Practice · Split the total gap into equal steps · Sensim Math

Variant 1 easy

The four numbers $-\frac{1}{5}$ , $x$ , $y$ , $1\frac{3}{5}$ are listed in order of size, and the gaps between neighboring numbers are all equal. Find the value of $x+y$ .

Your answer

Show solution

1 · Understandwhat's really being asked

Four numbers sit on a number line in order, evenly spaced. We know the two ends, $-\frac{1}{5}$ and $1\frac{3}{5}$ , and need the sum of the two middle numbers $x$ and $y$ .

Givens

The numbers in order: $-\frac{1}{5}$ , $x$ , $y$ , $1\frac{3}{5}$ .
The three gaps between neighbors are equal.

Unknowns

$x+y$

Constraints

Four numbers make three gaps, not four.

2 · Planchoose the strategy

#1 Draw a Diagram · also uses: #7 Identify Subproblems

Picture the four points on a number line. The distance from the first to the last is covered by three equal steps, so find the total distance, cut it into three, and step along from $-\frac{1}{5}$ .

3 · Execute4 carry out the plan

1Measure the whole stretch

#1 Draw a Diagram 7.NS.A.1

Write

1\frac{3}{5}

as

\frac{8}{5}

and subtract the left end. Subtracting a negative adds its size.

\frac{8}{5} - \left(-\frac{1}{5}\right) = \frac{8}{5} + \frac{1}{5} = \frac{9}{5}

The distance from the first point to the last is

\frac{9}{5}

.

2Find one step

#7 Identify Subproblems 7.NS.A.1

Four points have three equal gaps between them.

\frac{9}{5} \div 3 = \frac{3}{5}

Each gap is

\frac{3}{5}

.

3Step along to $x$ and $y$

#1 Draw a Diagram 6.NS.C.6

Start at

-\frac{1}{5}

and add one gap at a time.

x = -\frac{1}{5} + \frac{3}{5} = \frac{2}{5},\quad y = \frac{2}{5} + \frac{3}{5} = 1

One more step gives

1 + \frac{3}{5} = 1\frac{3}{5}

, the right end, as it should.

4Add the middle numbers

#7 Identify Subproblems 7.NS.A.1

Now add

x

and

y

.

\frac{2}{5} + 1 = \frac{7}{5}

So

x+y=\frac{7}{5}

.

Answer:

\frac{7}{5}

4 · Reviewdoes it hold up?

The middle pair is placed symmetrically, so $x+y$ must equal the sum of the two ends: $-\frac{1}{5} + \frac{8}{5} = \frac{7}{5}$ . It matches.

Another way: Use the symmetry directly: $x$ is one step right of the left end and $y$ is one step left of the right end, so the steps cancel and $x+y=-\frac{1}{5}+\frac{8}{5}=\frac{7}{5}$ without finding the step.

Standardsmin grade 7

6.NS.C.6 Rational numbers on the number line and coordinate plane — Placing the four rational numbers as evenly spaced points on a number line.
7.NS.A.1 Apply and extend understanding of addition and subtraction to rational numbers — Subtracting a negative fraction to get the distance and adding the step.

💡Takeaway. For evenly spaced numbers, the two middle ones add up to the same as the two ends.

Variant 2 easy

The four numbers $-\frac{3}{5}$ , $x$ , $y$ , $1\frac{4}{5}$ are listed in order of size, and the gaps between neighboring numbers are all equal. Find the value of $x+y$ .

Your answer

Show solution

1 · Understandwhat's really being asked

Four numbers sit on a number line in order, evenly spaced. We know the two ends, $-\frac{3}{5}$ and $1\frac{4}{5}$ , and need the sum of the two middle numbers $x$ and $y$ .

Givens

The numbers in order: $-\frac{3}{5}$ , $x$ , $y$ , $1\frac{4}{5}$ .
The three gaps between neighbors are equal.

Unknowns

$x+y$

Constraints

Four numbers make three gaps, not four.

2 · Planchoose the strategy

#1 Draw a Diagram · also uses: #7 Identify Subproblems

Picture the four points on a number line. The distance from the first to the last is covered by three equal steps, so find the total distance, cut it into three, and step along from $-\frac{3}{5}$ .

3 · Execute4 carry out the plan

1Measure the whole stretch

#1 Draw a Diagram 7.NS.A.1

Write

1\frac{4}{5}

as

\frac{9}{5}

and subtract the left end. Subtracting a negative adds its size.

\frac{9}{5} - \left(-\frac{3}{5}\right) = \frac{9}{5} + \frac{3}{5} = \frac{12}{5}

The distance from the first point to the last is

\frac{12}{5}

.

2Find one step

#7 Identify Subproblems 7.NS.A.1

Four points have three equal gaps between them.

