How can proportion be explained with a double number line?
A double number line makes corresponding values of two quantities visible. Moving from the price of three notebooks to one or five requires a paired operation on both lines. The assumption of a constant unit price should be stated and compared with non-proportional cases such as a starting charge or discount. Pupils can then see when a scaling rule is valid rather than applying it to any two numbers.
Drawing corresponding quantities
1. Identify the two quantities and units In “Three notebooks cost 24 lira; what do five cost?” the number of notebooks and the price are different kinds of quantity. Label the upper line notebooks and the lower line lira, aligning zero with zero and then three with 24. Explain that two parallel drawn lines do not themselves prove proportionality. Each notebook must have the same price. A discount or fixed delivery charge can change the relationship and require another model. The context determines whether a double number line is an appropriate representation rather than the diagram determining the context. 2. Keep corresponding points aligned If three notebooks above correspond to 24 lira below, one notebook corresponds to eight lira. Divide both quantities in the same relationship: 3 ÷ 3 = 1 and 24 ÷ 3 = 8. Then align five notebooks with 40 lira. A learner cannot jump three units on one line and draw an arbitrary interval on the other. Mathematical correspondence matters more than perfectly scaled artistic lengths, although the ordering of marks should remain consistent. Ask the student to describe each paired point aloud before writing an operation. 3. Compare unit rate with scaling For the first problem it is easy to find the price of one notebook. If four notebooks cost 28 lira and the price for eight is sought, multiplying both quantities by two is more direct. Both routes preserve the same relationship. Ask why a pupil cannot double the four while leaving 28 unchanged. Before introducing a mechanical cross-multiplication procedure, use paired scaling to show what a constant ratio means. Begin with divisible numbers, then use less convenient ones when the idea is secure. The diagram can display more than one valid path to an answer.
Testing the proportional assumption
4. Test additive reasoning carefully If three notebooks cost 24 lira, six can cost 48: two identical groups can be combined. But moving from three to five notebooks does not mean adding two lira to the price. That common mistake mixes the units of the two lines. Find the eight-lira price per notebook before moving to five. Have learners describe both the jump above and the corresponding jump below. Evaluate their explanation of the relationship, not only the multiplication 5 × 8. A student who knows an answer can still misunderstand why the scaling is valid. 5. Contrast a non-proportional situation A taxi may charge a starting fee in addition to a price per kilometer, so its price at zero kilometers need not be zero. This is not the simple proportional relation through zero used in the notebook example. Compare the two cases and state which assumption fails. A “buy three, pay for two” discount can likewise change the unit price. Not every context with two quantities is proportional. The model’s suitability depends on the meaning of the units, a constant ratio and the conditions described by the problem. 6. Justify the model on a fresh example Suppose a recipe uses two cups of flour for six cookies; ask how much flour the same recipe would use for ten. Label both lines, mark paired points and explain the intermediate step. State the assumption that the recipe can be scaled linearly; actual kitchen yield may vary. If a learner answers ten cups, locate the jump at which units or pairs were confused. Understanding includes explaining why a constant ratio can be maintained and recognizing a situation in which it cannot. A correct number alone does not demonstrate that the diagram was interpreted meaningfully.