# Gas Tank - TN Core

A car’s gas tank holds 16 gallons. After filling it for 20 seconds, the tank contains 2.5 gallons. a) How much gas is in the tank after 50 seconds? Describe how you determined that amount in words. b) If there is 8 gallons in the tank, how much time has passed? Explain your method using words or a model. c) Jeff says it would take 128 seconds to fill the tank. Explain why he is correct or incorrect using your own method.

Common Core State Standards for Mathematical Content Common Core State Standards for Mathematical Practice

7.RP.2. Recognize and represent proportional relationships a) c) between quantities.

Decide whether two quantities are in a proportional relationship.

Represent proportional relationships by equations.

1. Make sense of problems and persevere in solving them.

2. Reason abstractly and quantitatively.

3. Construct viable arguments and critique the reasoning of others.

4. Model with mathematics.

5. Use appropriate tools strategically.

6. Attend to precision.

7. Look for and make use of structure.

8. Look for and express regularity in repeated reasoning.

Essential Understandings/NCTM Resources

Reasoning with ratios involves attending to and coordinating two quantities.

A ratio is a multiplicative comparison of two quantities, or it is a joining of two quantities in a composed unit.

A number of mathematical connections link ratios and fractions: o Ratios are often expressed in fraction notation, although ratios and fractions do not have identical meaning. o Ratios are often used to make “part-part” comparisons, but fractions are not. o Ratios and fractions can be thought of as overlapping sets.

A proportion is a relationship of equality between two ratios. In a proportion, the ratio of two quantities remains constant as the corresponding values of the quantities change.

Proportional reasoning is complex and involves understanding that— o Equivalent ratios can be created by iterating and/or partitioning a composed unit; o If one quantity in a ratio is multiplied or divided by a particular factor, then the other quantity must be multiplied or divided by the same factor to maintain the proportional relationship o The two types of ratios – composed units and multiplicative comparisons – are related.

 A rate is a set of infinitely many equivalent ratios.

 Several ways of reasoning, all grounded in sense making, can be generalized into algorithms for solving proportion problems.

Explore Phase

Possible Solution Paths

Rate Approach a) Students may develop a rate from the problem noticing that

2.5 gallons filled the tank after 20 seconds. From this information, they can create a rate of 1 gallon per 8 seconds. This will create the ratio of

2.5 𝑔𝑎𝑙𝑙𝑜𝑛𝑠

20 𝑠𝑒𝑐𝑜𝑛𝑑𝑠

. When

1 𝑔𝑎𝑙𝑙𝑜𝑛 reduced, the rate becomes

8 𝑠𝑒𝑐𝑜𝑛𝑑𝑠

. From this, the student can develop the amount of gallons after 50 seconds by multiplying 1/8. This will yield 6.25 gallons.

Proportion a) Students may setup a proportion from the noticing that 2.5 gallons filled the tank after 20 seconds.

Suggested Proportions:

2.5 𝑔𝑎𝑙𝑙𝑜𝑛𝑠

20 𝑠𝑒𝑐𝑜𝑛𝑑𝑠

= 𝑥

50 𝑠𝑒𝑐𝑜𝑛𝑑𝑠

𝑜𝑟

20 𝑠𝑒𝑐𝑜𝑛𝑑𝑠

2.5 𝑔𝑎𝑙𝑙𝑜𝑛𝑠

=

50 𝑠𝑒𝑐𝑜𝑛𝑑𝑠 𝑥

Using cross products and solving for x, the student determines that the value of x is 6.25 gallons. b) Students may set up the following proportions:

2.5 𝑔𝑎𝑙𝑙𝑜𝑛𝑠

20 𝑠𝑒𝑐𝑜𝑛𝑑𝑠

=

Suggested Proportions:

8 𝑔𝑎𝑙𝑙𝑜𝑛𝑠

𝑜𝑟 𝑥

20 𝑠𝑒𝑐𝑜𝑛𝑑𝑠

2.5 𝑔𝑎𝑙𝑙𝑜𝑛𝑠

= 𝑥

8 𝑔𝑎𝑙𝑙𝑜𝑛𝑠

Using cross products and solving for x, the student determines that the value of x is 64 seconds. c) Students may setup a proportion and use cross products to prove proportionality.

