3

3.NF.A.2 Understand a fraction as a number on the number line — Placing each fraction at its tick and reading which lie between the endpoints
3.NF.A.3 Explain equivalence of fractions and compare fractions by reasoning — Rewriting the fractions as equivalent /12 to compare

💡 Put every fraction over the same 12 ticks, and you can see at a glance which ones sit in between!

Variant 5 answer: 1/2 and 4/6

Mark each given fraction on the number line, then write every one of them that lies between $\dfrac{5}{12}$ and $\dfrac{3}{4}$ .

$\dfrac{5}{6} \qquad \dfrac{11}{12} \qquad \dfrac{1}{2} \qquad \dfrac{4}{6}$

(figure) A number line from 0 to 1 is divided into 12 equal parts, with 0 labeled at the left end and 1 at the right end. Mark the given fractions $\dfrac{5}{6}$ , $\dfrac{11}{12}$ , $\dfrac{1}{2}$ , $\dfrac{4}{6}$ on this number line.

Show solution

Understand

On a number line from 0 to 1 divided into 12 equal parts, mark the fractions 5/6, 11/12, 1/2, 2/3, then list every one of them that lies strictly between 5/12 and 3/4.

Givens

The number line from 0 to 1 is split into 12 equal parts, so each tick is 1/12
The fractions to place are 5/6, 11/12, 1/2, 2/3
The target interval is between 5/12 and 3/4

Unknowns

Which of the fractions fall between 5/12 and 3/4

Constraints

'Between' means greater than 5/12 and less than 3/4
Comparisons are easiest when all fractions share the denominator 12

Plan

#2 Make a Systematic List · also uses: #1 Draw a Diagram

Execute

#1 Draw a Diagram 3.NF.A.3

Since the line has 12 equal parts, write the boundaries over 12: 5/12 is 5/12, and 3/4 is 9/12. So we want fractions between 5/12 and 9/12.

\dfrac{3}{4} = \dfrac{9}{12}

Matching the tick spacing makes every position a count of parts.

#2 Make a Systematic List 3.NF.A.3

Rewrite each fraction with denominator 12: 5/6 = 10/12, 11/12 = 11/12, 1/2 = 6/12, 2/3 = 8/12.

\dfrac{5}{6}=\dfrac{10}{12},\;\; \dfrac{1}{2}=\dfrac{6}{12},\;\; \dfrac{4}{6}=\dfrac{8}{12}

Equivalent fractions over 12 line up exactly with the marked ticks.

#2 Make a Systematic List 3.NF.A.2

Compare each in /12 to the range (5/12, 9/12): 10/12 (=5/6) too big, 11/12 (=11/12) too big, 6/12 (=1/2) yes, 8/12 (=2/3) yes. So 1/2 and 2/3 lie between.

5/12 < \text{kept} < 9/12

On the number line, a fraction is between two others exactly when its tick sits between their ticks.

Answer: 1/2 and 4/6

Review

In /12 the target window is 5/12 through 9/12; the kept fractions sit inside it and the others sit outside, and the marks on the number line agree.

Draw the diagram (tool 1) and physically mark all ticks: place the two endpoints, then read which of the marked points land between them, giving the same answer.

Standards · min grade 3

3.NF.A.2 Understand a fraction as a number on the number line — Placing each fraction at its tick and reading which lie between the endpoints
3.NF.A.3 Explain equivalence of fractions and compare fractions by reasoning — Rewriting the fractions as equivalent /12 to compare

💡 Put every fraction over the same 12 ticks, and you can see at a glance which ones sit in between!

Variant 6 answer: 1/2 and 3/8

Mark each given fraction on the number line, then write every one of them that lies between $\dfrac{2}{8}$ and $\dfrac{3}{4}$ .

