Practice · Each wrong graph still keeps one right coefficient · Sensim Math

Variant 1 easy

Liam and Mia each tried to graph the linear function $y=ax+b$ . Liam misread the value of $a$ , and Mia misread the value of $b$ , so they drew the two lines shown in the figure. Find the original linear function. ( $a$ and $b$ are constants.)

Your answer (check it yourself)

Show solution

1 · Understandwhat's really being asked

Two students each copied $y=ax+b$ with one mistake: Liam got $a$ wrong, Mia got $b$ wrong. From their two graphs, rebuild the correct equation.

Givens

Liam's line passes through $(0, 2)$ and $(2, 0)$ ; only his $a$ is wrong.
Mia's line passes through $(0, -2)$ and $(2, 0)$ ; only her $b$ is wrong.

Unknowns

The original equation $y=ax+b$ .

Constraints

In $y=ax+b$ , $a$ is the slope and $b$ is the $y$ -intercept, so each wrong graph still shows one correct number.

2 · Planchoose the strategy

#16 Count the Complement · also uses: #1 Draw a Diagram#13 Convert to Algebra

Change focus from what each student got wrong to what each got right. Liam read $b$ correctly, so his line's $y$ -intercept is the true $b$ . Mia read $a$ correctly, so her line's slope is the true $a$ . Read each off the figure and put them together.

3 · Execute3 carry out the plan

1Take b from Liam's line

#16 Count the Complement 8.F.A.3

Liam only misread

a

, so his

y

-intercept is right. His line crosses the

y

-axis at

(0, 2)

.

b=2

Ignore Liam's slope; keep his intercept.

2Take a from Mia's line

#1 Draw a Diagram 8.F.B.4

Mia only misread

b

, so her slope is right. Her line passes through

(0, -2)

and

(2, 0)

.

a=\frac{0-(-2)}{2-0}=\frac{2}{2}=1

Ignore Mia's intercept; keep her slope.

3Put the right pieces together

#13 Convert to Algebra 8.F.A.3

Use the correct slope and the correct intercept.

y=1\cdot x+2=x+2

Each wrong graph contributed exactly one true coefficient.

Answer:

y=x+2

4 · Reviewdoes it hold up?

Liam's line is $y=-x+2$ : same $b=2$ as the answer, only the slope differs. Mia's line is $y=x-2$ : same slope $1$ as the answer, only the intercept differs. Each student made exactly one mistake, as stated.

Another way: Write both drawn equations first, $y=-x+2$ and $y=x-2$ , then keep the coefficient each student is known to have read correctly: the constant term of the first and the $x$ -coefficient of the second.

Standardsmin grade 8

8.F.A.3 Interpret the equation y = mx + b as defining a linear function — Identifying $a$ as the slope and $b$ as the $y$-intercept of $y=ax+b$.
8.F.B.4 Construct a function to model a linear relationship between two quantities — Finding a slope from two points on a graph.

💡Takeaway. A wrong graph isn't useless -- ask which part the student got right, and keep exactly that part.

Variant 2 easy

Liam and Mia each tried to graph the linear function $y=ax+b$ . Liam misread the value of $a$ , and Mia misread the value of $b$ , so they drew the two lines shown in the figure. Find the original linear function. ( $a$ and $b$ are constants.)

Your answer (check it yourself)

Show solution

1 · Understandwhat's really being asked

Two students each copied $y=ax+b$ with one mistake: Liam got $a$ wrong, Mia got $b$ wrong. From their two graphs, rebuild the correct equation.

Givens

Liam's line passes through $(0, -1)$ and $(1, 0)$ ; only his $a$ is wrong.
Mia's line passes through $(0, 3)$ and $(3, 0)$ ; only her $b$ is wrong.

Unknowns

The original equation $y=ax+b$ .

Constraints

In $y=ax+b$ , $a$ is the slope and $b$ is the $y$ -intercept, so each wrong graph still shows one correct number.

2 · Planchoose the strategy

#16 Count the Complement · also uses: #1 Draw a Diagram#13 Convert to Algebra

Change focus from what each student got wrong to what each got right. Liam read $b$ correctly, so his line's $y$ -intercept is the true $b$ . Mia read $a$ correctly, so her line's slope is the true $a$ . Read each off the figure and put them together.

3 · Execute3 carry out the plan

1Take b from Liam's line

#16 Count the Complement 8.F.A.3

Liam only misread

a

, so his

y

-intercept is right. His line crosses the

y

-axis at

(0, -1)

.

b=-1

Ignore Liam's slope; keep his intercept.

