Mathematics
Grade 7
15 min
Identify proportional relationships
Identify proportional relationships
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1
Introduction & Learning Objectives
Learning Objectives
Define and differentiate between ratios, rates, and the constant of proportionality.
Determine if a relationship represented in a table of values is proportional by checking constant ratios.
Determine if a relationship represented by a graph is proportional by analyzing its shape and origin.
Determine if a relationship described by an equation is proportional by identifying the y=kx form.
Calculate the constant of proportionality (k) from tables, graphs, or equations.
Explain the key characteristics that define a proportional relationship.
Ever noticed how the cost of apples changes based on how many you buy? 🍎 Is there a consistent pattern in how the price relates to the quantity?
In this lesson, you'll learn to identify special relationships wher...
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Key Concepts & Vocabulary
TermDefinitionExample
RatioA comparison of two quantities, often expressed as a fraction (a/b) or using a colon (a:b).If there are 3 red apples and 2 green apples, the ratio of red to green apples is 3/2 or 3:2.
RateA ratio that compares two quantities with different units.Traveling 120 miles in 2 hours is a rate of 120 miles / 2 hours.
Unit RateA rate where the second quantity in the comparison is 1 unit.The unit rate for 120 miles in 2 hours is 60 miles per 1 hour (or 60 mph).
Proportional RelationshipA relationship between two quantities where their ratio is constant, or one quantity is a constant multiple of the other. This means as one quantity changes, the other changes by a consistent factor.The relationship between the number of hours worked and money earned at a fixed hourly wage...
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Core Formulas
Proportional Relationship Equation
$y = kx$
A relationship between two variables, x and y, is proportional if it can be expressed in this form, where 'k' is a non-zero constant (the constant of proportionality). This means y is always 'k' times x. If x=0, then y must also be 0.
Proportionality Test (Table)
$\frac{y}{x} = k$ (for all non-zero x values)
To check if a relationship shown in a table is proportional, calculate the ratio of the y-value to the x-value for each pair of data points (where x is not zero). If this ratio (k) is the same for all pairs, the relationship is proportional. Also, the point (0,0) must fit the pattern.
Proportionality Test (Graph)
A straight line passing through the origin $(0,0)$.
When graphed on a coordinate plane,...
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Challenging
A video game store sells new games for $60 each. A second store, a rental service, charges a $15 monthly membership fee plus $5 for each game rented. If you plan to get 4 games this month, which statement is true?
A.Store A is proportional and the better deal.
B.Store B is proportional and the better deal.
C.Store A is proportional, but Store B is the better deal.
D.Store B is proportional, but Store A is the better deal.
Challenging
The graph shows the cost of a gym membership over several months. The line passes through (0, 50) and (1, 80). How would the gym's pricing model need to change for the relationship to be proportional?
A.The gym would need to eliminate the monthly fee and only charge a one-time sign-up fee.
B.The gym would need to eliminate the one-time sign-up fee and only charge a constant monthly fee.
C.The gym would need to increase the monthly fee.
D.The gym would need to make the graph a curve instead of a line.
Challenging
A table shows the relationship between the side length of a square (x) and its area (y).
x (side) | y (area)
---|---
1 | 1
2 | 4
3 | 9
Why is the relationship between the side length and the area NOT a proportional relationship?
A.Because the area is always greater than the side length.
B.Because the graph of the relationship is a curve, not a straight line.
C.Because the table does not include the point (0,0).
D.Because the ratio of y/x is not constant.
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