Practice · Every difference quotient is the slope · Sensim Math

Variant 1 easy

A linear function $f(x)$ satisfies

$\frac{f(8)-f(1)}{7}+\frac{f(7)-f(2)}{5}+\frac{f(6)-f(3)}{3}+\frac{f(5)-f(4)}{1}=20$

Find the value of $f(26)-f(14)$ .

Your answer

Show solution

1 · Understandwhat's really being asked

For a linear function $f$ , a sum of four fractions equals $20$ . Use it to find $f(26)-f(14)$ .

Givens

$f$ is a linear function.
$\frac{f(8)-f(1)}{7}+\frac{f(7)-f(2)}{5}+\frac{f(6)-f(3)}{3}+\frac{f(5)-f(4)}{1}=20$

Unknowns

The value of $f(26)-f(14)$ .

Constraints

For a linear function, the change in $y$ divided by the change in $x$ is the same between any two points.

2 · Planchoose the strategy

#5 Look for a Pattern · also uses: #13 Convert to Algebra#16 Count the Complement

Look for the pattern in the denominators: $7=8-1$ , $5=7-2$ , $3=6-3$ , $1=5-4$ . Each one is exactly the gap between the two inputs on top, so every fraction is (change in $y$ )/(change in $x$ ) -- the slope. The scary sum is just four copies of the slope.

3 · Execute3 carry out the plan

1Spot the pattern in the denominators

#5 Look for a Pattern 8.EE.B.6

Each denominator equals the difference of the inputs in its numerator.

8-1=7,\ 7-2=5,\ 6-3=3,\ 5-4=1

So each fraction has the form

\frac{f(q)-f(p)}{q-p}

.

2Each fraction is the slope

#16 Count the Complement 8.EE.B.6

For a linear function,

\frac{f(q)-f(p)}{q-p}

is the slope

m

, no matter which two inputs you pick. So the sum is

4m

.

m+m+m+m=4m=20 \Rightarrow m=5

Four equal terms adding to

20

means each is

5

.

3Use the slope on the new pair

#13 Convert to Algebra 8.F.B.4

Apply the same rule to the inputs

14

and

26

.

\frac{f(26)-f(14)}{26-14}=5 \Rightarrow f(26)-f(14)=5\times 12=60

Change in

y

= slope

\times

change in

x

.

Answer:

60

4 · Reviewdoes it hold up?

Take a concrete example, $f(x)=5x$ : the fractions are $\frac{35}{7}=5$ , $\frac{25}{5}=5$ , $\frac{15}{3}=5$ , $\frac{5}{1}=5$ , total $20$ ; and $f(26)-f(14)=130-70=60$ .

Another way: Convert to Algebra directly: write $f(x)=mx+c$ . Then $f(q)-f(p)=m(q-p)$ , so each fraction simplifies to $m$ ; the constant $c$ cancels everywhere, giving $4m=20$ and $f(26)-f(14)=12m=60$ .

Standardsmin grade 8

8.EE.B.6 Use similar triangles to explain why the slope is the same between any two points — Knowing the slope is the same between any two points of a line.
8.F.B.4 Construct a function to model a linear relationship between two quantities — Using the rate of change to find a change in the function's value.

💡Takeaway. Before you grind through a messy sum, check whether every piece is secretly the same thing -- here each one was just the slope.

Variant 2 easy

A linear function $f(x)$ satisfies

$\frac{f(8)-f(1)}{7}+\frac{f(7)-f(2)}{5}+\frac{f(6)-f(3)}{3}+\frac{f(5)-f(4)}{1}=12$

Find the value of $f(30)-f(11)$ .

Your answer

Show solution

1 · Understandwhat's really being asked

For a linear function $f$ , a sum of four fractions equals $12$ . Use it to find $f(30)-f(11)$ .

Givens

$f$ is a linear function.
$\frac{f(8)-f(1)}{7}+\frac{f(7)-f(2)}{5}+\frac{f(6)-f(3)}{3}+\frac{f(5)-f(4)}{1}=12$

Unknowns

The value of $f(30)-f(11)$ .

Constraints

For a linear function, the change in $y$ divided by the change in $x$ is the same between any two points.

2 · Planchoose the strategy

#5 Look for a Pattern · also uses: #13 Convert to Algebra#16 Count the Complement

Look for the pattern in the denominators: $7=8-1$ , $5=7-2$ , $3=6-3$ , $1=5-4$ . Each one is exactly the gap between the two inputs on top, so every fraction is (change in $y$ )/(change in $x$ ) -- the slope. The scary sum is just four copies of the slope.

3 · Execute3 carry out the plan

1Spot the pattern in the denominators

#5 Look for a Pattern 8.EE.B.6

Each denominator equals the difference of the inputs in its numerator.

8-1=7,\ 7-2=5,\ 6-3=3,\ 5-4=1

So each fraction has the form

\frac{f(q)-f(p)}{q-p}

.

2Each fraction is the slope

#16 Count the Complement 8.EE.B.6

For a linear function,

\frac{f(q)-f(p)}{q-p}

is the slope

m

, no matter which two inputs you pick. So the sum is

4m

.

m+m+m+m=4m=12 \Rightarrow m=3

Four equal terms adding to

12

means each is

3

.