2Take a from Mia's line

#1 Draw a Diagram 8.F.B.4

Mia only misread

b

, so her slope is right. Her line passes through

(0, 3)

and

(3, 0)

.

a=\frac{0-3}{3-0}=\frac{-3}{3}=-1

Ignore Mia's intercept; keep her slope.

3Put the right pieces together

#13 Convert to Algebra 8.F.A.3

Use the correct slope and the correct intercept.

y=-1\cdot x-1=-x-1

Each wrong graph contributed exactly one true coefficient.

Answer:

y=-x-1

4 · Reviewdoes it hold up?

Liam's line is $y=x-1$ : same $b=-1$ as the answer, only the slope differs. Mia's line is $y=-x+3$ : same slope $-1$ as the answer, only the intercept differs. Each student made exactly one mistake, as stated.

Another way: Write both drawn equations first, $y=x-1$ and $y=-x+3$ , then keep the coefficient each student is known to have read correctly: the constant term of the first and the $x$ -coefficient of the second.

Standardsmin grade 8

8.F.A.3 Interpret the equation y = mx + b as defining a linear function — Identifying $a$ as the slope and $b$ as the $y$-intercept of $y=ax+b$.
8.F.B.4 Construct a function to model a linear relationship between two quantities — Finding a slope from two points on a graph.

💡Takeaway. A wrong graph isn't useless -- ask which part the student got right, and keep exactly that part.

Variant 3 easy

Liam and Mia each tried to graph the linear function $y=ax+b$ . Liam misread the value of $a$ , and Mia misread the value of $b$ , so they drew the two lines shown in the figure. Find the original linear function. ( $a$ and $b$ are constants.)

Your answer (check it yourself)

Show solution

1 · Understandwhat's really being asked

Two students each copied $y=ax+b$ with one mistake: Liam got $a$ wrong, Mia got $b$ wrong. From their two graphs, rebuild the correct equation.

Givens

Liam's line passes through $(0, 1)$ and $(-1, 0)$ ; only his $a$ is wrong.
Mia's line passes through $(0, -3)$ and $(-3, 0)$ ; only her $b$ is wrong.

Unknowns

The original equation $y=ax+b$ .

Constraints

In $y=ax+b$ , $a$ is the slope and $b$ is the $y$ -intercept, so each wrong graph still shows one correct number.

2 · Planchoose the strategy

#16 Count the Complement · also uses: #1 Draw a Diagram#13 Convert to Algebra

Change focus from what each student got wrong to what each got right. Liam read $b$ correctly, so his line's $y$ -intercept is the true $b$ . Mia read $a$ correctly, so her line's slope is the true $a$ . Read each off the figure and put them together.

3 · Execute3 carry out the plan

1Take b from Liam's line

#16 Count the Complement 8.F.A.3

Liam only misread

a

, so his

y

-intercept is right. His line crosses the

y

-axis at

(0, 1)

.

b=1

Ignore Liam's slope; keep his intercept.

2Take a from Mia's line

#1 Draw a Diagram 8.F.B.4

Mia only misread

b

, so her slope is right. Her line passes through

(0, -3)

and

(-3, 0)

.

a=\frac{0-(-3)}{-3-0}=\frac{3}{-3}=-1

Ignore Mia's intercept; keep her slope.

3Put the right pieces together

#13 Convert to Algebra 8.F.A.3

Use the correct slope and the correct intercept.

y=-1\cdot x+1=-x+1

Each wrong graph contributed exactly one true coefficient.

Answer:

y=-x+1

4 · Reviewdoes it hold up?

Liam's line is $y=x+1$ : same $b=1$ as the answer, only the slope differs. Mia's line is $y=-x-3$ : same slope $-1$ as the answer, only the intercept differs. Each student made exactly one mistake, as stated.

Another way: Write both drawn equations first, $y=x+1$ and $y=-x-3$ , then keep the coefficient each student is known to have read correctly: the constant term of the first and the $x$ -coefficient of the second.

Standardsmin grade 8

8.F.A.3 Interpret the equation y = mx + b as defining a linear function — Identifying $a$ as the slope and $b$ as the $y$-intercept of $y=ax+b$.
8.F.B.4 Construct a function to model a linear relationship between two quantities — Finding a slope from two points on a graph.

💡Takeaway. A wrong graph isn't useless -- ask which part the student got right, and keep exactly that part.

