Choosing a Method and Special Systems
I can decide which method fits a system, and tell no solution from infinitely many.
The problem this sheet solves
Students who reach 0 = 0 conclude that anything is a solution and will happily offer (0, 0) for a line that does not pass through it. The other half of this sheet is method choice: given three techniques and no instruction, students default to whichever was taught last rather than reading the system. This sheet gives a short decision rule for choosing a method, so the choice is made from the system's shape rather than from habit, and it treats the special cases as answers with precise meanings. Students are asked to name pairs that DO and do not satisfy an infinite-solution system, which is what exposes the everything-works misreading.
The key rule
Variable alone: substitute. Coefficients equal or opposite: eliminate. Graph only on grid lines. Both vanish: FALSE is none, TRUE is one line.
0 = 4 is no solution. 0 = 0 is one line, not anything.
Watch out
The mistake: A system gives 0 = 0, and a student writes: infinitely many solutions, so every ordered pair works.
The fix: Every pair ON THAT LINE works. (0, 1) and (4, 2) are solutions; (0, 0) is not, because 0 is not 1/4(0) + 1. Infinitely many, and still almost none of the plane.
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.