3Use the slope on the new pair

#13 Convert to Algebra 8.F.B.4

Apply the same rule to the inputs

11

and

30

.

\frac{f(30)-f(11)}{30-11}=3 \Rightarrow f(30)-f(11)=3\times 19=57

Change in

y

= slope

\times

change in

x

.

Answer:

57

4 · Reviewdoes it hold up?

Take a concrete example, $f(x)=3x$ : the fractions are $\frac{21}{7}=3$ , $\frac{15}{5}=3$ , $\frac{9}{3}=3$ , $\frac{3}{1}=3$ , total $12$ ; and $f(30)-f(11)=90-33=57$ .

Another way: Convert to Algebra directly: write $f(x)=mx+c$ . Then $f(q)-f(p)=m(q-p)$ , so each fraction simplifies to $m$ ; the constant $c$ cancels everywhere, giving $4m=12$ and $f(30)-f(11)=19m=57$ .

Standardsmin grade 8

8.EE.B.6 Use similar triangles to explain why the slope is the same between any two points — Knowing the slope is the same between any two points of a line.
8.F.B.4 Construct a function to model a linear relationship between two quantities — Using the rate of change to find a change in the function's value.

💡Takeaway. Before you grind through a messy sum, check whether every piece is secretly the same thing -- here each one was just the slope.

Variant 3 easy

A linear function $f(x)$ satisfies

$\frac{f(9)-f(1)}{8}+\frac{f(8)-f(2)}{6}+\frac{f(7)-f(3)}{4}+\frac{f(6)-f(4)}{2}=8$

Find the value of $f(35)-f(12)$ .

Your answer

Show solution

1 · Understandwhat's really being asked

For a linear function $f$ , a sum of four fractions equals $8$ . Use it to find $f(35)-f(12)$ .

Givens

$f$ is a linear function.
$\frac{f(9)-f(1)}{8}+\frac{f(8)-f(2)}{6}+\frac{f(7)-f(3)}{4}+\frac{f(6)-f(4)}{2}=8$

Unknowns

The value of $f(35)-f(12)$ .

Constraints

For a linear function, the change in $y$ divided by the change in $x$ is the same between any two points.

2 · Planchoose the strategy

#5 Look for a Pattern · also uses: #13 Convert to Algebra#16 Count the Complement

Look for the pattern in the denominators: $8=9-1$ , $6=8-2$ , $4=7-3$ , $2=6-4$ . Each one is exactly the gap between the two inputs on top, so every fraction is (change in $y$ )/(change in $x$ ) -- the slope. The scary sum is just four copies of the slope.

3 · Execute3 carry out the plan

1Spot the pattern in the denominators

#5 Look for a Pattern 8.EE.B.6

Each denominator equals the difference of the inputs in its numerator.

9-1=8,\ 8-2=6,\ 7-3=4,\ 6-4=2

So each fraction has the form

\frac{f(q)-f(p)}{q-p}

.

2Each fraction is the slope

#16 Count the Complement 8.EE.B.6

For a linear function,

\frac{f(q)-f(p)}{q-p}

is the slope

m

, no matter which two inputs you pick. So the sum is

4m

.

m+m+m+m=4m=8 \Rightarrow m=2

Four equal terms adding to

8

means each is

2

.

3Use the slope on the new pair

#13 Convert to Algebra 8.F.B.4

Apply the same rule to the inputs

12

and

35

.

\frac{f(35)-f(12)}{35-12}=2 \Rightarrow f(35)-f(12)=2\times 23=46

Change in

y

= slope

\times

change in

x

.

Answer:

46

4 · Reviewdoes it hold up?

Take a concrete example, $f(x)=2x$ : the fractions are $\frac{16}{8}=2$ , $\frac{12}{6}=2$ , $\frac{8}{4}=2$ , $\frac{4}{2}=2$ , total $8$ ; and $f(35)-f(12)=70-24=46$ .

Another way: Convert to Algebra directly: write $f(x)=mx+c$ . Then $f(q)-f(p)=m(q-p)$ , so each fraction simplifies to $m$ ; the constant $c$ cancels everywhere, giving $4m=8$ and $f(35)-f(12)=23m=46$ .

Standardsmin grade 8

8.EE.B.6 Use similar triangles to explain why the slope is the same between any two points — Knowing the slope is the same between any two points of a line.
8.F.B.4 Construct a function to model a linear relationship between two quantities — Using the rate of change to find a change in the function's value.

💡Takeaway. Before you grind through a messy sum, check whether every piece is secretly the same thing -- here each one was just the slope.

Variant 4 easy

A linear function $f(x)$ satisfies

$\frac{f(9)-f(1)}{8}+\frac{f(8)-f(2)}{6}+\frac{f(7)-f(3)}{4}+\frac{f(6)-f(4)}{2}=12$

Find the value of $f(44)-f(25)$ .

