Rate of Change and Slope from a Graph
I can find a line's rate of change by counting rise over run between two grid corners.
The problem this sheet solves
Inversion is the error here, and it survives because the two numbers are both correct. A student who counts the triangle properly and then puts the larger value on top gets a slope of 2 for a gentle line, and nothing about the arithmetic looks wrong. Counting from points that are not on exact grid corners is the companion problem, and it makes every answer approximate. This sheet insists on two points at exact grid corners, then keeps rise on top every time, so the fraction is built the same way in every problem. The slope triangle is drawn on the graph rather than described, and the sheet asks students to sanity check steepness against the number they wrote, which catches inversion immediately.
The key rule
Pick two points where the line crosses exact grid corners, count the rise and the run between them, then write slope = rise over run — rise is always on top.
Rise on top. Run underneath. Never the other way.
Watch out
The mistake: A student counts rise 1 and run 2, then writes slope = 2/1 because 2 is the bigger number.
The fix: Rise goes on top: slope = 1/2. A slope of 2 would call this gentle line steep.
What is included
- Reference page — student-friendly definitions, a worked example, a Watch Out error pair, and the key rule
- Practice page — six problems, building from foundational to multi-step
- Apply page — a word problem, a reasoning prompt, and an error-analysis task
- Assess — a half-page exit ticket, two to a sheet, with a score box
- Full answer key — every item worked, plus a grading guide that says what each wrong answer tells you
- Ink-saver edition of all 7 pages
- 14-slide teacher deck (PowerPoint) with presenter notes on every slide
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.