1. Unit 5. Day 14.

2. Modeling Proportional Relationships
Traditional
Alternate

3. Official Math Standard 7.RP.2.c
Represent proportional relationships by equations. For example, if total cost t is proportional to the number n of items purchased at a constant price p, the relationship between the total cost and the number of items can be expressed as t = pn.
$ ℎ𝑎𝑡
$ ℎ𝑎𝑡 6 24 ÷4
Four hats cost $24. 4 ÷4 𝑐 𝑡 = 6 𝑛 ∙ ℎ
𝑥:ℎ𝑎𝑡𝑠
𝑦:𝑐𝑜𝑠𝑡
𝑦 = 6𝑥

4. Official Math Standard 7.RP.2.c
Represent proportional relationships by equations. For example, if total cost t is proportional to the number n of items purchased at a constant price p, the relationship between the total cost and the number of items can be expressed as t = pn.
Six sodas cost $9.
9 6
3 2
$ 𝑠𝑜𝑑𝑎 9
$ 𝑠𝑜𝑑𝑎
$ 𝑠𝑜𝑑𝑎 9
$ 𝑠𝑜𝑑𝑎
1.50 ÷6 ÷6 6 6 ÷6 ÷6
𝑥:𝑠𝑜𝑑𝑎𝑠
𝑦:𝑐𝑜𝑠𝑡
3 2 𝑡 𝑐 𝑦 = 𝑥 𝑛 ∙ 𝑠 𝑐 𝑦 𝑡 =
1.5 ∙ 𝑥 𝑛 𝑠

5. Official Math Standard 7.RP.2.c
Represent proportional relationships by equations. For example, if total cost t is proportional to the number n of items purchased at a constant price p, the relationship between the total cost and the number of items can be expressed as t = pn.
12 miles in 5 hours
12 5 12
𝑚𝑖 ℎ𝑟
𝑚𝑖 1 ℎ𝑟
𝑚𝑖 ℎ𝑟
2.4 12
𝑚𝑖 1 ℎ𝑟 ÷5 ÷5 5 5 ÷5 ÷5
𝑥:𝑡𝑖𝑚𝑒 (ℎ𝑟𝑠)
𝑦:𝑚𝑖𝑙𝑒𝑠
12 5 𝑦 𝑑 = ∙ 𝑥 𝑡 𝑦 𝑑 =
2.4 ∙ 𝑥 𝑡

6. Official Math Standard 7.RP.2.c
Represent proportional relationships by equations. For example, if total cost t is proportional to the number n of items purchased at a constant price p, the relationship between the total cost and the number of items can be expressed as t = pn.
4 pizzas were ordered to feed 10 teens
4 10
2 5
𝑝𝑖𝑧𝑧𝑎 𝑡𝑒𝑒𝑛 4
𝑝𝑖𝑧𝑧𝑎 1 𝑡𝑒𝑒𝑛
𝑝𝑖𝑧𝑧𝑎 𝑡𝑒𝑒𝑛 4
0.4
𝑝𝑖𝑧𝑧𝑎 1 𝑡𝑒𝑒𝑛
÷10
÷10 10 10
÷10
÷10
𝑥:𝑡𝑒𝑒𝑛𝑠
𝑦:𝑝𝑖𝑧𝑧𝑎
2 5 𝑝 = ∙ 𝑡 𝑝 𝑦 =
0.4 ∙ 𝑦 𝑥 𝑥 𝑡

7. 𝑦 = 𝑟 𝑥 𝑦 = 6𝑥 𝑐 = 6ℎ 𝑡 = 6𝑛 𝑐 = 3 2 𝑠 𝑦 = 3 2 𝑥 𝑡 = 3 2 𝑛 𝑑 = 12 5 𝑡
$ 6 1 ℎ𝑎𝑡
𝑦 = 6𝑥
𝑐 = 6ℎ
𝑡 = 6𝑛
$ 𝑠𝑜𝑑𝑎
𝑐 = 3 2 𝑠
𝑦 = 3 2 𝑥
𝑡 = 3 2 𝑛 𝑦 = 𝑟 𝑥
Wait a minute!
12 5 𝑚𝑖 1 ℎ𝑟
𝑑 = 𝑡
𝑦 = 𝑥
2 5 𝑝𝑖𝑧𝑧𝑎 1 𝑡𝑒𝑒𝑛
𝑝 = 2 5 𝑡
𝑦 = 2 5 𝑥

8. Four hats cost $24. 24 4 = 𝑦 𝑥 𝑑=6ℎ 𝑥:ℎ𝑎𝑡𝑠 𝑦:𝑐𝑜𝑠𝑡 24 𝑦 4 𝑥
Example A: Please model with an equation.
Four hats cost $24.
$ ℎ𝑎𝑡 = $ ℎ𝑎𝑡 24 𝑦 4 𝑥
24 4 = 𝑦 𝑥
4 𝑦 =
24 𝑥
𝑑=6ℎ 4𝑦 =
24𝑥 4 4
𝑥:ℎ𝑎𝑡𝑠
𝑦:𝑐𝑜𝑠𝑡 𝑦 = 6𝑥

