L3 · One Constant, Many Names: Unit Rate, Scale Factor, Slope

Learning goal

“I can recognize the constant of proportionality in its three working costumes - unit rate (per one unit), scale factor (drawing compared to actual), and slope (rise over run on a graph through the origin) - and translate a problem into whichever name it is wearing.”

Unit rate — k stated with a denominator of ONE unit: $4.50 per 1 kg, 15 L per 1 min. "Per" is the sound of a unit rate.

Scale factor — the k of a drawing or model: scale factor = drawing length ÷ actual length (same units). A scale factor of 2.5 enlarges; 1/30 shrinks.

Slope — on a graph, the steepness number: slope = rise ÷ run between two points. On a proportional graph (a line through the origin), slope is exactly k.

Worked example

Worked example 1 (unit rate). 500 g of nuts cost $7.50. Price p is proportional to mass m: k = p/m = 7.50/500 = $0.015 per gram. Stated per kilogram instead: $15.00 per kg. A UNIT RATE is k with the second quantity normalized to 1 - the "per" number a shopper actually compares. Worked example 2 (scale factor). A model car's length is 8 cm; the real car is 320 cm. Scale factor = drawing ÷ actual = 8/320 = 1/40 = 0.025. Read it as a k statement: model length = 0.025 × actual length. Equivalently the real car is 40 × the model (320/8 = 40) - the reciprocal direction, as always.

Check yourself — explain it

Why are a unit rate and a scale factor both 'k'?

Paired problem — example still visible

A photo is enlarged: original width 10 cm becomes a print width of 25 cm. Find the scale factor (print ÷ original). -> 25/10 = 2.5 - an enlargement, since k > 1.

Faded completion — last step is yours

"Unit rate: 240 words typed in 4 min → ___ words per min, and that number is the ___ of the relationship." -> 60; constant of proportionality (k)

Ready to practise?

10 scored questions with a 3-rung hint ladder. Mastery needs 8/10 plus both gate questions.

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