Cogito
Integrated Math 1 · Chapter 2 · Lesson 1
Slope and Rate of Change
How fast one thing changes with another.
12 problems · about 21 minutes · F-IF.B.6, S-ID.C.7
What this lesson teaches
The student computes slope from points, graphs and tables, and interprets it in context.
- Slope is rise over run, or (y₂ − y₁) ÷ (x₂ − x₁).
- Its sign gives the direction; zero is horizontal and vertical is undefined.
- A relationship is linear exactly when the rate of change is constant.
Warm Up
Straightforward practice. Get the method working first.
5 problemsThrough (2, 4) and (6, 16). What is the slope?
Answer 3
Why 3.
What makes a relationship linear?
Answer The rate of change never varies.
Why A constant rate of change.
Through (0, 0) and (2, 6). Slope?
Answer 3
Why 6 ÷ 2.
Through (1, 5) and (3, 5). Slope?
Answer 0
Why No change in y.
Through (2, 8) and (5, 2). Slope?
Answer -2
Why −6 ÷ 3.
Build It Up
The same ideas with more to keep track of.
3 problemsThrough (−1, 2) and (3, 10). Slope?
Answer 2
Why 8 ÷ 4.
A horizontal line. What is its slope?
Answer 0
Why y never changes.
Through (4, 1) and (4, 9). Is the slope defined? 1 yes, 0 no.
Answer 0
Why The run is zero.
Stretch Yourself
Mixed problems. Work out what kind of question it is before you start.
4 problemsSort each slope by the direction of the line.
Answer Rises left to right: Slope 4, Slope 0.5 · Falls left to right: Slope −3, Slope −0.2
Why The sign alone decides the direction.
Through (0, 7) and (4, 3). Slope?
Answer -1
Why −4 ÷ 4.
The Two Points: Through (1, 3) and (4, 12). What is the slope?
Answer 3
Why 3.
The Taxi: $8 for 2 miles and $20 for 8 miles. Cost per mile, in dollars?
Answer 2 dollars
Why $2 per mile.