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Calculating Volumes of Compound Objects
Before the
  • Assessment Task: Glasses (15 minutes)
    • Have students do the assessment 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 find out the kinds of difficulties that students have with it. You will then be able to target your help more effectively in the follow-up lesson.

      • Give each student a copy of the Glasses task sheet. Introduce the task briefly and help the class to understand the problem and its context.

        • Spend 15 minutes working individually to answer these questions.
        • Show all your work on the worksheet.

      It is important that, as far as possible, students answer the questions without assistance.

      Students who sit together often produce similar answers and then when they come to compare their work, they have little to discuss. For this reason, we suggest that if students do the assessment task in class, you ask them to move to different seats. Then at the beginning of the formative assessment lesson, allow them to return to their usual places. Experience has shown that this produces more profitable discussions.

      Students should not worry too much if they cannot understand or do everything because there will be a lesson using this task which should help them. Explain to students that by the end of that lesson they should expect to answer questions such as these with confidence. 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. The purpose of this is to forewarn you of issues that will arise during the lesson itself, so that you may prepare carefully.

      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 list 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 your own lists of questions, based on your own students' work, using the ideas below. 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 help the majority of students. These can be written on the board at the beginning of the lesson.

      Common Issues: Suggested Questions and Prompts:

      Student does not discriminate between length, area, and volume formulas.

      • For example: The student multiplies too many or too few numbers together to calculate a volume.
      • Or: The student chooses an incorrect formula that involves the square of the height (Q3).
      • What is the difference between a length, an area, and a volume formula?
      • Compare the formulas for the cylinder and cone. What is their common base area? How does that show in the formulas?

      Student has difficulty in identifying the values to substitute for variables in the formula.

      • For example: The student does not match the variables in the formula with measures on the figure when applying a formula, using diameter rather than radius, or multiplying arbitrary numbers (Q1).
      • Or: The student uses slant height rather than the Pythagorean Theorem to find perpendicular height (Q1c).
      • What measures do the variables in your formula stand for? Draw these in on the diagram.

      Student makes calculation errors.

      • For example: The student makes a numerical error in calculation, such as doubling rather than squaring.
      • How can you check your answers?

      Student chooses wrong formula.

      • For example: The student chooses the pyramid formula for a prism (Q1).
      • What is the difference between a prism and a pyramid?
      • Is a cylinder a prism or a pyramid? Explain your answer.

      Student has difficulty decomposing a 3D shape.

      • For example: The student does not calculate the volume of a compound shape using appropriate formulas for constituent parts. There is no attempt, use of a single incorrect formula, or an incorrect decomposition.
      • Imagine you can take this 3D shape to pieces. What pieces would you make in order to calculate the volume using formulas you know?
      • For Glass 3, the bowl of the glass goes down into the stem.

      Student assumes proportionality.

      • For example: The student assumes that halving the volume also halves the height, giving a response of 3 cm (Q2).
      • Look back at Q1b. Are the volumes of the two parts of the figure the same? How does this affect your answer?

      Student answers all questions correctly.

      • The student needs an extension task.
      • Write down how someone could decide which formulas represent volumes.
      • Find the height of the liquid in Glass 3 when it is half full. Explain your answer.
      • Show step-by-step how the formula given in question 4 was derived.
      • Make up a challenging glass volume problem of your own and solve it.
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