← Apply a defined symbol to compute with fractions · Apply a Newly Defined Operation

Apply a defined symbol to compute with fractions · 10 practice problems

4.NF.B.3

Generated variants — 10

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

Variant 1 easy answer: 1

For any two numbers $a$ and $b$ , define a new operation $\star$ as follows:

$a \star b = \dfrac{a + b}{a - b}$

Find the difference between $6 \star 2$ and $8 \star 4$ .

Show solution

1 · Understandwhat's really being asked

A made-up operation a star b means (a+b) divided by (a-b). I must compute 6 star 2 and 8 star 4 and find the difference between the two results.

Givens

a star b = (a+b)/(a-b).
We need 6 star 2 and 8 star 4.
We want the difference between the two results.

Unknowns

The difference between 6 star 2 and 8 star 4.

Constraints

Substitute the given numbers into the definition exactly as written.

2 · Planchoose the strategy

#7 Identify Subproblems · also uses: #6 Guess and Check

Break the task into two subproblems -- evaluate each star expression by plugging into the rule -- then subtract the two fractions. The rule is a recipe to follow step by step.

3 · Execute3 carry out the plan

1Compute 6 star 2

#7 Identify Subproblems 4.OA.A.3

Substitute a=6, b=2 into (a+b)/(a-b): the top is 6+2 = 8 and the bottom is 6-2 = 4, so 6 star 2 = 8/4.

6\star 2=\dfrac{6+2}{6-2}=\dfrac{8}{4}

Following a defined recipe is just doing the add and subtract it tells you to do.

2Compute 8 star 4

#7 Identify Subproblems 4.OA.A.3

Substitute a=8, b=4: the top is 8+4 = 12 and the bottom is 8-4 = 4, so 8 star 4 = 12/4.

8\star 4=\dfrac{8+4}{8-4}=\dfrac{12}{4}

Same recipe with different numbers.

3Find the difference

#7 Identify Subproblems 4.NF.B.3

Both fractions have denominator 4, so subtract the numerators: 12/4 - 8/4 = 1.

3-2=1

Same denominator means you just subtract the top numbers -- a Grade 4 fraction subtraction.

Answer: 1

4 · Reviewdoes it hold up?

The larger result is 8 star 4 and the smaller is 6 star 2; subtracting the smaller from the larger gives the positive difference 1, which is reasonable since both results are close in size.

Another way: Notice both bottoms are 4, so only the tops 12 and 8 matter; 12 - 8 over 4 gives 1 directly.

Standardsmin grade 4

4.OA.A.3 Solve multi-step word problems using four operations with whole numbers — Evaluating each defined star expression by performing the whole-number add and subtract.
4.NF.B.3 Understand a fraction with numerator greater than one as sum of unit fractions — Subtracting the two fractions to get 1.

💡Takeaway. This only needs Grade 4 skills -- just follow the rule, then subtract the fractions!

Variant 2 easy answer: 2

For any two numbers $a$ and $b$ , define a new operation $\star$ as follows:

$a \star b = \dfrac{a + b}{a - b}$

Find the difference between $5 \star 2$ and $8 \star 5$ .

Show solution

1 · Understandwhat's really being asked

A made-up operation a star b means (a+b) divided by (a-b). I must compute 5 star 2 and 8 star 5 and find the difference between the two results.

Givens

a star b = (a+b)/(a-b).
We need 5 star 2 and 8 star 5.
We want the difference between the two results.

Unknowns

The difference between 5 star 2 and 8 star 5.

Constraints

Substitute the given numbers into the definition exactly as written.

2 · Planchoose the strategy

#7 Identify Subproblems · also uses: #6 Guess and Check

Break the task into two subproblems -- evaluate each star expression by plugging into the rule -- then subtract the two fractions. The rule is a recipe to follow step by step.

3 · Execute3 carry out the plan

1Compute 5 star 2

#7 Identify Subproblems 4.OA.A.3

Substitute a=5, b=2 into (a+b)/(a-b): the top is 5+2 = 7 and the bottom is 5-2 = 3, so 5 star 2 = 7/3.

