Comparing Functions Across Representations
I can compare two functions' rates of change and initial values when they are shown in different forms.
What students learn
Students learn to find the rate of change and the initial value of a function given as a rule, a table, a graph or words, and compare two functions feature by feature.
See inside
The key rule
Find the same feature in each function, then compare: a rate needs two points; an initial value is the output at x = 0.
rate = (change in y) ÷ (change in x)
Changes faster means the bigger size of the change in y for each 1 in x, whichever way it goes.
Watch out
A: y = 4x − 1. B has the pairs (0, 1), (1, 4), (2, 7). Which has the greater rate of change?
The mistake: B's rate is 4, from (1, 4): the same as A4 is one y-value, not a change.
The fix: B: (7 − 1) ÷ (2 − 0) = 3; 4 > 3, so AA rate needs two points.
The mistakes it targets
- Reads one y-value as the rate of change
- Swaps the rate of change and the initial value
- Compares initial values when the question asks about rates
- Forgets to divide the change in y by the change in x
- Reads a minus sign in the rule as a decreasing function
The interactive Teacher Deck (32 slides, PowerPoint)
- The Learn page taught on screen: 8 fill-in blanks, each filled on its own click as students write it
- The worked example one slide per step (four steps), each with the reason for the step
- Six practice and three apply problems: a your-turn slide with a think-time bar, then the solution one line per click
- Jump buttons on every slide: go straight to any problem, the answer bank or the exit ticket
- Speaker notes on each problem: the question to ask, the worked answer and the common error to listen for
- A completed deck PDF (19 pages) with every slide fully built, to print or post
The student packet (9 pages)
- Learn: guided notes with four definitions, the key rule, a worked example and a Watch Out error pair
- Practice: two warm-up, two standard and two stretch problems, with an answer bank for self-checking that hides three common-mistake decoys
- Apply: one word problem, a reasoning prompt and one error analysis
- Assess: an exit ticket of four items, two per page, with an I-can self-rating
- Homework (page 5): eight new problems, two warm-up, four standard and two stretch
- Answer key (pages 6–7): every problem worked step by step, five common mistakes with what to do about each, and a grading guide, plus the homework key (page 8)
- Color and ink-saver editions of the whole packet
Standard codes refer to the Common Core State Standards for Mathematics. Math Class 678 is an independent publisher, not affiliated with, sponsored by, or endorsed by NGA Center or CCSSO. Every product is sold on Teachers Pay Teachers.