Lesson 9 — Proportional Relationships as y = kx

Summary

Learners express a proportional relationship in the compact form y = kx, where k is the constant of proportionality — the amount of y for each 1 of x. They find k from a table, graph, or story, write the equation, and use it to predict a new value.

Objectives

• Express a proportional relationship in the form y = kx and interpret what k means. (D05.S2.07.02)

Connection

A bicycle moves at a steady 3 metres every second. Then the distance y is always 3 times the time x: y = 3x. The number 3 is the constant of proportionality k — the metres per second, the unit rate. Write it as y = kx and you can predict anything: after 10 seconds, y = 3 × 10 = 30 metres. Every fixed price, speed, or dose compresses to one tidy equation, and that equation is a machine for answering “what if” — a tool used by navigators, engineers, and cooks in every culture where quantity is measured.

Materials

• y = kx matching cards
• Equation writing page
• Counters

Preparation

• Copy matching cards and equation writing pages.
• Have worked examples ready: y = 3x from a table (1,3), (2,6), (3,9); and a story (3 metres per second) → k = 3 → y = 3x.
• Recall from Lesson 8: constant of proportionality and straight lines through the origin.

Facilitator note

This lesson is written to the learner (“you”). The ideas to land: (1) a proportional relationship can be written y = kx, where k is the constant of proportionality (the unit rate); (2) k is found by dividing any y by its x (k = y/x); and (3) once you have y = kx you can predict a new value by multiplying. Because writing y = kx is a foundational skill, use explicit instruction, worked examples, and guided practice before independent work (Kirschner, Sweller & Clark, 2006). Watch for the slips of putting k on the wrong side (writing x = ky) and of computing k as x/y instead of y/x. Retrieval: ask learners to recall Lesson 8’s tests — y = kx is the equation form of the same relationship; its graph is a straight line through the origin with steepness k. The technology lens: with a calculator or graphing tool, changing k and watching the line steepen is a quick way to feel the rate. The egalitarian lens (docs/philosophy.md §4): writing a rule as y = kx lets anyone predict fairly and check a claim — the equation is a compact, portable truth, not a secret. The global lens: expressing a relationship as a single equation generalizes the proportional “per one” thinking recorded across traditions (e.g., the Rhind Papyrus’s unit divisions) into one modern, universal form.

Procedure

1. Gather (5 min). Last time you saw proportionality in tables and graphs. Today you write it as one short equation and use it to predict.
2. Meet y = kx (10 min). In a proportional relationship, y is always k times x: y = kx. The k is the constant of proportionality — the amount of y for each 1 of x. Find it by dividing: k = y ÷ x.
3. Worked example — from a table (10 min). The table shows (1, 3), (2, 6), (3, 9). Check the ratio: 3÷1 = 3, 6÷2 = 3, 9÷3 = 3. So k = 3, and the equation is y = 3x.
4. Worked example — from a story, then predict (10 min). A bicycle covers 3 metres per second. Here k = 3 (metres per second), so y = 3x. Predict: after 10 seconds, y = 3 × 10 = 30 metres. The equation is a prediction machine.
5. Practice with a partner (10 min). Take your matching cards: pair each table, graph, and sentence with its equation y = kx, find k, and use it to predict one new value. Trade and check each other’s k.
6. Close (5 min). A proportional relationship is y = kx. Find k by dividing (k = y/x), write the equation, and multiply to predict. One tidy equation holds the whole relationship.

Differentiation

• Support: Use only whole-number k with a ready table; find k by counting “per one” with counters; describe each step aloud so learners with low vision can follow by listening.
• Extension: Given two relationships, compare their k values and explain which grows faster; write y = kx for a real situation of the learner’s own and use it to predict.

Assessment

• Formative (observation/self): Can the learner find k from a table, graph, or story, write y = kx, and use it to predict a new value with a correct interpretation of k?
• Self-check: The learner asks, “Did I compute k as y ÷ x (not x ÷ y)? Does my equation y = kx match the table and graph, and did my prediction come from multiplying x by k?”

Home connection

At home, find a steady rate (price per item, distance per trip, food per person), write it as y = kx, and use it to predict the total for a larger amount.

Resources

• The form y = kx and the constant of proportionality are standard in middle-grades mathematics; see John A. Van de Walle, Elementary and Middle School Mathematics (10th ed., 2019). The note that “per one” proportional reasoning appears in the Rhind Papyrus (Egypt, c. 1650 BCE) is documented history (MacTutor, “Mathematics in Egyptian Papyri,” S-288); treating all traditions as worthy is a value (docs/philosophy.md §4). The claim that a compact equation makes prediction open to everyone is a value commitment.