← 4-2 · 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 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

Understand

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.

Plan

#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.

Execute

#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.

#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.

#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

Review

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.

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

Standards · min 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.

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

Variant 2 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

Understand

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.

Plan

#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.

Execute

#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.

#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.

#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

Review

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.

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

Standards · min 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.

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

Variant 3 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

Understand

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.

Plan

#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.

Execute

#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.

#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.

#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

Review

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.

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

Standards · min 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.

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

Variant 4 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

Understand

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.

Plan

#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.

Execute

#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.

#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.

#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

Review

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.

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

Standards · min 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.

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

Variant 5 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

Understand

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.

Plan

#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.

Execute

#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.

#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.

#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

Review

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.

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

Standards · min 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.

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

Variant 6 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

Understand

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.

Plan

#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.

Execute

#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.

#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.

#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

Review

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.

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

Standards · min 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.

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

Variant 7 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

Understand

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.

Plan

#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.

Execute

#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.

#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.

#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

Review

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.

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

Standards · min 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.

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

Variant 8 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

Understand

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.

Plan

#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.

Execute

#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.

#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.

#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

Review

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.

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

Standards · min 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.

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

Variant 9 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

Understand

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.

Plan

#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.

Execute

#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.

#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.

#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

Review

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.

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

Standards · min 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.

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

Variant 10 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

Understand

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.

Plan

#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.

Execute

#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.

#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.

#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

Review

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.

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

Standards · min 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.

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