Variant 4 easy

Liam and Mia each tried to graph the linear function $y=ax+b$ . Liam misread the value of $a$ , and Mia misread the value of $b$ , so they drew the two lines shown in the figure. Find the original linear function. ( $a$ and $b$ are constants.)

Your answer (check it yourself)

Show solution

1 · Understandwhat's really being asked

Two students each copied $y=ax+b$ with one mistake: Liam got $a$ wrong, Mia got $b$ wrong. From their two graphs, rebuild the correct equation.

Givens

Liam's line passes through $(0, -3)$ and $(-3, 0)$ ; only his $a$ is wrong.
Mia's line passes through $(0, 1)$ and $(-1, 0)$ ; only her $b$ is wrong.

Unknowns

The original equation $y=ax+b$ .

Constraints

In $y=ax+b$ , $a$ is the slope and $b$ is the $y$ -intercept, so each wrong graph still shows one correct number.

2 · Planchoose the strategy

#16 Count the Complement · also uses: #1 Draw a Diagram#13 Convert to Algebra

Change focus from what each student got wrong to what each got right. Liam read $b$ correctly, so his line's $y$ -intercept is the true $b$ . Mia read $a$ correctly, so her line's slope is the true $a$ . Read each off the figure and put them together.

3 · Execute3 carry out the plan

1Take b from Liam's line

#16 Count the Complement 8.F.A.3

Liam only misread

a

, so his

y

-intercept is right. His line crosses the

y

-axis at

(0, -3)

.

b=-3

Ignore Liam's slope; keep his intercept.

2Take a from Mia's line

#1 Draw a Diagram 8.F.B.4

Mia only misread

b

, so her slope is right. Her line passes through

(0, 1)

and

(-1, 0)

.

a=\frac{0-1}{-1-0}=\frac{-1}{-1}=1

Ignore Mia's intercept; keep her slope.

3Put the right pieces together

#13 Convert to Algebra 8.F.A.3

Use the correct slope and the correct intercept.

y=1\cdot x-3=x-3

Each wrong graph contributed exactly one true coefficient.

Answer:

y=x-3

4 · Reviewdoes it hold up?

Liam's line is $y=-x-3$ : same $b=-3$ as the answer, only the slope differs. Mia's line is $y=x+1$ : same slope $1$ as the answer, only the intercept differs. Each student made exactly one mistake, as stated.

Another way: Write both drawn equations first, $y=-x-3$ and $y=x+1$ , then keep the coefficient each student is known to have read correctly: the constant term of the first and the $x$ -coefficient of the second.

Standardsmin grade 8

8.F.A.3 Interpret the equation y = mx + b as defining a linear function — Identifying $a$ as the slope and $b$ as the $y$-intercept of $y=ax+b$.
8.F.B.4 Construct a function to model a linear relationship between two quantities — Finding a slope from two points on a graph.

💡Takeaway. A wrong graph isn't useless -- ask which part the student got right, and keep exactly that part.

Variant 5 hard

Liam and Mia each tried to graph the linear function $y=ax+b$ . Liam misread the value of $a$ , and Mia misread the value of $b$ , so they drew the two lines shown in the figure. Find the original linear function. ( $a$ and $b$ are constants.)

Your answer (check it yourself)

Show solution

1 · Understandwhat's really being asked

Two students each copied $y=ax+b$ with one mistake: Liam got $a$ wrong, Mia got $b$ wrong. From their two graphs, rebuild the correct equation.

Givens

Liam's line passes through $(0, 3)$ and $(-1, 0)$ ; only his $a$ is wrong.
Mia's line passes through $(0, -2)$ and $(-4, 0)$ ; only her $b$ is wrong.

Unknowns

The original equation $y=ax+b$ .

Constraints

In $y=ax+b$ , $a$ is the slope and $b$ is the $y$ -intercept, so each wrong graph still shows one correct number.

2 · Planchoose the strategy

#16 Count the Complement · also uses: #1 Draw a Diagram#13 Convert to Algebra

Change focus from what each student got wrong to what each got right. Liam read $b$ correctly, so his line's $y$ -intercept is the true $b$ . Mia read $a$ correctly, so her line's slope is the true $a$ . Read each off the figure and put them together.

3 · Execute3 carry out the plan

1Take b from Liam's line

#16 Count the Complement 8.F.A.3

Liam only misread

a

, so his

y

-intercept is right. His line crosses the

y

-axis at

(0, 3)

.

b=3

Ignore Liam's slope; keep his intercept.