$\dfrac{1}{2} \qquad \dfrac{7}{8} \qquad \dfrac{1}{4} \qquad \dfrac{3}{8}$

(figure) A number line from 0 to 1 is divided into 8 equal parts, with 0 labeled at the left end and 1 at the right end. Mark the given fractions $\dfrac{1}{2}$ , $\dfrac{7}{8}$ , $\dfrac{1}{4}$ , $\dfrac{3}{8}$ on this number line.

Show solution

Understand

On a number line from 0 to 1 divided into 8 equal parts, mark the fractions 1/2, 7/8, 1/4, 3/8, then list every one of them that lies strictly between 1/4 and 3/4.

Givens

The number line from 0 to 1 is split into 8 equal parts, so each tick is 1/8
The fractions to place are 1/2, 7/8, 1/4, 3/8
The target interval is between 1/4 and 3/4

Unknowns

Which of the fractions fall between 1/4 and 3/4

Constraints

'Between' means greater than 1/4 and less than 3/4
Comparisons are easiest when all fractions share the denominator 8

Plan

#2 Make a Systematic List · also uses: #1 Draw a Diagram

Rewriting every fraction over the common denominator 8 (matching the number-line ticks) lets us list each one's position and read off which lie between the two endpoints. The number line is the diagram that anchors the comparison.

Execute

#1 Draw a Diagram 3.NF.A.3

Since the line has 8 equal parts, write the boundaries over 8: 1/4 is 2/8, and 3/4 is 6/8. So we want fractions between 2/8 and 6/8.

\dfrac{3}{4} = \dfrac{6}{8}

Matching the tick spacing makes every position a count of parts.

#2 Make a Systematic List 3.NF.A.3

Rewrite each fraction with denominator 8: 1/2 = 4/8, 7/8 = 7/8, 1/4 = 2/8, 3/8 = 3/8.

\dfrac{1}{2}=\dfrac{4}{8},\;\; \dfrac{1}{4}=\dfrac{2}{8}

Equivalent fractions over 8 line up exactly with the marked ticks.

#2 Make a Systematic List 3.NF.A.2

Compare each in /8 to the range (2/8, 6/8): 4/8 (=1/2) yes, 7/8 (=7/8) too big, 2/8 (=1/4) too small, 3/8 (=3/8) yes. So 1/2 and 3/8 lie between.

2/8 < \text{kept} < 6/8

On the number line, a fraction is between two others exactly when its tick sits between their ticks.

Answer: 1/2 and 3/8

Review

In /8 the target window is 2/8 through 6/8; the kept fractions sit inside it and the others sit outside, and the marks on the number line agree.

Draw the diagram (tool 1) and physically mark all ticks: place the two endpoints, then read which of the marked points land between them, giving the same answer.

Standards · min grade 3

3.NF.A.2 Understand a fraction as a number on the number line — Placing each fraction at its tick and reading which lie between the endpoints
3.NF.A.3 Explain equivalence of fractions and compare fractions by reasoning — Rewriting the fractions as equivalent /8 to compare

💡 Put every fraction over the same 8 ticks, and you can see at a glance which ones sit in between!

Variant 7 answer: 2/3 and 5/6

Mark each given fraction on the number line, then write every one of them that lies between $\dfrac{1}{2}$ and $\dfrac{1}{1}$ .

$\dfrac{2}{3} \qquad \dfrac{5}{6} \qquad \dfrac{1}{3} \qquad \dfrac{1}{6}$

(figure) A number line from 0 to 1 is divided into 6 equal parts, with 0 labeled at the left end and 1 at the right end. Mark the given fractions $\dfrac{2}{3}$ , $\dfrac{5}{6}$ , $\dfrac{1}{3}$ , $\dfrac{1}{6}$ on this number line.

Show solution

Understand

On a number line from 0 to 1 divided into 6 equal parts, mark the fractions 2/3, 5/6, 1/3, 1/6, then list every one of them that lies strictly between 1/2 and 1/1.