5\star 2=\dfrac{5+2}{5-2}=\dfrac{7}{3}

Following a defined recipe is just doing the add and subtract it tells you to do.

2Compute 8 star 5

#7 Identify Subproblems 4.OA.A.3

Substitute a=8, b=5: the top is 8+5 = 13 and the bottom is 8-5 = 3, so 8 star 5 = 13/3.

8\star 5=\dfrac{8+5}{8-5}=\dfrac{13}{3}

Same recipe with different numbers.

3Find the difference

#7 Identify Subproblems 4.NF.B.3

Both fractions have denominator 3, so subtract the numerators: 13/3 - 7/3 = 2.

\dfrac{13}{3}-\dfrac{7}{3}=2

Same denominator means you just subtract the top numbers -- a Grade 4 fraction subtraction.

Answer: 2

4 · Reviewdoes it hold up?

The larger result is 8 star 5 and the smaller is 5 star 2; subtracting the smaller from the larger gives the positive difference 2, which is reasonable since both results are close in size.

Another way: Notice both bottoms are 3, so only the tops 13 and 7 matter; 13 - 7 over 3 gives 2 directly.

Standardsmin grade 4

4.OA.A.3 Solve multi-step word problems using four operations with whole numbers — Evaluating each defined star expression by performing the whole-number add and subtract.
4.NF.B.3 Understand a fraction with numerator greater than one as sum of unit fractions — Subtracting the two fractions to get 2.

💡Takeaway. This only needs Grade 4 skills -- just follow the rule, then subtract the fractions!

Variant 3 easy answer: 2/5

For any two numbers $a$ and $b$ , define a new operation $\star$ as follows:

$a \star b = \dfrac{a + b}{a - b}$

Find the difference between $8 \star 3$ and $9 \star 4$ .

Show solution

1 · Understandwhat's really being asked

A made-up operation a star b means (a+b) divided by (a-b). I must compute 8 star 3 and 9 star 4 and find the difference between the two results.

Givens

a star b = (a+b)/(a-b).
We need 8 star 3 and 9 star 4.
We want the difference between the two results.

Unknowns

The difference between 8 star 3 and 9 star 4.

Constraints

Substitute the given numbers into the definition exactly as written.

2 · Planchoose the strategy

#7 Identify Subproblems · also uses: #6 Guess and Check

Break the task into two subproblems -- evaluate each star expression by plugging into the rule -- then subtract the two fractions. The rule is a recipe to follow step by step.

3 · Execute3 carry out the plan

1Compute 8 star 3

#7 Identify Subproblems 4.OA.A.3

Substitute a=8, b=3 into (a+b)/(a-b): the top is 8+3 = 11 and the bottom is 8-3 = 5, so 8 star 3 = 11/5.

8\star 3=\dfrac{8+3}{8-3}=\dfrac{11}{5}

Following a defined recipe is just doing the add and subtract it tells you to do.

2Compute 9 star 4

#7 Identify Subproblems 4.OA.A.3

Substitute a=9, b=4: the top is 9+4 = 13 and the bottom is 9-4 = 5, so 9 star 4 = 13/5.

9\star 4=\dfrac{9+4}{9-4}=\dfrac{13}{5}

Same recipe with different numbers.

3Find the difference

#7 Identify Subproblems 4.NF.B.3

Both fractions have denominator 5, so subtract the numerators: 13/5 - 11/5 = 2/5.

\dfrac{13}{5}-\dfrac{11}{5}=\dfrac{2}{5}

Same denominator means you just subtract the top numbers -- a Grade 4 fraction subtraction.

Answer: 2/5

4 · Reviewdoes it hold up?

The larger result is 9 star 4 and the smaller is 8 star 3; subtracting the smaller from the larger gives the positive difference 2/5, which is reasonable since both results are close in size.

Another way: Notice both bottoms are 5, so only the tops 13 and 11 matter; 13 - 11 over 5 gives 2/5 directly.