Your answer

Show solution

1 · Understandwhat's really being asked

For a linear function $f$ , a sum of four fractions equals $12$ . Use it to find $f(44)-f(25)$ .

Givens

$f$ is a linear function.
$\frac{f(9)-f(1)}{8}+\frac{f(8)-f(2)}{6}+\frac{f(7)-f(3)}{4}+\frac{f(6)-f(4)}{2}=12$

Unknowns

The value of $f(44)-f(25)$ .

Constraints

For a linear function, the change in $y$ divided by the change in $x$ is the same between any two points.

2 · Planchoose the strategy

#5 Look for a Pattern · also uses: #13 Convert to Algebra#16 Count the Complement

Look for the pattern in the denominators: $8=9-1$ , $6=8-2$ , $4=7-3$ , $2=6-4$ . Each one is exactly the gap between the two inputs on top, so every fraction is (change in $y$ )/(change in $x$ ) -- the slope. The scary sum is just four copies of the slope.

3 · Execute3 carry out the plan

1Spot the pattern in the denominators

#5 Look for a Pattern 8.EE.B.6

Each denominator equals the difference of the inputs in its numerator.

9-1=8,\ 8-2=6,\ 7-3=4,\ 6-4=2

So each fraction has the form

\frac{f(q)-f(p)}{q-p}

.

2Each fraction is the slope

#16 Count the Complement 8.EE.B.6

For a linear function,

\frac{f(q)-f(p)}{q-p}

is the slope

m

, no matter which two inputs you pick. So the sum is

4m

.

m+m+m+m=4m=12 \Rightarrow m=3

Four equal terms adding to

12

means each is

3

.

3Use the slope on the new pair

#13 Convert to Algebra 8.F.B.4

Apply the same rule to the inputs

25

and

44

.

\frac{f(44)-f(25)}{44-25}=3 \Rightarrow f(44)-f(25)=3\times 19=57

Change in

y

= slope

\times

change in

x

.

Answer:

57

4 · Reviewdoes it hold up?

Take a concrete example, $f(x)=3x$ : the fractions are $\frac{24}{8}=3$ , $\frac{18}{6}=3$ , $\frac{12}{4}=3$ , $\frac{6}{2}=3$ , total $12$ ; and $f(44)-f(25)=132-75=57$ .

Another way: Convert to Algebra directly: write $f(x)=mx+c$ . Then $f(q)-f(p)=m(q-p)$ , so each fraction simplifies to $m$ ; the constant $c$ cancels everywhere, giving $4m=12$ and $f(44)-f(25)=19m=57$ .

Standardsmin grade 8

8.EE.B.6 Use similar triangles to explain why the slope is the same between any two points — Knowing the slope is the same between any two points of a line.
8.F.B.4 Construct a function to model a linear relationship between two quantities — Using the rate of change to find a change in the function's value.

💡Takeaway. Before you grind through a messy sum, check whether every piece is secretly the same thing -- here each one was just the slope.

Variant 5 medium

A linear function $f(x)$ satisfies

$\frac{f(8)-f(1)}{7}+\frac{f(7)-f(2)}{5}+\frac{f(6)-f(3)}{3}+\frac{f(5)-f(4)}{1}=28$

Find the value of $f(57)-f(21)$ .

Your answer

Show solution

1 · Understandwhat's really being asked

For a linear function $f$ , a sum of four fractions equals $28$ . Use it to find $f(57)-f(21)$ .

Givens

$f$ is a linear function.
$\frac{f(8)-f(1)}{7}+\frac{f(7)-f(2)}{5}+\frac{f(6)-f(3)}{3}+\frac{f(5)-f(4)}{1}=28$

Unknowns

The value of $f(57)-f(21)$ .

Constraints

For a linear function, the change in $y$ divided by the change in $x$ is the same between any two points.

2 · Planchoose the strategy

#5 Look for a Pattern · also uses: #13 Convert to Algebra#16 Count the Complement

Look for the pattern in the denominators: $7=8-1$ , $5=7-2$ , $3=6-3$ , $1=5-4$ . Each one is exactly the gap between the two inputs on top, so every fraction is (change in $y$ )/(change in $x$ ) -- the slope. The scary sum is just four copies of the slope.

3 · Execute3 carry out the plan

1Spot the pattern in the denominators

#5 Look for a Pattern 8.EE.B.6

Each denominator equals the difference of the inputs in its numerator.

8-1=7,\ 7-2=5,\ 6-3=3,\ 5-4=1

So each fraction has the form

\frac{f(q)-f(p)}{q-p}

.

2Each fraction is the slope

#16 Count the Complement 8.EE.B.6

For a linear function,

\frac{f(q)-f(p)}{q-p}

is the slope

m

, no matter which two inputs you pick. So the sum is

4m

.

m+m+m+m=4m=28 \Rightarrow m=7

Four equal terms adding to

28

means each is

7

.

3Use the slope on the new pair

#13 Convert to Algebra 8.F.B.4

Apply the same rule to the inputs

21

and

57

.

\frac{f(57)-f(21)}{57-21}=7 \Rightarrow f(57)-f(21)=7\times 36=252

Change in

y

= slope

\times

change in

x

.