Givens

The number line from 0 to 1 is split into 6 equal parts, so each tick is 1/6
The fractions to place are 2/3, 5/6, 1/3, 1/6
The target interval is between 1/2 and 1/1

Unknowns

Which of the fractions fall between 1/2 and 1/1

Constraints

'Between' means greater than 1/2 and less than 1/1
Comparisons are easiest when all fractions share the denominator 6

Plan

#2 Make a Systematic List · also uses: #1 Draw a Diagram

Execute

#1 Draw a Diagram 3.NF.A.3

Since the line has 6 equal parts, write the boundaries over 6: 1/2 is 3/6, and 1/1 is 6/6. So we want fractions between 3/6 and 6/6.

\dfrac{1}{1} = \dfrac{6}{6}

Matching the tick spacing makes every position a count of parts.

#2 Make a Systematic List 3.NF.A.3

Rewrite each fraction with denominator 6: 2/3 = 4/6, 5/6 = 5/6, 1/3 = 2/6, 1/6 = 1/6.

\dfrac{2}{3}=\dfrac{4}{6},\;\; \dfrac{1}{3}=\dfrac{2}{6}

Equivalent fractions over 6 line up exactly with the marked ticks.

#2 Make a Systematic List 3.NF.A.2

Compare each in /6 to the range (3/6, 6/6): 4/6 (=2/3) yes, 5/6 (=5/6) yes, 2/6 (=1/3) too small, 1/6 (=1/6) too small. So 2/3 and 5/6 lie between.

3/6 < \text{kept} < 6/6

On the number line, a fraction is between two others exactly when its tick sits between their ticks.

Answer: 2/3 and 5/6

Review

In /6 the target window is 3/6 through 6/6; the kept fractions sit inside it and the others sit outside, and the marks on the number line agree.

Draw the diagram (tool 1) and physically mark all ticks: place the two endpoints, then read which of the marked points land between them, giving the same answer.

Standards · min grade 3

3.NF.A.2 Understand a fraction as a number on the number line — Placing each fraction at its tick and reading which lie between the endpoints
3.NF.A.3 Explain equivalence of fractions and compare fractions by reasoning — Rewriting the fractions as equivalent /6 to compare

💡 Put every fraction over the same 6 ticks, and you can see at a glance which ones sit in between!

Variant 8 answer: 1/3 and 5/9

Mark each given fraction on the number line, then write every one of them that lies between $\dfrac{2}{9}$ and $\dfrac{2}{3}$ .

$\dfrac{1}{3} \qquad \dfrac{5}{9} \qquad \dfrac{1}{9} \qquad \dfrac{8}{9}$

(figure) A number line from 0 to 1 is divided into 9 equal parts, with 0 labeled at the left end and 1 at the right end. Mark the given fractions $\dfrac{1}{3}$ , $\dfrac{5}{9}$ , $\dfrac{1}{9}$ , $\dfrac{8}{9}$ on this number line.

Show solution

Understand

On a number line from 0 to 1 divided into 9 equal parts, mark the fractions 1/3, 5/9, 1/9, 8/9, then list every one of them that lies strictly between 2/9 and 2/3.

Givens

The number line from 0 to 1 is split into 9 equal parts, so each tick is 1/9
The fractions to place are 1/3, 5/9, 1/9, 8/9
The target interval is between 2/9 and 2/3

Unknowns

Which of the fractions fall between 2/9 and 2/3

Constraints

'Between' means greater than 2/9 and less than 2/3
Comparisons are easiest when all fractions share the denominator 9

Plan

#2 Make a Systematic List · also uses: #1 Draw a Diagram

Rewriting every fraction over the common denominator 9 (matching the number-line ticks) lets us list each one's position and read off which lie between the two endpoints. The number line is the diagram that anchors the comparison.

Execute

#1 Draw a Diagram 3.NF.A.3

Since the line has 9 equal parts, write the boundaries over 9: 2/9 is 2/9, and 2/3 is 6/9. So we want fractions between 2/9 and 6/9.

\dfrac{2}{3} = \dfrac{6}{9}

Matching the tick spacing makes every position a count of parts.