Standardsmin grade 4

4.OA.A.3 Solve multi-step word problems using four operations with whole numbers — Evaluating each defined star expression by performing the whole-number add and subtract.
4.NF.B.3 Understand a fraction with numerator greater than one as sum of unit fractions — Subtracting the two fractions to get 2/5.

💡Takeaway. This only needs Grade 4 skills -- just follow the rule, then subtract the fractions!

Variant 4 medium answer: 4/3

For any two numbers $a$ and $b$ , define a new operation $\star$ as follows:

$a \star b = \dfrac{a + b}{a - b}$

Find the difference between $7 \star 4$ and $9 \star 6$ .

Show solution

1 · Understandwhat's really being asked

A made-up operation a star b means (a+b) divided by (a-b). I must compute 7 star 4 and 9 star 6 and find the difference between the two results.

Givens

a star b = (a+b)/(a-b).
We need 7 star 4 and 9 star 6.
We want the difference between the two results.

Unknowns

The difference between 7 star 4 and 9 star 6.

Constraints

Substitute the given numbers into the definition exactly as written.

2 · Planchoose the strategy

#7 Identify Subproblems · also uses: #6 Guess and Check

Break the task into two subproblems -- evaluate each star expression by plugging into the rule -- then subtract the two fractions. The rule is a recipe to follow step by step.

3 · Execute3 carry out the plan

1Compute 7 star 4

#7 Identify Subproblems 4.OA.A.3

Substitute a=7, b=4 into (a+b)/(a-b): the top is 7+4 = 11 and the bottom is 7-4 = 3, so 7 star 4 = 11/3.

7\star 4=\dfrac{7+4}{7-4}=\dfrac{11}{3}

Following a defined recipe is just doing the add and subtract it tells you to do.

2Compute 9 star 6

#7 Identify Subproblems 4.OA.A.3

Substitute a=9, b=6: the top is 9+6 = 15 and the bottom is 9-6 = 3, so 9 star 6 = 15/3.

9\star 6=\dfrac{9+6}{9-6}=\dfrac{15}{3}

Same recipe with different numbers.

3Find the difference

#7 Identify Subproblems 4.NF.B.3

Both fractions have denominator 3, so subtract the numerators: 15/3 - 11/3 = 4/3.

5-\dfrac{11}{3}=\dfrac{4}{3}

Same denominator means you just subtract the top numbers -- a Grade 4 fraction subtraction.

Answer: 4/3

4 · Reviewdoes it hold up?

The larger result is 9 star 6 and the smaller is 7 star 4; subtracting the smaller from the larger gives the positive difference 4/3, which is reasonable since both results are close in size.

Another way: Notice both bottoms are 3, so only the tops 15 and 11 matter; 15 - 11 over 3 gives 4/3 directly.

Standardsmin grade 4

4.OA.A.3 Solve multi-step word problems using four operations with whole numbers — Evaluating each defined star expression by performing the whole-number add and subtract.
4.NF.B.3 Understand a fraction with numerator greater than one as sum of unit fractions — Subtracting the two fractions to get 4/3.

💡Takeaway. This only needs Grade 4 skills -- just follow the rule, then subtract the fractions!

Variant 5 medium answer: 6/5

For any two numbers $a$ and $b$ , define a new operation $\star$ as follows:

$a \star b = \dfrac{a + b}{a - b}$

Find the difference between $7 \star 2$ and $10 \star 5$ .

Show solution

1 · Understandwhat's really being asked

A made-up operation a star b means (a+b) divided by (a-b). I must compute 7 star 2 and 10 star 5 and find the difference between the two results.

Givens

a star b = (a+b)/(a-b).
We need 7 star 2 and 10 star 5.
We want the difference between the two results.

Unknowns

The difference between 7 star 2 and 10 star 5.

Constraints

Substitute the given numbers into the definition exactly as written.

