Grade 7 · Grade 7: Proportional Reasoning and Rational Numbers · Proportional Relationships and Percent
Proportional Relationships and the Constant of Proportionality
Test whether two quantities are proportional, find k, and write y = kx from tables, graphs and words.
Learning objectives
- Compute unit rates including ratios of fractions.
- Decide whether a relationship is proportional using tables and graphs.
- Identify the constant of proportionality in every representation.
- Write and use the equation y = kx.
AERO Mathematics alignment
AERO.M7.NS.1
Compute unit rates associated with ratios of fractions, including ratios of lengths, areas and other quantities measured in like or different units.
AERO.M7.NS.2
Decide whether two quantities are in a proportional relationship by testing equivalent ratios in a table or graphing on a coordinate plane.
AERO.M7.NS.3
Identify the constant of proportionality (unit rate) in tables, graphs, equations, diagrams and verbal descriptions, and represent it as y = kx.
MYP criteria
- Criterion A — Knowing and understanding
- Criterion B — Investigating patterns
- Criterion C — Communicating
- Criterion D — Applying mathematics in real-life contexts
ENGAGE
Start here
One table shows a taxi charging 2 QAR per km. Another shows a taxi charging 5 QAR plus 2 QAR per km.
Think about it
Both graphs are straight lines. Why is only one of them proportional?
Hint: Look at what happens when the distance is zero.
EXPLAIN
Testing for proportionality
Two quantities are proportional when every pair has the same ratio, y/x = k. That constant k is called the constant of proportionality, and the relationship can always be written y = kx.
| Representation | Where k appears |
|---|---|
| Table | Divide each y by its x; all answers equal k |
| Graph | The line passes through the origin; k is the steepness |
| Equation | y = kx; k is the multiplier |
| Words | The per-one rate, such as 2 riyals per kilometre |
Unit rates with fractions
If ½ litre of paint covers ⅔ of a square metre, the unit rate is (⅔) ÷ (½) = 4/3 square metres per litre. Dividing by a fraction is multiplying by its reciprocal.
Common misconception
A straight-line graph is not automatically proportional. The 5 QAR flag-fall means the second taxi costs money at zero distance, so the line misses the origin and y/x is not constant.
INVESTIGATION
Try it yourself
Time how long it takes you to walk 20, 40 and 60 metres. Make a table, calculate y/x for each row and decide whether your walking is proportional. Graph it and mark the constant of proportionality.
Watch
Introduction to Proportional Relationships
Khan Academy · 6:00
Shows the table-and-graph test for proportionality you will use throughout.
Before you watch: Does a straight-line graph always mean a proportional relationship?
- What is k in y = 4.5x, and what does it mean?
- Why must a proportional graph pass through the origin?
Interactive simulation · PhET
Graphing Lines
Set the intercept to zero and change the slope. Record each equation as y = kx.
While you explore
- What is always true about the graph of a proportional relationship?
- Where can you see the constant of proportionality on the graph?
Key vocabulary
- Origin
- The point where both axes cross.
- Constant of proportionality
- The fixed number you multiply x by to get y.
Practice questions
0/1 correct
Level 1 · Criterion A
Which equation represents a proportional relationship?
MYP criterion tasks
Level 2 · Criterion B
A table shows (2, 5), (4, 10), (6, 15). Is it proportional? Find k and write the equation.
Reflect & track
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