#2 Make a Systematic List 3.NF.A.3

Rewrite each fraction with denominator 9: 1/3 = 3/9, 5/9 = 5/9, 1/9 = 1/9, 8/9 = 8/9.

\dfrac{1}{3}=\dfrac{3}{9}

Equivalent fractions over 9 line up exactly with the marked ticks.

#2 Make a Systematic List 3.NF.A.2

Compare each in /9 to the range (2/9, 6/9): 3/9 (=1/3) yes, 5/9 (=5/9) yes, 1/9 (=1/9) too small, 8/9 (=8/9) too big. So 1/3 and 5/9 lie between.

2/9 < \text{kept} < 6/9

On the number line, a fraction is between two others exactly when its tick sits between their ticks.

Answer: 1/3 and 5/9

Review

In /9 the target window is 2/9 through 6/9; the kept fractions sit inside it and the others sit outside, and the marks on the number line agree.

Draw the diagram (tool 1) and physically mark all ticks: place the two endpoints, then read which of the marked points land between them, giving the same answer.

Standards · min grade 3

3.NF.A.2 Understand a fraction as a number on the number line — Placing each fraction at its tick and reading which lie between the endpoints
3.NF.A.3 Explain equivalence of fractions and compare fractions by reasoning — Rewriting the fractions as equivalent /9 to compare

💡 Put every fraction over the same 9 ticks, and you can see at a glance which ones sit in between!

Variant 9 answer: 1/4 and 3/4 and 1/2 and 3/8

Mark each given fraction on the number line, then write every one of them that lies between $\dfrac{1}{8}$ and $\dfrac{7}{8}$ .

$\dfrac{1}{4} \qquad \dfrac{3}{4} \qquad \dfrac{1}{2} \qquad \dfrac{3}{8}$

(figure) A number line from 0 to 1 is divided into 8 equal parts, with 0 labeled at the left end and 1 at the right end. Mark the given fractions $\dfrac{1}{4}$ , $\dfrac{3}{4}$ , $\dfrac{1}{2}$ , $\dfrac{3}{8}$ on this number line.

Show solution

Understand

On a number line from 0 to 1 divided into 8 equal parts, mark the fractions 1/4, 3/4, 1/2, 3/8, then list every one of them that lies strictly between 1/8 and 7/8.

Givens

The number line from 0 to 1 is split into 8 equal parts, so each tick is 1/8
The fractions to place are 1/4, 3/4, 1/2, 3/8
The target interval is between 1/8 and 7/8

Unknowns

Which of the fractions fall between 1/8 and 7/8

Constraints

'Between' means greater than 1/8 and less than 7/8
Comparisons are easiest when all fractions share the denominator 8

Plan

#2 Make a Systematic List · also uses: #1 Draw a Diagram

Execute

#1 Draw a Diagram 3.NF.A.3

Since the line has 8 equal parts, write the boundaries over 8: 1/8 is 1/8, and 7/8 is 7/8. So we want fractions between 1/8 and 7/8.

\dfrac{7}{8}

Matching the tick spacing makes every position a count of parts.

#2 Make a Systematic List 3.NF.A.3

Rewrite each fraction with denominator 8: 1/4 = 2/8, 3/4 = 6/8, 1/2 = 4/8, 3/8 = 3/8.

\dfrac{1}{4}=\dfrac{2}{8},\;\; \dfrac{3}{4}=\dfrac{6}{8},\;\; \dfrac{1}{2}=\dfrac{4}{8}

Equivalent fractions over 8 line up exactly with the marked ticks.

#2 Make a Systematic List 3.NF.A.2

Compare each in /8 to the range (1/8, 7/8): 2/8 (=1/4) yes, 6/8 (=3/4) yes, 4/8 (=1/2) yes, 3/8 (=3/8) yes. So 1/4 and 3/4 and 1/2 and 3/8 lie between.