\frac{12}{5} \div 3 = \frac{4}{5}

Each gap is

\frac{4}{5}

.

3Step along to $x$ and $y$

#1 Draw a Diagram 6.NS.C.6

Start at

-\frac{3}{5}

and add one gap at a time.

x = -\frac{3}{5} + \frac{4}{5} = \frac{1}{5},\quad y = \frac{1}{5} + \frac{4}{5} = 1

One more step gives

1 + \frac{4}{5} = 1\frac{4}{5}

, the right end, as it should.

4Add the middle numbers

#7 Identify Subproblems 7.NS.A.1

Now add

x

and

y

.

\frac{1}{5} + 1 = \frac{6}{5}

So

x+y=\frac{6}{5}

.

Answer:

\frac{6}{5}

4 · Reviewdoes it hold up?

The middle pair is placed symmetrically, so $x+y$ must equal the sum of the two ends: $-\frac{3}{5} + \frac{9}{5} = \frac{6}{5}$ . It matches.

Another way: Use the symmetry directly: $x$ is one step right of the left end and $y$ is one step left of the right end, so the steps cancel and $x+y=-\frac{3}{5}+\frac{9}{5}=\frac{6}{5}$ without finding the step.

Standardsmin grade 7

6.NS.C.6 Rational numbers on the number line and coordinate plane — Placing the four rational numbers as evenly spaced points on a number line.
7.NS.A.1 Apply and extend understanding of addition and subtraction to rational numbers — Subtracting a negative fraction to get the distance and adding the step.

💡Takeaway. For evenly spaced numbers, the two middle ones add up to the same as the two ends.

Variant 3 easy

The four numbers $-\frac{1}{7}$ , $x$ , $y$ , $1\frac{4}{7}$ are listed in order of size, and the gaps between neighboring numbers are all equal. Find the value of $x+y$ .

Your answer

Show solution

1 · Understandwhat's really being asked

Four numbers sit on a number line in order, evenly spaced. We know the two ends, $-\frac{1}{7}$ and $1\frac{4}{7}$ , and need the sum of the two middle numbers $x$ and $y$ .

Givens

The numbers in order: $-\frac{1}{7}$ , $x$ , $y$ , $1\frac{4}{7}$ .
The three gaps between neighbors are equal.

Unknowns

$x+y$

Constraints

Four numbers make three gaps, not four.

2 · Planchoose the strategy

#1 Draw a Diagram · also uses: #7 Identify Subproblems

Picture the four points on a number line. The distance from the first to the last is covered by three equal steps, so find the total distance, cut it into three, and step along from $-\frac{1}{7}$ .

3 · Execute4 carry out the plan

1Measure the whole stretch

#1 Draw a Diagram 7.NS.A.1

Write

1\frac{4}{7}

as

\frac{11}{7}

and subtract the left end. Subtracting a negative adds its size.

\frac{11}{7} - \left(-\frac{1}{7}\right) = \frac{11}{7} + \frac{1}{7} = \frac{12}{7}

The distance from the first point to the last is

\frac{12}{7}

.

2Find one step

#7 Identify Subproblems 7.NS.A.1

Four points have three equal gaps between them.

\frac{12}{7} \div 3 = \frac{4}{7}

Each gap is

\frac{4}{7}

.

3Step along to $x$ and $y$

#1 Draw a Diagram 6.NS.C.6

Start at

-\frac{1}{7}

and add one gap at a time.

x = -\frac{1}{7} + \frac{4}{7} = \frac{3}{7},\quad y = \frac{3}{7} + \frac{4}{7} = 1

One more step gives

1 + \frac{4}{7} = 1\frac{4}{7}

, the right end, as it should.

4Add the middle numbers

#7 Identify Subproblems 7.NS.A.1

Now add

x

and

y

.

\frac{3}{7} + 1 = \frac{10}{7}

So

x+y=\frac{10}{7}

.

Answer:

\frac{10}{7}

4 · Reviewdoes it hold up?

The middle pair is placed symmetrically, so $x+y$ must equal the sum of the two ends: $-\frac{1}{7} + \frac{11}{7} = \frac{10}{7}$ . It matches.

Another way: Use the symmetry directly: $x$ is one step right of the left end and $y$ is one step left of the right end, so the steps cancel and $x+y=-\frac{1}{7}+\frac{11}{7}=\frac{10}{7}$ without finding the step.

Standardsmin grade 7

6.NS.C.6 Rational numbers on the number line and coordinate plane — Placing the four rational numbers as evenly spaced points on a number line.
7.NS.A.1 Apply and extend understanding of addition and subtraction to rational numbers — Subtracting a negative fraction to get the distance and adding the step.

💡Takeaway. For evenly spaced numbers, the two middle ones add up to the same as the two ends.

