Learning Math: Measurement
Angle Measurement Homework
Session 4, Homework
Draw any quadrilateral. Then draw a point anywhere inside the quadrilateral, and connect that point to each of the vertices, as shown below:
Now answer the following questions:
- How many triangles have been formed?
- What is the total sum of the angle measures of all the triangles?
- How much of the total sum from part (b) is represented by the angles around the center point (i.e., what is their sum)?
- How much of the total sum from part (b) is represented by the interior angles of the quadrilateral?
- Repeat the activity with a five-sided polygon and an eight-sided polygon, and then attempt to generalize your result to an n-sided polygon. Note 8
Estimate the number of degrees between two adjacent legs of the starfish below. Then, using a protractor, measure one of the angles. (You may want to print the PDF image in order to do this.) How close was your estimate?
Print the figures below from the PDF document. For each figure, cut out a and e. When are a and e congruent? What other angles have the same measure as ∠a? As ∠e?
- How many angles can be formed with the rays below?Look at all the possible combinations. For example, three rays can form three distinct angles. Do you see them?
- Predict the number of angles formed with seven rays and with 10 rays. Can you generalize your prediction to n rays?
Write a series of commands in the style used in the Interactive Activity (or Geo-Logo commands) to draw a regular decagon (a 10-sided figure). You can test your commands using the Non-Interactive Activity. How many total degrees did you turn to make the decagon?
A sled got lost in the darkness of a polar night. Mayday emergency calls were received all night, but the darkness prohibited a search. The next morning, planes searched the area, and the pilots saw these tracks made by the sled:
- Use turns to describe the route of the sled as if you had been in it.
- If the sled continued in the same way, it might have returned to its starting point. How many turns would the sled have had to make to return to its starting point?The pilot described the track as follows: “It looks like the sled made three equal turns to the right. The four parts of the track seem to be equally long, and the resulting angle between each part measures about 150 degrees.”
- What do you think the pilot means by “the resulting angle”?
- How does the pilot’s description differ from your own?
- If you were to make a 40-degree turn on the sled, what would the resulting angle be? If the sled track forms a 130-degree resulting angle, what is the size of the turn?
You are interested in making a quilt like the one shown below. In the center, a star is made from six pieces of material:
- Is it possible to make the star with the piece below? Why or why not?
Print and cut out several copies of this image from the PDF document.
- What about the highlighted piece below — could it be used to make the star in the quilt? Without measuring, determine the measures of angles A, B, and C, and explain how you arrived at your solution.
In the activities in the text, you determined the rule for finding the sum of the interior angles of a polygon, 180 • (n – 2), where n is the number of sides of the polygon. Problem H1 presents an alternative approach for finding the sum of the interior angles in a polygon. Think about the two rules and explain why they are the same.
This is similar to the well-known handshake problem: If each person shakes hands with another person, how many handshakes will there be for a total of n people?
There is a great deal of mathematics involved in designing quilts. If you’re working in a group, you may want to discuss Problem H7.
- Four triangles are formed.
- The total sum of the angles in the four triangles is 4 • 180° = 720°.
- The sum of the angles at the center is 360 degrees (a full circle).
- The interior angles sum to 360 degrees (or 720° – 360°). This shows that a quadrilateral has 360 degrees in its angles.
- A five-sided polygon: Five triangles are formed. The sum of the angles is 5 • 180°.The sum of the angles at the center point is still 360 degrees. The sum of the interior angles is (5 • 180°) – 360° = 180°(5 – 2) = 540°.
An eight-sided polygon: Eight triangles are formed. The sum of the angles is 8 • 180°. The sum of the angles at the center point is 360 degrees. The sum of the interior angles is (8 • 180°) – 360° = 180°(8 – 2) = 1,080°.
An n-sided polygon: n triangles are formed. The sum of the angles is n • 180°. The sum of the angles at the center point is 360 degrees. The sum of the interior angles, then, is (n • 180°) – 360°, or 180°(n – 2).
A good estimate is 72 degrees, since there are five angles roughly equally spaced around a circle (72 = 360 5). Depending on which angle you actually measure, it may be slightly larger or smaller than 72 degrees.
Angles ∠a and ∠e are congruent only when the two lines are parallel. Angles ∠d and ∠h are congruent to ∠a and ∠e when the lines are parallel, as are ∠b, ∠c, ∠f, and ∠g to one another. (In Figure 3, all the angles are congruent.) If the lines are not parallel, ∠a and ∠d are still congruent, as are ∠e and ∠h, but they are not congruent to one another.
- Two rays form one angle. Three rays form three angles. Four rays form six angles. Five rays form 10 angles. Six rays form 15 angles.b.
- With each new ray, the number of angles seems to grow by one less than the total number of rays, so seven rays should form 21 angles (15 + [7 – 1]). Using this information we can make the following table:
Rays 2 3 4 5 6 7 8 9 10 Angles 1 3 6 10 15 21 28 36 45
A formula is harder to find. One way to do it is to recognize that the first ray in an angle can be picked out of any of the n rays; the second ray can be picked out of (n – 1) rays. Multiplying the two will give us double the count of all the angles that can be formed (try some of the numbers from the table above to test this). So there are a total of n(n – 1)/2 angles that can be formed.
Here is one way to do it: Repeat 5; Go Forward 4; Rotate Right 36; End Repeat; Repeat 5; Go Forward 4; Rotate Right 36; End Repeat.
Notice that we’ve done this as a two-step process, following the same sequence twice. Alternatively, the sequence can also be written as the following: Repeat 2; Repeat 5; Go Forward 4; Rotate Right 36; End Repeat; End Repeat.
The total turn was 360 degrees.
- The sled turns 30 degrees to the right, three times.
- Since it takes 360 degrees to turn all the way around, there would have to be 12 total turns.
- The resulting angle is the angle between the turns — the interior angle.
- The pilot’s description focused on interior angles rather than exterior angles.
- The resulting angle for a 40-degree turn would be 140 degrees. If the sled track forms a 130-degree resulting angle, the turn was 50 degrees.
- No. The acute angle must be exactly 60 degrees if six of them are to fit together at the center of the quilt.
- Yes. Angle B (and therefore ∠A) must be 120 degrees. We know this because the angle shown is 30 degrees, meaning that the angle on the other side of ∠B is also 30 degrees, leaving 120 degrees for ∠B (since they form a straight line). Adding ∠A and ∠B gives us 240 degrees, so ∠C (and its opposite angle) is 60 degrees, which is the required angle for the quilt.
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.