Answer:

252

4 · Reviewdoes it hold up?

Take a concrete example, $f(x)=7x$ : the fractions are $\frac{49}{7}=7$ , $\frac{35}{5}=7$ , $\frac{21}{3}=7$ , $\frac{7}{1}=7$ , total $28$ ; and $f(57)-f(21)=399-147=252$ .

Another way: Convert to Algebra directly: write $f(x)=mx+c$ . Then $f(q)-f(p)=m(q-p)$ , so each fraction simplifies to $m$ ; the constant $c$ cancels everywhere, giving $4m=28$ and $f(57)-f(21)=36m=252$ .

Standardsmin grade 8

8.EE.B.6 Use similar triangles to explain why the slope is the same between any two points — Knowing the slope is the same between any two points of a line.
8.F.B.4 Construct a function to model a linear relationship between two quantities — Using the rate of change to find a change in the function's value.

💡Takeaway. Before you grind through a messy sum, check whether every piece is secretly the same thing -- here each one was just the slope.

Variant 6 medium

A linear function $f(x)$ satisfies

$\frac{f(10)-f(1)}{9}+\frac{f(9)-f(2)}{7}+\frac{f(8)-f(3)}{5}+\frac{f(7)-f(4)}{3}+\frac{f(6)-f(5)}{1}=20$

Find the value of $f(33)-f(19)$ .

Your answer

Show solution

1 · Understandwhat's really being asked

For a linear function $f$ , a sum of five fractions equals $20$ . Use it to find $f(33)-f(19)$ .

Givens

$f$ is a linear function.
$\frac{f(10)-f(1)}{9}+\frac{f(9)-f(2)}{7}+\frac{f(8)-f(3)}{5}+\frac{f(7)-f(4)}{3}+\frac{f(6)-f(5)}{1}=20$

Unknowns

The value of $f(33)-f(19)$ .

Constraints

For a linear function, the change in $y$ divided by the change in $x$ is the same between any two points.

2 · Planchoose the strategy

#5 Look for a Pattern · also uses: #13 Convert to Algebra#16 Count the Complement

Look for the pattern in the denominators: $9=10-1$ , $7=9-2$ , $5=8-3$ , $3=7-4$ , $1=6-5$ . Each one is exactly the gap between the two inputs on top, so every fraction is (change in $y$ )/(change in $x$ ) -- the slope. The scary sum is just five copies of the slope.

3 · Execute3 carry out the plan

1Spot the pattern in the denominators

#5 Look for a Pattern 8.EE.B.6

Each denominator equals the difference of the inputs in its numerator.

10-1=9,\ 9-2=7,\ 8-3=5,\ 7-4=3,\ 6-5=1

So each fraction has the form

\frac{f(q)-f(p)}{q-p}

.

2Each fraction is the slope

#16 Count the Complement 8.EE.B.6

For a linear function,

\frac{f(q)-f(p)}{q-p}

is the slope

m

, no matter which two inputs you pick. So the sum is

5m

.

m+m+m+m+m=5m=20 \Rightarrow m=4

Five equal terms adding to

20

means each is

4

.

3Use the slope on the new pair

#13 Convert to Algebra 8.F.B.4

Apply the same rule to the inputs

19

and

33

.

\frac{f(33)-f(19)}{33-19}=4 \Rightarrow f(33)-f(19)=4\times 14=56

Change in

y

= slope

\times

change in

x

.

Answer:

56

4 · Reviewdoes it hold up?

Take a concrete example, $f(x)=4x$ : the fractions are $\frac{36}{9}=4$ , $\frac{28}{7}=4$ , $\frac{20}{5}=4$ , $\frac{12}{3}=4$ , $\frac{4}{1}=4$ , total $20$ ; and $f(33)-f(19)=132-76=56$ .

Another way: Convert to Algebra directly: write $f(x)=mx+c$ . Then $f(q)-f(p)=m(q-p)$ , so each fraction simplifies to $m$ ; the constant $c$ cancels everywhere, giving $5m=20$ and $f(33)-f(19)=14m=56$ .

Standardsmin grade 8

8.EE.B.6 Use similar triangles to explain why the slope is the same between any two points — Knowing the slope is the same between any two points of a line.
8.F.B.4 Construct a function to model a linear relationship between two quantities — Using the rate of change to find a change in the function's value.

💡Takeaway. Before you grind through a messy sum, check whether every piece is secretly the same thing -- here each one was just the slope.

Variant 7 medium

A linear function $f(x)$ satisfies

$\frac{f(11)-f(1)}{10}+\frac{f(10)-f(2)}{8}+\frac{f(9)-f(3)}{6}+\frac{f(8)-f(4)}{4}+\frac{f(7)-f(5)}{2}=10$

Find the value of $f(38)-f(16)$ .

Your answer

Show solution

1 · Understandwhat's really being asked

For a linear function $f$ , a sum of five fractions equals $10$ . Use it to find $f(38)-f(16)$ .