2.5 𝑔𝑎𝑙𝑙𝑜𝑛𝑠 16 𝑔𝑎𝑙𝑙𝑜𝑛𝑠

=

20 𝑠𝑒𝑐𝑜𝑛𝑑𝑠 128 𝑠𝑒𝑐𝑜𝑛𝑑𝑠

When creating the cross products, the student gets 320 =

320. The student needs to explain that since the cross products are equal, the proportions are equivalent.

Assessing Questions:

Why did you setup the ratio of gallons to seconds?

How did you use the ratio to find the amount of gas after 50 seconds?

Can you draw a picture/model that represents the situation?

 Could you create a table that extends the pattern to prove that the relationship is proportional?

Assessing Questions:

Does it matter which unit goes on the top in the proportion?

Why is the x in the denominator/numerator this time?

Is your proportion in part c equivalent? Do the ratios simplify to the same rate?

How could figuring out the scale factor between the first ratio (20 seconds) to the second ratio (50 seconds) help you find the missing amount of gallons?

Is there another way to find the time in Part B?

What does the value 320 mean in the sense of the problem?

(What is the label that goes with it?)

Scaling/Equivalent Ratios a) Students make the connection between 20 seconds and 50 seconds. The multiple is 2.5. Therefore, they use 2.5 and multiply 2.5 gallons by 2.5. This gives them 6.25 gallons as the amount at 50 seconds. c) If you simplify the rate of 2.5/20 and 16/128, they both reduce to 1/8.

Possible Student Misconceptions

Incorrect Number used in Rate Approach

Students may incorrectly multiply by the rate. For example, students may multiply the 50 seconds by 8.

Assessing Questions:

 Why did you multiply by 2.5?

 How can you use this connection to find other gallons? (Offer the time of 85 seconds or a like number)

Incorrect Setup of Proportion

ALL PARTS

Students may make several errors that create an incorrect proportion.

Entry/Extensions

If students can’t get started….

If students finish early….

Assessing Questions:

Does the number you got make sense compared to the size of the tank?

 If you multiply by the rate you got, will you get 2.5 gallons?

Assessing Questions:

Does keeping the units the same in each ratio matter? a.

Should I flip the gallons and seconds?

If setting up each ratio with the units in the same place is important, how does this change your answer?

Assessing Questions:

 What do we know about the size of the tank?

 If we doubled the amount of time (20 seconds), what would happen to the gallons of gas?

How could you compare Jeff’s statement to your work?

Assessing Questions:

Can you solve the problem without a proportion? o Can you make a model/picture/table?

Is there a way to find out how much gas would be in the tank at: o 42 seconds? 100 seconds? n seconds?

 Can we write an equation based off of the problem to find the amount of gas at any time?

Discuss/Analyze

Whole Group Questions

Part A

 The students should be able to create either a proportion or develop a rate based off of the knowledge that the tank is filling at a rate of

1 gallon per 8 seconds. o What is the rate? o How did you setup your proportion? o Did you have to find rate to solve? o Is there a way to extend the pattern to other times?

Part B

The students need to find the connection between the rate or use a proportion to solve for the amount of time that has passed. o Can you use the rate for part a to solve part b?

 How? o Does the order of the units in the proportion matter? o Is there another way to solve this problem?

 Is there an equation method to solve with?

Part C

 The students need to make a connection between the two amounts and determine that the ratios are proportional. There are several ways to determine this. They could simplify the ratios. They could set them in proportion and prove using cross products. o Do both the ratios simplify to the same equivalent ratio? o Are the ratios equivalent? What does that mean?