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Calculating Volumes of Compound Objects
During the
  • Improve Individual Solutions to the Glasses Task (15 minutes)
      • Return students' written responses to the Glasses problem.
      • Give each student a mini-whiteboard, pen, and eraser.
        • Recall what we were working on previously. What was the task?
      • Ask students to reread the Glasses task and their own work.
      • Draw their attention to the questions you have about their work.
      • If you have not added questions to students' work, write a short list of your most common questions on the board. Students can then select a few questions appropriate to their own work and begin answering them.

        • I have read your solutions and have some questions about your work.
        • I would like you to work on your own to answer my questions for 10 minutes.
        • Write your answers on your mini-whiteboard, so I can see what you're thinking about as I walk around.

      The aim of this activity is to encourage students to re-engage with their work and reflect on what they have done. Often students comment that they find even their own reasoning hard to follow! This is a sure sign that they need to work on the quality of their explanations.

  • Collaborative Small-Group Work on the Glasses Task (10 minutes)
      • Organize the students into small groups of two or three.
        • Now I would like you to work in small groups.
        • Take turns to explain what you did on the problems and where you got stuck.
        • Together, try to produce a solution that is better than your individual solutions.

      As students work, you have two tasks: to note students' approaches to the task, and to support student problem solving.

      Note students' approaches to the task.

      Listen to students, and identify the issues they are discussing. In particular, listen to see whether they are addressing the difficulties outlines in the table, Common Issues.

      Which parts of the task do they struggle with most?

      You can use this information to focus the whole-class discussion later in the lesson.

      Support student problem solving.

      From time to time, intervene and ask questions to help students clarify their thinking.

      Try not to help students too much by "taking over" their work. Instead, encourage students to explain to one another. If several students in the class are struggling with the same issue, write a relevant question on the board.

      • If a group of students continues to struggle with identifying the missing lengths and constructions in Q2 and Q3 of Glasses, offer them a copy of the Student Hint sheet.
      • For students who are succeeding with the task, suggest that they try to find the height of liquid in Glass 3 when it is half full. This is a more demanding problem.
  • Collaborative Analysis of Sample Responses to Discuss (15 minutes)
      • Give each small group of students a copy of each of the Sample Responses to Discuss.
        • None of the sample responses shows the correct answer for the volume.
        • Work together to find and explain the errors each student made.
        • Explain what the student could do to complete his or her solution correctly.

      The sample responses show some of the common errors made by students on Q1c and Q2 in trials. This task will give students an opportunity to address those common misconceptions without providing a complete solution strategy.

  • Whole-Class Discussion (15 minutes)
      • Organize a whole-class discussion.
      • The intention is for you to focus on getting students to understand the different methods of working out the answers, to articulate common misconceptions, and to share their reasoning, rather than to check numerical solutions.

        • Let's stop and talk about the different errors the students made.
        • Look at Logan and Isaac's solutions to Question 1c.
        • Which student produced the better solution? Explain your response.
        • Which height do you need to calculate this volume?
        • Zsa-Zsa, your group wrote an explanation of that. Could you explain for us, please?
        • Put Logan's/Isaac's explanation into your own words.
        • Look at Yasmin and Brianna's solutions to Question 1.
        • Which student produced the better solution? Explain your response.
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