2Take a from Mia's line

#1 Draw a Diagram 8.F.B.4

Mia only misread

b

, so her slope is right. Her line passes through

(0, -2)

and

(-4, 0)

.

a=\frac{0-(-2)}{-4-0}=\frac{2}{-4}=-\frac{1}{2}

Ignore Mia's intercept; keep her slope.

3Put the right pieces together

#13 Convert to Algebra 8.F.A.3

Use the correct slope and the correct intercept.

y=-\frac{1}{2}\cdot x+3=-\frac{1}{2}x+3

Each wrong graph contributed exactly one true coefficient.

Answer:

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

4 · Reviewdoes it hold up?

Liam's line is $y=3x+3$ : same $b=3$ as the answer, only the slope differs. Mia's line is $y=-\frac{1}{2}x-2$ : same slope $-\frac{1}{2}$ as the answer, only the intercept differs. Each student made exactly one mistake, as stated.

Another way: Write both drawn equations first, $y=3x+3$ and $y=-\frac{1}{2}x-2$ , then keep the coefficient each student is known to have read correctly: the constant term of the first and the $x$ -coefficient of the second.

Standardsmin grade 8

8.F.A.3 Interpret the equation y = mx + b as defining a linear function — Identifying $a$ as the slope and $b$ as the $y$-intercept of $y=ax+b$.
8.F.B.4 Construct a function to model a linear relationship between two quantities — Finding a slope from two points on a graph.

💡Takeaway. A wrong graph isn't useless -- ask which part the student got right, and keep exactly that part.

Variant 6 medium

Liam and Mia each tried to graph the linear function $y=ax+b$ . Liam misread the value of $a$ , and Mia misread the value of $b$ , so they drew the two lines shown in the figure. Find the original linear function. ( $a$ and $b$ are constants.)

Your answer (check it yourself)

Show solution

1 · Understandwhat's really being asked

Two students each copied $y=ax+b$ with one mistake: Liam got $a$ wrong, Mia got $b$ wrong. From their two graphs, rebuild the correct equation.

Givens

Liam's line passes through $(0, 3)$ and $(3, 0)$ ; only his $a$ is wrong.
Mia's line passes through $(0, -4)$ and $(2, 0)$ ; only her $b$ is wrong.

Unknowns

The original equation $y=ax+b$ .

Constraints

In $y=ax+b$ , $a$ is the slope and $b$ is the $y$ -intercept, so each wrong graph still shows one correct number.

2 · Planchoose the strategy

#16 Count the Complement · also uses: #1 Draw a Diagram#13 Convert to Algebra

Change focus from what each student got wrong to what each got right. Liam read $b$ correctly, so his line's $y$ -intercept is the true $b$ . Mia read $a$ correctly, so her line's slope is the true $a$ . Read each off the figure and put them together.

3 · Execute3 carry out the plan

1Take b from Liam's line

#16 Count the Complement 8.F.A.3

Liam only misread

a

, so his

y

-intercept is right. His line crosses the

y

-axis at

(0, 3)

.

b=3

Ignore Liam's slope; keep his intercept.

2Take a from Mia's line

#1 Draw a Diagram 8.F.B.4

Mia only misread

b

, so her slope is right. Her line passes through

(0, -4)

and

(2, 0)

.

a=\frac{0-(-4)}{2-0}=\frac{4}{2}=2

Ignore Mia's intercept; keep her slope.

3Put the right pieces together

#13 Convert to Algebra 8.F.A.3

Use the correct slope and the correct intercept.

y=2\cdot x+3=2x+3

Each wrong graph contributed exactly one true coefficient.

Answer:

y=2x+3

4 · Reviewdoes it hold up?

Liam's line is $y=-x+3$ : same $b=3$ as the answer, only the slope differs. Mia's line is $y=2x-4$ : same slope $2$ as the answer, only the intercept differs. Each student made exactly one mistake, as stated.

Another way: Write both drawn equations first, $y=-x+3$ and $y=2x-4$ , then keep the coefficient each student is known to have read correctly: the constant term of the first and the $x$ -coefficient of the second.

Standardsmin grade 8

8.F.A.3 Interpret the equation y = mx + b as defining a linear function — Identifying $a$ as the slope and $b$ as the $y$-intercept of $y=ax+b$.
8.F.B.4 Construct a function to model a linear relationship between two quantities — Finding a slope from two points on a graph.

💡Takeaway. A wrong graph isn't useless -- ask which part the student got right, and keep exactly that part.