Variant 4 easy

The four numbers $-\frac{3}{7}$ , $x$ , $y$ , $1\frac{5}{7}$ are listed in order of size, and the gaps between neighboring numbers are all equal. Find the value of $x+y$ .

Your answer

Show solution

1 · Understandwhat's really being asked

Four numbers sit on a number line in order, evenly spaced. We know the two ends, $-\frac{3}{7}$ and $1\frac{5}{7}$ , and need the sum of the two middle numbers $x$ and $y$ .

Givens

The numbers in order: $-\frac{3}{7}$ , $x$ , $y$ , $1\frac{5}{7}$ .
The three gaps between neighbors are equal.

Unknowns

$x+y$

Constraints

Four numbers make three gaps, not four.

2 · Planchoose the strategy

#1 Draw a Diagram · also uses: #7 Identify Subproblems

Picture the four points on a number line. The distance from the first to the last is covered by three equal steps, so find the total distance, cut it into three, and step along from $-\frac{3}{7}$ .

3 · Execute4 carry out the plan

1Measure the whole stretch

#1 Draw a Diagram 7.NS.A.1

Write

1\frac{5}{7}

as

\frac{12}{7}

and subtract the left end. Subtracting a negative adds its size.

\frac{12}{7} - \left(-\frac{3}{7}\right) = \frac{12}{7} + \frac{3}{7} = \frac{15}{7}

The distance from the first point to the last is

\frac{15}{7}

.

2Find one step

#7 Identify Subproblems 7.NS.A.1

Four points have three equal gaps between them.

\frac{15}{7} \div 3 = \frac{5}{7}

Each gap is

\frac{5}{7}

.

3Step along to $x$ and $y$

#1 Draw a Diagram 6.NS.C.6

Start at

-\frac{3}{7}

and add one gap at a time.

x = -\frac{3}{7} + \frac{5}{7} = \frac{2}{7},\quad y = \frac{2}{7} + \frac{5}{7} = 1

One more step gives

1 + \frac{5}{7} = 1\frac{5}{7}

, the right end, as it should.

4Add the middle numbers

#7 Identify Subproblems 7.NS.A.1

Now add

x

and

y

.

\frac{2}{7} + 1 = \frac{9}{7}

So

x+y=\frac{9}{7}

.

Answer:

\frac{9}{7}

4 · Reviewdoes it hold up?

The middle pair is placed symmetrically, so $x+y$ must equal the sum of the two ends: $-\frac{3}{7} + \frac{12}{7} = \frac{9}{7}$ . It matches.

Another way: Use the symmetry directly: $x$ is one step right of the left end and $y$ is one step left of the right end, so the steps cancel and $x+y=-\frac{3}{7}+\frac{12}{7}=\frac{9}{7}$ without finding the step.

Standardsmin grade 7

6.NS.C.6 Rational numbers on the number line and coordinate plane — Placing the four rational numbers as evenly spaced points on a number line.
7.NS.A.1 Apply and extend understanding of addition and subtraction to rational numbers — Subtracting a negative fraction to get the distance and adding the step.

💡Takeaway. For evenly spaced numbers, the two middle ones add up to the same as the two ends.

Variant 5 medium

The four numbers $-\frac{6}{7}$ , $x$ , $y$ , $1\frac{2}{7}$ are listed in order of size, and the gaps between neighboring numbers are all equal. Find the value of $x+y$ .

Your answer

Show solution

1 · Understandwhat's really being asked

Four numbers sit on a number line in order, evenly spaced. We know the two ends, $-\frac{6}{7}$ and $1\frac{2}{7}$ , and need the sum of the two middle numbers $x$ and $y$ .

Givens

The numbers in order: $-\frac{6}{7}$ , $x$ , $y$ , $1\frac{2}{7}$ .
The three gaps between neighbors are equal.

Unknowns

$x+y$

Constraints

Four numbers make three gaps, not four.

2 · Planchoose the strategy

#1 Draw a Diagram · also uses: #7 Identify Subproblems

Picture the four points on a number line. The distance from the first to the last is covered by three equal steps, so find the total distance, cut it into three, and step along from $-\frac{6}{7}$ .

3 · Execute4 carry out the plan

1Measure the whole stretch

#1 Draw a Diagram 7.NS.A.1

Write

1\frac{2}{7}

as

\frac{9}{7}

and subtract the left end. Subtracting a negative adds its size.

\frac{9}{7} - \left(-\frac{6}{7}\right) = \frac{9}{7} + \frac{6}{7} = \frac{15}{7}

The distance from the first point to the last is

\frac{15}{7}

.

2Find one step

#7 Identify Subproblems 7.NS.A.1

Four points have three equal gaps between them.

\frac{15}{7} \div 3 = \frac{5}{7}

Each gap is

\frac{5}{7}

.

