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Increasing and Decreasing Quantities by a Percent
Before the
Lesson
  • Assessment Task: Percent Changes (15 minutes)
    • Have the students do this task, in class or for homework, a day or more before the formative assessment lesson. This will give you an opportunity to assess the work, and to find out the kinds of difficulties students have with it. You will be able to target your help more effectively in the follow-up lesson.

      • Give each student a copy of the assessment task Percent Changes.
        • Read through the questions, and try to answer them as carefully as you can. The example at the top of the page should help you understand how to write out your answers.

      It is important that, as far as possible, students are allowed to answer the questions without your assistance.

      Students should not worry too much if they cannot understand or do everything, because in the next lesson they will engage in a similar task, which should help them. Explain to students that by the end of the next lesson, they should expect to answer questions such as these confidently. This is their goal.

  • Assessing Students' Responses
    • Collect students' responses to the task. Make some notes on what their work reveals about their current levels of understanding, and their different problem solving approaches.

      We suggest that you do not score students' work. The research shows that this will be counterproductive, as it will encourage students to compare their scores and distract their attention from what they can do to improve their mathematics.

      Instead, help students to make further progress by summarizing their difficulties as a series of questions. Some suggestions for these are given on the next page. These have been drawn from common difficulties observed in trials of this unit.

      We suggest that you write a list of your own questions, based on your students' work, using the ideas that follow. You may choose to write questions on each student's work. If you do not have time to do this, select a few questions that will be of help the majority of students. These can be written on the board at the end of the lesson.

      The solution to all these difficulties is not to teach algorithms by rote, but rather to work meaningfully on the powerful idea that all percent changes are just multiplications by a scale factor.

      Common Issues: Suggested Questions and Prompts:

      Student makes the incorrect assumption that a percentage increase means the calculation must include an addition.

      • For example: 40.85 + 0.6 or 40.85 + 1.6. (Q1.)
      • A single multiplication by 1.06 is enough.
      • Does your answer make sense? Can you check that it is correct?
      • "Compared to last year, 50% more people attended the festival." What does this mean? Describe in words how you can work out how many people attended the festival this year. Give me an example.
      • Can you express the increase as a single multiplication?

      Student makes the incorrect assumption that a percentage decrease means the calculation must include a subtraction.

      • For example: 56.99 − 0.45 or 56.99 − 1.45. (Q2.)
      • A single multiplication by 0.55 is enough.
      • Does your answer make sense? Can you check that it is correct?
      • In a sale, an item is marked "50% off." What does this mean? Describe in words how you calculate the price of an item in the sale. Give me an example.
      • Can you express the decrease as a single multiplication?

      Student converts the percentage to a decimal incorrectly.

      • For example: 40.85 × 0.6. (Q1.)
      • How can you write 50% as a decimal? How can you write 5% as a decimal?

      Student uses inefficient method.

      • For example: First the student calculates 1%, then multiplies by 6 to find 6%, and then adds this answer on:

        (40.85 ÷ 100) × 6 + 40.85. (Q1.)

      • Or: 56.99 × 0.45 = ANS, then 56.99 − ANS (Q2.)
      • A single multiplication is enough.
      • Can you think of a method that reduces the number of calculator key presses?
      • How can you show your calculation with just one step?

      Student is unable to calculate percentage change.

      • For example: 450 − 350 = 100% (Q3.)
      • Or: The difference is calculated, then the student does not know how to proceed or he/she divides by 450. (Q3.)
      • The calculation (450 −350) ÷ 350 × 100 is correct.
      • Are you calculating the percentage change to the amount $350 or to the amount $450?
      • If the price of a t-shirt increased by $6, describe in words how you could calculate the percentage change. Give me an example. Use the same method in Q3.

      Student subtracts percentages.

      • For example: 25 − 20 = 5%. (Q4.)
      • Because we are combining multipliers (0.8 × 1.25 = 1), there is no overall change in prices.
      • Make up the price of an item and check to see if your answer is correct.

      Student fails to use brackets in the calculation.

      • For example: 450 − 350 ÷ 350 × 100. (Q4.)
      • In your problem, what operation will the calculator carry out first?

      Student misinterprets what needs to be included the answer.

      • For example: The answer is just operator symbols.
      • If you just entered these symbols into your calculator, would you get the correct answer?
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