Scale Drawing Errors: Three Students, One Ratio
Ms. Okafor's Grade 7 class worked a scale-drawing problem: a map uses the scale 1.5 cm : 8 km, and two towns sit 12.75 cm apart on the map. She collected four responses. Aiden set up 1.5/8 = 12.75/x and got 68 km. Bianca multiplied 12.75 by 8 and got 102 km. Chen divided 12.75 by 8, then multiplied by 1.5, and got about 2.4 km. Dara subtracted 12.75 minus 1.5 to get 11.25, then added 8, and got 19.25 km. On last week's exit ticket, Bianca correctly priced 7 pencils at 3 for $1.20, and Chen correctly halved a recipe. Dara's work on unit rates with whole numbers has been accurate all term. Ms. Okafor has 25 minutes of small-group time tomorrow and can pull three students. She wants to know which errors are worth reteaching to the whole class and which are individual slips.
1. Classify each incorrect response as a conceptual gap in multiplicative reasoning, a setup error, or a slip. Justify each classification using the specific numbers in the student's work.
Read the worked answer
Bianca, 102 km. Her answer equals 12.75 x 8, which is exactly what a scale of 1 cm : 8 km would produce. She used the 8 and dropped the 1.5, so she applied an incomplete ratio rather than an inverted or additive one. Her correct pencil pricing shows she can build a unit rate when one term is 1. Likely a setup error on a scale where neither term is 1.
Chen, about 2.4 km. He computed 12.75 x (1.5 / 8), the reciprocal of the correct factor (8 / 1.5). Dividing by the real-world quantity when converting map to ground is a structural error about which number is 'per' which. Halving a recipe does not test that, because a factor of one half has an obvious direction.
Dara, 19.25 km. She treated the relationship as a difference: 11.25 cm of extra map distance plus 8 km. That is additive reasoning applied to a multiplicative comparison, the deepest gap of the three. Whole-number unit rates can mask it because the numbers often cooperate.
Aiden set up the proportion correctly and should be used publicly as a model, not just praised privately.
2. Design two short items that would separate a conceptual gap from a setup slip before committing tomorrow's small group. Explain what each item reveals.
Read the worked answer
Item 1, double number line with a non-unit first term. Scale: 4 cm : 10 km. A trail is 10 cm on the map. Ask for the real distance and require a diagram. Correct answer 25 km. Bianca-style students who ignore the first term will multiply 10 by 10 and get 100; Dara-style students will add 6 and get 16. One item surfaces both patterns with friendly numbers.
Item 2, prediction without computing. Scale: 3 cm : 20 km. Two towns are 9 cm apart, so is the real distance more or less than 60 km, and how do you know? A student who reasons multiplicatively sees 9 is three times 3, so the distance triples to 60 exactly. Additive thinkers cannot answer without computing and usually guess. This item also distinguishes reasoning from answer-getting.
Score the items by the written explanation, not the number. A student who writes 'the map distance is three times 3, so the real distance is three times 20' has the concept and only needs a careful setup routine.
3. Recommend how to use tomorrow's 25 minutes: who is in the small group, what the whole class does, and what Bianca gets.
Read the worked answer
Whole-class warm-up, 5 minutes. One scale where neither term is 1, done on a double number line, with two students explaining why multiplying by the second term alone fails. This surfaces the Bianca error for anyone who made it silently and gives Aiden a legitimate audience.
Small group, 20 minutes: Dara and Chen. Both need the same core idea, that a scale compares two quantities and you scale both by the same factor. Start with a double number line, then a ratio table, then return to 12.75 cm and let them find the error themselves. Do not reteach cross-multiplication yet; it lets students skip the reasoning that is missing.
Bianca, 3-minute check at the teacher table. Give her two items with a non-unit first term and ask her to say the scale aloud before computing. If both are correct, she needs a routine, not a group: read the scale, write both terms as a ratio, label units, then solve. Monitor her next three assignments rather than pulling her from instruction.
Reserve the last 2 minutes for all three to state the difference between adding and scaling.