3Step along to $x$ and $y$

#1 Draw a Diagram 6.NS.C.6

Start at

-\frac{6}{7}

and add one gap at a time.

x = -\frac{6}{7} + \frac{5}{7} = -\frac{1}{7},\quad y = -\frac{1}{7} + \frac{5}{7} = \frac{4}{7}

One more step gives

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

, the right end, as it should.

4Add the middle numbers

#7 Identify Subproblems 7.NS.A.1

Now add

x

and

y

.

-\frac{1}{7} + \frac{4}{7} = \frac{3}{7}

So

x+y=\frac{3}{7}

.

Answer:

\frac{3}{7}

4 · Reviewdoes it hold up?

The middle pair is placed symmetrically, so $x+y$ must equal the sum of the two ends: $-\frac{6}{7} + \frac{9}{7} = \frac{3}{7}$ . It matches.

Another way: Use the symmetry directly: $x$ is one step right of the left end and $y$ is one step left of the right end, so the steps cancel and $x+y=-\frac{6}{7}+\frac{9}{7}=\frac{3}{7}$ without finding the step.

Standardsmin grade 7

6.NS.C.6 Rational numbers on the number line and coordinate plane — Placing the four rational numbers as evenly spaced points on a number line.
7.NS.A.1 Apply and extend understanding of addition and subtraction to rational numbers — Subtracting a negative fraction to get the distance and adding the step.

💡Takeaway. For evenly spaced numbers, the two middle ones add up to the same as the two ends.

Variant 6 medium

The four numbers $-\frac{5}{8}$ , $x$ , $y$ , $2\frac{3}{8}$ are listed in order of size, and the gaps between neighboring numbers are all equal. Find the value of $x+y$ .

Your answer

Show solution

1 · Understandwhat's really being asked

Four numbers sit on a number line in order, evenly spaced. We know the two ends, $-\frac{5}{8}$ and $2\frac{3}{8}$ , and need the sum of the two middle numbers $x$ and $y$ .

Givens

The numbers in order: $-\frac{5}{8}$ , $x$ , $y$ , $2\frac{3}{8}$ .
The three gaps between neighbors are equal.

Unknowns

$x+y$

Constraints

Four numbers make three gaps, not four.

2 · Planchoose the strategy

#1 Draw a Diagram · also uses: #7 Identify Subproblems

Picture the four points on a number line. The distance from the first to the last is covered by three equal steps, so find the total distance, cut it into three, and step along from $-\frac{5}{8}$ .

3 · Execute4 carry out the plan

1Measure the whole stretch

#1 Draw a Diagram 7.NS.A.1

Write

2\frac{3}{8}

as

\frac{19}{8}

and subtract the left end. Subtracting a negative adds its size.

\frac{19}{8} - \left(-\frac{5}{8}\right) = \frac{19}{8} + \frac{5}{8} = 3

The distance from the first point to the last is

3

.

2Find one step

#7 Identify Subproblems 7.NS.A.1

Four points have three equal gaps between them.

3 \div 3 = 1

Each gap is

1

.

3Step along to $x$ and $y$

#1 Draw a Diagram 6.NS.C.6

Start at

-\frac{5}{8}

and add one gap at a time.

x = -\frac{5}{8} + 1 = \frac{3}{8},\quad y = \frac{3}{8} + 1 = 1\frac{3}{8}

One more step gives

1\frac{3}{8} + 1 = 2\frac{3}{8}

, the right end, as it should.

4Add the middle numbers

#7 Identify Subproblems 7.NS.A.1

Now add

x

and

y

.

\frac{3}{8} + 1\frac{3}{8} = \frac{7}{4}

So

x+y=\frac{7}{4}

.

Answer:

\frac{7}{4}

4 · Reviewdoes it hold up?

The middle pair is placed symmetrically, so $x+y$ must equal the sum of the two ends: $-\frac{5}{8} + \frac{19}{8} = \frac{7}{4}$ . It matches.

Another way: Use the symmetry directly: $x$ is one step right of the left end and $y$ is one step left of the right end, so the steps cancel and $x+y=-\frac{5}{8}+\frac{19}{8}=\frac{7}{4}$ without finding the step.

Standardsmin grade 7

6.NS.C.6 Rational numbers on the number line and coordinate plane — Placing the four rational numbers as evenly spaced points on a number line.
7.NS.A.1 Apply and extend understanding of addition and subtraction to rational numbers — Subtracting a negative fraction to get the distance and adding the step.

💡Takeaway. For evenly spaced numbers, the two middle ones add up to the same as the two ends.

Variant 7 medium

The four numbers $-\frac{3}{8}$ , $x$ , $y$ , $1\frac{7}{8}$ are listed in order of size, and the gaps between neighboring numbers are all equal. Find the value of $x+y$ .