9. Six sodas cost $9. 9 6 = 𝑦 𝑥 𝑑= 3 2 𝑠 𝑥:𝑠𝑜𝑑𝑎 𝑦:𝑐𝑜𝑠𝑡 9 𝑦 6 𝑥
Example B: Please model with an equation.
Six sodas cost $9.
$ 𝑠𝑜𝑑𝑎 = $ 𝑠𝑜𝑑𝑎 9 𝑦 6 𝑥
9 6 = 𝑦 𝑥
6 𝑦 =
9 𝑥
𝑑= 3 2 𝑠 6𝑦 = 9𝑥 6 6
𝑥:𝑠𝑜𝑑𝑎
𝑦:𝑐𝑜𝑠𝑡
9 6 𝑥
3 2 𝑥 𝑦 =

10. 12 miles in 5 hours 12 5 = 𝑦 𝑥 𝑑= 12 5 𝑡 𝑥:𝑡𝑖𝑚𝑒 (ℎ𝑟𝑠) 𝑦:𝑑𝑖𝑠𝑡𝑎𝑛𝑐𝑒 (𝑚𝑖)
Example C: Please model with an equation.
12 miles in 5 hours 12 𝑦
𝑚𝑖 ℎ𝑟 = 𝑚𝑖 ℎ𝑟 5 𝑥
= 𝑦 𝑥
5 𝑦 =
12 𝑥
𝑑= 12 5 𝑡 5𝑦 =
12𝑥 5 5
𝑥:𝑡𝑖𝑚𝑒 (ℎ𝑟𝑠)
𝑦:𝑑𝑖𝑠𝑡𝑎𝑛𝑐𝑒 (𝑚𝑖)
12 5 𝑥 𝑦 =

11. 4 pizzas were ordered to feed 10 teens
Example D: Please model with an equation.
4 pizzas were ordered to feed 10 teens
𝑝𝑖𝑧𝑧𝑎 𝑡𝑒𝑒𝑛 = 𝑝𝑖𝑧𝑧𝑎 𝑡𝑒𝑒𝑛 4 𝑦 10 𝑥
= 𝑦 𝑥
10 𝑦 =
4 𝑥
𝑝= 2 5 𝑡
10𝑦 = 4𝑥 10 10
𝑥:𝑡𝑒𝑒𝑛𝑠
𝑦:𝑝𝑖𝑧𝑧𝑎
2 5 𝑥
4 10 𝑥 𝑦 =

12. If we have a proportional relationship
And we want to represent it with an equation (model)
We have two options to create our equation.
Find Unit Rate
Cross Multiply

13. 𝑦= 𝑥 𝑟 16 cups of flour are needed to produce 6 cakes 6 16 = 𝑦 𝑥
Example E:
16 cups of flour are needed to produce 6 cakes
Find Unit Rate
Cross Multiply 6
𝑐𝑎𝑘𝑒 𝑓𝑙𝑜𝑢𝑟 = 𝑐𝑎𝑘𝑒 𝑓𝑙𝑜𝑢𝑟 𝑦 16 𝑥
6 16
3 8 6
𝑐𝑎𝑘𝑒 𝑓𝑙𝑜𝑢𝑟
÷16
𝑐𝑎𝑘𝑒 1 𝑓𝑙𝑜𝑢𝑟
= 𝑦 𝑥 16
÷16
16 𝑦 =
6 𝑥
3 8
𝑦= 𝑥 𝑟
16𝑦 = 6𝑥 16 16
𝑥:𝑓𝑙𝑜𝑢𝑟
𝑦:𝑐𝑎𝑘𝑒𝑠
𝑐= 3 8 𝑓
6 16 𝑥
3 8 𝑥 𝑦 =

14. 𝑦= 𝑥 𝑟 A car uses 1 1 2 gallons to travel 45 miles. 𝑑=30𝑔 𝑥:𝑔𝑎𝑙𝑙𝑜𝑛𝑠
Example F:
Find Unit Rate
Cross Multiply 45
𝑚𝑖 𝑔𝑎𝑙 = 𝑚𝑖 𝑔𝑎𝑙 𝑦
3 2 𝑥
÷ 3 2
90 3 30 45
𝑚𝑖 𝑔𝑎𝑙
𝑚𝑖 1 𝑔𝑎𝑙
= 𝑦 𝑥
1 1 2
3 2
÷ 3 2
3 2 𝑦 =
45 𝑥
3 2 𝑦
𝑦= 𝑥 𝑟 =
45𝑥 30
3 2
3 2
𝑑=30𝑔
𝑥:𝑔𝑎𝑙𝑙𝑜𝑛𝑠
𝑦:𝑑𝑖𝑠𝑡𝑎𝑛𝑐𝑒 (𝑚𝑖)
90 3 𝑥 𝑦 =
30𝑥