Cross-Multiplying Without Quantities
A seventh-grade teacher has 28 students in a unit on proportional relationships. On the quiz, 21 of 28 students correctly solve the equation 3/4 = x/6 for x. But when asked, A recipe uses 3 cups of flour for every 4 cups of milk. How much flour is needed for 6 cups of milk? only 9 students answer correctly; 14 answer 4.5 or 8, and several write 3/6 = x/4 before cross-multiplying. When the teacher interviews five students, all five say the method is cross-multiply and two cannot explain what the 3 and the 4 represent in the problem. The department's next unit is on scale drawings, which assumes students can set up and interpret proportions from context.
1. Diagnose what the quiz results reveal about student understanding, citing specific evidence.
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The gap between 21 of 28 on the equation and 9 of 28 on the word problem is the central finding: students own a procedure without owning the quantities.
The evidence points to three related problems:
- No quantity reasoning. Two of five interviewed students cannot say what 3 and 4 represent. If the numbers are not attached to flour and milk, any arrangement of them looks acceptable-which explains writing 3/6 = x/4.
- Treating the equation as a template to fill in order of appearance, not as a statement of equal ratios. That produces 4.5 (halving the wrong quantity or scaling incorrectly) and 8.
- No multiplicative comparison. Students are not asking whether the answer should be larger or smaller than 3, which a rough estimate would catch.
Crucially, this is not a cross-multiplication error. The students who cross-multiplied a correctly set-up equation did fine. The failure occurs upstream, at representing the situation. That distinction determines what to teach next.
2. Design a two-day instructional sequence to address the diagnosed problem, and explain why each element is placed where it is.
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Day 1 - Build the quantities before the symbols.
- Sort and notice (10 min): pairs of ratio statements, some equivalent and some additive (3 to 4 vs. 3 more than 4). Students justify with diagrams.
- Ratio tables and double number lines (25 min): the flour-and-milk problem drawn as a table, scaled by 1.5. Students see that both quantities scale by the same factor.
- Error analysis (10 min): present 3/6 = x/4 and 4.5 as student work; students identify what each student assumed about the quantities.
Day 2 - Connect representations to the equation.
- Tape diagrams to equations (15 min): students write the proportion from a diagram, then check it against a table.
- Estimation first (10 min): before solving, predict whether the answer is greater or less than 3 and why.
- Return to the algorithm (20 min): cross-multiplying as one efficient strategy among several, with a check against the table.
Why this order: the procedure is placed last because introducing it first is what allowed students to succeed at equations while failing at situations. Keeping the table and diagram available means the algorithm is anchored to meaning.
3. The teacher wants a short assessment that distinguishes procedural skill from understanding. What should it contain, and how should results be used before starting the scale-drawing unit?
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The assessment should separate the two constructs so the teacher can tell whether the gap has closed.
Part A - Procedure (2 items): solve 3/4 = x/6 and one similar equation. These should remain easy; a drop here would signal that the conceptual work displaced fluency.
Part B - Setup from context (2 items): word problems that do not state the proportion, including one where the given numbers appear in an order that invites a wrong setup.
Part C - Justify (1–2 items): Is 4.5 a reasonable answer to the flour problem? Explain using the table or diagram. Also give a table and ask for the equation it represents.
Decision rule before scale drawings:
- If Part A is strong but Part B still fails, continue context-to-equation work and do not begin scale drawings on schedule; scale drawings will reproduce the same failure at a harder level.
- If Part C shows students can reject an unreasonable answer but Part B is shaky, a one-day bridge focused on setup is likely enough.
- If both parts improve, proceed but include setup tasks in the first scale-drawing lesson as a check.