Givens

$f$ is a linear function.
$\frac{f(11)-f(1)}{10}+\frac{f(10)-f(2)}{8}+\frac{f(9)-f(3)}{6}+\frac{f(8)-f(4)}{4}+\frac{f(7)-f(5)}{2}=10$

Unknowns

The value of $f(38)-f(16)$ .

Constraints

For a linear function, the change in $y$ divided by the change in $x$ is the same between any two points.

2 · Planchoose the strategy

#5 Look for a Pattern · also uses: #13 Convert to Algebra#16 Count the Complement

Look for the pattern in the denominators: $10=11-1$ , $8=10-2$ , $6=9-3$ , $4=8-4$ , $2=7-5$ . Each one is exactly the gap between the two inputs on top, so every fraction is (change in $y$ )/(change in $x$ ) -- the slope. The scary sum is just five copies of the slope.

3 · Execute3 carry out the plan

1Spot the pattern in the denominators

#5 Look for a Pattern 8.EE.B.6

Each denominator equals the difference of the inputs in its numerator.

11-1=10,\ 10-2=8,\ 9-3=6,\ 8-4=4,\ 7-5=2

So each fraction has the form

\frac{f(q)-f(p)}{q-p}

.

2Each fraction is the slope

#16 Count the Complement 8.EE.B.6

For a linear function,

\frac{f(q)-f(p)}{q-p}

is the slope

m

, no matter which two inputs you pick. So the sum is

5m

.

m+m+m+m+m=5m=10 \Rightarrow m=2

Five equal terms adding to

10

means each is

2

.

3Use the slope on the new pair

#13 Convert to Algebra 8.F.B.4

Apply the same rule to the inputs

16

and

38

.

\frac{f(38)-f(16)}{38-16}=2 \Rightarrow f(38)-f(16)=2\times 22=44

Change in

y

= slope

\times

change in

x

.

Answer:

44

4 · Reviewdoes it hold up?

Take a concrete example, $f(x)=2x$ : the fractions are $\frac{20}{10}=2$ , $\frac{16}{8}=2$ , $\frac{12}{6}=2$ , $\frac{8}{4}=2$ , $\frac{4}{2}=2$ , total $10$ ; and $f(38)-f(16)=76-32=44$ .

Another way: Convert to Algebra directly: write $f(x)=mx+c$ . Then $f(q)-f(p)=m(q-p)$ , so each fraction simplifies to $m$ ; the constant $c$ cancels everywhere, giving $5m=10$ and $f(38)-f(16)=22m=44$ .

Standardsmin grade 8

8.EE.B.6 Use similar triangles to explain why the slope is the same between any two points — Knowing the slope is the same between any two points of a line.
8.F.B.4 Construct a function to model a linear relationship between two quantities — Using the rate of change to find a change in the function's value.

💡Takeaway. Before you grind through a messy sum, check whether every piece is secretly the same thing -- here each one was just the slope.

Variant 8 medium

A linear function $f(x)$ satisfies

$\frac{f(10)-f(1)}{9}+\frac{f(9)-f(2)}{7}+\frac{f(8)-f(3)}{5}+\frac{f(7)-f(4)}{3}+\frac{f(6)-f(5)}{1}=15$

Find the value of $f(40)-f(17)$ .

Your answer

Show solution

1 · Understandwhat's really being asked

For a linear function $f$ , a sum of five fractions equals $15$ . Use it to find $f(40)-f(17)$ .

Givens

$f$ is a linear function.
$\frac{f(10)-f(1)}{9}+\frac{f(9)-f(2)}{7}+\frac{f(8)-f(3)}{5}+\frac{f(7)-f(4)}{3}+\frac{f(6)-f(5)}{1}=15$

Unknowns

The value of $f(40)-f(17)$ .

Constraints

For a linear function, the change in $y$ divided by the change in $x$ is the same between any two points.

2 · Planchoose the strategy

#5 Look for a Pattern · also uses: #13 Convert to Algebra#16 Count the Complement

Look for the pattern in the denominators: $9=10-1$ , $7=9-2$ , $5=8-3$ , $3=7-4$ , $1=6-5$ . Each one is exactly the gap between the two inputs on top, so every fraction is (change in $y$ )/(change in $x$ ) -- the slope. The scary sum is just five copies of the slope.

3 · Execute3 carry out the plan

1Spot the pattern in the denominators

#5 Look for a Pattern 8.EE.B.6

Each denominator equals the difference of the inputs in its numerator.

10-1=9,\ 9-2=7,\ 8-3=5,\ 7-4=3,\ 6-5=1

So each fraction has the form

\frac{f(q)-f(p)}{q-p}

.

2Each fraction is the slope

#16 Count the Complement 8.EE.B.6

For a linear function,

\frac{f(q)-f(p)}{q-p}

is the slope

m

, no matter which two inputs you pick. So the sum is

5m

.

m+m+m+m+m=5m=15 \Rightarrow m=3

Five equal terms adding to

15

means each is

3

.

3Use the slope on the new pair

#13 Convert to Algebra 8.F.B.4

Apply the same rule to the inputs

17

and

40

.

