Cogito
Integrated Math 1 · Chapter 2 · Lesson 3
Modelling with Linear Functions
Turning a situation into an equation.
12 problems · about 21 minutes · F-LE.A.1, F-LE.B.5
What this lesson teaches
The student builds linear models from context and interprets slope and intercept in that context.
- A linear model is rate times units plus a fixed amount.
- The slope is the per-unit rate; the intercept is the starting value.
- Context restricts which inputs make sense.
Warm Up
Straightforward practice. Get the method working first.
5 problems$40 fee plus $6 an hour. Cost for 5 hours, in dollars?
Answer 70
Why $70.
In a cost model, what does the y-intercept represent?
Answer The fixed amount charged before any units are used.
Why The starting or fixed amount.
C = 5x + 12. What is the fixed amount?
Answer 12
Why The constant.
C = 5x + 12. What is the rate per unit?
Answer 5
Why The coefficient.
C = 5x + 12 at x = 8. What is C?
Answer 52
Why 40 + 12.
Build It Up
The same ideas with more to keep track of.
3 problems$25 joining fee plus $10 a month. Cost after 4 months, in dollars?
Answer 65
Why 25 + 40.
A tank holds 100 L and drains 4 L a minute. Litres after 10 minutes?
Answer 60
Why 100 − 40.
For a model counting tickets sold, is x = 2.5 in the domain? 1 yes, 0 no.
Answer 0
Why Tickets come whole.
Stretch Yourself
Mixed problems. Work out what kind of question it is before you start.
4 problemsPut the modelling steps in order.
Answer 1. Identify the fixed starting amount. 2. Identify the rate per unit. 3. Write the equation combining both. 4. State which input values make sense.
Why The domain is stated once the model exists.
C = 2x + 40 at x = 15. What is C?
Answer 70
Why 30 + 40.
The Plan: C = 0.10m + 20. What is the cost for 300 minutes, in dollars?
Answer 50 dollars
Why $50.
The Fee: A gym charges $30 to join plus $15 a month. Cost after 6 months, in dollars?
Answer 120 dollars
Why $120.