Practice · Find fractions between two on the number line · Sensim Math

Variant 1 easy answer:

\frac{1}{2}

and

\frac{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

1 · Understandwhat's really being asked

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

Givens

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

Unknowns

Which of the fractions fall between $\frac{1}{6}$ and $\frac{2}{3}$

Constraints

'Between' means greater than $\frac{1}{6}$ and less than $\frac{2}{3}$
Comparisons are easiest when all fractions share the denominator 6

2 · Planchoose the strategy

#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.

3 · Execute3 carry out the plan

1Express the endpoints in the common denominator

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

Since the line has 6 equal parts, write the boundaries over 6:

\frac{1}{6}

is

\frac{1}{6}

, and

\frac{2}{3}

is

\frac{4}{6}

. So we want fractions between

\frac{1}{6}

and

\frac{4}{6}

.

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

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

2Convert the fractions to /6

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

Rewrite each fraction with denominator 6:

\frac{1}{2}

=

\frac{3}{6}

,

\frac{5}{6}

=

\frac{5}{6}

,

\frac{1}{3}

=

\frac{2}{6}

,

\frac{1}{6}

=

\frac{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.

3List which fall between $\frac{1}{6}$ and $\frac{4}{6}$

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

Compare each in /6 to the range (

\frac{1}{6}

,

\frac{4}{6}

):

\frac{3}{6}

(=

\frac{1}{2}

) yes,

\frac{5}{6}

(=

\frac{5}{6}

) too big,

\frac{2}{6}

(=

\frac{1}{3}

) yes,

\frac{1}{6}

(=

\frac{1}{6}

) too small. So

\frac{1}{2}

and

\frac{1}{3}

lie between.

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

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

Answer:

\frac{1}{2}

and

\frac{1}{3}

4 · Reviewdoes it hold up?

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

Another way: 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.

Standardsmin 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

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

Variant 2 easy answer:

\frac{2}{3}

and

\frac{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

1 · Understandwhat's really being asked

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

Givens

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

Unknowns

Which of the fractions fall between $\frac{1}{2}$ and $\frac{1}{1}$

Constraints

'Between' means greater than $\frac{1}{2}$ and less than $\frac{1}{1}$
Comparisons are easiest when all fractions share the denominator 6

2 · Planchoose the strategy

#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.

3 · Execute3 carry out the plan

1Express the endpoints in the common denominator

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

Since the line has 6 equal parts, write the boundaries over 6:

\frac{1}{2}

is

\frac{3}{6}

, and

\frac{1}{1}

is

\frac{6}{6}

. So we want fractions between

\frac{3}{6}

and

\frac{6}{6}

.

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

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

2Convert the fractions to /6

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

Rewrite each fraction with denominator 6:

\frac{2}{3}

=

\frac{4}{6}

,

\frac{5}{6}

=

\frac{5}{6}

,

\frac{1}{3}

=

\frac{2}{6}

,

\frac{1}{6}

=

\frac{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.

3List which fall between $\frac{3}{6}$ and $\frac{6}{6}$

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

Compare each in /6 to the range (

\frac{3}{6}

,

\frac{6}{6}

):

\frac{4}{6}

(=

\frac{2}{3}

) yes,

\frac{5}{6}

(=

\frac{5}{6}

) yes,

\frac{2}{6}

(=

\frac{1}{3}

) too small,

\frac{1}{6}

(=

\frac{1}{6}

) too small. So

\frac{2}{3}

and

\frac{5}{6}

lie between.

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

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

Answer:

\frac{2}{3}

and

\frac{5}{6}

4 · Reviewdoes it hold up?

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

Another way: 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.

Standardsmin 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

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

Variant 3 easy answer:

\frac{1}{2}

and

\frac{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

1 · Understandwhat's really being asked

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

Givens

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

Unknowns

Which of the fractions fall between $\frac{1}{4}$ and $\frac{3}{4}$

Constraints

'Between' means greater than $\frac{1}{4}$ and less than $\frac{3}{4}$
Comparisons are easiest when all fractions share the denominator 8

2 · Planchoose the strategy

#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.

3 · Execute3 carry out the plan

1Express the endpoints in the common denominator

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

Since the line has 8 equal parts, write the boundaries over 8:

\frac{1}{4}

is

\frac{2}{8}

, and

\frac{3}{4}

is

\frac{6}{8}

. So we want fractions between

\frac{2}{8}

and

\frac{6}{8}

.

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

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

2Convert the fractions to /8

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

Rewrite each fraction with denominator 8:

\frac{1}{2}

=

\frac{4}{8}

,

\frac{7}{8}

=

\frac{7}{8}

,

\frac{1}{4}

=

\frac{2}{8}

,

\frac{3}{8}

=

\frac{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.