\frac{f(40)-f(17)}{40-17}=3 \Rightarrow f(40)-f(17)=3\times 23=69

Change in

y

= slope

\times

change in

x

.

Answer:

69

4 · Reviewdoes it hold up?

Take a concrete example, $f(x)=3x$ : the fractions are $\frac{27}{9}=3$ , $\frac{21}{7}=3$ , $\frac{15}{5}=3$ , $\frac{9}{3}=3$ , $\frac{3}{1}=3$ , total $15$ ; and $f(40)-f(17)=120-51=69$ .

Another way: Convert to Algebra directly: write $f(x)=mx+c$ . Then $f(q)-f(p)=m(q-p)$ , so each fraction simplifies to $m$ ; the constant $c$ cancels everywhere, giving $5m=15$ and $f(40)-f(17)=23m=69$ .

Standardsmin grade 8

8.EE.B.6 Use similar triangles to explain why the slope is the same between any two points — Knowing the slope is the same between any two points of a line.
8.F.B.4 Construct a function to model a linear relationship between two quantities — Using the rate of change to find a change in the function's value.

💡Takeaway. Before you grind through a messy sum, check whether every piece is secretly the same thing -- here each one was just the slope.

Variant 9 hard

A linear function $f(x)$ satisfies

$\frac{f(10)-f(1)}{9}+\frac{f(9)-f(2)}{7}+\frac{f(8)-f(3)}{5}+\frac{f(7)-f(4)}{3}+\frac{f(6)-f(5)}{1}=10$

Find the value of $f(50)-f(24)$ .

Your answer

Show solution

1 · Understandwhat's really being asked

For a linear function $f$ , a sum of five fractions equals $10$ . Use it to find $f(50)-f(24)$ .

Givens

$f$ is a linear function.
$\frac{f(10)-f(1)}{9}+\frac{f(9)-f(2)}{7}+\frac{f(8)-f(3)}{5}+\frac{f(7)-f(4)}{3}+\frac{f(6)-f(5)}{1}=10$

Unknowns

The value of $f(50)-f(24)$ .

Constraints

For a linear function, the change in $y$ divided by the change in $x$ is the same between any two points.

2 · Planchoose the strategy

#5 Look for a Pattern · also uses: #13 Convert to Algebra#16 Count the Complement

Look for the pattern in the denominators: $9=10-1$ , $7=9-2$ , $5=8-3$ , $3=7-4$ , $1=6-5$ . Each one is exactly the gap between the two inputs on top, so every fraction is (change in $y$ )/(change in $x$ ) -- the slope. The scary sum is just five copies of the slope.

3 · Execute3 carry out the plan

1Spot the pattern in the denominators

#5 Look for a Pattern 8.EE.B.6

Each denominator equals the difference of the inputs in its numerator.

10-1=9,\ 9-2=7,\ 8-3=5,\ 7-4=3,\ 6-5=1

So each fraction has the form

\frac{f(q)-f(p)}{q-p}

.

2Each fraction is the slope

#16 Count the Complement 8.EE.B.6

For a linear function,

\frac{f(q)-f(p)}{q-p}

is the slope

m

, no matter which two inputs you pick. So the sum is

5m

.

m+m+m+m+m=5m=10 \Rightarrow m=2

Five equal terms adding to

10

means each is

2

.

3Use the slope on the new pair

#13 Convert to Algebra 8.F.B.4

Apply the same rule to the inputs

24

and

50

.

\frac{f(50)-f(24)}{50-24}=2 \Rightarrow f(50)-f(24)=2\times 26=52

Change in

y

= slope

\times

change in

x

.

Answer:

52

4 · Reviewdoes it hold up?

Take a concrete example, $f(x)=2x$ : the fractions are $\frac{18}{9}=2$ , $\frac{14}{7}=2$ , $\frac{10}{5}=2$ , $\frac{6}{3}=2$ , $\frac{2}{1}=2$ , total $10$ ; and $f(50)-f(24)=100-48=52$ .

Another way: Convert to Algebra directly: write $f(x)=mx+c$ . Then $f(q)-f(p)=m(q-p)$ , so each fraction simplifies to $m$ ; the constant $c$ cancels everywhere, giving $5m=10$ and $f(50)-f(24)=26m=52$ .

Standardsmin grade 8

8.EE.B.6 Use similar triangles to explain why the slope is the same between any two points — Knowing the slope is the same between any two points of a line.
8.F.B.4 Construct a function to model a linear relationship between two quantities — Using the rate of change to find a change in the function's value.

💡Takeaway. Before you grind through a messy sum, check whether every piece is secretly the same thing -- here each one was just the slope.

Variant 10 hard

A linear function $f(x)$ satisfies

$\frac{f(11)-f(1)}{10}+\frac{f(10)-f(2)}{8}+\frac{f(9)-f(3)}{6}+\frac{f(8)-f(4)}{4}+\frac{f(7)-f(5)}{2}=20$

Find the value of $f(60)-f(33)$ .

Your answer

Show solution

1 · Understandwhat's really being asked

For a linear function $f$ , a sum of five fractions equals $20$ . Use it to find $f(60)-f(33)$ .

