Unit 5. Day 14.

Modeling Proportional Relationships Traditional Alternate

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 $ 1 ℎ𝑎𝑡 6 24 ÷4 Four hats cost $24. 4 ÷4 𝑐 𝑡 = 6 𝑛 ∙ ℎ 𝑥:ℎ𝑎𝑡𝑠 𝑦:𝑐𝑜𝑠𝑡 𝑦 = 6𝑥

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 $ 1 𝑠𝑜𝑑𝑎 $ 𝑠𝑜𝑑𝑎 9 $ 1 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

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

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

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 $ 3 2 1 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 = 12 5 𝑡 𝑦 = 12 5 𝑥 2 5 𝑝𝑖𝑧𝑧𝑎 1 𝑡𝑒𝑒𝑛 𝑝 = 2 5 𝑡 𝑦 = 2 5 𝑥

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

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 =

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 12 5 = 𝑦 𝑥 5 𝑦 = 12 𝑥 𝑑= 12 5 𝑡 5𝑦 = 12𝑥 5 5 𝑥:𝑡𝑖𝑚𝑒 (ℎ𝑟𝑠) 𝑦:𝑑𝑖𝑠𝑡𝑎𝑛𝑐𝑒 (𝑚𝑖) 12 5 𝑥 𝑦 =

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 4 10 = 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 =

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

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 6 16 = 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 =

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 45 3 2 = 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