Variant 7 medium

Liam and Mia each tried to graph the linear function $y=ax+b$ . Liam misread the value of $a$ , and Mia misread the value of $b$ , so they drew the two lines shown in the figure. Find the original linear function. ( $a$ and $b$ are constants.)

Your answer (check it yourself)

Show solution

1 · Understandwhat's really being asked

Two students each copied $y=ax+b$ with one mistake: Liam got $a$ wrong, Mia got $b$ wrong. From their two graphs, rebuild the correct equation.

Givens

Liam's line passes through $(0, 4)$ and $(-2, 0)$ ; only his $a$ is wrong.
Mia's line passes through $(0, 2)$ and $(1, 0)$ ; only her $b$ is wrong.

Unknowns

The original equation $y=ax+b$ .

Constraints

In $y=ax+b$ , $a$ is the slope and $b$ is the $y$ -intercept, so each wrong graph still shows one correct number.

2 · Planchoose the strategy

#16 Count the Complement · also uses: #1 Draw a Diagram#13 Convert to Algebra

Change focus from what each student got wrong to what each got right. Liam read $b$ correctly, so his line's $y$ -intercept is the true $b$ . Mia read $a$ correctly, so her line's slope is the true $a$ . Read each off the figure and put them together.

3 · Execute3 carry out the plan

1Take b from Liam's line

#16 Count the Complement 8.F.A.3

Liam only misread

a

, so his

y

-intercept is right. His line crosses the

y

-axis at

(0, 4)

.

b=4

Ignore Liam's slope; keep his intercept.

2Take a from Mia's line

#1 Draw a Diagram 8.F.B.4

Mia only misread

b

, so her slope is right. Her line passes through

(0, 2)

and

(1, 0)

.

a=\frac{0-2}{1-0}=\frac{-2}{1}=-2

Ignore Mia's intercept; keep her slope.

3Put the right pieces together

#13 Convert to Algebra 8.F.A.3

Use the correct slope and the correct intercept.

y=-2\cdot x+4=-2x+4

Each wrong graph contributed exactly one true coefficient.

Answer:

y=-2x+4

4 · Reviewdoes it hold up?

Liam's line is $y=2x+4$ : same $b=4$ as the answer, only the slope differs. Mia's line is $y=-2x+2$ : same slope $-2$ as the answer, only the intercept differs. Each student made exactly one mistake, as stated.

Another way: Write both drawn equations first, $y=2x+4$ and $y=-2x+2$ , then keep the coefficient each student is known to have read correctly: the constant term of the first and the $x$ -coefficient of the second.

Standardsmin grade 8

8.F.A.3 Interpret the equation y = mx + b as defining a linear function — Identifying $a$ as the slope and $b$ as the $y$-intercept of $y=ax+b$.
8.F.B.4 Construct a function to model a linear relationship between two quantities — Finding a slope from two points on a graph.

💡Takeaway. A wrong graph isn't useless -- ask which part the student got right, and keep exactly that part.

Variant 8 medium

Liam and Mia each tried to graph the linear function $y=ax+b$ . Liam misread the value of $a$ , and Mia misread the value of $b$ , so they drew the two lines shown in the figure. Find the original linear function. ( $a$ and $b$ are constants.)

Your answer (check it yourself)

Show solution

1 · Understandwhat's really being asked

Two students each copied $y=ax+b$ with one mistake: Liam got $a$ wrong, Mia got $b$ wrong. From their two graphs, rebuild the correct equation.

Givens

Liam's line passes through $(0, -2)$ and $(2, 0)$ ; only his $a$ is wrong.
Mia's line passes through $(0, 4)$ and $(2, 0)$ ; only her $b$ is wrong.

Unknowns

The original equation $y=ax+b$ .

Constraints

In $y=ax+b$ , $a$ is the slope and $b$ is the $y$ -intercept, so each wrong graph still shows one correct number.

2 · Planchoose the strategy

#16 Count the Complement · also uses: #1 Draw a Diagram#13 Convert to Algebra

Change focus from what each student got wrong to what each got right. Liam read $b$ correctly, so his line's $y$ -intercept is the true $b$ . Mia read $a$ correctly, so her line's slope is the true $a$ . Read each off the figure and put them together.

3 · Execute3 carry out the plan

1Take b from Liam's line

#16 Count the Complement 8.F.A.3

Liam only misread

a

, so his

y

-intercept is right. His line crosses the

y

-axis at

(0, -2)

.

b=-2

Ignore Liam's slope; keep his intercept.

