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Evaluating Statements About Enlargements (2D and 3D)
Before the
Lesson
  • Assessment Task: A Fair Price (15 minutes)
    • Set this task, in class or for homework, a few days before the formative assessment lesson. This will give you an opportunity to assess the work to find out the kinds of difficulties students have with it. Then you will be able to target your help more effectively in the follow-up lesson.

      • Give each student a copy of A Fair Price.
      • Briefly introduce the task and help the class to understand the problems and their context.
        • Read through the questions and try to answer them as carefully as you can.
        • Show all your work so that I can understand your reasoning.
      • Explain to the students what "a fair price" means.
        • In the questions, the term "a fair price" means that the amount you get should be in proportion to the amount you pay.
        • So, for example, if a pound of cookies costs $3, a fair price for two pounds will be $6.

      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 that should help them. Explain to students that by the end of the next lesson, they should expect to be able 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 will 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 below. 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.

      Write a list of questions, applicable to your own class. If you have enough time, add appropriate questions to each piece of your students' work.

      Common Issues: Suggested Questions and Prompts:

      Student assumes the diagrams are accurate representations.

      • For example: The student writes "I've counted the candy. The larger circle has more than twice the amount of candy that the smaller one has."
      • Or: The student writes "Three small pizzas fit into the large one."
      • The pictures are not accurate.
      • How can you use math to check that your answer is accurate?

      Student fails to mention scale.

      • For example: The areas of the two pizzas are calculated but not the scale of increase.
      • How can you figure out the scale of increase in area/volume using your answers?

      Student focuses on non-mathematical issues.

      • For example: The student writes "It takes longer to make three small pizzas than one large one. The large one should cost $8."
      • Now consider a fair price from the point of view of the customer.
      • Are three small pizzas equivalent to one big one? How do you know?

      Student makes a technical error.

      • For example: The student substitutes the diameter into the formula instead of the radius.
      • Or: The student makes a mistake when calculating an area or volume.
      • What does r in the formula represent?
      • Check your calculations.

      Student simply triples the price of the pizza or doubles the price of a cone of popcorn.

      • Do you really get three times as much pizza?
      • Do you really get twice as much popcorn?

      Student correctly answers all the questions.

      • Student needs an extension task.
      • If a pizza is made that has a diameter four times bigger (ten times/n times), what should its price be? How do you know? Can you use algebra to explain your answer?
      • If a cone of popcorn has a diameter and height four times bigger (ten times/n times), what should its price be? How do you decide? Can you use algebra to explain your answer?
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