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(Times are approximate and will depend on the needs of the students.)
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.
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.
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.
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
5. "If you double the radius of a circle then you double its area" is false.
If the radius is a, A = πa^{2}
If the radius is 2a, A = 4πa^{2}
6. "If you double the radius of a sphere then you double its surface area" is false.
If the radius is a, A = 4πa^{2}
If the radius is 2a, A = 16πa^{2}
7. "If you double the radius of a sphere then you double its volume" is false.
If the radius is a, V = ^{4}⁄_{3}πa^{3}
If the radius is 2a, V = ^{32}⁄_{3}πa^{3}
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.
9. "If you double just the height of a cylinder then you double its volume" is true.
If the height is b, V = πr^{2}b
If the height is 2b, V = 2πr^{2}b
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 = πa^{2}b
If the radius is 2a and the height is 2b, V = 8πa^{2}b
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 = ⅓πa^{2}h
If the base radius is 2a, V = ^{4}⁄_{3}πa^{2}h
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 = ⅓πa^{2}b
If the base radius is 2a, and the height is 2b, V = ^{8}⁄_{3}πa^{2}b