We used this meeting to explore a problem from Mathematical Mindsets by Jo Boaler. I had worked on it a few weeks ago as part of an online book group with LINCS. I decided not to give out all the questions in the task at once, but you can look at the problem URL above to see the whole thing.
We started by looking at different rectangles…
Imagine a rectangle with an area of 20 sq. cm. What could its length and width be? List at least five different combinations.
The instructions threw us a bit. The whole number factors of 20 are 1 x 20, 2 x 10, and 4 x 5. That’s only three rectangles. Did they consider 1 x 20 different than 20 x 1? That didn’t seem right. One is just a rotation of the other. Then they must be including fractional factors (is that the right terminology?).
Solange and Linda found 2 1/2 x 8, 3 1/3 x 6 and 1 1/4 x 16. We cut the rectangles out of graph paper and put them on the board.
Solange noticed that if you start with 2 factors (for example, 4 x 5), multiply the first factor by 2 and divide the second factor by 2, you will get the same product. So, 4 x 2 = 8 and 5/2 = 2.5, so 8 x 2.5 = 20. We used this method to find other factors of 20, including fractions.
This brought us to a question: How does Solange’s pattern work in this series of multiplication problems? What do you think? Is it the same as the change from 4 x 5 to 8 x 2 1/2 to 16 x 1 1/4?
4 x 5 = 20 6 x 3 1/3 = 20 8 x 2 1/2 = 20
If you enlarge each of your rectangles by a scale factor of 2, what would their new dimensions be? What would their areas be? What do you notice?
We talked a little about what it means to enlarge a rectangle by a scale factor of 2. We decided it meant to double the height and the width. We discovered that the area of the second rectangle was always four times the are of the original rectangle.
What happens when you enlarge rectangles with different areas by a scale factor of 2? What if you enlarge them by a scale factor of 3? Or 4? Or 5 …? Or k? What if k is a fraction?
We found that tripling the height and width of a rectangle increased the area by 9 times the original area. Davida wrote a conjecture based on the scale factors of 2 and 3.
Davida’s conjecture: The amount of the increase of the dimensions correlates with the increase in size with relation to the original area. The increase amount is squared and multiplied by the original area to obtain the area of the new rectangle.
A scale factor of 2 resulted in an increase of 4 times the original area. A scale factor of 3 resulted in an increase of 9 times the original area. What about scale factor of 1.5?
Solange and Linda’s general rule: The scale factor squared times the original area gives you the new area.
Do your conclusions apply to plane shapes other than rectangles?
It looked like the general rule applied to triangles and circles.
Solange and Linda made a new conjecture: The general rule above applies to rectangles, triangles, circles, parallelograms, etc.
This brought up some questions:
- Will it work for irregular shapes?
- Will it work for a cross, an X, a T-shaped figure?
Now explore what happens to the surface area and volume of different cuboids when they are enlarged by different scale factors. Do your conclusions apply to solids other than cuboids?
We had time to explore the volume of rectangular prisms, but didn’t get to surface area or what would happen with other 3-dimensional figures (We really wished we had manipulatives such as snap cubes at this point in the meeting. Actually, tiles would have been really helpful when exploring area as well.)
In summary, we talk about how this lesson would be useful to help students understand conversions of square yards to square feet or square feet to square inches. For example, how many square inches are in a square foot? Our first instinct is to say, 12, of course. However, if you draw a square foot and then break it into square inches, you will count 144 square inches. In the terminology we used today, we might say that the scale factor was 12, so the increase of a 2-dimensional object would be 122 times the original number. In truth, the figure is staying the same size, but the numbers used in the dimensions are increase by 12, so the calculation is the same. How many cubic inches are there in a cubic foot?
Location: NYCCT, 25 Chapel St.
Attended: Eric, Davida. Linda, Solange
Programs represented: BMCC, LAGCC, NYCCT