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Proportional Reasoning

Unit

Proportional Reasoning

Objectives

Students will apply proportional reasoning to solve problems. Students will:

  • use proportions to solve problems, and identify and interpret the unit rate or constant of proportionality.
  • determine if quantities are proportionally related.
  • interpret the points on the graph of a proportional relationship.

Essential Questions

  • How is mathematics used to quantify, compare, represent, and model numbers?
  • How are relationships represented mathematically?
  • How can expressions, equations and inequalities be used to quantify, solve, model and/or analyze mathematical situations?

Related Unit and Lesson Plans

Related Materials & Resources

The possible inclusion of commercial websites below is not an implied endorsement of their products, which are not free, and are not required for this lesson plan.

http://ca.ixl.com/math/grade-7/solve-proportions-word-problems

  • This activity allows students to find the constant of proportionality from different graphs.

http://ca.ixl.com/math/grade-7/constant-rate-of-change

  • This Web page provides examples of solving proportions, as well as instruction on setting up proportions to solve problems. The page also explains how to check for proportionality.

https://www.purplemath.com/modules/ratio2.htm

  • This activity gives students an opportunity to determine whether different ratios form a proportion. Ratios are given in different forms.
    http://ca.ixl.com/math/grade-7/do-the-ratios-form-a-proportion
  • This activity gives students an opportunity to examine verbal descriptions for presence of proportionality.

http://ca.ixl.com/math/grade-7/do-the-ratios-form-a-proportion-word-problems

  • This activity asks students to find the constant of variation or unit rate, given some graphs.

http://ca.ixl.com/math/grade-8/find-the-constant-of-variation-graphs

  • This activity asks students to graph proportional relationships by clicking points on a graph.

http://ca.ixl.com/math/grade-8/graph-a-proportional-relationship

  • This activity provides students additional practice with finding the unit rate, given a written description.

http://ca.ixl.com/math/grade-8/unit-rates

Formative Assessment

  • View

    Multiple-Choice Items:

    1. Given the proportion , what is the unit rate?
    1. 3
    2. 6
    3. 9
    4. 12

     

    1. 104 oz = ___ c ? (Hint: There are 8 ounces in 1 cup.)
    1. 12 c
    2. 13 c
    3. 14 c
    4. 15 c

     

    1. Natalie writes 8 articles every 5 days. According to this rate, how many articles will she write in 30 days?
    1. 33
    2. 36
    3. 42
    4. 48

     

     

     

     

    1. Which of the following shows a proportional relationship?

     

    C.

    D.

     

    1. Which of the following forms a proportion with 2:3?
    1. 4:6
    2. 4:9
    3. 6:6
    4. 6:12

     

     

     

     

     

     

    1. Which of the following tables shows a proportional relationship?

    1. Given the graph below, what is the unit rate?

    1. 1 ounce/cup
    2. 6 ounces/cup
    3. 8 ounces/cup
    4. 16 ounces/cup

     

     

     

    1. Given the graph below, which point represents 55 miles run after a total of 5 weeks?

    1. (5, 55)
    2. (50, 5)
    3. (55, 5)
    4. (0, 55)

     

     

     

    1. According to the graph below, how many laps will a swimmer have completed in 19 weeks?

    A.153

    B.162

    C.171

    D.180

     

     

    Multiple-Choice Answer Key:

     

    1. A

    2. B

    3. D

    4. C

    5. A

    6. A

    7. C

    8. A

    9. C

     

     

    Short-Answer Items:

    1. Adrian spends $400 on groceries every 5 weeks. How many weeks will pass before she has spent a total of $1,200? Set up a proportion that may be used to solve the problem and solve.

     

    1. There are 5 math teachers for every 80 students. There are 9 math teachers for every 144 students. Does this describe a proportional relationship? Explain.

     

    1. Given the graph below, identify the point that represents the unit rate. Explain the meaning of the unit rate, as related to the context shown in the graph.

     

    Short-Answer Key and Scoring Rubrics:

    1. Adrian spends $400 on groceries every 5 weeks. How many weeks will pass before she has spent a total of $1,200? Set up a proportion that may be used to solve the problem and solve.

    A correct proportion is . Solving for x gives x = 15. Thus, 15 weeks will pass before she has spent $1,200 on groceries.

    Points

    Description

    2
    • The student provides an accurate proportion and solution.

    1
    • The student provides an accurate proportion or solution.

    0
    • The student does not provide an accurate proportion or solution.

     

     

     

     

     

    1. There are 5 math teachers for every 80 students. There are 9 math teachers for every 144 students. Does this describe a proportional relationship? Explain.

    The correct answer is yes. An accurate explanation is as follows: The ratios,  and , are equal. Thus, the relationship shows a proportional relationship.

    Points

    Description

    2
    • The student provides a correct answer and explanation.

    1
    • The student provides a correct answer or explanation.

    0
    • The student does not provide a correct answer or explanation.

     

     

     

     

     

     

    1. Given the graph below, identify the point that represents the unit rate. Explain the meaning of the unit rate, as related to the context shown in the graph.

