Constant of Proportionality (K)
What is the general formula used to write an equation for a proportional relationship between x and y?
y = kx
Solve using long division: 180/ 4.
45
x (Hours) y (Miles)
2 10
4 20
6 30
Is this table proportional? If so, what is k?
Yes, it is proportional. k = 5.
A double number line shows 4 yards of yarn aligning vertically with 5. What is the ratio of yards of yarn to cost in dollars?
5
(4 yards for every 5 dollars).
What algebraic equation do you use to calculate the constant of proportionality (k) when you are given values for x and y?
k = y/x
Solve using long division or decimal strategies: 4.50 / 6.
0.75
In a proportional table, why is finding the row where x = 1 (the unit rate) helpful for filling in missing numbers?
Because 1 x anything = that number. Once you know the value for 1 unit, you can easily multiply it by any target number.
To make a shade of orange paint, you mix 4 mL of red paint with 12 mL of yellow paint. If you scale this down on a double number line to use only 1 mL of red paint, how many mL of yellow paint do you need?
3 mL of yellow paint (4 / 4 = 1 mL red)
12 / 4 = 3 mL yellow
A garden hose dispenses 18 gallons of water every 3 minutes. What is the constant of proportionality (k) in gallons per minute, and what is the equation representing total gallons (g) in (m) minutes?
k = 6 gallons per minute (18/3 = 6).
Equation: g = 6m.
Solve 0.32 × 3,000 by ignoring the zeros first, then adjusting the decimal place. Explain your decimal shift strategy.
First, multiply 0.32 x 3 = 0.96. Then shift the decimal 3 places to the right to account for the 3 zeros in 3,000.
Lin earns $90 for 5 hours of work. Use a unit rate strategy to find how much she earns for 8 hours of work and how much she earns for 2.1 hours of work.
Unit rate = $18/ hour (90 / 5 = 18).
For 8 hours: 18 x 8 = $144.
For 2.1 hours: 18 x 2.1 = $37.80.
A recipe uses 3 cups of flour for every 2 cups of milk. If a double number line is scaled up so that you use 12 cups of flour, how many cups of milk align with it?
8 cups of milk (Scale factor is x4)
3 x 4= 12 cups flour
2 x 4 = 8 cups milk
If the constant of proportionality for converting feet to inches is 12, what is the constant of proportionality for converting inches to feet?
The reciprocal of 12/1, which is 1/12!
An administrative assistant types 180 words in 4 minutes. Use long division to find their typing rate in words per minute.
45 words per minute
x y
2 3
4 5
6 7
Is this table proportional? Explain why or why not using ratios.
No, it is not proportional. 3/2 = 1.5, but 5/4 = 1.25. Because y/x is not constant, it is not proportional.
Why might a ratio table be better to use than a double number line when you are scaling down to extremely small decimal values (like finding the cost of 1 GB of memory from a 128 GB card)?
Double number lines become cluttered, crowded, and hard to draw when tick marks get extremely close together. Tables can hold tiny decimal numbers without running out of physical room.
A theater sold 400 tickets and brought in $3,200 in total revenue. Write two different equations for this relationship: one showing revenue (r) as a function of tickets (t), and one showing tickets (t) as a function of revenue (r).
r = 8t (since $3,200/ 400 = $8 per ticket)
t = 1/8r or t = 0.125r (since 1/ 8 = 0.125 tickets per dollar)
Solve 0.0004 x 20,000 using place value adjustments. Show how the decimal places and zeros interact.
Multiply 4 x 2 = 8. Moving 4 places left for 0.0004 and 4 places right for 20,000 cancels out completely, leaving 8.
Noah bought 4 tacos for $6.00. Jada bought 50 tacos for $72.00. Create a small table for each person to find their unit price per taco. Did they pay the same rate?
Noah's unit price: $6.00 / 4 = $1.50 per taco.
Jada's unit price: $72.00 / 50 = $1.44 per taco.
No, they did not pay the same rate (Jada paid less per taco).
Lin’s drink recipe uses 3 L of orange juice for every 4 L of soda water (3:4). Noah’s recipe uses 4 L of orange juice for every 5 L of soda water (4:5). Using equivalent double number line scaling, determine whose drink tastes more "orange-y" when both recipes use 12 L of orange juice.
Lin (3:4 scaled up by 4): 12 L of orange juice needs 16 L soda water.
Noah (4:5 scaled up by 3): 12 L orange juice needs 15 L soda water.
Noah uses less soda water for the same amount of orange juice, making it stronger.
Store A sells a 4-pack of juice for $6.00. Store B sells a 6-pack of the same energy drink for $10.00. Which store offers the better unit price, and by how much per drink?
Store A Unit Price: $6.00 / 4 = $1.50 per drink.
Store B Unit Price: $10.00 / 6 = $1.67 per drink.
Difference: $1.67 - $1.50 = $0.17 savings per drink at Store A.
Store A offers the better deal by $0.17 per drink