1. Proportional Relationships
Lesson 4.2.1
Proportional Relationships
- Tables -
Obj. 7.RP.2a,b

2. Paradise Island
A Heaven On Earth!

3. The Bahamas is made up of 700 islands, where only 30 are inhabited by people. The Bahamas is known for it's sandy white beaches, its crystal clear water, coral reefs, and it's very pleasing tropical surroundings.
The Bahamas main tourist attraction is the capital city of Nassau also know as
“Paradise Island”.

4. If you ever decide to vacation in the Bahamas, you will certainly want to stay at the “Atlantis Hotel ! ”
Why?
Well, let’s take a look at some of the exciting things you can do while you are there…

5. The Water Park
In the water park there are water slides and river slides. One famous water slide is called the Mayan Temple. The Mayan Temple has a 60-ft. drop from the top to the bottom. Then, as one falls down the long slide the person falls into a tunnel. The tunnel is underground and one can see sharks in the tunnel. There is a glass blocking the sharks, preventing any harm to the visitors.

6. Swim with the Dolphins
Feed the Stingrays

7. Jet Skiing

8. If you’re not afraid of heights you may want to try…
Parasailing

9. In todays lesson, we will use some data from a couple of the fun activities at Paradise Island.
We will be using what we learned from yesterdays lesson about unit rates to determine if a “set” of data is proportional!

10. Section 1:
What makes data from a TABLE
proportional?

11. Things you need to know first:
A fancy word for Unit Rate is the Constant of Proportionality.
They mean the exact same thing! It is a rate of ONE of something.
In 7th grade math, the Constant of Proportionality (unit rate) is often, represented by the variable, k.
The Constant of Proportionality (unit rate) is the KEY to determining if
data in a table is PROPORTIONAL or NOT PROPORTIONAL.

12. Costs to Jet Ski
Minutes, x
(independent)
Cost, y
(dependent) 8
$18 12
$27 20
$45 36
$81 45
$101.25
As seen in the table, the cost to rent a jet ski at Paradise Island depends upon the how many minutes you use the jet ski.
A Table in math shows a relationship between two quantities.
In a table, the independent variable is represented by the letter x. Remember, independent variables can stand on their own. They have control of a situation.
The dependent variable is represented by the letter y.
Their value depends or relies upon the value of x.

13. The data in the table may be related, but how do we determine if the data is proportional?
Here’s where we need to find the constant of proportionality (unit rate).
Constant of Proportionality = 𝑑𝑒𝑝𝑒𝑛𝑑𝑒𝑛𝑡 (𝑦) 𝑖𝑛𝑑𝑒𝑝𝑒𝑛𝑑𝑒𝑛𝑡 (𝑥)
Costs to Jet Ski
Formula:
k = 𝑦 𝑥
Minutes, x
(independent)
Cost, y
(dependent) 8
$18 12
$27 20
$45 36
$81 45
$101.25
𝟏𝟖 𝟖 =𝟐.𝟐𝟓
Find the constant of proportionality (k) for each x, y pair in the table by setting up a ratio, 𝒚 𝒙 .
Then divide with your calculator.
The first one is done for you!
What do you notice about the constant of
proportionality (unit rates) for each x, y pair?

14. Costs to Jet Ski What makes data from a TABLE proportional? k = 𝒚 𝒙
Minutes, x
(independent)
Cost, y
(dependent) 8
$18 12
$27 20
$45 36
$81 45
$101.25
k = 𝒚 𝒙
What makes data from a TABLE proportional?
18 8 =2.25
Answer:
Every row in the table will have the same constant of proportionality (unit rate)!
27 12 =2.25
45 20 =2.25
81 36 =2.25
=2.25

15. Section 2:
What about non-proportional
relationships?

16. Parasailing k = 𝑦 𝑥 Formula:
Number of Riders x
Cost y 1
$90 2
$150 3
$210 4
$270
Formula:
k = 𝑦 𝑥
𝟗𝟎 𝟏 =𝟗𝟎
𝟏𝟓𝟎 𝟐 =𝟕𝟓
The cost to go parasailing depends upon the number of riders going.
Find the constant of proportionality for each x, y pair.
What do you notice about the unit rates for the parasailing table verse the unit rates for the jet ski table?
𝟐𝟏𝟎 𝟑 =𝟕𝟎
𝟐𝟕𝟎 𝟒 =𝟔𝟕.𝟓
Answer: The unit rates are not the same for each row! Because of this, the relationship between number of rides and cost is NOT PROPORTIONAL.

17.
You Try!
Determine if each table is proportional or non-proportional. IF the table is proportional, then state the constant of proportionality.
1) ) X Y 1 2 6 3 10 4 14 X Y 2 1 4 6 3 8
1) Non-Proportional
2) Proportional (k = .5)

18. Section 3: How can you use the Constant of
Proportionality to find missing values?

19. If we know that a table is proportional, then the constant of proportionality can be used to find missing values.
Let’s look at our Jet Ski table which we found to be proportional…

20. Costs to Jet Ski There are two missing y values in the table.
Minutes, x
(independent)
Cost, y
(dependent) 8
$18 12
$27 20
$45 36
$81 45
$101.25 52 60
There are two missing y values in the table.
Take a minute to try and figure out how the constant of proportionality (or unit rate) can be used to find the missing values.
•𝟐.𝟐𝟓
𝑘=2.25
•𝟐.𝟐𝟓
𝑘=2.25
•𝟐.𝟐𝟓
𝑘=2.25
•𝟐.𝟐𝟓
𝑘=2.25
Once you determine that the data in the table is proportional, you can find ANY y value by using the constant of proportionality.
How? Multiple the x value and the constant of proportionality, k to get the y value.
•𝟐.𝟐𝟓
𝑘=2.25
Give students a minute or two to see if they can determine the cost for 52 minutes and the cost for 60 minutes. Then click for explanation.
•𝟐.𝟐𝟓
$117
•𝟐.𝟐𝟓
$135

21. Costs to Jet Ski
All proportional relationships can be expressed as an equation in the format
𝑦=𝒌𝑥 where k is the constant
Minutes, x
(independent)
Cost, y
(dependent) 8
$18 12
$27 20
$45 36
$81 45
$101.25 52 60
𝒌=2.25
•𝟐.𝟐𝟓
•𝟐.𝟐𝟓
•𝟐.𝟐𝟓
The equation below can be used to represent our table.
𝑦=2.25𝑥
Constant of
Proportionality
X is the minutes
Y is the cost
•𝟐.𝟐𝟓
•𝟐.𝟐𝟓
•𝟐.𝟐𝟓
$117
•𝟐.𝟐𝟓
$135

22. Number of Tickets Sold (X)
You Try!
𝒌= 𝒚 𝒙
𝒚=𝒌𝒙
Band Concert Ticket Sales
The table represents a proportional relationship. 1) Find the constant of proportionality for the table. 2) Write an equation in 𝑦=𝑘𝑥 format that represents the table. 3) Use the equation to determine how much money will be collected if 503 tickets are sold.
Number of Tickets Sold (X)
Money Collected (Y) 5 10 20 15 30 40 25 50
𝒌=𝟐
𝒚=𝟐𝒙
𝒚=𝟐(𝟓𝟎𝟑)
=$𝟏,𝟎𝟎𝟔

23. End of PowerPoint