2 · Planchoose the strategy

#7 Identify Subproblems · also uses: #6 Guess and Check

Break the task into two subproblems -- evaluate each star expression by plugging into the rule -- then subtract the two fractions. The rule is a recipe to follow step by step.

3 · Execute3 carry out the plan

1Compute 7 star 2

#7 Identify Subproblems 4.OA.A.3

Substitute a=7, b=2 into (a+b)/(a-b): the top is 7+2 = 9 and the bottom is 7-2 = 5, so 7 star 2 = 9/5.

7\star 2=\dfrac{7+2}{7-2}=\dfrac{9}{5}

Following a defined recipe is just doing the add and subtract it tells you to do.

2Compute 10 star 5

#7 Identify Subproblems 4.OA.A.3

Substitute a=10, b=5: the top is 10+5 = 15 and the bottom is 10-5 = 5, so 10 star 5 = 15/5.

10\star 5=\dfrac{10+5}{10-5}=\dfrac{15}{5}

Same recipe with different numbers.

3Find the difference

#7 Identify Subproblems 4.NF.B.3

Both fractions have denominator 5, so subtract the numerators: 15/5 - 9/5 = 6/5.

3-\dfrac{9}{5}=\dfrac{6}{5}

Same denominator means you just subtract the top numbers -- a Grade 4 fraction subtraction.

Answer: 6/5

4 · Reviewdoes it hold up?

The larger result is 10 star 5 and the smaller is 7 star 2; subtracting the smaller from the larger gives the positive difference 6/5, which is reasonable since both results are close in size.

Another way: Notice both bottoms are 5, so only the tops 15 and 9 matter; 15 - 9 over 5 gives 6/5 directly.

Standardsmin grade 4

4.OA.A.3 Solve multi-step word problems using four operations with whole numbers — Evaluating each defined star expression by performing the whole-number add and subtract.
4.NF.B.3 Understand a fraction with numerator greater than one as sum of unit fractions — Subtracting the two fractions to get 6/5.

💡Takeaway. This only needs Grade 4 skills -- just follow the rule, then subtract the fractions!

Variant 6 medium answer: 1

For any two numbers $a$ and $b$ , define a new operation $\star$ as follows:

$a \star b = \dfrac{a + b}{a - b}$

Find the difference between $9 \star 5$ and $11 \star 7$ .

Show solution

1 · Understandwhat's really being asked

A made-up operation a star b means (a+b) divided by (a-b). I must compute 9 star 5 and 11 star 7 and find the difference between the two results.

Givens

a star b = (a+b)/(a-b).
We need 9 star 5 and 11 star 7.
We want the difference between the two results.

Unknowns

The difference between 9 star 5 and 11 star 7.

Constraints

Substitute the given numbers into the definition exactly as written.

2 · Planchoose the strategy

#7 Identify Subproblems · also uses: #6 Guess and Check

Break the task into two subproblems -- evaluate each star expression by plugging into the rule -- then subtract the two fractions. The rule is a recipe to follow step by step.

3 · Execute3 carry out the plan

1Compute 9 star 5

#7 Identify Subproblems 4.OA.A.3

Substitute a=9, b=5 into (a+b)/(a-b): the top is 9+5 = 14 and the bottom is 9-5 = 4, so 9 star 5 = 14/4.

9\star 5=\dfrac{9+5}{9-5}=\dfrac{14}{4}

Following a defined recipe is just doing the add and subtract it tells you to do.

2Compute 11 star 7

#7 Identify Subproblems 4.OA.A.3

Substitute a=11, b=7: the top is 11+7 = 18 and the bottom is 11-7 = 4, so 11 star 7 = 18/4.

11\star 7=\dfrac{11+7}{11-7}=\dfrac{18}{4}

Same recipe with different numbers.

3Find the difference

#7 Identify Subproblems 4.NF.B.3

Both fractions have denominator 4, so subtract the numerators: 18/4 - 14/4 = 1.

\dfrac{9}{2}-\dfrac{7}{2}=1

Same denominator means you just subtract the top numbers -- a Grade 4 fraction subtraction.

