Learning Math: Measurement
Area Part C: Scaling the Area (35 minutes)
Session 6, Part C
In This Part
- Similar Figures
- Scaling Polygons
What happens to the area of a figure if we scale it up or down (i.e., enlarge or reduce it)? In Part C, we review the concept of similarity and examine the relationship between a scale factor and the resulting area of the similar figure. Previously we only explored similar triangles, but in this section we will use a variety of shapes.
When we enlarge or reduce a figure, we are using an important mathematical idea: similarity. Similar figures have the same shape but are not necessarily the same size. More formally, we state that two figures are similar if and only if two things are true: (1) The corresponding angles have the same measure, and (2) the corresponding segments are in proportion. Enlarging or reducing a figure produces two figures that are similar.
The second attribute means that when we are building a similar figure, we must increase or decrease the sides multiplicatively by the scale factor. What happens to the length of each side when we enlarge a figure, say, by a scale factor of 2? Well, since in similar figures the corresponding sides are in proportion, each of the sides of the enlarged similar figure is twice as long as the corresponding side of the original figure. So, for example, in the enlargement of the trapezoid shown below on the left, the enlarged trapezoid is similar to the small trapezoid because the angles are congruent and each of the sides is proportionally larger (twice as long):
Building similar figures, however, is not always so straightforward! For example, the trapezoid below is not similar to the original trapezoid. The angles are congruent, but the corresponding sides are not proportional — some of the sides have been “stretched” more than others:
Print several copies of the sample polygons (PDF) to use in the problems that follow. You’ll need to cut out each of the polygons in order to manipulate them. (If you have access to Power Polygons, you can use them for these problems.)
- In this problem, you will build similar figures by enlarging Triangle N. Use multiple copies of Triangle N to build the enlarged triangles. First use a scale factor of 2, then a scale factor of 3, and then a scale factor of 4. Check to make sure that all corresponding sides are proportionally larger than the original polygon. Sketch your enlargements.
- What happens to the area of a figure that you enlarge by a scale factor of 2? Is the area of the enlarged figure twice that of the original?
Use Rectangle C, Parallelogram M, and Trapezoid K to build similar figures with scale factors of 2, 3, and 4, respectively. Then calculate the area of each enlargement in terms of the original polygon, and record your results in the table below.
For example, it takes four copies of the original trapezoid to make a similar shape with a scale factor of 2, so the enlarged trapezoid has an area that is four times greater than the original:
You do not have to build the enlargements of each shape using only that type of polygon, but you will need to determine the area of each enlargement in terms of the original polygon.
Scale Factor of 2: Area of the Enlargement in Terms of the Original Shape
Scale Factor of 3: Area of the Enlargement in Terms of the Original Shape
Scale Factor of 4: Area of the Enlargement in Terms of the Original Shape
Examine your enlargements. What is the relationship between the scale factor and the number of copies of the original shape needed to make a larger similar shape?
What is the relationship between the scale factor and the area of the enlarged figure?
If the area of a polygon were 8 cm2, what would the area of an enlargement with a scale factor of 3 be?
You may want to sketch a rectangle that has an area of 8 cm2 and the enlargement of that rectangle using a scale factor of 3.
If the scale factor of an enlargement is k, explain why the enlarged area is k2 times greater than the original area.
Take It Further
A rep-tile is a shape whose copies can be put together to make a larger similar shape. Look at Polygons B, C, F, L, and M. Which of these are rep-tiles? What do you notice about all of the rep-tile shapes?
You can further explore the notion of similarity in Learning Math: Geometry, Session 8.
If you are working in a group, you can divide the tasks in this problem. You may also want to try scaling up Polygons B, F, H, and L.
In this problem, you use the polygons to create similar figures. You may use more than one type of shape to build the similar figures, but you do not have to. Be sure that the figures you create satisfy both of the requirements for similarity — you may need to measure the sides and angles of the figures to check.
Polygon H, the hexagon, poses a challenge. You can use hexagons, trapezoids, and equilateral triangles to build the similar hexagons in this problem.
- All sketches produce a similar triangle:
- No, it is four times larger. In all cases, the area of a doubled polygon is four times the area of the original. Tripling the side lengths results in a polygon nine times larger in area. Quadrupling the side lengths results in a polygon 16 times larger in area — and so on.
If Ao is the area of the original polygon, then we can write the following:
The number of copies needed is the square of the scale factor. For example, making a copy that is three times larger in each direction will take nine copies of the original shape.
