Practice · Apply a defined symbol to compute with fractions · Sensim Math

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

$a \star b$ = $\frac{a+b}{a-b}$ .
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

\frac{a+b}{a-b}

: the top is 6+2 = 8 and the bottom is 6-2 = 4, so

6 \star 2

=

\frac{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

=

\frac{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:

\frac{12}{4} - \frac{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

$a \star b$ = $\frac{a+b}{a-b}$ .
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

\frac{a+b}{a-b}

: the top is 5+2 = 7 and the bottom is 5-2 = 3, so

5 \star 2

=

\frac{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

=

\frac{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:

\frac{13}{3} - \frac{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: $\frac{2}{5}$

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

$a \star b$ = $\frac{a+b}{a-b}$ .
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

\frac{a+b}{a-b}

: the top is 8+3 = 11 and the bottom is 8-3 = 5, so

8 \star 3

=

\frac{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

=

\frac{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:

\frac{13}{5} - \frac{11}{5} = \frac{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:

\frac{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 $\frac{2}{5}$ , 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 $\frac{2}{5}$ 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 $\frac{2}{5}$.

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

Variant 4 medium answer: $\frac{4}{3}$

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

$a \star b$ = $\frac{a+b}{a-b}$ .
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

\frac{a+b}{a-b}

: the top is 7+4 = 11 and the bottom is 7-4 = 3, so

7 \star 4

=

\frac{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

=

\frac{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:

\frac{15}{3} - \frac{11}{3} = \frac{4}{3}

.

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

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

Answer:

\frac{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 $\frac{4}{3}$ , 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 $\frac{4}{3}$ 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 $\frac{4}{3}$.

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

Variant 5 medium answer: $\frac{6}{5}$

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

$a \star b$ = $\frac{a+b}{a-b}$ .
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

\frac{a+b}{a-b}

: the top is 7+2 = 9 and the bottom is 7-2 = 5, so

7 \star 2

=

\frac{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

=

\frac{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:

\frac{15}{5} - \frac{9}{5} = \frac{6}{5}

.

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

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

Answer:

\frac{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 $\frac{6}{5}$ , 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 $\frac{6}{5}$ 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 $\frac{6}{5}$.

💡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

$a \star b$ = $\frac{a+b}{a-b}$ .
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

\frac{a+b}{a-b}

: the top is 9+5 = 14 and the bottom is 9-5 = 4, so

9 \star 5

=

\frac{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

=

\frac{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:

\frac{18}{4} - \frac{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: $\frac{6}{7}$

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

$a \star b$ = $\frac{a+b}{a-b}$ .
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

\frac{a+b}{a-b}

: the top is 8+1 = 9 and the bottom is 8-1 = 7, so

8 \star 1

=

\frac{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

=

\frac{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:

\frac{15}{7} - \frac{9}{7} = \frac{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:

\frac{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 $\frac{6}{7}$ , 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 $\frac{6}{7}$ 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 $\frac{6}{7}$.

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

Variant 8 hard answer: $\frac{4}{7}$

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

$a \star b$ = $\frac{a+b}{a-b}$ .
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

\frac{a+b}{a-b}

: the top is 10+3 = 13 and the bottom is 10-3 = 7, so

10 \star 3

=

\frac{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

=

\frac{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:

\frac{17}{7}

-

\frac{13}{7}

=

\frac{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:

\frac{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 $\frac{4}{7}$ , 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 $\frac{4}{7}$ 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 $\frac{4}{7}$.

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

Variant 9 hard answer: $\frac{2}{5}$

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

$a \star b$ = $\frac{a+b}{a-b}$ .
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

\frac{a+b}{a-b}

: the top is 11+6 = 17 and the bottom is 11-6 = 5, so

11 \star 6

=

\frac{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

=

\frac{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:

\frac{19}{5} - \frac{17}{5} = \frac{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:

\frac{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 $\frac{2}{5}$ , 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 $\frac{2}{5}$ 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 $\frac{2}{5}$.

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

Variant 10 hard answer: $\frac{8}{7}$

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

$a \star b$ = $\frac{a+b}{a-b}$ .
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

\frac{a+b}{a-b}

: the top is 9+2 = 11 and the bottom is 9-2 = 7, so

9 \star 2

=

\frac{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

=

\frac{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:

\frac{19}{7} - \frac{11}{7} = \frac{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:

\frac{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 $\frac{8}{7}$ , 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 $\frac{8}{7}$ 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 $\frac{8}{7}$.

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

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

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