Answer: 1

4 · Reviewdoes it hold up?

The larger result is 11 star 7 and the smaller is 9 star 5; subtracting the smaller from the larger gives the positive difference 1, which is reasonable since both results are close in size.

Another way: Notice both bottoms are 4, so only the tops 18 and 14 matter; 18 - 14 over 4 gives 1 directly.

Standardsmin grade 4

4.OA.A.3 Solve multi-step word problems using four operations with whole numbers — Evaluating each defined star expression by performing the whole-number add and subtract.
4.NF.B.3 Understand a fraction with numerator greater than one as sum of unit fractions — Subtracting the two fractions to get 1.

💡Takeaway. This only needs Grade 4 skills -- just follow the rule, then subtract the fractions!

Variant 7 medium answer: 6/7

For any two numbers $a$ and $b$ , define a new operation $\star$ as follows:

$a \star b = \dfrac{a + b}{a - b}$

Find the difference between $8 \star 1$ and $11 \star 4$ .

Show solution

1 · Understandwhat's really being asked

A made-up operation a star b means (a+b) divided by (a-b). I must compute 8 star 1 and 11 star 4 and find the difference between the two results.

Givens

a star b = (a+b)/(a-b).
We need 8 star 1 and 11 star 4.
We want the difference between the two results.

Unknowns

The difference between 8 star 1 and 11 star 4.

Constraints

Substitute the given numbers into the definition exactly as written.

2 · Planchoose the strategy

#7 Identify Subproblems · also uses: #6 Guess and Check

Break the task into two subproblems -- evaluate each star expression by plugging into the rule -- then subtract the two fractions. The rule is a recipe to follow step by step.

3 · Execute3 carry out the plan

1Compute 8 star 1

#7 Identify Subproblems 4.OA.A.3

Substitute a=8, b=1 into (a+b)/(a-b): the top is 8+1 = 9 and the bottom is 8-1 = 7, so 8 star 1 = 9/7.

8\star 1=\dfrac{8+1}{8-1}=\dfrac{9}{7}

Following a defined recipe is just doing the add and subtract it tells you to do.

2Compute 11 star 4

#7 Identify Subproblems 4.OA.A.3

Substitute a=11, b=4: the top is 11+4 = 15 and the bottom is 11-4 = 7, so 11 star 4 = 15/7.

11\star 4=\dfrac{11+4}{11-4}=\dfrac{15}{7}

Same recipe with different numbers.

3Find the difference

#7 Identify Subproblems 4.NF.B.3

Both fractions have denominator 7, so subtract the numerators: 15/7 - 9/7 = 6/7.

\dfrac{15}{7}-\dfrac{9}{7}=\dfrac{6}{7}

Same denominator means you just subtract the top numbers -- a Grade 4 fraction subtraction.

Answer: 6/7

4 · Reviewdoes it hold up?

The larger result is 11 star 4 and the smaller is 8 star 1; subtracting the smaller from the larger gives the positive difference 6/7, which is reasonable since both results are close in size.

Another way: Notice both bottoms are 7, so only the tops 15 and 9 matter; 15 - 9 over 7 gives 6/7 directly.

Standardsmin grade 4

4.OA.A.3 Solve multi-step word problems using four operations with whole numbers — Evaluating each defined star expression by performing the whole-number add and subtract.
4.NF.B.3 Understand a fraction with numerator greater than one as sum of unit fractions — Subtracting the two fractions to get 6/7.

💡Takeaway. This only needs Grade 4 skills -- just follow the rule, then subtract the fractions!

Variant 8 hard answer: 4/7

For any two numbers $a$ and $b$ , define a new operation $\star$ as follows:

$a \star b = \dfrac{a + b}{a - b}$

Find the difference between $10 \star 3$ and $12 \star 5$ .

Show solution

1 · Understandwhat's really being asked

A made-up operation a star b means (a+b) divided by (a-b). I must compute 10 star 3 and 12 star 5 and find the difference between the two results.