    The unit rate is represented by the point (1, 25). This indicates that after 1 month, she has spent $25 on medicine. In other words, she spends $25 per month on medicine.

    Points

    Description

    2
    • The student provides an accurate point and explanation.

    1
    • The student provides an accurate point or explanation.

    0
    • The student does not provide an accurate point or explanation.

     

     

     

     

    Performance Assessment:

    1. Arielle’s current cellphone service costs $58 per month. What will be her cumulative costs at the end of one year? Write a proportion that may be used to solve the problem. Solve.

     

     

     

    1. Suppose the maximum amount Arielle may spend on cumulative costs per year is $525. Considering the service will cost a flat rate each month, write an equation that may represent the amount she can afford per month.

     

     

     

    1. Compare the unit rate for problems #13 and #14 above. Describe the appearance of the graphs that would represent the situations in problems #13 and #14.

     

     

     

    1. Arielle analyzes her budget and realizes that she may be able to spend a maximum of $600 each year on cellphone service. What flat rate, per month, would she be able to afford? Write a proportion that may be used to solve the problem. Solve.

     

     

     

    1. Arielle’s friend, Jason, shows her a graph of the relationship between number of months and cumulative cellphone service costs. The unit rate is (1, 45). Does this graph represent a proportional relationship that would fit the budget from problem #14 above? Explain.

     

     

     

    1. A competing phone company is offering a special promotion that involves a flat rate of $30 per month, plus a cost of $0.17 for each text message sent. Is this relationship proportional? Explain. If Arielle sends a minimum of 250 text messages per month, how will the cost of this offer compare to the costs per month for the other monthly rates described above?

     

     

    Performance Assessment Key:

    1. Arielle’s current cellphone service costs $58 per month. What will be her cumulative costs at the end of one year? Write a proportion that may be used to solve the problem. Solve.

    58 × 12 = $696                x = $696

     

    1. Suppose the maximum amount Arielle may spend on cumulative costs per year is $525. Considering the service will cost a flat rate each month, write an equation that may represent the amount she can afford per month.

    525 ÷ 12 = $43.75

     

    1. Compare the unit rate for problems #13 and #14 above. Describe the appearance of the graphs that would represent the situations in problems #13 and #14.

    The unit rate for the first graph is . The unit rate for the second graph is .

    Both would be straight lines going through the origin, (0, 0). The first line would have a steeper incline and it would go through the points (1, 58) and (12, 696). The second line would have a flatter incline and it would go through the points (1, 43.75) and (12, 525).

     

    1. Arielle analyzes her budget and realizes that she may be able to spend a maximum of $600 each year on cellphone service. What flat rate, per month, would she be able to afford? Write a proportion that may be used to solve the problem. Solve.

               x = $50

     

    1. Arielle’s friend, Jason, shows her a graph of the relationship between number of months and cumulative cellphone service costs. The unit rate is (1, 45). Does this graph represent a proportional relationship that would fit the budget from problem #14 above? Explain.

    Yes. Arielle can afford $50 per month or less. The unit rate of $45/month is less than $50/month. This plan is in Arielle’s budget.

     

    1. A competing phone company is offering a special promotion that involves a flat rate of $30 per month, plus a cost of $0.17 for each text message sent. Is this relationship proportional? Explain. If Arielle sends a minimum of 250 text messages per month, how will the cost of this offer compare to the costs per month for the other monthly rates described above?

    No. The flat rate is added on to the constant rate which means the graph of the line would not go through the origin. The y-intercept would be (0, 30), not (0, 0). The annual cost would be $402.50. The monthly cost would average $33.54. This is less expensive than the plans mentioned above.

    Performance Assessment Scoring Rubric:

     

    Points

    Description

    4
    • All answers are complete and correct.
    • Work is shown and goes beyond the requirements of the problem.
    • Demonstrates advanced understanding of modeling word problems with proportions and equations, solving, applying unit rates to make decisions, comparing plans, and providing explanations for answers.

    3
    • Most answers are complete and correct.
    • Work is shown for most problems and all requirements of the problems are met.
    • Demonstrates a strong understanding of modeling word problems with proportions and equations, solving, applying unit rates to make decisions, comparing plans, and providing explanations for answers.

    2
    • Many answers are complete and correct.
    • Little work is shown and it fails to meet all the requirements of the problems.
    • Demonstrates a limited understanding of modeling word problems with proportions and equations, solving, applying unit rates to make decisions, comparing plans, and providing explanations for answers.

    1
    • Some answers are complete and correct.
    • Very little correct work is shown.
    • Demonstrates a very limited understanding of modeling word problems with proportions and equations, solving, applying unit rates to make decisions, comparing plans, and providing explanations for answers.

    0
    • Very few or no answers are correct.
    • No work attempted or requirements of problems not even partially met.
    • Demonstrates no understanding of modeling word problems with proportions and equations, solving, applying unit rates to make decisions, comparing plans, and providing explanations for answers.
Final 05/17/2013
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