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Evaluating Statements About Enlargements (2D and 3D)
During the
Lesson
  • Whole-Class Introduction (10 minutes)
    • This introduction will provide students with a model about how they should work during the collaborative tasks.

      • Give each student a mini-whiteboard, a pen, and an eraser. Use the projector resource, Enlarging Rectangles
        • Decide whether each statement is true or false.
        • Write a convincing explanation.
        • If you think a statement is false, then replace it with a correct one.
      • After a few minutes, ask two or three students for their answers. Encourage them to write their explanations on the board.
      • If students are struggling to provide convincing arguments, you could ask the following questions:
        • Can you use a diagram to convince me? Show me.
        • How do you know for sure your answer is correct for all rectangles?
        • Can you use algebra to convince me? Show me.
        • If the area is not doubled, then what scaling is taking place? Why is this?

      Students may find it easiest to start by considering specific examples.

      • Encourage students to consider the statements more generally.

      In this way, students may see that the first statement is true, but the second should be revised.

        • If you double the length and width of a rectangle, then you multiply its area by 4.
      • You may then want to look at different scale factors:
        • If the two measurements are multiplied by 3 instead of doubled, what happens to the perimeter and area?
        • If the two measurements are increased by a scale factor of 10/n, what happens to the area?
  • Collaborative Activity: Scaling Up (20 minutes)
      • To introduce this task you may want to use the projector resource 3D Shapes.
      • You could also show the class real examples of these 3D shapes.
      • Organize the students into groups of two or three.
      • Give each group the cut-up cards Scaling Up, a copy of the Formula Sheet, a large sheet of paper, and a glue stick.
        • The cards show rectangular prisms, circles, spheres, cylinders, and cones.
        • Sort the cards into these five different mathematical objects.
        • Your task is to decide whether each statement is true or false.
        • If you think a statement is false, change the second part of the statement to make it true.
        • Try to figure out what it is about the formula for the shape's area or volume that makes the statement true or false.
        • Show calculations, draw diagrams, and use algebra to convince yourself that you have made a correct decision.
        • When everyone in your group agrees with the decision for one object, place the statements on the poster and write your explanations around it.
        • Begin by working with the statements on rectangular prisms.
        • You may not have time to consider all twelve statements.
        • It is better for you to explain your reasoning fully for a few statements than to rush through trying to decide whether all the statements are true or false.
      • If you think your class will understand the notion of leaving π in the answer, then do not give out calculators as their use can prevent students noticing the factor of the increase.

      You have two tasks during the small group work: to make a note of student approaches to the task, and to support student problem solving.

      Make a note of student approaches to the task.

      Listen and watch students carefully. In particular, listen to see whether students are addressing the difficulties outlined in the Common Issues table. You can then use this information to focus a whole-class discussion towards the end of the lesson.

      Support student problem solving.

      Try not to do the thinking for students but rather help them to reason for themselves. Encourage students to engage with each others' explanations, and take responsibility for each others' understanding.

        • Judith, why do you think this statement is true/false?
        • James, do you agree with Judith? Can you put her explanation into your own words?
      • If students are struggling to get started on the task:
        • What formula can you use to check if the statement is correct? What values can you put into this formula?
        • If it is not twice as big, by what factor has the area/volume increased? How do you know?
      • At first, you may want to focus your questioning on the cards about rectangular prisms:
        • Are any of these statements true? What is it about the formula that makes the formula true?
        • What has the volume of the rectangular prism increased by for this statement? How does this increase relate to the formula?

      Students often prefer to multiply out π. This means they may not notice the factor of increase.

        • Can you express these two areas/volumes for this statement as multiples of π? How does this help?

      Students often do not recognize the relationship between the formula and the factor of the increase.

        • Show me two statements that are correct. What has doubled in each formula? What has remained the same?
        • Show me two statements where the area or volume has increased by a factor of four. Look at the two formulas and figure out why the area(s)/volume(s) have increased by the same factor.
        • Show me two statements where just the radius is doubled but the factors of increase are different. Look at the two formulas and figure out why the area(s)/volume(s) have increased by a different factor?
        • When the radius/height of this shape is doubled, what variable will change in the formula? [E.g., r, r2, or h.] How does this affect the area/volume?
      • If a lot of students are struggling on the same issue you may want to hold a brief whole-class discussion.
      • Encourage students who work through the task more quickly to think about how they can explain the scaling in general terms. They may use algebra in their explanation, or simply highlight the properties of a formula that determine the scaling.
        • Can you use algebra to show you are correct? If the radius has a length of n, what is double its length? How can you use this in the formula?
        • Can you figure out if the statement is true or not just by looking at the formula? Why? Why not?
      • If students finish early, have them consider what happens if the word "multiply by 3" replaces the word "double."

      Extending the Activities Over Two Lessons

      You may decide to spread the work over two lessons. If so, ask students to stop working on the task 10 minutes before the end of the lesson.

      • Ask students to glue the cards that they have worked on to the large sheet of paper. Remind them that there should be an explanation accompanying each card. Students can then use a paperclip to attach any remaining cards to their posters.
      • Hold a short whole-class discussion. Ask a representative from each group to use thier group poster to explain their thinking about one statement to the whole class. Encourage the rest of the class to challenge their explanations, but avoid intervening too much yourself.

      You can then re-start the lesson with more poster work or by sharing posters (immediately below), as you see fit.

  • Sharing Posters (10 minutes)
      • When a group has completed all the statements about one object, ask the students to compare their reasoning with that of a neighboring group.
        • Check which answers are different.
        • A member of each group needs to explain their reasoning for these answers. If anything is unclear, ask for clarification.
        • Then together consider if you should change any of your answers.
        • It is important that everyone in both groups understands the math. You are responsible for each other's learning.
  • Whole-Class Discussion (20 minutes)
      • Discuss as a class how the structure of a formula determines the increase.
        • Find me a card where the statement is correct. How does the formula relate to an increase by a factor of two? Find me another. What do the formulas have in common?
        • Find me a card that uses a formula involving r2. Is the statement correct? Why/Why not? By what factor has the area/volume increased? Is it the same increase for all cards that use r2? Why/Why not?
        • Find me a card that uses a formula involving r3. Is the statement correct? Why/Why not? By what factor has the volume increased?
      • Now try to extend some of these generalizations. Use the projector resource Is It Correct?
        • I want you to decide if any of these statements are true.
        • If you think a statement is not true, then change the last part of the statement to make it true.
      • Students may have difficulties making decisions without using specific dimensions. Encourage those students who progressed well in the lesson to think of a general explanation.
        • If the statement is correct, how do you know?
        • If the statement is not correct, then by what factor do the perimeter/area/volume increase? How do you know?
        • How does the increase relate to the formula?
        • If all the dimensions increase by five/ten/n times, what happens to the perimeter/area and volume?
      • Ask two or three students to explain their answer.

      Some students will begin to see that for similar shapes the area scale factor is the square of the scale (i.e. 32, 52, 102, or n2) and the volume scale factor is the cube of the scale (i.e. 33, 53, 103, or n3).

      • If you have time you may want to consider the same task but use a different shape, such as a circle, cylinder, or cone.
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