Givens

a star b = (a+b)/(a-b).
We need 10 star 3 and 12 star 5.
We want the difference between the two results.

Unknowns

The difference between 10 star 3 and 12 star 5.

Constraints

Substitute the given numbers into the definition exactly as written.

2 · Planchoose the strategy

#7 Identify Subproblems · also uses: #6 Guess and Check

Break the task into two subproblems -- evaluate each star expression by plugging into the rule -- then subtract the two fractions. The rule is a recipe to follow step by step.

3 · Execute3 carry out the plan

1Compute 10 star 3

#7 Identify Subproblems 4.OA.A.3

Substitute a=10, b=3 into (a+b)/(a-b): the top is 10+3 = 13 and the bottom is 10-3 = 7, so 10 star 3 = 13/7.

10\star 3=\dfrac{10+3}{10-3}=\dfrac{13}{7}

Following a defined recipe is just doing the add and subtract it tells you to do.

2Compute 12 star 5

#7 Identify Subproblems 4.OA.A.3

Substitute a=12, b=5: the top is 12+5 = 17 and the bottom is 12-5 = 7, so 12 star 5 = 17/7.

12\star 5=\dfrac{12+5}{12-5}=\dfrac{17}{7}

Same recipe with different numbers.

3Find the difference

#7 Identify Subproblems 4.NF.B.3

Both fractions have denominator 7, so subtract the numerators: 17/7 - 13/7 = 4/7.

\dfrac{17}{7}-\dfrac{13}{7}=\dfrac{4}{7}

Same denominator means you just subtract the top numbers -- a Grade 4 fraction subtraction.

Answer: 4/7

4 · Reviewdoes it hold up?

The larger result is 12 star 5 and the smaller is 10 star 3; subtracting the smaller from the larger gives the positive difference 4/7, which is reasonable since both results are close in size.

Another way: Notice both bottoms are 7, so only the tops 17 and 13 matter; 17 - 13 over 7 gives 4/7 directly.

Standardsmin grade 4

4.OA.A.3 Solve multi-step word problems using four operations with whole numbers — Evaluating each defined star expression by performing the whole-number add and subtract.
4.NF.B.3 Understand a fraction with numerator greater than one as sum of unit fractions — Subtracting the two fractions to get 4/7.

💡Takeaway. This only needs Grade 4 skills -- just follow the rule, then subtract the fractions!

Variant 9 hard answer: 2/5

For any two numbers $a$ and $b$ , define a new operation $\star$ as follows:

$a \star b = \dfrac{a + b}{a - b}$

Find the difference between $11 \star 6$ and $12 \star 7$ .

Show solution

1 · Understandwhat's really being asked

A made-up operation a star b means (a+b) divided by (a-b). I must compute 11 star 6 and 12 star 7 and find the difference between the two results.

Givens

a star b = (a+b)/(a-b).
We need 11 star 6 and 12 star 7.
We want the difference between the two results.

Unknowns

The difference between 11 star 6 and 12 star 7.

Constraints

Substitute the given numbers into the definition exactly as written.

2 · Planchoose the strategy

#7 Identify Subproblems · also uses: #6 Guess and Check

Break the task into two subproblems -- evaluate each star expression by plugging into the rule -- then subtract the two fractions. The rule is a recipe to follow step by step.

3 · Execute3 carry out the plan

1Compute 11 star 6

#7 Identify Subproblems 4.OA.A.3

Substitute a=11, b=6 into (a+b)/(a-b): the top is 11+6 = 17 and the bottom is 11-6 = 5, so 11 star 6 = 17/5.

11\star 6=\dfrac{11+6}{11-6}=\dfrac{17}{5}

Following a defined recipe is just doing the add and subtract it tells you to do.

2Compute 12 star 7

#7 Identify Subproblems 4.OA.A.3

Substitute a=12, b=7: the top is 12+7 = 19 and the bottom is 12-7 = 5, so 12 star 7 = 19/5.

