Linear functions and slope
Function rules, constant rate of change, slope, y-intercept and comparing representations
About four lessons of 45 to 60 minutes
A line is a story with a steady rate
A taxi starts with a $4 fee and adds $2 per mile. A plant is 5 cm tall and grows 3 cm each week. A streaming service charges a flat monthly fee plus a cost per rental. These situations are linear because the change is steady.
A linear function connects a rule, a table, a graph and a story. The slope tells the repeated change, and the y-intercept tells the starting value. Once students can move between those representations, linear graphs stop being pictures and become readable models.
- Taxi fare$4 plus $2 per mile is y = 2x + 4
- Plant growthsteady growth makes a straight-line pattern
- Phone planstarting fee plus cost per unit
- Graph of a lineslope is the rate of change
What students will be able to do
Students will identify functions, distinguish linear from nonlinear relationships, find slope and y-intercept from tables and equations, construct linear function rules, and compare functions across tables, equations, graphs and stories.
- I can tell whether a relationship is a function.
- I can identify a constant rate of change in a table.
- I can interpret slope as rise over run and as a unit rate.
- I can write a linear function in the form y = mx + b.
- I can compare two functions shown in different forms.
Standards this unit teaches
- 8.EE.B.5Common Core (US)Graph proportional relationships
Graph proportional relationships and interpret the unit rate as the slope of the graph.
- 8.EE.B.6Common Core (US)Slope and the equation of a line
Use similar triangles to explain why slope is constant and derive y = mx and y = mx + b.
- 8.F.A.1Common Core (US)Understand functions
Understand that a function assigns exactly one output to each input.
- 8.F.A.2Common Core (US)Compare functions
Compare properties of functions represented in different ways, such as equations, tables and graphs.
- 8.F.A.3Common Core (US)Linear versus nonlinear
Recognize that y = mx + b defines a linear function and identify functions that are not linear.
- 8.F.B.4Common Core (US)Construct linear functions
Construct a linear function to model a relationship and interpret its rate of change and initial value.
- 8.F.B.5Common Core (US)Describe functional relationships
Describe how a function behaves from its graph, including where it increases, decreases or stays constant.
Prior knowledge
This unit builds on skills students should already have met. Revisit any that are shaky first.
Words to teach and display
- Function
- a rule that gives exactly one output for each input
- Input
- the value that goes into a function, often x
- Output
- the value a function gives back, often y
- Slope
- the rate of change of a line, found as rise over run
- Y-intercept
- the value of y when x is 0, where the line crosses the y-axis
- Linear function
- a function with a constant rate of change, whose graph is a straight line
Teach it: concrete, pictorial, abstract
The lesson moves from things students can hold, to pictures and diagrams, to the written maths. The diagrams below are drawn from data, so they are accurate and print cleanly. Teach straight from them.
1. What makes a relationship a function?
ConcreteA function is a dependable rule: each input has exactly one output. The same input cannot point to two different outputs.
Is this table a function: x values 1, 2, 2, 3 with y values 4, 5, 8, 9?
- Check repeated inputs.
- The input 2 appears twice.
- It has two different outputs, 5 and 8.
Answer: No. It is not a function because input 2 has two outputs.
- Can two different inputs have the same output?
- What is forbidden in a function table?
2. Constant rate of change
PictorialA linear table changes by the same amount each time x increases by 1. That repeated change is the rate of change, and on a graph it becomes the slope.
A table has x: 0, 1, 2, 3 and y: 4, 6, 8, 10. Find the rate of change.
- As x increases by 1, y goes 4, 6, 8, 10.
- Each y-value increases by 2.
- So the rate of change is 2 per 1.
Answer: The rate of change is 2.
- What would make a table nonlinear?
- Why must you compare equal x-steps?
3. Slope and y-intercept
AbstractIn y = mx + b, m is the slope and b is the y-intercept. The slope is the repeated change; the intercept is the starting value when x is 0.
Write a rule for a line with slope 3 and y-intercept 5.
- Start with y = mx + b.
- Replace m with 3.
- Replace b with 5.
Answer: y = 3x + 5.
- In y = 4x - 2, what is the slope?
- What does the y-intercept mean in a real story?
4. Comparing functions across representations
AbstractTwo functions can be compared even when one is a table and one is an equation. Find the same features in both: slope, starting value and whether the relationship is linear.
Function A is y = 2x + 1. Function B has values (0, 5), (1, 8), (2, 11). Which has the greater rate of change?
- Function A's slope is 2.
- Function B's y-values increase by 3 each time x increases by 1, so its slope is 3.
- Compare 3 and 2.
Answer: Function B has the greater rate of change.
- Which feature compares steepness?
- Which feature compares starting value?
Common misconceptions and how to address them
MisconceptionAny table with x and y values is automatically a function.
Why it happens: Students see an input-output layout and assume the rule is valid.
How to address it: Check repeated inputs. If one input has two different outputs, the relationship is not a function.
MisconceptionThe y-intercept is the last value in the table.
Why it happens: Students look for a visible endpoint instead of x = 0.
How to address it: The y-intercept is always the y-value when x is 0. If x = 0 is not shown, work backward using the slope.
MisconceptionA graph that goes upward is always linear.
Why it happens: Increasing and linear get confused.
How to address it: Linear means constant rate of change and a straight graph. A curved graph can increase but still be nonlinear.
Guided practice (with answers)
1. Is x: 1, 2, 3 and y: 5, 5, 5 a function?
Answer: Yes. Each input has exactly one output.
2. Find the slope from points (0, 2) and (3, 11).
Answer: 3, because rise/run = (11 - 2)/(3 - 0) = 9/3 = 3.
3. In y = -2x + 7, name the slope and y-intercept.
Answer: Slope = -2, y-intercept = 7.
4. A table has y-values 10, 13, 16, 19 as x increases by 1. Is it linear?
Answer: Yes, because the rate of change is constantly +3.
5. Write a rule for slope 4 and y-intercept -1.
Answer: y = 4x - 1.
Independent practice worksheets
Practise function tables, linear equations and coordinate reasoning with live worksheet sets.
Differentiation
- Start with tables where x increases by 1 before using wider gaps.
- Use color to mark equal x-steps and matching y-changes.
- Keep the template y = mx + b visible and label m as slope, b as start.
- Compare two functions when neither is given as an equation.
- Use negative slopes and interpret them as decreasing relationships.
- Ask students to create a table, equation and story for the same line.
Assessment: exit ticket
A short exit ticket covering function definition, slope and a linear rule.
1. Why is x: 1, 1 with y: 2, 3 not a function?
Answer: The input 1 has two different outputs.
2. Find the slope from (0, 4) and (2, 10).
Answer: 3, because (10 - 4)/(2 - 0) = 6/2 = 3.
3. Write the equation with slope 5 and y-intercept 2.
Answer: y = 5x + 2.
Teacher notes and timings
- Rough timing: Lesson 1 functions, Lesson 2 constant rate and slope, Lesson 3 y = mx + b, Lesson 4 comparing representations and exit ticket.
- This single unit intentionally covers the connected Grade 8 Common Core function cluster: 8.EE.B.5, 8.EE.B.6, 8.F.A.1-3 and 8.F.B.4-5.