3List which fall between $\frac{2}{8}$ and $\frac{6}{8}$

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

Compare each in /8 to the range (

\frac{2}{8}

,

\frac{6}{8}

):

\frac{4}{8}

(=

\frac{1}{2}

) yes,

\frac{7}{8}

(=

\frac{7}{8}

) too big,

\frac{2}{8}

(=

\frac{1}{4}

) too small,

\frac{3}{8}

(=

\frac{3}{8}

) yes. So

\frac{1}{2}

and

\frac{3}{8}

lie between.

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

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

Answer:

\frac{1}{2}

and

\frac{3}{8}

4 · Reviewdoes it hold up?

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

Another way: 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.

Standardsmin 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

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

Variant 4 medium answer:

\frac{1}{4}

and

\frac{3}{4}

and

\frac{1}{2}

and

\frac{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

1 · Understandwhat's really being asked

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

Givens

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

Unknowns

Which of the fractions fall between $\frac{1}{8}$ and $\frac{7}{8}$

Constraints

'Between' means greater than $\frac{1}{8}$ and less than $\frac{7}{8}$
Comparisons are easiest when all fractions share the denominator 8

2 · Planchoose the strategy

#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.

3 · Execute3 carry out the plan

1Express the endpoints in the common denominator

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

Since the line has 8 equal parts, write the boundaries over 8:

\frac{1}{8}

is

\frac{1}{8}

, and

\frac{7}{8}

is

\frac{7}{8}

. So we want fractions between

\frac{1}{8}

and

\frac{7}{8}

.

\dfrac{7}{8}

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

2Convert the fractions to /8

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

Rewrite each fraction with denominator 8:

\frac{1}{4}

=

\frac{2}{8}

,

\frac{3}{4}

=

\frac{6}{8}

,

\frac{1}{2}

=

\frac{4}{8}

,

\frac{3}{8}

=

\frac{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.

3List which fall between $\frac{1}{8}$ and $\frac{7}{8}$

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

Compare each in /8 to the range (

\frac{1}{8}

,

\frac{7}{8}

):

\frac{2}{8}

(=

\frac{1}{4}

) yes,

\frac{6}{8}

(=

\frac{3}{4}

) yes,

\frac{4}{8}

(=

\frac{1}{2}

) yes,

\frac{3}{8}

(=

\frac{3}{8}

) yes. So

\frac{1}{4}

and

\frac{3}{4}

and

\frac{1}{2}

and

\frac{3}{8}

lie between.

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

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

Answer:

\frac{1}{4}

and

\frac{3}{4}

and

\frac{1}{2}

and

\frac{3}{8}

4 · Reviewdoes it hold up?

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

Another way: 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.

Standardsmin 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

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

Variant 5 medium answer:

\frac{1}{3}

and

\frac{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

1 · Understandwhat's really being asked

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

Givens

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

Unknowns

Which of the fractions fall between $\frac{2}{9}$ and $\frac{2}{3}$

Constraints

'Between' means greater than $\frac{2}{9}$ and less than $\frac{2}{3}$
Comparisons are easiest when all fractions share the denominator 9

2 · Planchoose the strategy

#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.

3 · Execute3 carry out the plan

1Express the endpoints in the common denominator

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

Since the line has 9 equal parts, write the boundaries over 9:

\frac{2}{9}

is

\frac{2}{9}

, and

\frac{2}{3}

is

\frac{6}{9}

. So we want fractions between

\frac{2}{9}

and

\frac{6}{9}

.

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

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

2Convert the fractions to /9

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

Rewrite each fraction with denominator 9:

\frac{1}{3}

=

\frac{3}{9}

,

\frac{5}{9}

=

\frac{5}{9}

,

\frac{1}{9}

=

\frac{1}{9}

,

\frac{8}{9}

=

\frac{8}{9}

.

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

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

3List which fall between $\frac{2}{9}$ and $\frac{6}{9}$

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

Compare each in /9 to the range (

\frac{2}{9}

,

\frac{6}{9}

):

\frac{3}{9}

(=

\frac{1}{3}

) yes,

\frac{5}{9}

(=

\frac{5}{9}

) yes,

\frac{1}{9}

(=

\frac{1}{9}

) too small,

\frac{8}{9}

(=

\frac{8}{9}

) too big. So

\frac{1}{3}

and

\frac{5}{9}

lie between.

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

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

Answer:

\frac{1}{3}

and

\frac{5}{9}

4 · Reviewdoes it hold up?

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

Another way: 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.