2Take a from Mia's line

#1 Draw a Diagram 8.F.B.4

Mia only misread

b

, so her slope is right. Her line passes through

(0, 4)

and

(2, 0)

.

a=\frac{0-4}{2-0}=\frac{-4}{2}=-2

Ignore Mia's intercept; keep her slope.

3Put the right pieces together

#13 Convert to Algebra 8.F.A.3

Use the correct slope and the correct intercept.

y=-2\cdot x-2=-2x-2

Each wrong graph contributed exactly one true coefficient.

Answer:

y=-2x-2

4 · Reviewdoes it hold up?

Liam's line is $y=x-2$ : same $b=-2$ as the answer, only the slope differs. Mia's line is $y=-2x+4$ : same slope $-2$ as the answer, only the intercept differs. Each student made exactly one mistake, as stated.

Another way: Write both drawn equations first, $y=x-2$ and $y=-2x+4$ , then keep the coefficient each student is known to have read correctly: the constant term of the first and the $x$ -coefficient of the second.

Standardsmin grade 8

8.F.A.3 Interpret the equation y = mx + b as defining a linear function — Identifying $a$ as the slope and $b$ as the $y$-intercept of $y=ax+b$.
8.F.B.4 Construct a function to model a linear relationship between two quantities — Finding a slope from two points on a graph.

💡Takeaway. A wrong graph isn't useless -- ask which part the student got right, and keep exactly that part.

Variant 9 medium

Liam and Mia each tried to graph the linear function $y=ax+b$ . Liam misread the value of $a$ , and Mia misread the value of $b$ , so they drew the two lines shown in the figure. Find the original linear function. ( $a$ and $b$ are constants.)

Your answer (check it yourself)

Show solution

1 · Understandwhat's really being asked

Two students each copied $y=ax+b$ with one mistake: Liam got $a$ wrong, Mia got $b$ wrong. From their two graphs, rebuild the correct equation.

Givens

Liam's line passes through $(0, -4)$ and $(2, 0)$ ; only his $a$ is wrong.
Mia's line passes through $(0, 1)$ and $(2, 0)$ ; only her $b$ is wrong.

Unknowns

The original equation $y=ax+b$ .

Constraints

In $y=ax+b$ , $a$ is the slope and $b$ is the $y$ -intercept, so each wrong graph still shows one correct number.

2 · Planchoose the strategy

#16 Count the Complement · also uses: #1 Draw a Diagram#13 Convert to Algebra

Change focus from what each student got wrong to what each got right. Liam read $b$ correctly, so his line's $y$ -intercept is the true $b$ . Mia read $a$ correctly, so her line's slope is the true $a$ . Read each off the figure and put them together.

3 · Execute3 carry out the plan

1Take b from Liam's line

#16 Count the Complement 8.F.A.3

Liam only misread

a

, so his

y

-intercept is right. His line crosses the

y

-axis at

(0, -4)

.

b=-4

Ignore Liam's slope; keep his intercept.

2Take a from Mia's line

#1 Draw a Diagram 8.F.B.4

Mia only misread

b

, so her slope is right. Her line passes through

(0, 1)

and

(2, 0)

.

a=\frac{0-1}{2-0}=\frac{-1}{2}=-\frac{1}{2}

Ignore Mia's intercept; keep her slope.

3Put the right pieces together

#13 Convert to Algebra 8.F.A.3

Use the correct slope and the correct intercept.

y=-\frac{1}{2}\cdot x-4=-\frac{1}{2}x-4

Each wrong graph contributed exactly one true coefficient.

Answer:

y=-\frac{1}{2}x-4

4 · Reviewdoes it hold up?

Liam's line is $y=2x-4$ : same $b=-4$ as the answer, only the slope differs. Mia's line is $y=-\frac{1}{2}x+1$ : same slope $-\frac{1}{2}$ as the answer, only the intercept differs. Each student made exactly one mistake, as stated.

Another way: Write both drawn equations first, $y=2x-4$ and $y=-\frac{1}{2}x+1$ , then keep the coefficient each student is known to have read correctly: the constant term of the first and the $x$ -coefficient of the second.

Standardsmin grade 8

8.F.A.3 Interpret the equation y = mx + b as defining a linear function — Identifying $a$ as the slope and $b$ as the $y$-intercept of $y=ax+b$.
8.F.B.4 Construct a function to model a linear relationship between two quantities — Finding a slope from two points on a graph.

💡Takeaway. A wrong graph isn't useless -- ask which part the student got right, and keep exactly that part.