1/8 < \text{kept} < 7/8

On the number line, a fraction is between two others exactly when its tick sits between their ticks.

Answer: 1/4 and 3/4 and 1/2 and 3/8

Review

In /8 the target window is 1/8 through 7/8; the kept fractions sit inside it and the others sit outside, and the marks on the number line agree.

Draw the diagram (tool 1) and physically mark all ticks: place the two endpoints, then read which of the marked points land between them, giving the same answer.

Standards · min grade 3

3.NF.A.2 Understand a fraction as a number on the number line — Placing each fraction at its tick and reading which lie between the endpoints
3.NF.A.3 Explain equivalence of fractions and compare fractions by reasoning — Rewriting the fractions as equivalent /8 to compare

💡 Put every fraction over the same 8 ticks, and you can see at a glance which ones sit in between!

Variant 10 answer: 1/2 and 2/5

Mark each given fraction on the number line, then write every one of them that lies between $\dfrac{3}{10}$ and $\dfrac{4}{5}$ .

$\dfrac{1}{2} \qquad \dfrac{9}{10} \qquad \dfrac{2}{5} \qquad \dfrac{1}{10}$

(figure) A number line from 0 to 1 is divided into 10 equal parts, with 0 labeled at the left end and 1 at the right end. Mark the given fractions $\dfrac{1}{2}$ , $\dfrac{9}{10}$ , $\dfrac{2}{5}$ , $\dfrac{1}{10}$ on this number line.

Show solution

Understand

On a number line from 0 to 1 divided into 10 equal parts, mark the fractions 1/2, 9/10, 2/5, 1/10, then list every one of them that lies strictly between 3/10 and 4/5.

Givens

The number line from 0 to 1 is split into 10 equal parts, so each tick is 1/10
The fractions to place are 1/2, 9/10, 2/5, 1/10
The target interval is between 3/10 and 4/5

Unknowns

Which of the fractions fall between 3/10 and 4/5

Constraints

'Between' means greater than 3/10 and less than 4/5
Comparisons are easiest when all fractions share the denominator 10

Plan

#2 Make a Systematic List · also uses: #1 Draw a Diagram

Execute

#1 Draw a Diagram 3.NF.A.3

Since the line has 10 equal parts, write the boundaries over 10: 3/10 is 3/10, and 4/5 is 8/10. So we want fractions between 3/10 and 8/10.

\dfrac{4}{5} = \dfrac{8}{10}

Matching the tick spacing makes every position a count of parts.

#2 Make a Systematic List 3.NF.A.3

Rewrite each fraction with denominator 10: 1/2 = 5/10, 9/10 = 9/10, 2/5 = 4/10, 1/10 = 1/10.

\dfrac{1}{2}=\dfrac{5}{10},\;\; \dfrac{2}{5}=\dfrac{4}{10}

Equivalent fractions over 10 line up exactly with the marked ticks.

#2 Make a Systematic List 3.NF.A.2

Compare each in /10 to the range (3/10, 8/10): 5/10 (=1/2) yes, 9/10 (=9/10) too big, 4/10 (=2/5) yes, 1/10 (=1/10) too small. So 1/2 and 2/5 lie between.

3/10 < \text{kept} < 8/10

On the number line, a fraction is between two others exactly when its tick sits between their ticks.

Answer: 1/2 and 2/5

Review

In /10 the target window is 3/10 through 8/10; the kept fractions sit inside it and the others sit outside, and the marks on the number line agree.

Draw the diagram (tool 1) and physically mark all ticks: place the two endpoints, then read which of the marked points land between them, giving the same answer.

Standards · min grade 3

3.NF.A.2 Understand a fraction as a number on the number line — Placing each fraction at its tick and reading which lie between the endpoints
3.NF.A.3 Explain equivalence of fractions and compare fractions by reasoning — Rewriting the fractions as equivalent /10 to compare

💡 Put every fraction over the same 10 ticks, and you can see at a glance which ones sit in between!