Your answer

Show solution

1 · Understandwhat's really being asked

Four numbers sit on a number line in order, evenly spaced. We know the two ends, $-\frac{3}{8}$ and $1\frac{7}{8}$ , and need the sum of the two middle numbers $x$ and $y$ .

Givens

The numbers in order: $-\frac{3}{8}$ , $x$ , $y$ , $1\frac{7}{8}$ .
The three gaps between neighbors are equal.

Unknowns

$x+y$

Constraints

Four numbers make three gaps, not four.

2 · Planchoose the strategy

#1 Draw a Diagram · also uses: #7 Identify Subproblems

Picture the four points on a number line. The distance from the first to the last is covered by three equal steps, so find the total distance, cut it into three, and step along from $-\frac{3}{8}$ .

3 · Execute4 carry out the plan

1Measure the whole stretch

#1 Draw a Diagram 7.NS.A.1

Write

1\frac{7}{8}

as

\frac{15}{8}

and subtract the left end. Subtracting a negative adds its size.

\frac{15}{8} - \left(-\frac{3}{8}\right) = \frac{15}{8} + \frac{3}{8} = \frac{9}{4}

The distance from the first point to the last is

\frac{9}{4}

.

2Find one step

#7 Identify Subproblems 7.NS.A.1

Four points have three equal gaps between them.

\frac{9}{4} \div 3 = \frac{3}{4}

Each gap is

\frac{3}{4}

.

3Step along to $x$ and $y$

#1 Draw a Diagram 6.NS.C.6

Start at

-\frac{3}{8}

and add one gap at a time.

x = -\frac{3}{8} + \frac{3}{4} = \frac{3}{8},\quad y = \frac{3}{8} + \frac{3}{4} = 1\frac{1}{8}

One more step gives

1\frac{1}{8} + \frac{3}{4} = 1\frac{7}{8}

, the right end, as it should.

4Add the middle numbers

#7 Identify Subproblems 7.NS.A.1

Now add

x

and

y

.

\frac{3}{8} + 1\frac{1}{8} = \frac{3}{2}

So

x+y=\frac{3}{2}

.

Answer:

\frac{3}{2}

4 · Reviewdoes it hold up?

The middle pair is placed symmetrically, so $x+y$ must equal the sum of the two ends: $-\frac{3}{8} + \frac{15}{8} = \frac{3}{2}$ . It matches.

Another way: Use the symmetry directly: $x$ is one step right of the left end and $y$ is one step left of the right end, so the steps cancel and $x+y=-\frac{3}{8}+\frac{15}{8}=\frac{3}{2}$ without finding the step.

Standardsmin grade 7

6.NS.C.6 Rational numbers on the number line and coordinate plane — Placing the four rational numbers as evenly spaced points on a number line.
7.NS.A.1 Apply and extend understanding of addition and subtraction to rational numbers — Subtracting a negative fraction to get the distance and adding the step.

💡Takeaway. For evenly spaced numbers, the two middle ones add up to the same as the two ends.

Variant 8 hard

The four numbers $-\frac{4}{9}$ , $x$ , $y$ , $1\frac{8}{9}$ are listed in order of size, and the gaps between neighboring numbers are all equal. Find the value of $x+y$ .

Your answer

Show solution

1 · Understandwhat's really being asked

Four numbers sit on a number line in order, evenly spaced. We know the two ends, $-\frac{4}{9}$ and $1\frac{8}{9}$ , and need the sum of the two middle numbers $x$ and $y$ .

Givens

The numbers in order: $-\frac{4}{9}$ , $x$ , $y$ , $1\frac{8}{9}$ .
The three gaps between neighbors are equal.

Unknowns

$x+y$

Constraints

Four numbers make three gaps, not four.

2 · Planchoose the strategy

#1 Draw a Diagram · also uses: #7 Identify Subproblems

Picture the four points on a number line. The distance from the first to the last is covered by three equal steps, so find the total distance, cut it into three, and step along from $-\frac{4}{9}$ .

3 · Execute4 carry out the plan

1Measure the whole stretch

#1 Draw a Diagram 7.NS.A.1

Write

1\frac{8}{9}

as

\frac{17}{9}

and subtract the left end. Subtracting a negative adds its size.

\frac{17}{9} - \left(-\frac{4}{9}\right) = \frac{17}{9} + \frac{4}{9} = \frac{7}{3}

The distance from the first point to the last is

\frac{7}{3}

.

2Find one step

#7 Identify Subproblems 7.NS.A.1

Four points have three equal gaps between them.

\frac{7}{3} \div 3 = \frac{7}{9}

Each gap is

\frac{7}{9}

.

