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Learning Math: Measurement

What Does It Mean To Measure? Part C: Nonstandard Units (30 minutes)

Session 1, Part C

Print this PDF and cut out the small and medium triangles to fit into the shapes.

Problem C1

Using two small triangles and one medium triangle, build each of these polygons. Note 10

  • Square
  • Rectangle that is not a square
  • Parallelogram that is not a rectangle
  • Triangle
  • Trapezoid

 


Problem C2

Without using measuring tools, can you determine which of the polygons has the greatest area? Explain. Note 11

 


Problem C3

Without using measuring tools, place the polygons in order from least to greatest by the length of their perimeters. How did you determine which shape has the smallest perimeter? Note 12

Notice that the length of the hypotenuse of the small triangle and the length of the legs of the medium triangle are equal. This length can be used as one of your “nonstandard” units. What other nonstandard unit can be established by examining the lengths of the sides? Use these nonstandard units to establish the perimeters of the shapes.

 


Problem C4

If we can measure without using standard units, why do we need standard units?

 

Video Segment

In this video segment, Lori demonstrates the method her group used to figure out the perimeter of the five different figures, using a nonstandard unit of measure. As you watch the segment, think about the possible difficulties one may encounter when using nonstandard units of measure.

Would Lori’s answer be different if she had chosen a different side or a different triangle to measure with?

You can find this segment on the session video approximately 22 minutes and 29 seconds after the Annenberg Media logo.

 


Take It Further

Problem C5

Use the Pythagorean theorem to determine the lengths of the sides of the small and medium triangles. Recall that the Pythagorean theorem states that a2 + b2= c2 for the legs (a and b) and hypotenuse (c) of a right triangle. If the length of a leg of one of the small triangles is assigned a value of 1 unit, determine the length of the hypotenuse of the small triangle and the lengths of the legs and hypotenuse of the medium triangle. Use this information to determine the perimeters of the polygons in Problem C1.

Notes

Note 10

If you are working in a group, first try working individually to make each of the five polygons (using all three triangles each time). Provide hints to colleagues who are having difficulty making a particular shape by starting the puzzle for them; put two of the three triangles in place and let them figure out where the third triangle belongs. Resist simply showing someone a finished puzzle. Once the five polygons have been completed, discuss Problems C2 and C3.

 

 


Note 11

It may appear to some people that the areas of the five shapes are different. Others will argue that the areas must be the same since each polygon is constructed from the same identical pieces. Reflect on this question, or discuss it if you are working in a group.

 

 


Note 12

There are different ways to order the polygons by perimeter, all of which require us to first establish a nonstandard unit of measure (e.g., the side of one of the triangles; the length of an eraser) and then to take some measurements in terms of the nonstandard unit. If you are working in a group, share your approaches. Be sure that everyone agrees on the order in which to place the polygons (in terms of perimeter), and that individuals justify their conclusions.

 

 

Solutions

Problem C1

 

 

 

 

 

 

 


Problem C2

All five polygons have the same area, since they are made up of the same three smaller polygons. Since there is no overlapping, the area of all five is the same as the sum of the area of the three triangles.

 


Problem C3

First, you need to arbitrarily choose a unit with which to measure the perimeters. One way to do this is to choose the length of a leg of the smaller triangle as your unit. Whatever unit you choose, you will discover that the square has the smallest perimeter. The rectangle’s perimeter is slightly larger. The parallelogram, triangle, and trapezoid are tied for the largest perimeters.

 


Problem C4

Standard units provide a frame of reference that can always be relied on. Although we can say, “This one is three times longer than that one,” standard units provide everyone with an equivalent value for “that one.” With standard units, anyone measuring the same item could say, “This one is six times longer than 1 cm.”

 

 


Problem C5

Small triangle: Using the Pythagorean theorem to determine the length of the hypotenuse, we can write the following:

c2 = a2 + b2 = 12 + 12 = 2, or
c = √2

The length of the hypotenuse is √2, or approximately 1.414 units.

Medium triangle: The length of the legs is equal to the hypotenuse of the smaller triangle, or √2. So, to determine the hypotenuse, we can write the following:

c2 = a2 + b2 =√22 +√22 = 4
So the length c is equal to 2 units.

Remember, our unit is the length of one leg of the small triangle. So the perimeters are as follows:

Square:
4 •√2, or approximately 5.656 units

Rectangle that is not a square:
2 • 1 + 2 • 2 = 6 units

Parallelogram that is not a rectangle:
2 •√2 + 2 • 2, or approximately 6.828 units

Triangle:
2 + 2 + (2 •√2), or approximately 6.828 units

Trapezoid:
2 •√2 + 1 + 3, or approximately 6.828 units

 

 

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