Standardsmin 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

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

Variant 6 medium answer:

\frac{1}{2}

and

\frac{2}{5}

and

\frac{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

1 · Understandwhat's really being asked

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

Givens

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

Unknowns

Which of the fractions fall between $\frac{1}{5}$ and $\frac{7}{10}$

Constraints

'Between' means greater than $\frac{1}{5}$ and less than $\frac{7}{10}$
Comparisons are easiest when all fractions share the denominator 10

2 · Planchoose the strategy

#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.

3 · Execute3 carry out the plan

1Express the endpoints in the common denominator

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

Since the line has 10 equal parts, write the boundaries over 10:

\frac{1}{5}

is

\frac{2}{10}

, and

\frac{7}{10}

is

\frac{7}{10}

. So we want fractions between

\frac{2}{10}

and

\frac{7}{10}

.

\dfrac{7}{10}

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

2Convert the fractions to /10

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

Rewrite each fraction with denominator 10:

\frac{1}{2}

=

\frac{5}{10}

,

\frac{2}{5}

=

\frac{4}{10}

,

\frac{3}{10}

=

\frac{3}{10}

,

\frac{9}{10}

=

\frac{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.

3List which fall between $\frac{2}{10}$ and $\frac{7}{10}$

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

Compare each in /10 to the range (

\frac{2}{10}

,

\frac{7}{10}

):

\frac{5}{10}

(=

\frac{1}{2}

) yes,

\frac{4}{10}

(=

\frac{2}{5}

) yes,

\frac{3}{10}

(=

\frac{3}{10}

) yes,

\frac{9}{10}

(=

\frac{9}{10}

) too big. So

\frac{1}{2}

and

\frac{2}{5}

and

\frac{3}{10}

lie between.

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

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

Answer:

\frac{1}{2}

and

\frac{2}{5}

and

\frac{3}{10}

4 · Reviewdoes it hold up?

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

Another way: 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.

Standardsmin 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

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

Variant 7 medium answer:

\frac{1}{2}

and

\frac{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

1 · Understandwhat's really being asked

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

Givens

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

Unknowns

Which of the fractions fall between $\frac{3}{10}$ and $\frac{4}{5}$

Constraints

'Between' means greater than $\frac{3}{10}$ and less than $\frac{4}{5}$
Comparisons are easiest when all fractions share the denominator 10

2 · Planchoose the strategy

#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.

3 · Execute3 carry out the plan

1Express the endpoints in the common denominator

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

Since the line has 10 equal parts, write the boundaries over 10:

\frac{3}{10}

is

\frac{3}{10}

, and

\frac{4}{5}

is

\frac{8}{10}

. So we want fractions between

\frac{3}{10}

and

\frac{8}{10}

.

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

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

2Convert the fractions to /10

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

Rewrite each fraction with denominator 10:

\frac{1}{2}

=

\frac{5}{10}

,

\frac{9}{10}

=

\frac{9}{10}

,

\frac{2}{5}

=

\frac{4}{10}

,

\frac{1}{10}

=

\frac{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.

3List which fall between $\frac{3}{10}$ and $\frac{8}{10}$

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

Compare each in /10 to the range (

\frac{3}{10}

,

\frac{8}{10}

):

\frac{5}{10}

(=

\frac{1}{2}

) yes,

\frac{9}{10}

(=

\frac{9}{10}

) too big,

\frac{4}{10}

(=

\frac{2}{5}

) yes,

\frac{1}{10}

(=

\frac{1}{10}

) too small. So

\frac{1}{2}

and

\frac{2}{5}

lie between.

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

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

Answer:

\frac{1}{2}

and

\frac{2}{5}

4 · Reviewdoes it hold up?

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

Another way: 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.

Standardsmin 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

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

Variant 8 hard answer:

\frac{1}{2}

and

\frac{5}{12}

and

\frac{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

1 · Understandwhat's really being asked

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

Givens

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

Unknowns

Which of the fractions fall between $\frac{1}{3}$ and $\frac{5}{6}$

Constraints

'Between' means greater than $\frac{1}{3}$ and less than $\frac{5}{6}$
Comparisons are easiest when all fractions share the denominator 12

2 · Planchoose the strategy

#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.

3 · Execute3 carry out the plan

1Express the endpoints in the common denominator

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

Since the line has 12 equal parts, write the boundaries over 12:

\frac{1}{3}

is

\frac{4}{12}

, and

\frac{5}{6}

is

\frac{10}{12}

. So we want fractions between

\frac{4}{12}

and

\frac{10}{12}

.