Givens

$f$ is a linear function.
$\frac{f(11)-f(1)}{10}+\frac{f(10)-f(2)}{8}+\frac{f(9)-f(3)}{6}+\frac{f(8)-f(4)}{4}+\frac{f(7)-f(5)}{2}=20$

Unknowns

The value of $f(60)-f(33)$ .

Constraints

For a linear function, the change in $y$ divided by the change in $x$ is the same between any two points.

2 · Planchoose the strategy

#5 Look for a Pattern · also uses: #13 Convert to Algebra#16 Count the Complement

Look for the pattern in the denominators: $10=11-1$ , $8=10-2$ , $6=9-3$ , $4=8-4$ , $2=7-5$ . Each one is exactly the gap between the two inputs on top, so every fraction is (change in $y$ )/(change in $x$ ) -- the slope. The scary sum is just five copies of the slope.

3 · Execute3 carry out the plan

1Spot the pattern in the denominators

#5 Look for a Pattern 8.EE.B.6

Each denominator equals the difference of the inputs in its numerator.

11-1=10,\ 10-2=8,\ 9-3=6,\ 8-4=4,\ 7-5=2

So each fraction has the form

\frac{f(q)-f(p)}{q-p}

.

2Each fraction is the slope

#16 Count the Complement 8.EE.B.6

For a linear function,

\frac{f(q)-f(p)}{q-p}

is the slope

m

, no matter which two inputs you pick. So the sum is

5m

.

m+m+m+m+m=5m=20 \Rightarrow m=4

Five equal terms adding to

20

means each is

4

.

3Use the slope on the new pair

#13 Convert to Algebra 8.F.B.4

Apply the same rule to the inputs

33

and

60

.

\frac{f(60)-f(33)}{60-33}=4 \Rightarrow f(60)-f(33)=4\times 27=108

Change in

y

= slope

\times

change in

x

.

Answer:

108

4 · Reviewdoes it hold up?

Take a concrete example, $f(x)=4x$ : the fractions are $\frac{40}{10}=4$ , $\frac{32}{8}=4$ , $\frac{24}{6}=4$ , $\frac{16}{4}=4$ , $\frac{8}{2}=4$ , total $20$ ; and $f(60)-f(33)=240-132=108$ .

Another way: Convert to Algebra directly: write $f(x)=mx+c$ . Then $f(q)-f(p)=m(q-p)$ , so each fraction simplifies to $m$ ; the constant $c$ cancels everywhere, giving $5m=20$ and $f(60)-f(33)=27m=108$ .

Standardsmin grade 8

8.EE.B.6 Use similar triangles to explain why the slope is the same between any two points — Knowing the slope is the same between any two points of a line.
8.F.B.4 Construct a function to model a linear relationship between two quantities — Using the rate of change to find a change in the function's value.

💡Takeaway. Before you grind through a messy sum, check whether every piece is secretly the same thing -- here each one was just the slope.

Variant 11 hard

A linear function $f(x)$ satisfies

$\frac{f(12)-f(1)}{11}+\frac{f(11)-f(2)}{9}+\frac{f(10)-f(3)}{7}+\frac{f(9)-f(4)}{5}+\frac{f(8)-f(5)}{3}+\frac{f(7)-f(6)}{1}=18$

Find the value of $f(45)-f(20)$ .

Your answer

Show solution

1 · Understandwhat's really being asked

For a linear function $f$ , a sum of six fractions equals $18$ . Use it to find $f(45)-f(20)$ .

Givens

$f$ is a linear function.
$\frac{f(12)-f(1)}{11}+\frac{f(11)-f(2)}{9}+\frac{f(10)-f(3)}{7}+\frac{f(9)-f(4)}{5}+\frac{f(8)-f(5)}{3}+\frac{f(7)-f(6)}{1}=18$

Unknowns

The value of $f(45)-f(20)$ .

Constraints

For a linear function, the change in $y$ divided by the change in $x$ is the same between any two points.

2 · Planchoose the strategy

#5 Look for a Pattern · also uses: #13 Convert to Algebra#16 Count the Complement

Look for the pattern in the denominators: $11=12-1$ , $9=11-2$ , $7=10-3$ , $5=9-4$ , $3=8-5$ , $1=7-6$ . Each one is exactly the gap between the two inputs on top, so every fraction is (change in $y$ )/(change in $x$ ) -- the slope. The scary sum is just six copies of the slope.

3 · Execute3 carry out the plan

1Spot the pattern in the denominators

#5 Look for a Pattern 8.EE.B.6

Each denominator equals the difference of the inputs in its numerator.

12-1=11,\ 11-2=9,\ 10-3=7,\ 9-4=5,\ 8-5=3,\ 7-6=1

So each fraction has the form

\frac{f(q)-f(p)}{q-p}

.

2Each fraction is the slope

#16 Count the Complement 8.EE.B.6

For a linear function,

\frac{f(q)-f(p)}{q-p}

is the slope

m

, no matter which two inputs you pick. So the sum is

6m

.

m+m+m+m+m+m=6m=18 \Rightarrow m=3

Six equal terms adding to

18

means each is

3

.