3Step along to $x$ and $y$

#1 Draw a Diagram 6.NS.C.6

Start at

-\frac{4}{9}

and add one gap at a time.

x = -\frac{4}{9} + \frac{7}{9} = \frac{1}{3},\quad y = \frac{1}{3} + \frac{7}{9} = 1\frac{1}{9}

One more step gives

1\frac{1}{9} + \frac{7}{9} = 1\frac{8}{9}

, the right end, as it should.

4Add the middle numbers

#7 Identify Subproblems 7.NS.A.1

Now add

x

and

y

.

\frac{1}{3} + 1\frac{1}{9} = \frac{13}{9}

So

x+y=\frac{13}{9}

.

Answer:

\frac{13}{9}

4 · Reviewdoes it hold up?

The middle pair is placed symmetrically, so $x+y$ must equal the sum of the two ends: $-\frac{4}{9} + \frac{17}{9} = \frac{13}{9}$ . It matches.

Another way: Use the symmetry directly: $x$ is one step right of the left end and $y$ is one step left of the right end, so the steps cancel and $x+y=-\frac{4}{9}+\frac{17}{9}=\frac{13}{9}$ without finding the step.

Standardsmin grade 7

6.NS.C.6 Rational numbers on the number line and coordinate plane — Placing the four rational numbers as evenly spaced points on a number line.
7.NS.A.1 Apply and extend understanding of addition and subtraction to rational numbers — Subtracting a negative fraction to get the distance and adding the step.

💡Takeaway. For evenly spaced numbers, the two middle ones add up to the same as the two ends.

Variant 9 hard

The four numbers $-\frac{5}{9}$ , $x$ , $y$ , $1\frac{7}{9}$ are listed in order of size, and the gaps between neighboring numbers are all equal. Find the value of $x+y$ .

Your answer

Show solution

1 · Understandwhat's really being asked

Four numbers sit on a number line in order, evenly spaced. We know the two ends, $-\frac{5}{9}$ and $1\frac{7}{9}$ , and need the sum of the two middle numbers $x$ and $y$ .

Givens

The numbers in order: $-\frac{5}{9}$ , $x$ , $y$ , $1\frac{7}{9}$ .
The three gaps between neighbors are equal.

Unknowns

$x+y$

Constraints

Four numbers make three gaps, not four.

2 · Planchoose the strategy

#1 Draw a Diagram · also uses: #7 Identify Subproblems

Picture the four points on a number line. The distance from the first to the last is covered by three equal steps, so find the total distance, cut it into three, and step along from $-\frac{5}{9}$ .

3 · Execute4 carry out the plan

1Measure the whole stretch

#1 Draw a Diagram 7.NS.A.1

Write

1\frac{7}{9}

as

\frac{16}{9}

and subtract the left end. Subtracting a negative adds its size.

\frac{16}{9} - \left(-\frac{5}{9}\right) = \frac{16}{9} + \frac{5}{9} = \frac{7}{3}

The distance from the first point to the last is

\frac{7}{3}

.

2Find one step

#7 Identify Subproblems 7.NS.A.1

Four points have three equal gaps between them.

\frac{7}{3} \div 3 = \frac{7}{9}

Each gap is

\frac{7}{9}

.

3Step along to $x$ and $y$

#1 Draw a Diagram 6.NS.C.6

Start at

-\frac{5}{9}

and add one gap at a time.

x = -\frac{5}{9} + \frac{7}{9} = \frac{2}{9},\quad y = \frac{2}{9} + \frac{7}{9} = 1

One more step gives

1 + \frac{7}{9} = 1\frac{7}{9}

, the right end, as it should.

4Add the middle numbers

#7 Identify Subproblems 7.NS.A.1

Now add

x

and

y

.

\frac{2}{9} + 1 = \frac{11}{9}

So

x+y=\frac{11}{9}

.

Answer:

\frac{11}{9}

4 · Reviewdoes it hold up?

The middle pair is placed symmetrically, so $x+y$ must equal the sum of the two ends: $-\frac{5}{9} + \frac{16}{9} = \frac{11}{9}$ . It matches.

Another way: Use the symmetry directly: $x$ is one step right of the left end and $y$ is one step left of the right end, so the steps cancel and $x+y=-\frac{5}{9}+\frac{16}{9}=\frac{11}{9}$ without finding the step.

Standardsmin grade 7

6.NS.C.6 Rational numbers on the number line and coordinate plane — Placing the four rational numbers as evenly spaced points on a number line.
7.NS.A.1 Apply and extend understanding of addition and subtraction to rational numbers — Subtracting a negative fraction to get the distance and adding the step.

💡Takeaway. For evenly spaced numbers, the two middle ones add up to the same as the two ends.

Variant 10 medium

The four numbers $-\frac{2}{9}$ , $x$ , $y$ , $1\frac{4}{9}$ are listed in order of size, and the gaps between neighboring numbers are all equal. Find the value of $x+y$ .