12\star 7=\dfrac{12+7}{12-7}=\dfrac{19}{5}

Same recipe with different numbers.

3Find the difference

#7 Identify Subproblems 4.NF.B.3

Both fractions have denominator 5, so subtract the numerators: 19/5 - 17/5 = 2/5.

\dfrac{19}{5}-\dfrac{17}{5}=\dfrac{2}{5}

Same denominator means you just subtract the top numbers -- a Grade 4 fraction subtraction.

Answer: 2/5

4 · Reviewdoes it hold up?

The larger result is 12 star 7 and the smaller is 11 star 6; subtracting the smaller from the larger gives the positive difference 2/5, which is reasonable since both results are close in size.

Another way: Notice both bottoms are 5, so only the tops 19 and 17 matter; 19 - 17 over 5 gives 2/5 directly.

Standardsmin grade 4

4.OA.A.3 Solve multi-step word problems using four operations with whole numbers — Evaluating each defined star expression by performing the whole-number add and subtract.
4.NF.B.3 Understand a fraction with numerator greater than one as sum of unit fractions — Subtracting the two fractions to get 2/5.

💡Takeaway. This only needs Grade 4 skills -- just follow the rule, then subtract the fractions!

Variant 10 hard answer: 8/7

For any two numbers $a$ and $b$ , define a new operation $\star$ as follows:

$a \star b = \dfrac{a + b}{a - b}$

Find the difference between $9 \star 2$ and $13 \star 6$ .

Show solution

1 · Understandwhat's really being asked

A made-up operation a star b means (a+b) divided by (a-b). I must compute 9 star 2 and 13 star 6 and find the difference between the two results.

Givens

a star b = (a+b)/(a-b).
We need 9 star 2 and 13 star 6.
We want the difference between the two results.

Unknowns

The difference between 9 star 2 and 13 star 6.

Constraints

Substitute the given numbers into the definition exactly as written.

2 · Planchoose the strategy

#7 Identify Subproblems · also uses: #6 Guess and Check

Break the task into two subproblems -- evaluate each star expression by plugging into the rule -- then subtract the two fractions. The rule is a recipe to follow step by step.

3 · Execute3 carry out the plan

1Compute 9 star 2

#7 Identify Subproblems 4.OA.A.3

Substitute a=9, b=2 into (a+b)/(a-b): the top is 9+2 = 11 and the bottom is 9-2 = 7, so 9 star 2 = 11/7.

9\star 2=\dfrac{9+2}{9-2}=\dfrac{11}{7}

Following a defined recipe is just doing the add and subtract it tells you to do.

2Compute 13 star 6

#7 Identify Subproblems 4.OA.A.3

Substitute a=13, b=6: the top is 13+6 = 19 and the bottom is 13-6 = 7, so 13 star 6 = 19/7.

13\star 6=\dfrac{13+6}{13-6}=\dfrac{19}{7}

Same recipe with different numbers.

3Find the difference

#7 Identify Subproblems 4.NF.B.3

Both fractions have denominator 7, so subtract the numerators: 19/7 - 11/7 = 8/7.

\dfrac{19}{7}-\dfrac{11}{7}=\dfrac{8}{7}

Same denominator means you just subtract the top numbers -- a Grade 4 fraction subtraction.

Answer: 8/7

4 · Reviewdoes it hold up?

The larger result is 13 star 6 and the smaller is 9 star 2; subtracting the smaller from the larger gives the positive difference 8/7, which is reasonable since both results are close in size.

Another way: Notice both bottoms are 7, so only the tops 19 and 11 matter; 19 - 11 over 7 gives 8/7 directly.

Standardsmin grade 4

4.OA.A.3 Solve multi-step word problems using four operations with whole numbers — Evaluating each defined star expression by performing the whole-number add and subtract.
4.NF.B.3 Understand a fraction with numerator greater than one as sum of unit fractions — Subtracting the two fractions to get 8/7.

💡Takeaway. This only needs Grade 4 skills -- just follow the rule, then subtract the fractions!