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

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

2Convert the fractions to /12

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

Rewrite each fraction with denominator 12:

\frac{1}{2}

=

\frac{6}{12}

,

\frac{11}{12}

=

\frac{11}{12}

,

\frac{5}{12}

=

\frac{5}{12}

,

\frac{3}{4}

=

\frac{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.

3List which fall between $\frac{4}{12}$ and $\frac{10}{12}$

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

Compare each in /12 to the range (

\frac{4}{12}

,

\frac{10}{12}

):

\frac{6}{12}

(=

\frac{1}{2}

) yes,

\frac{11}{12}

(=

\frac{11}{12}

) too big,

\frac{5}{12}

(=

\frac{5}{12}

) yes,

\frac{9}{12}

(=

\frac{3}{4}

) yes. So

\frac{1}{2}

and

\frac{5}{12}

and

\frac{3}{4}

lie between.

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

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

Answer:

\frac{1}{2}

and

\frac{5}{12}

and

\frac{3}{4}

4 · Reviewdoes it hold up?

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

Another way: 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.

Standardsmin 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

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

Variant 9 hard answer:

\frac{5}{12}

and

\frac{1}{2}

and

\frac{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

1 · Understandwhat's really being asked

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

Givens

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

Unknowns

Which of the fractions fall between $\frac{1}{4}$ and $\frac{2}{3}$

Constraints

'Between' means greater than $\frac{1}{4}$ and less than $\frac{2}{3}$
Comparisons are easiest when all fractions share the denominator 12

2 · Planchoose the strategy

#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.

3 · Execute3 carry out the plan

1Express the endpoints in the common denominator

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

Since the line has 12 equal parts, write the boundaries over 12:

\frac{1}{4}

is

\frac{3}{12}

, and

\frac{2}{3}

is

\frac{8}{12}

. So we want fractions between

\frac{3}{12}

and

\frac{8}{12}

.

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

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

2Convert the fractions to /12

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

Rewrite each fraction with denominator 12:

\frac{5}{12}

=

\frac{5}{12}

,

\frac{1}{2}

=

\frac{6}{12}

,

\frac{7}{12}

=

\frac{7}{12}

,

\frac{11}{12}

=

\frac{11}{12}

.

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

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

3List which fall between $\frac{3}{12}$ and $\frac{8}{12}$

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

Compare each in /12 to the range (

\frac{3}{12}

,

\frac{8}{12}

):

\frac{5}{12}

(=

\frac{5}{12}

) yes,

\frac{6}{12}

(=

\frac{1}{2}

) yes,

\frac{7}{12}

(=

\frac{7}{12}

) yes,

\frac{11}{12}

(=

\frac{11}{12}

) too big. So

\frac{5}{12}

and

\frac{1}{2}

and

\frac{7}{12}

lie between.

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

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

Answer:

\frac{5}{12}

and

\frac{1}{2}

and

\frac{7}{12}

4 · Reviewdoes it hold up?

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

Another way: 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.

Standardsmin 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

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

Variant 10 hard answer:

\frac{1}{2}

and

\frac{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

1 · Understandwhat's really being asked

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

Givens

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

Unknowns

Which of the fractions fall between $\frac{5}{12}$ and $\frac{3}{4}$

Constraints

'Between' means greater than $\frac{5}{12}$ and less than $\frac{3}{4}$
Comparisons are easiest when all fractions share the denominator 12

2 · Planchoose the strategy

#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.

3 · Execute3 carry out the plan

1Express the endpoints in the common denominator

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

Since the line has 12 equal parts, write the boundaries over 12:

\frac{5}{12}

is

\frac{5}{12}

, and

\frac{3}{4}

is

\frac{9}{12}

. So we want fractions between

\frac{5}{12}

and

\frac{9}{12}

.

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

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

2Convert the fractions to /12

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

Rewrite each fraction with denominator 12:

\frac{5}{6}

=

\frac{10}{12}

,

\frac{11}{12}

=

\frac{11}{12}

,

\frac{1}{2}

=

\frac{6}{12}

,

\frac{2}{3}

=

\frac{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.

3List which fall between $\frac{5}{12}$ and $\frac{9}{12}$

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

Compare each in /12 to the range (

\frac{5}{12}

,

\frac{9}{12}

):

\frac{10}{12}

(=

\frac{5}{6}

) too big,

\frac{11}{12}

(=

\frac{11}{12}

) too big,

\frac{6}{12}

(=

\frac{1}{2}

) yes,

\frac{8}{12}

(=

\frac{2}{3}

) yes. So

\frac{1}{2}

and

\frac{2}{3}

lie between.

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

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

Answer:

\frac{1}{2}

and

\frac{4}{6}

4 · Reviewdoes it hold up?

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

Another way: 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.

Standardsmin 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

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

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

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