Your answer

Show solution

1 · Understandwhat's really being asked

Four numbers sit on a number line in order, evenly spaced. We know the two ends, $-\frac{2}{9}$ and $1\frac{4}{9}$ , and need the sum of the two middle numbers $x$ and $y$ .

Givens

The numbers in order: $-\frac{2}{9}$ , $x$ , $y$ , $1\frac{4}{9}$ .
The three gaps between neighbors are equal.

Unknowns

$x+y$

Constraints

Four numbers make three gaps, not four.

2 · Planchoose the strategy

#1 Draw a Diagram · also uses: #7 Identify Subproblems

Picture the four points on a number line. The distance from the first to the last is covered by three equal steps, so find the total distance, cut it into three, and step along from $-\frac{2}{9}$ .

3 · Execute4 carry out the plan

1Measure the whole stretch

#1 Draw a Diagram 7.NS.A.1

Write

1\frac{4}{9}

as

\frac{13}{9}

and subtract the left end. Subtracting a negative adds its size.

\frac{13}{9} - \left(-\frac{2}{9}\right) = \frac{13}{9} + \frac{2}{9} = \frac{5}{3}

The distance from the first point to the last is

\frac{5}{3}

.

2Find one step

#7 Identify Subproblems 7.NS.A.1

Four points have three equal gaps between them.

\frac{5}{3} \div 3 = \frac{5}{9}

Each gap is

\frac{5}{9}

.

3Step along to $x$ and $y$

#1 Draw a Diagram 6.NS.C.6

Start at

-\frac{2}{9}

and add one gap at a time.

x = -\frac{2}{9} + \frac{5}{9} = \frac{1}{3},\quad y = \frac{1}{3} + \frac{5}{9} = \frac{8}{9}

One more step gives

\frac{8}{9} + \frac{5}{9} = 1\frac{4}{9}

, the right end, as it should.

4Add the middle numbers

#7 Identify Subproblems 7.NS.A.1

Now add

x

and

y

.

\frac{1}{3} + \frac{8}{9} = \frac{11}{9}

So

x+y=\frac{11}{9}

.

Answer:

\frac{11}{9}

4 · Reviewdoes it hold up?

The middle pair is placed symmetrically, so $x+y$ must equal the sum of the two ends: $-\frac{2}{9} + \frac{13}{9} = \frac{11}{9}$ . It matches.

Another way: Use the symmetry directly: $x$ is one step right of the left end and $y$ is one step left of the right end, so the steps cancel and $x+y=-\frac{2}{9}+\frac{13}{9}=\frac{11}{9}$ without finding the step.

Standardsmin grade 7

6.NS.C.6 Rational numbers on the number line and coordinate plane — Placing the four rational numbers as evenly spaced points on a number line.
7.NS.A.1 Apply and extend understanding of addition and subtraction to rational numbers — Subtracting a negative fraction to get the distance and adding the step.

💡Takeaway. For evenly spaced numbers, the two middle ones add up to the same as the two ends.

Variant 11 hard

The four numbers $-\frac{3}{10}$ , $x$ , $y$ , $2\frac{1}{10}$ are listed in order of size, and the gaps between neighboring numbers are all equal. Find the value of $x+y$ .

Your answer

Show solution

1 · Understandwhat's really being asked

Four numbers sit on a number line in order, evenly spaced. We know the two ends, $-\frac{3}{10}$ and $2\frac{1}{10}$ , and need the sum of the two middle numbers $x$ and $y$ .

Givens

The numbers in order: $-\frac{3}{10}$ , $x$ , $y$ , $2\frac{1}{10}$ .
The three gaps between neighbors are equal.

Unknowns

$x+y$

Constraints

Four numbers make three gaps, not four.

2 · Planchoose the strategy

#1 Draw a Diagram · also uses: #7 Identify Subproblems

Picture the four points on a number line. The distance from the first to the last is covered by three equal steps, so find the total distance, cut it into three, and step along from $-\frac{3}{10}$ .

3 · Execute4 carry out the plan

1Measure the whole stretch

#1 Draw a Diagram 7.NS.A.1

Write

2\frac{1}{10}

as

\frac{21}{10}

and subtract the left end. Subtracting a negative adds its size.

\frac{21}{10} - \left(-\frac{3}{10}\right) = \frac{21}{10} + \frac{3}{10} = \frac{12}{5}

The distance from the first point to the last is

\frac{12}{5}

.

2Find one step

#7 Identify Subproblems 7.NS.A.1

Four points have three equal gaps between them.

\frac{12}{5} \div 3 = \frac{4}{5}

Each gap is

\frac{4}{5}

.

