Eighth Grade · Math · week 12

Compare functions across forms

Compare the rate of change and initial value of two linear functions given in different forms (equation, table, or words).

Functions can arrive as an equation, a table, or a story -- but the same two questions always work: what is the rate of change, and what is the initial value? Pull those two numbers out of each representation first, then compare.

Same rate but different initial values means two parallel lines that never cross. Same rate AND same initial value means it is really the same function, just dressed differently.

📋 For the teacher
Teach it (10–15 min):
  • No matter the form, the two numbers worth pulling out first are always the same: rate of change and initial value.
  • Line the two functions up side by side (a simple two-column comparison) before answering "which is bigger" -- comparing from memory invites mistakes.
  • Same rate + different initial value = parallel lines, never crossing -- this previews systems of equations in a few weeks.
  • Same rate AND same initial value means the two representations describe the exact same function.
Talk about it:
  • What two numbers do you pull out of ANY function, no matter which form it is given in?
  • If two functions have the same rate but different initial values, will their graphs ever cross? Why?
  • How would you prove a table and an equation describe the exact same function?

Mastery looks like: Given two functions in different forms, they extract rate and initial value from each and correctly compare them.

Common stumbles: Assuming a table's first listed row is automatically input 0; comparing raw numbers without first identifying which is rate and which is initial value.

Homework for this lessonType it, write it with a stylus, or print it for pencil & paper.open →
Records attendance + 15 min of Math for this learner.
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