Mathematics
Grade 7
15 min
Constant rate of change
Constant rate of change
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1
Introduction & Learning Objectives
Learning Objectives
Define and identify a constant rate of change.
Calculate the constant rate of change (unit rate) from given information.
Represent constant rate of change using tables, graphs, and equations.
Distinguish between constant and non-constant rates of change.
Solve real-world problems involving constant rates of change.
Explain the relationship between constant rate of change and proportional relationships.
Have you ever noticed how some things change steadily, like a car cruising at the same speed or water filling a bucket at a steady flow? 🚗💧
In this lesson, you'll explore what it means for something to change at a 'constant rate' and how to identify and use it in everyday situations. This skill is crucial for understanding how quantities r...
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Key Concepts & Vocabulary
TermDefinitionExample
RateA ratio that compares two quantities measured in different units.If you drive 120 miles in 2 hours, the rate is 120 miles / 2 hours.
Rate of ChangeA measure of how one quantity changes in relation to another quantity.The change in temperature over time, or the change in distance over time.
Constant Rate of ChangeWhen the ratio of the change in the dependent variable to the change in the independent variable remains the same throughout a relationship.A car traveling at a steady speed of 60 miles per hour has a constant rate of change.
Unit RateA rate where the second quantity in the comparison is one unit.If you earn $150 for 10 hours of work, the unit rate is $15 per hour.
Proportional RelationshipA relationship between two quantities where their ratio is constan...
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Core Formulas
Rate of Change Formula
$\text{Rate of Change} = \frac{\text{Change in Dependent Variable}}{\text{Change in Independent Variable}}$
This formula helps you calculate how much one quantity changes for every unit change in another quantity. It's the foundation for finding a constant rate.
Unit Rate Calculation
$\text{Unit Rate} = \frac{\text{Total Quantity 1}}{\text{Total Quantity 2 (where Quantity 2 is the unit quantity)}}$
To find the unit rate, divide the first quantity by the second quantity. This simplifies the rate to 'per one' of the second quantity.
Proportional Relationship Equation
$y = kx$
This equation represents a proportional relationship where 'y' is the dependent variable, 'x' is the independent variable, and 'k...
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Challenging
Consider the relationship in the table. Which statement is true?
| x | 1 | 2 | 3 |
|---|---|---|---|
| y | 5 | 8 | 11 |
A.It is a proportional relationship because the rate of change is constant.
B.It is not a proportional relationship because the ratio y/x is not constant.
C.It is a proportional relationship because y increases as x increases.
D.It is not a proportional relationship because the rate of change is not constant.
Challenging
Company A's pricing is shown in the table. Company B's pricing is given by the equation y = 5.5x, where y is the total cost for x items. Which company has a higher cost per item, and by how much?
Company A:
| Items (x) | Cost (y) |
|-----------|----------|
| 2 | $12 |
| 5 | $30 |
A.Company A has a higher cost per item, by $0.50.
B.Company B has a higher cost per item, by $0.50.
C.Company A has a higher cost per item, by $1.00.
D.Both companies have the same cost per item.
Challenging
A train left a station and traveled at a steady speed. After 90 minutes, it had traveled 135 miles. How far from the station will the train be after 4 hours of travel?
A.270 miles
B.540 miles
C.360 miles
D.315 miles
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