Lesson Plan: Representing Proportional Relationships
Lesson Objectives
This lesson can be completed in one 50-minute class period. However, additional practice may be beneficial and could extend into a second class period if time allows.
- Identify proportional relationships in tables, graphs, and equations
- Understand the constant of proportionality (unit rate)
- Represent proportional relationships using equations in the form y = kx
- Interpret points on graphs of proportional relationships
- Solve real-world problems involving proportional relationships
Florida BEST Standards
- MA.7.AR.3.1: Apply previous understanding of percentages and ratios to solve multi-step real-world problems.
Prerequisite Skills
- Understanding of proportional relationships
- Basic algebraic concepts
- Familiarity with coordinate plane
- Ability to interpret ratios and unit rates
Key Vocabulary
- Equation
- Coordinate plane
- y-intercept
- Slope
- Unit rate
- Proportional relationship
Warm-up Activity (10 minutes)
Students will complete an equivalent ratio table. This activity will be conducted using an interactive whiteboard or handouts.
Example: A recipe calls for 2 cups of flour for every 3 cups of milk. Complete the equivalent ratio table:
Flour (cups) | 2 | 4 | □ | 8 | 10 |
---|---|---|---|---|---|
Milk (cups) | 3 | □ | 9 | □ | 15 |
Solution:
Flour (cups) | 2 | 4 | 6 | 8 | 10 |
---|---|---|---|---|---|
Milk (cups) | 3 | 6 | 9 | 12 | 15 |
Explanation: To fill in the missing values, we use the relationship between flour and milk. For every 2 cups of flour, we need 3 cups of milk. So, we can multiply both the flour and milk amounts by the same factor to find equivalent ratios.
For the second column: 2 × 2 = 4 cups of flour, 3 × 2 = 6 cups of milk
For the third column: 2 × 3 = 6 cups of flour, 3 × 3 = 9 cups of milk
For the fourth column: 2 × 4 = 8 cups of flour, 3 × 4 = 12 cups of milk
Resource: https://www.media4math.com/library/card/equivalent-ratio-tables
Teach (25 minutes)
Definitions (5 minutes)
Equation: A mathematical statement that shows two expressions are equal
Coordinate plane: A two-dimensional plane formed by the intersection of a vertical line called y-axis and a horizontal line called x-axis
y-intercept: The point where a line crosses the y-axis
Slope: The steepness of a line, calculated as the change in y divided by the change in x
Unit rate: A rate where the denominator is 1
Proportional relationship: A relationship between two quantities where one quantity is a constant multiple of the other
Instruction (20 minutes)
This slide show provides an overview of proportions:
https://www.media4math.com/library/slideshow/overview-proportions
This slide show provides multiple examples of solving proportions algebraically. Review several of these examples before moving on to the more detailed examples below:
https://www.media4math.com/library/slideshow/math-examples-solving-proportions-algebraically
Example 1: Distance and Time
Problem: A car travels at a constant speed of 60 miles per hour. Express the relationship between distance traveled (y) and time (x) as an equation.
Solution:
• The unit rate is 60 miles per hour
• Let y represent the distance traveled and x represent the time
• Equation: y = 60x
Example 2: Circle Geometry
Problem: Provide a data table and derive the equation showing that the slope is π. This could also be a hands-on activity in which students are measuring the diameters and circumferences for different circular objects. (Using a string, wrap it around the circular shape and then measure the length of the string to find the circumference.)
Solution:
Diameter (d) | Circumference (C) | C/d |
---|---|---|
1 | 3.14 | 3.14 |
2 | 6.28 | 3.14 |
3 | 9.42 | 3.14 |
4 | 12.56 | 3.14 |
• The ratio C/d is constant and approximately equal to 3.14 (π)
• This suggests a proportional relationship: C = πd
• The equation is y = πx, where y is the circumference and x is the diameter
• The slope of this line is π, which is the constant of proportionality
Example 3: Hourly Wages
Problem: Start with a data table, derive the equation, and solve a proportion.
Solution:
Hours worked (x) | Wages earned ($) (y) |
---|---|
2 | 30 |
4 | 60 |
6 | 90 |
8 | 120 |
• Calculate the unit rate: 120 / 8 = $15 per hour
• The equation is y = 15x, where y is wages earned and x is hours worked
Solving a proportion: If someone works for 10 hours, how much will they earn?
• Use the equation: y = 15(10) = $150
Review (10 minutes)
Students will practice solving real-world problems using ratio tables and proportions.
Example 1
A store sells 5 notebooks for \$7. How many notebooks can be bought for \$42?
Solution using ratio table:
Notebooks | 5 | 10 | 15 | 20 | 25 | 30 |
---|---|---|---|---|---|---|
Cost ($) | 7 | 14 | 21 | 28 | 35 | 42 |
We can see that 30 notebooks can be bought for \$42.
Example 2
A car travels 210 miles in 3 hours. At this rate, how long will it take to travel 350 miles?
Solution using proportion:
• Set up the proportion: 210/3 = 350/x, where x is the time in hours
• Cross multiply: 210x = 3 * 350
• Solve for x: 210x = 1050
x = 1050/210 = 5
• Therefore, it will take 5 hours to travel 350 miles at this rate.
Assess (5 minutes)
Use this 10-question quiz to assess student understanding.
Quiz
- Write an equation to represent a proportional relationship where y is 3 times x.
- Graph the proportional relationship y = 2.5x.
- What does the point (0,0) represent in a proportional relationship?
- In the equation y = kx, what does k represent?
- If a graph of a proportional relationship passes through the point (4,10), what is the unit rate?
- A car travels 240 miles in 4 hours at a constant speed. Write an equation to represent this relationship.
- In a proportional relationship, what does the point (1,r) represent?
- Graph the equation y = 0.5x.
- A recipe calls for 2 cups of flour for every 3 cups of milk. Write an equation to represent the relationship between flour (y) and milk (x).
- If 4 oranges cost $2, how much would 10 oranges cost? Set up and solve a proportion.
Answer Key
- y = 3x
- ]
- The starting point or initial value of the relationship
- The constant of proportionality or unit rate
- 2.5 (10 ÷ 4 = 2.5)
- y = 60x, where y is distance and x is time
- The unit rate of the proportional relationship
- y = (2/3)x
- 4/2 = 10/x, 4x = 20, x = 5. 10 oranges would cost $5.
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