1. Unit 5. Day 15.

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.
$ 𝑎𝑝𝑝𝑙𝑒
$ 𝑎𝑝𝑝𝑙𝑒
0.30
1.20 ÷4
Three apples cost $1.20 4 ÷4 𝑐 𝑡 =
0.3 𝑛 ∙ 𝑎
𝑥:𝑎𝑝𝑝𝑙𝑒
𝑦:𝑐𝑜𝑠𝑡
𝑦 =0.3𝑥

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.
4 shirts cost $50.
25 2
50 4
$ 𝑠ℎ𝑖𝑟𝑡 50
$ 𝑠ℎ𝑖𝑟𝑡
$ 𝑠ℎ𝑖𝑟𝑡 50
$ 𝑠ℎ𝑖𝑟𝑡
12.50 ÷4 ÷4 4 4 ÷4 ÷4
𝑥:𝑠ℎ𝑖𝑟𝑡𝑠
𝑦:𝑐𝑜𝑠𝑡
25 2 𝑡 𝑐 𝑦 = 𝑥 𝑛 ∙ 𝑠 𝑐 𝑡 𝑦 =
12.5 ∙ 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.
20 kilometers in 6 hours
20 6
10 3 20
𝑚𝑖 1 ℎ𝑟
3. 3
𝑘𝑚 ℎ𝑟
𝑘𝑚 ℎ𝑟 20 ÷6
𝑘𝑚 1 ℎ𝑟 ÷6 6 6 ÷6 ÷6
𝑥:𝑡𝑖𝑚𝑒 (ℎ𝑟𝑠)
𝑦:𝑚𝑖𝑙𝑒𝑠
10 3 𝑦 𝑑 = ∙ 𝑥 𝑡 𝑑 𝑦 =
3. 3 ∙ 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. Modeling Proportional Relationships
Traditional
Alternate

8. 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

9. 𝑦= 𝑥 𝑟 Three apples cost $1.20 1.2 3 = 𝑦 𝑥 𝑥:𝑎𝑝𝑝𝑙𝑒𝑠 𝑦:𝑐𝑜𝑠𝑡 𝑐=0.4𝑎 1.20
Example A:
Three apples cost $1.20
Find Unit Rate
Cross Multiply
$ 𝑎𝑝𝑝𝑙𝑒 = $ 𝑎𝑝𝑝𝑙𝑒
1.20 𝑦 3 𝑥
1.20
0.40
$ 𝑎𝑝𝑝𝑙𝑒 ÷3
$ 𝑎𝑝𝑝𝑙𝑒
= 𝑦 𝑥 3 ÷3
3 𝑦 =
1.2 𝑥
𝑦= 𝑥 𝑟
0.4 3𝑦 =
1.2𝑥 3 3
𝑥:𝑎𝑝𝑝𝑙𝑒𝑠
𝑦:𝑐𝑜𝑠𝑡
𝑐=0.4𝑎 𝑦 =
0.4𝑥

10. 𝑦= 𝑥 𝑟 We walked 10 miles in 4 hours. 10 4 = 𝑦 𝑥 𝑥:𝑡𝑖𝑚𝑒 𝑑= 5 2 𝑡
Example B:
We walked 10 miles in 4 hours.
Find Unit Rate
Cross Multiply
𝑚𝑖 ℎ𝑟 = 𝑚𝑖 ℎ𝑟 10 𝑦 4 𝑥
10 4
5 2 10
𝑚𝑖 ℎ𝑟 ÷4
𝑚𝑖 1 ℎ𝑟
= 𝑦 𝑥 4 ÷4
4 𝑦 =
10 𝑥
5 2
𝑦= 𝑥 𝑟 4𝑦 =
10𝑥 4 4
𝑥:𝑡𝑖𝑚𝑒
𝑦:𝑑𝑖𝑠𝑡𝑎𝑛𝑐𝑒
𝑑= 5 2 𝑡
10 4 𝑥
5 2 𝑥 𝑦 =

11. Example C*: Mr. Jordan paid $6 for eight donuts.
Model each situation with an equation
Example C*: Mr. Jordan paid $6 for eight donuts.
Example D*: We rode a bike miles in hour.
Example E*: A truck uses gallons to travel 45 miles.

12. 𝑦= 𝑥 𝑟 Mr. Jordan paid $6 for eight donuts. 6 8 = 𝑦 𝑥 𝑥:𝑑𝑜𝑛𝑢𝑡 𝑐= 3 4 𝑑
Example C*:
Mr. Jordan paid $6 for eight donuts.
Find Unit Rate
Cross Multiply 6
𝑐𝑜𝑠𝑡 𝑑𝑜𝑛𝑢𝑡 = 𝑐𝑜𝑠𝑡 𝑑𝑜𝑛𝑢𝑡 𝑦 8 𝑥
6 8
3 4 6
$ 𝑑𝑜𝑛𝑢𝑡 ÷8
$ 𝑑𝑜𝑛𝑢𝑡
6 8 = 𝑦 𝑥 8 ÷8
8 𝑦 =
6 𝑥
3 4
𝑦= 𝑥 𝑟 8𝑦 = 6𝑥 8 8
𝑥:𝑑𝑜𝑛𝑢𝑡
𝑦:𝑐𝑜𝑠𝑡
𝑐= 3 4 𝑑
3 4 𝑥
6 8 𝑥 𝑦 =

13. 𝑦= 𝑥 𝑟 We rode a bike 3 1 4 miles in 1 2 hour. 𝑑= 13 2 𝑡 𝑥:𝑡𝑖𝑚𝑒
Example D*:
Find Unit Rate
Cross Multiply
13 4 𝑦
𝑚𝑖 ℎ𝑟 = 𝑚𝑖 ℎ𝑟 𝑥
1 2
3 1 4
13 4
÷ 1 2
26 4
13 2
𝑚𝑖 ℎ𝑟
𝑚𝑖 1 ℎ𝑟
= 𝑦 𝑥
1 2
÷ 1 2
1 2 𝑦 𝑥 =
1 2 𝑦
13 4 𝑥
13 2
𝑦= 𝑥 𝑟 =
1 2
1 2
𝑑= 13 2 𝑡
𝑥:𝑡𝑖𝑚𝑒
𝑦:𝑑𝑖𝑠𝑡𝑎𝑛𝑐𝑒 (𝑚𝑖)
13 2 𝑥
26 4 𝑥 𝑦 =

14. 𝑦= 𝑥 𝑟 A truck uses 2 1 4 gallons to travel 45 miles. 𝑑=20𝑔 𝑥:𝑔𝑎𝑙𝑙𝑜𝑛𝑠
Example E*:
Find Unit Rate
Cross Multiply 45
𝑚𝑖 𝑔𝑎𝑙 = 𝑚𝑖 𝑔𝑎𝑙 𝑦
9 4 𝑥
÷ 9 4
180 9 20 45
𝑚𝑖 𝑔𝑎𝑙
𝑚𝑖 1 𝑔𝑎𝑙
= 𝑦 𝑥
2 1 4
9 4
÷ 9 4
9 4 𝑦 =
45 𝑥
9 4 𝑦
𝑦= 𝑥 𝑟 =
45𝑥 20
9 4
9 4
𝑑=20𝑔
𝑥:𝑔𝑎𝑙𝑙𝑜𝑛𝑠
𝑦:𝑑𝑖𝑠𝑡𝑎𝑛𝑐𝑒 (𝑚𝑖)
180 9 𝑥 𝑦 =
20𝑥