3Step along to $x$ and $y$

#1 Draw a Diagram 6.NS.C.6

Start at

-\frac{3}{10}

and add one gap at a time.

x = -\frac{3}{10} + \frac{4}{5} = \frac{1}{2},\quad y = \frac{1}{2} + \frac{4}{5} = 1\frac{3}{10}

One more step gives

1\frac{3}{10} + \frac{4}{5} = 2\frac{1}{10}

, the right end, as it should.

4Add the middle numbers

#7 Identify Subproblems 7.NS.A.1

Now add

x

and

y

.

\frac{1}{2} + 1\frac{3}{10} = \frac{9}{5}

So

x+y=\frac{9}{5}

.

Answer:

\frac{9}{5}

4 · Reviewdoes it hold up?

The middle pair is placed symmetrically, so $x+y$ must equal the sum of the two ends: $-\frac{3}{10} + \frac{21}{10} = \frac{9}{5}$ . It matches.

Another way: Use the symmetry directly: $x$ is one step right of the left end and $y$ is one step left of the right end, so the steps cancel and $x+y=-\frac{3}{10}+\frac{21}{10}=\frac{9}{5}$ without finding the step.

Standardsmin grade 7

6.NS.C.6 Rational numbers on the number line and coordinate plane — Placing the four rational numbers as evenly spaced points on a number line.
7.NS.A.1 Apply and extend understanding of addition and subtraction to rational numbers — Subtracting a negative fraction to get the distance and adding the step.

💡Takeaway. For evenly spaced numbers, the two middle ones add up to the same as the two ends.

Variant 12 hard

The four numbers $-\frac{4}{11}$ , $x$ , $y$ , $1\frac{3}{11}$ are listed in order of size, and the gaps between neighboring numbers are all equal. Find the value of $x+y$ .

Your answer

Show solution

1 · Understandwhat's really being asked

Four numbers sit on a number line in order, evenly spaced. We know the two ends, $-\frac{4}{11}$ and $1\frac{3}{11}$ , and need the sum of the two middle numbers $x$ and $y$ .

Givens

The numbers in order: $-\frac{4}{11}$ , $x$ , $y$ , $1\frac{3}{11}$ .
The three gaps between neighbors are equal.

Unknowns

$x+y$

Constraints

Four numbers make three gaps, not four.

2 · Planchoose the strategy

#1 Draw a Diagram · also uses: #7 Identify Subproblems

Picture the four points on a number line. The distance from the first to the last is covered by three equal steps, so find the total distance, cut it into three, and step along from $-\frac{4}{11}$ .

3 · Execute4 carry out the plan

1Measure the whole stretch

#1 Draw a Diagram 7.NS.A.1

Write

1\frac{3}{11}

as

\frac{14}{11}

and subtract the left end. Subtracting a negative adds its size.

\frac{14}{11} - \left(-\frac{4}{11}\right) = \frac{14}{11} + \frac{4}{11} = \frac{18}{11}

The distance from the first point to the last is

\frac{18}{11}

.

2Find one step

#7 Identify Subproblems 7.NS.A.1

Four points have three equal gaps between them.

\frac{18}{11} \div 3 = \frac{6}{11}

Each gap is

\frac{6}{11}

.

3Step along to $x$ and $y$

#1 Draw a Diagram 6.NS.C.6

Start at

-\frac{4}{11}

and add one gap at a time.

x = -\frac{4}{11} + \frac{6}{11} = \frac{2}{11},\quad y = \frac{2}{11} + \frac{6}{11} = \frac{8}{11}

One more step gives

\frac{8}{11} + \frac{6}{11} = 1\frac{3}{11}

, the right end, as it should.

4Add the middle numbers

#7 Identify Subproblems 7.NS.A.1

Now add

x

and

y

.

\frac{2}{11} + \frac{8}{11} = \frac{10}{11}

So

x+y=\frac{10}{11}

.

Answer:

\frac{10}{11}

4 · Reviewdoes it hold up?

The middle pair is placed symmetrically, so $x+y$ must equal the sum of the two ends: $-\frac{4}{11} + \frac{14}{11} = \frac{10}{11}$ . It matches.

Another way: Use the symmetry directly: $x$ is one step right of the left end and $y$ is one step left of the right end, so the steps cancel and $x+y=-\frac{4}{11}+\frac{14}{11}=\frac{10}{11}$ without finding the step.

Standardsmin grade 7

6.NS.C.6 Rational numbers on the number line and coordinate plane — Placing the four rational numbers as evenly spaced points on a number line.
7.NS.A.1 Apply and extend understanding of addition and subtraction to rational numbers — Subtracting a negative fraction to get the distance and adding the step.

💡Takeaway. For evenly spaced numbers, the two middle ones add up to the same as the two ends.

Split the total gap into equal steps · 12 practice problems

Generated variants — 12

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