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Evaluating Statements About Enlargements (2D and 3D)
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Homework
  • Improve Individual Solutions to the Assessment Task (10 minutes)
      • Return to students their original assessment task A Fair Price, as well as a second blank copy of the task.
        • Look at your original responses, and think about what you have learned in this lesson.
        • Using what you have learned, try to improve your work.
      • If you have not added questions to individual pieces of work, then write your list of questions on the board. Students are to select from this list only the questions they think are appropriate to their own work.
  • Solutions: A Fair Price
      1. Jasmina is correct: if the radius doubles so will the circumference. This is a linear relationship.

        The scale factor is 2.

        The circumference of the small candy ring is 4π.

        The circumference of the large candy ring is 8π.

        A fair price for the large candy ring is 80 cents, double the price of the small one.

      2. Jasmina is incorrect.

        The area of a small pizza is 9π and that of a large one is 36π. The area of four small pizzas is equal to the area of one large pizza.

        The areas of similar figures are related by the square of the scale factor: 22 = 4.

        A correct statement might be, "I get the same amount of pizza from four small ones as one large one."

        A fair price for the large pizza would be $12, or 4 times $3.

      3. Jasmina is incorrect.

        The volume of the small cone is 8π and that of the large cone is 64π. The scale factor of the radius and height is 2.

        Volumes of similar figures are related by the cube of the scale factor: 23 = 8.

        A correct statement might be, "I get the same amount of popcorn from eight small ones as one large one."

        A fair price would be 8 × $1.20 = $9.60.

  • Solutions: Scaling Up
    • Below are the general solutions obtained by reasoning algebraically. Students will also provide solutions using on specific value.

      1. "If you double just the width of a rectangular prism then you double its volume" is true.

      If the width is a, V = alh

      If the width is 2a, V = 2alh

      An increase by a factor of 2.

      2. "If you double just the width and height of a rectangular prism then you double its volume" is false.

      If the width is a and the height is b, V = alb

      If the width is 2a and the height is 2b, V = 4alb.

      An increase by a factor of 4.

      3. "If you double the width, height, and length of a rectangular prism then you double its volume" is false.

      If the width is a, the length is b and the height is c, V = abc

      If the width is 2a, the length is 2b and the height is 2c, V = 8abc.

      An increase by a factor of 8.

      4. "If you double the radius of a circle then you double its circumference" is true.

      If the radius is a, C = 2πa

      If the radius is 2a, C = 4πa

      An increase by a factor of 2.

      5. "If you double the radius of a circle then you double its area" is false.

      If the radius is a, A = πa2

      If the radius is 2a, A = 4πa2

      An increase by a factor of 4.

      6. "If you double the radius of a sphere then you double its surface area" is false.

      If the radius is a, A = 4πa2

      If the radius is 2a, A = 16πa2

      An increase by a factor of 4.

      7. "If you double the radius of a sphere then you double its volume" is false.

      If the radius is a, V = 43πa3

      If the radius is 2a, V = 323πa3

      An increase by a factor of 8.

      8. "If you double just the radius of a cylinder then you double its curved surface area" is true.

      If the radius is a, A = 2πah

      If the radius is 2a, A = 4πah.

      An increase by a factor of 2.

      9. "If you double just the height of a cylinder then you double its volume" is true.

      If the height is b, V = πr2b

      If the height is 2b, V = 2πr2b

      An increase by a factor of 2.

      10. "If you double both the radius and height of a cylinder then you double its volume" is false.

      If the radius is a and the height is b, V = πa2b

      If the radius is 2a and the height is 2b, V = 8πa2b

      An increase by a factor of 8.

      11. "If you double just the base radius of a cone then you double its volume" is false.

      If the base radius is a, V = ⅓πa2h

      If the base radius is 2a, V = 43πa2h

      An increase by a factor of 4.

      12. "If you double both the height and base radius of a cone then you double its volume" is false.

      If the base radius is a, and the height is b, V = ⅓πa2b

      If the base radius is 2a, and the height is 2b, V = 83πa2b

      An increase by a factor of 8.

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