Variant 10 hard

Liam and Mia each tried to graph the linear function $y=ax+b$ . Liam misread the value of $a$ , and Mia misread the value of $b$ , so they drew the two lines shown in the figure. Find the original linear function. ( $a$ and $b$ are constants.)

Your answer (check it yourself)

Show solution

1 · Understandwhat's really being asked

Two students each copied $y=ax+b$ with one mistake: Liam got $a$ wrong, Mia got $b$ wrong. From their two graphs, rebuild the correct equation.

Givens

Liam's line passes through $(0, 5)$ and $(5, 0)$ ; only his $a$ is wrong.
Mia's line passes through $(0, 2)$ and $(-2, 0)$ ; only her $b$ is wrong.

Unknowns

The original equation $y=ax+b$ .

Constraints

In $y=ax+b$ , $a$ is the slope and $b$ is the $y$ -intercept, so each wrong graph still shows one correct number.

2 · Planchoose the strategy

#16 Count the Complement · also uses: #1 Draw a Diagram#13 Convert to Algebra

Change focus from what each student got wrong to what each got right. Liam read $b$ correctly, so his line's $y$ -intercept is the true $b$ . Mia read $a$ correctly, so her line's slope is the true $a$ . Read each off the figure and put them together.

3 · Execute3 carry out the plan

1Take b from Liam's line

#16 Count the Complement 8.F.A.3

Liam only misread

a

, so his

y

-intercept is right. His line crosses the

y

-axis at

(0, 5)

.

b=5

Ignore Liam's slope; keep his intercept.

2Take a from Mia's line

#1 Draw a Diagram 8.F.B.4

Mia only misread

b

, so her slope is right. Her line passes through

(0, 2)

and

(-2, 0)

.

a=\frac{0-2}{-2-0}=\frac{-2}{-2}=1

Ignore Mia's intercept; keep her slope.

3Put the right pieces together

#13 Convert to Algebra 8.F.A.3

Use the correct slope and the correct intercept.

y=1\cdot x+5=x+5

Each wrong graph contributed exactly one true coefficient.

Answer:

y=x+5

4 · Reviewdoes it hold up?

Liam's line is $y=-x+5$ : same $b=5$ as the answer, only the slope differs. Mia's line is $y=x+2$ : same slope $1$ as the answer, only the intercept differs. Each student made exactly one mistake, as stated.

Another way: Write both drawn equations first, $y=-x+5$ and $y=x+2$ , then keep the coefficient each student is known to have read correctly: the constant term of the first and the $x$ -coefficient of the second.

Standardsmin grade 8

8.F.A.3 Interpret the equation y = mx + b as defining a linear function — Identifying $a$ as the slope and $b$ as the $y$-intercept of $y=ax+b$.
8.F.B.4 Construct a function to model a linear relationship between two quantities — Finding a slope from two points on a graph.

💡Takeaway. A wrong graph isn't useless -- ask which part the student got right, and keep exactly that part.

Variant 11 hard

Liam and Mia each tried to graph the linear function $y=ax+b$ . Liam misread the value of $a$ , and Mia misread the value of $b$ , so they drew the two lines shown in the figure. Find the original linear function. ( $a$ and $b$ are constants.)

Your answer (check it yourself)

Show solution

1 · Understandwhat's really being asked

Two students each copied $y=ax+b$ with one mistake: Liam got $a$ wrong, Mia got $b$ wrong. From their two graphs, rebuild the correct equation.

Givens

Liam's line passes through $(0, 2)$ and $(4, 0)$ ; only his $a$ is wrong.
Mia's line passes through $(0, -6)$ and $(3, 0)$ ; only her $b$ is wrong.

Unknowns

The original equation $y=ax+b$ .

Constraints

In $y=ax+b$ , $a$ is the slope and $b$ is the $y$ -intercept, so each wrong graph still shows one correct number.

2 · Planchoose the strategy

#16 Count the Complement · also uses: #1 Draw a Diagram#13 Convert to Algebra

Change focus from what each student got wrong to what each got right. Liam read $b$ correctly, so his line's $y$ -intercept is the true $b$ . Mia read $a$ correctly, so her line's slope is the true $a$ . Read each off the figure and put them together.

3 · Execute3 carry out the plan

1Take b from Liam's line

#16 Count the Complement 8.F.A.3

Liam only misread

a

, so his

y

-intercept is right. His line crosses the

y

-axis at

(0, 2)

.

b=2

Ignore Liam's slope; keep his intercept.