The area of the enlarged figure is the original area multiplied by the square of the scale factor.
Because the scale factor is 3, the area is nine times larger. Therefore, the area of the enlarged figure is 72 cm2.
For example, suppose the original figure were a 4-by-2 rectangle (with an area of 8 cm2). The new shape would then be 12 by 6, with an area of 72 cm2 — nine times the original area. Here’s how it breaks down:
A = 12 • 6
A = (4 • 3) • (2 • 3)
A = 4 • (3 • 2) • 3associative property
A = 4 • (2 • 3) • 3commutative property
A = (4 • 2) • (3 • 3)associative property
One way to think about it is that enlarging an object will require k copies of that object in each direction: k copies in one direction, multiplied by k copies in the other direction, for a total of k2.
All of these polygons are rep-tiles. Most rep-tiles have side lengths that have a common factor, but this is not a requirement.
Session 1 What Does It Mean To Measure?
Explore what can be measured and what it means to measure. Identify measurable properties such as weight, surface area, and volume, and discuss which metric units are more appropriate for measuring these properties. Refine your use of precision instruments, and learn about alternate methods such as displacement. Explore approximation techniques, and reason about how to make better approximations.
Session 2 Fundamentals of Measurement
Investigate the difference between a count and a measure, and examine essential ideas such as unit iteration, partitioning, and the compensatory principle. Learn about the many uses of ratio in measurement and how scale models help us understand relative sizes. Investigate the constant of proportionality in isosceles right triangles, and learn about precision and accuracy in measurement.
Session 3 The Metric System
Learn about the relationships between units in the metric system and how to represent quantities using different units. Estimate and measure quantities of length, mass, and capacity, and solve measurement problems.
Session 4 Angle Measurement
Review appropriate notation for angle measurement, and describe angles in terms of the amount of turn. Use reasoning to determine the measures of angles in polygons based on the idea that there are 360 degrees in a complete turn. Learn about the relationships among angles within shapes, and generalize a formula for finding the sum of the angles in any n-gon. Use activities based on GeoLogo to explore the differences among interior, exterior, and central angles.
Session 5 Indirect Measurement and Trigonometry
Learn how to use the concept of similarity to measure distance indirectly, using methods involving similar triangles, shadows, and transits. Apply basic right-angle trigonometry to learn about the relationships among steepness, angle of elevation, and height-to-distance ratio. Use trigonometric ratios to solve problems involving right triangles.
Session 6 Area
Learn that area is a measure of how much surface is covered. Explore the relationship between the size of the unit used and the resulting measurement. Find the area of irregular shapes by counting squares or subdividing the figure into sections. Learn how to approximate the area more accurately by using smaller and smaller units. Relate this counting approach to the standard area formulas for triangles, trapezoids, and parallelograms.
Session 7 Circles and Pi (π)
Investigate the circumference and area of a circle. Examine what underlies the formulas for these measures, and learn how the features of the irrational number pi (π) affect both of these measures.
Session 8 Volume
Explore several methods for finding the volume of objects, using both standard cubic units and non-standard measures. Explore how volume formulas for solid objects such as spheres, cylinders, and cones are derived and related.
Session 9 Measurement Relationships
Examine the relationships between area and perimeter when one measure is fixed. Determine which shapes maximize area while minimizing perimeter, and vice versa. Explore the proportional relationship between surface area and volume. Construct open-box containers, and use graphs to approximate the dimensions of the resulting rectangular prism that holds the maximum volume.
Session 10 Classroom Case Studies, K-2
Watch this program in the 10th session for K-2 teachers. Explore how the concepts developed in this course can be applied through case studies of K-2 teachers (former course participants who have adapted their new knowledge to their classrooms), as well as a set of typical measurement problems for K-2 students.
Session 11 Classroom Case Studies, 3-5
Watch this program in the 10th session for grade 3-5 teachers. Explore how the concepts developed in this course can be applied through case studies of grade 3-5 teachers (former course participants who have adapted their new knowledge to their classrooms), as well as a set of typical measurement problems for grade 3-5 students.
Session 12 Classroom Case Studies, 6-8
Watch this program in the 10th session for grade 6-8 teachers. Explore how the concepts developed in this course can be applied through case studies of grade 6-8 teachers (former course participants who have adapted their new knowledge to their classrooms), as well as a set of typical measurement problems for grade 6-8 students.