Embedded Files
Learning Targets
When there is a constant rate, I can identify the two quantities that are in a proportional relationship.
I can use a table with 2 rows and 2 columns to find an unknown value in a proportional relationship.
Notes
Notes
Let’s use constants of proportionality to solve more problems.
If we identify two quantities in a problem and one is proportional to the other, then we can calculate the constant of proportionality and use it to answer other questions about the situation. For example, Andre runs at a constant speed, 5 meters every 2 seconds. How long does it take him to run 91 meters at this rate?
In this problem there are two quantities, time (in seconds) and distance (in meters). Since Andre is running at a constant speed, time is proportional to distance. We can make a table with distance and time as column headers and fill in the given information.
To find the value in the right column, we multiply the value in the left column by ⅖ because ⅖ ⋅ 5 = 2. This means that it takes Andre ⅖ seconds to run one meter.
At this rate, it would take Andre ⅖ ⋅ 91 = (182/5), or 36.4 seconds to walk 91 meters. In general, if t is the time it takes to walk d meters at that pace, then t = ⅖ d.
Activities
Activities
3.2 The Price of Rope
3.2 The Price of Rope
Two students are solving the same problem: At a hardware store, they can cut a length of rope off of a big roll, so you can buy any length you like. The cost for 6 feet of rope is $7.50. How much would you pay for 50 feet of rope, at this rate?
Kiran knows he can solve the problem this way. Looking at his table, what you Kiran's answer be?
2. Kiran wants to know if there is a more efficient way of solving the problem. Priya says she can solve the problem with only 2 rows in the table. What do you think Priya's method is?
Scale Factor Method (Kiran)
Scale Factor Method (Kiran)
Constant of Proportionality Method (Lin)
Constant of Proportionality Method (Lin)
Add to Your Notes
Add to Your Notes
Which method do you prefer?
Although either method will work, there are reasons to prefer using the constant of proportionality to approach problems like these. First, the constant of proportionality 1.25 means something important in the problem—it’s the price of 1 foot of rope. Because of that, the 1.25 could be easily used to compute the price of any length of rope. If no students bring it up, point out that the equation y=1.25x could be used to relate any length of rope, x, to its price, y.
3.3 Swimming, Manufacturing, and Painting
3.3 Swimming, Manufacturing, and Painting
1. Tyler swims at a constant speed, 5 meters every 4 seconds. How long does it take him to swim 114 meters?
2. A factory produces 3 bottles of sparkling water for every 8 bottles of plain water. How many bottles of sparkling water does the company produce when it produces 600 bottles of plain water?
3. A certain shade of light blue paint is made by mixing 112 quarts of blue paint with 5 quarts of white paint. How much white paint would you need to mix with 4 quarts of blue paint?
4. For each of the previous three situations, write an equation to represent the proportional relationship in this form: y = k x. Remember, you only substitute a number for k.
Summary
Summary
Assignment
Assignment
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