Two quantities are directly proportional when their ratio stays constant under the stated model. This Secondary 1 Mathematics guide for Seletar families connects a per-unit rate, a table of values and an equation.
If identical notebooks cost $3 each with no fixed fee or discount, the cost of n notebooks is C = 3n. Doubling the quantity doubles the cost, and the ratio C/n is three for every positive quantity in this model.
Continue your Secondary 1 Mathematics journey: Bukit Timah Secondary 1 Mathematics programme · Parent reading guide · Word-problem translation. For the 3-pax teaching approach, class fit and current arrangements, begin with the programme page.
Identify the two quantities and their units · Find the constant from one valid pair · Test a table using ratios, not just increases · Recognise the effect of a fixed fee · Scale both quantities by the same factor · Use a graph only within the model’s domain · Try a short constant-rate check · Use the relationship to guide support
Identify the two quantities and their units
In the notebook example, n counts notebooks and C measures cost in dollars. The constant three has the meaning three dollars per notebook.
Naming units helps prevent an inverted rate. Dividing cost by quantity gives dollars per notebook; dividing quantity by cost gives notebooks per dollar.
Ask which rate the equation needs. C = 3n uses cost per notebook because multiplying it by a notebook count produces a cost.
Find the constant from one valid pair
If four identical items cost $10 under a constant-price model, the cost per item is 10 ÷ 4 = $2.50. The equation is C = 2.5n.
For seven items, the model gives C = 2.5 × 7 = $17.50. Check that the calculation uses the same item and unchanged price conditions.
A unitary method expresses the same reasoning: first find the amount for one unit, then multiply by the required number of units.
Test a table using ratios, not just increases
Consider quantities 2, 4 and 6 with corresponding costs 6, 12 and 18. Each cost divided by quantity equals three. The table is consistent with C = 3n.
Now consider quantities 2, 4 and 6 with costs 8, 14 and 20. The costs increase by six each time, but the ratios are 4, 3.5 and 10/3. They are not constant.
A constant increase for equal steps shows a linear pattern, but direct proportion additionally requires a zero fixed part. In the second table, C = 3n + 2 fits the values.
Recognise the effect of a fixed fee
Suppose a hypothetical order costs $3 per item plus a fixed $2 packing charge per order. For positive order quantities, the model is C = 3n + 2.
Two items cost eight dollars and four items cost fourteen dollars. Doubling the number of items does not double the total cost because the fixed charge is counted once.
Read the pricing conditions before assuming proportion. The model is an invented classroom example, not a statement about any actual seller’s prices.
Scale both quantities by the same factor
If a mixture uses two cups of ingredient A for every five cups of ingredient B, multiplying both amounts by three gives six cups and fifteen cups. The ratio remains 2:5.
To find the amount of B for four cups of A, the scale factor from two to four is two, so B becomes ten cups.
This reasoning assumes the same recipe proportions and compatible units. Adding the same number to both amounts generally does not preserve the ratio: 2:5 and 4:7 are different.
Use a graph only within the model’s domain
A direct-proportion equation y = kx has a straight-line graph through the origin. For y = 3x, the plotted pairs include (0, 0), (1, 3) and (2, 6).
In a notebook-count context, only non-negative whole-number quantities make sense. A continuous line can show the mathematical relationship, but it does not mean that every fractional notebook order is meaningful.
A graph’s appearance is supporting evidence, not a substitute for reading its scale and the given rule. Check the ratio or equation as well.
Try a short constant-rate check
- Five identical pens cost $8 with no additional charge. Find the cost of eight pens at the same unit price.
- A direct-proportion model has y = kx and the pair x = 4, y = 14. Find k and y when x = 10.
- Does C = 4n + 5 represent direct proportion between C and n? Explain.
- A ratio is 3:7. If the first quantity is 12, find the second.
Answers: $12.80; k = 3.5 and y = 35; no, because the fixed five prevents a constant C/n ratio; and 28.
Ask the student to identify the unchanged quantity in their method. It should be the per-unit rate or ratio, not merely the difference between two values.
Use the relationship to guide support
For Seletar families, bring a question where the student chose addition instead of scaling, inverted a rate or ignored a fixed fee. The wording and quantity labels reveal where the translation needs work.
The Bukit Timah programme page explains the 3-pax approach and current arrangements. Match examples to the student’s school topic before extending to other kinds of proportion.
Choose the next useful reading step
Word-problem translation · PSLE to algebra · Diagnose learning gaps
For a conversation about suitable support, continue to Secondary 1 Mathematics tuition in Bukit Timah. Bring recent marked work and the student’s current school topic list so the starting point is clear.