Learn · Illustrative Mathematics · Grade 6 · Unit 6
Short lessons you can play with. Match equations to tape diagrams, keep a hanger balanced to solve equations, write expressions where letters stand for numbers, split rectangles to see the distributive property, build squares and cubes for exponents, and connect tables, equations, and graphs, then check yourself. When you’re ready, practice in Balance Lab.
Nine chapters follow the unit’s lessons in order. Each step shows one idea with something to change, then a quick check. A wrong answer names the likely mistake and gives one more try. Fraction answers can be typed as 7/3, 2 1/3, or a close decimal.
Tape diagrams, hanger diagrams, area diagrams (rectangles split into parts), squares and cubes, tables, and graphs. The pictures come first, and the equation beside each one changes with it, so the symbols always have a picture to fall back on.
Many students read = as “now write the answer,” which makes equations like 12 = x + 5 or 8 + 4 = ? + 5 confusing. Research links seeing = as a balance between two equal amounts to doing much better at solving equations. So equations here are written both ways, and the hanger tips whenever the two sides stop being equal.
A letter is always defined as a number (“n is the number of markers in one pack,” “c is the cost in dollars”), never as a thing (“n is markers”). Treating letters as labels is a well-known stumbling block in early algebra.
Steps say “subtract 3 from each side,” not “move it over” or “cancel,” so the reason stays visible. Every solution is checked by putting it back into the equation. A move that only changes one side tips the hanger.
Quick checks name common misconceptions: 2³ as 6, 3x with x = 4 read as 34, 3(x + 4) as 3x + 4, 3 · 4² worked as (3 · 4)², 3x = 7 solved as 3/7, “7 less than t” as 7 − t, two expressions called equivalent because they agree at one value, and a table rule that fits only the first row. Some steps show a solved problem with a mistake to find. Studying solved problems, including wrong ones, is one of the main recommendations in the U.S. Department of Education’s practice guide for teaching algebra.
Several steps show two strategies at once: reasoning from a tape versus undoing on a hanger, multiplying 6 · 45 in one block or in parts, and guessing and checking versus working backward for an exponent. Comparing strategies helps students choose one on purpose. Order of operations is explained with reasons rather than a memorized word like PEMDAS, which leads many students to multiply before dividing.
Each chapter ends with a link to the matching station in Balance Lab, the practice game for this unit. Progress is saved on this device.
The unit’s partner activities (info gaps and card sorts) and the open-ended parts of Lesson 19 work better in class. Negative numbers come in Unit 7, so every answer here is positive.