3Use the slope on the new pair

#13 Convert to Algebra 8.F.B.4

Apply the same rule to the inputs

20

and

45

.

\frac{f(45)-f(20)}{45-20}=3 \Rightarrow f(45)-f(20)=3\times 25=75

Change in

y

= slope

\times

change in

x

.

Answer:

75

4 · Reviewdoes it hold up?

Take a concrete example, $f(x)=3x$ : the fractions are $\frac{33}{11}=3$ , $\frac{27}{9}=3$ , $\frac{21}{7}=3$ , $\frac{15}{5}=3$ , $\frac{9}{3}=3$ , $\frac{3}{1}=3$ , total $18$ ; and $f(45)-f(20)=135-60=75$ .

Another way: Convert to Algebra directly: write $f(x)=mx+c$ . Then $f(q)-f(p)=m(q-p)$ , so each fraction simplifies to $m$ ; the constant $c$ cancels everywhere, giving $6m=18$ and $f(45)-f(20)=25m=75$ .

Standardsmin grade 8

8.EE.B.6 Use similar triangles to explain why the slope is the same between any two points — Knowing the slope is the same between any two points of a line.
8.F.B.4 Construct a function to model a linear relationship between two quantities — Using the rate of change to find a change in the function's value.

💡Takeaway. Before you grind through a messy sum, check whether every piece is secretly the same thing -- here each one was just the slope.

Variant 12 hard

A linear function $f(x)$ satisfies

$\frac{f(12)-f(1)}{11}+\frac{f(11)-f(2)}{9}+\frac{f(10)-f(3)}{7}+\frac{f(9)-f(4)}{5}+\frac{f(8)-f(5)}{3}+\frac{f(7)-f(6)}{1}=24$

Find the value of $f(70)-f(41)$ .

Your answer

Show solution

1 · Understandwhat's really being asked

For a linear function $f$ , a sum of six fractions equals $24$ . Use it to find $f(70)-f(41)$ .

Givens

$f$ is a linear function.
$\frac{f(12)-f(1)}{11}+\frac{f(11)-f(2)}{9}+\frac{f(10)-f(3)}{7}+\frac{f(9)-f(4)}{5}+\frac{f(8)-f(5)}{3}+\frac{f(7)-f(6)}{1}=24$

Unknowns

The value of $f(70)-f(41)$ .

Constraints

For a linear function, the change in $y$ divided by the change in $x$ is the same between any two points.

2 · Planchoose the strategy

#5 Look for a Pattern · also uses: #13 Convert to Algebra#16 Count the Complement

Look for the pattern in the denominators: $11=12-1$ , $9=11-2$ , $7=10-3$ , $5=9-4$ , $3=8-5$ , $1=7-6$ . Each one is exactly the gap between the two inputs on top, so every fraction is (change in $y$ )/(change in $x$ ) -- the slope. The scary sum is just six copies of the slope.

3 · Execute3 carry out the plan

1Spot the pattern in the denominators

#5 Look for a Pattern 8.EE.B.6

Each denominator equals the difference of the inputs in its numerator.

12-1=11,\ 11-2=9,\ 10-3=7,\ 9-4=5,\ 8-5=3,\ 7-6=1

So each fraction has the form

\frac{f(q)-f(p)}{q-p}

.

2Each fraction is the slope

#16 Count the Complement 8.EE.B.6

For a linear function,

\frac{f(q)-f(p)}{q-p}

is the slope

m

, no matter which two inputs you pick. So the sum is

6m

.

m+m+m+m+m+m=6m=24 \Rightarrow m=4

Six equal terms adding to

24

means each is

4

.

3Use the slope on the new pair

#13 Convert to Algebra 8.F.B.4

Apply the same rule to the inputs

41

and

70

.

\frac{f(70)-f(41)}{70-41}=4 \Rightarrow f(70)-f(41)=4\times 29=116

Change in

y

= slope

\times

change in

x

.

Answer:

116

4 · Reviewdoes it hold up?

Take a concrete example, $f(x)=4x$ : the fractions are $\frac{44}{11}=4$ , $\frac{36}{9}=4$ , $\frac{28}{7}=4$ , $\frac{20}{5}=4$ , $\frac{12}{3}=4$ , $\frac{4}{1}=4$ , total $24$ ; and $f(70)-f(41)=280-164=116$ .

Another way: Convert to Algebra directly: write $f(x)=mx+c$ . Then $f(q)-f(p)=m(q-p)$ , so each fraction simplifies to $m$ ; the constant $c$ cancels everywhere, giving $6m=24$ and $f(70)-f(41)=29m=116$ .

Standardsmin grade 8

8.EE.B.6 Use similar triangles to explain why the slope is the same between any two points — Knowing the slope is the same between any two points of a line.
8.F.B.4 Construct a function to model a linear relationship between two quantities — Using the rate of change to find a change in the function's value.

💡Takeaway. Before you grind through a messy sum, check whether every piece is secretly the same thing -- here each one was just the slope.

Every difference quotient is the slope · 12 practice problems

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