2Take a from Mia's line

#1 Draw a Diagram 8.F.B.4

Mia only misread

b

, so her slope is right. Her line passes through

(0, -6)

and

(3, 0)

.

a=\frac{0-(-6)}{3-0}=\frac{6}{3}=2

Ignore Mia's intercept; keep her slope.

3Put the right pieces together

#13 Convert to Algebra 8.F.A.3

Use the correct slope and the correct intercept.

y=2\cdot x+2=2x+2

Each wrong graph contributed exactly one true coefficient.

Answer:

y=2x+2

4 · Reviewdoes it hold up?

Liam's line is $y=-\frac{1}{2}x+2$ : same $b=2$ as the answer, only the slope differs. Mia's line is $y=2x-6$ : same slope $2$ as the answer, only the intercept differs. Each student made exactly one mistake, as stated.

Another way: Write both drawn equations first, $y=-\frac{1}{2}x+2$ and $y=2x-6$ , then keep the coefficient each student is known to have read correctly: the constant term of the first and the $x$ -coefficient of the second.

Standardsmin grade 8

8.F.A.3 Interpret the equation y = mx + b as defining a linear function — Identifying $a$ as the slope and $b$ as the $y$-intercept of $y=ax+b$.
8.F.B.4 Construct a function to model a linear relationship between two quantities — Finding a slope from two points on a graph.

💡Takeaway. A wrong graph isn't useless -- ask which part the student got right, and keep exactly that part.

Variant 12 hard

Liam and Mia each tried to graph the linear function $y=ax+b$ . Liam misread the value of $a$ , and Mia misread the value of $b$ , so they drew the two lines shown in the figure. Find the original linear function. ( $a$ and $b$ are constants.)

Your answer (check it yourself)

Show solution

1 · Understandwhat's really being asked

Two students each copied $y=ax+b$ with one mistake: Liam got $a$ wrong, Mia got $b$ wrong. From their two graphs, rebuild the correct equation.

Givens

Liam's line passes through $(0, -2)$ and $(-4, 0)$ ; only his $a$ is wrong.
Mia's line passes through $(0, 6)$ and $(-2, 0)$ ; only her $b$ is wrong.

Unknowns

The original equation $y=ax+b$ .

Constraints

In $y=ax+b$ , $a$ is the slope and $b$ is the $y$ -intercept, so each wrong graph still shows one correct number.

2 · Planchoose the strategy

#16 Count the Complement · also uses: #1 Draw a Diagram#13 Convert to Algebra

Change focus from what each student got wrong to what each got right. Liam read $b$ correctly, so his line's $y$ -intercept is the true $b$ . Mia read $a$ correctly, so her line's slope is the true $a$ . Read each off the figure and put them together.

3 · Execute3 carry out the plan

1Take b from Liam's line

#16 Count the Complement 8.F.A.3

Liam only misread

a

, so his

y

-intercept is right. His line crosses the

y

-axis at

(0, -2)

.

b=-2

Ignore Liam's slope; keep his intercept.

2Take a from Mia's line

#1 Draw a Diagram 8.F.B.4

Mia only misread

b

, so her slope is right. Her line passes through

(0, 6)

and

(-2, 0)

.

a=\frac{0-6}{-2-0}=\frac{-6}{-2}=3

Ignore Mia's intercept; keep her slope.

3Put the right pieces together

#13 Convert to Algebra 8.F.A.3

Use the correct slope and the correct intercept.

y=3\cdot x-2=3x-2

Each wrong graph contributed exactly one true coefficient.

Answer:

y=3x-2

4 · Reviewdoes it hold up?

Liam's line is $y=-\frac{1}{2}x-2$ : same $b=-2$ as the answer, only the slope differs. Mia's line is $y=3x+6$ : same slope $3$ as the answer, only the intercept differs. Each student made exactly one mistake, as stated.

Another way: Write both drawn equations first, $y=-\frac{1}{2}x-2$ and $y=3x+6$ , then keep the coefficient each student is known to have read correctly: the constant term of the first and the $x$ -coefficient of the second.

Standardsmin grade 8

8.F.A.3 Interpret the equation y = mx + b as defining a linear function — Identifying $a$ as the slope and $b$ as the $y$-intercept of $y=ax+b$.
8.F.B.4 Construct a function to model a linear relationship between two quantities — Finding a slope from two points on a graph.

💡Takeaway. A wrong graph isn't useless -- ask which part the student got right, and keep exactly that part.

Each wrong graph still keeps one right coefficient · 12 practice problems

Generated variants — 12

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