PBS Teachers™

Multimedia resources & professional development for America's preK-12 educators.

Neighborhood Math: Answers

Math at the Mall

1. You may want to choose a medium-sized mall or shopping area. That gives you the range and sizes of shops. If it is a very large mall you may want to divide the mall up into regions and give groups of students a region.
2. The people who manage your mall may be able to supply you with a list of the shops alphabetically or by location. This way your students can role play being the owners of the mall and making the decisions about the categories and where each shop fits.
3. The numbers will vary depending on your mall and categories.
4. The charts and graphs will vary depending on your mall and categories.
5. The following are some questions that might get students thinking. What types of shops occur the most? What shops occur the least? Why do you think this it is this way? Are the shops for a specific type of person or for the general population? Are there shops designed to attract young people? Are there shops designed to attract men? How do the numbers compare? What type of person is the mall trying to attract?
6. A scale drawing of the mall should be available in a flyer or through the mall management. One way to figure out scale is to measure the length in real life and relate that to the length on the drawing. Ultimately students will need to be able to say something like 1 cm on the map is the same as 2 meters in real life. If you use metric measures the scale question is easier. For this example the scale would be 1:200 since 1 cm on the map is equal to 200 cm or 2 meters in real life.
7. Using the map and the scaling factor you can calculate fairly good estimates of the areas of the different shops. Also note that if you calculate the area on the map, you have to multiply it by the scaling factor squared because the factor is used in both dimensions. If we use our previous estimate, a 1 cm by 1 cm square on the map is equivalent to a 2 meter by 2 meter square of floor at the mall, which is the same as 4 square meters or 40,000 square cm.
8. With this question students need to take into account the open space in the mall where people walk and sit. It also could account for space designated for maintenance and other non-shop related activities.
9. The charts and graphs will vary depending on the mall and categories.
10. The following are some questions that might get students thinking. What types of shops take the most space? What types of shops have the least floor space? Why do you think it is this way? What determines how much space a shop needs?
11. Answers will vary. Many malls have a few large stores to anchor the mall, and then a lot of smaller specialty shops used to target certain types of people.

Math in the Park or City

1. Most bricks are rectangles.
2. Rectangles tessellate nicely in that you can place them together and build a surface without any gaps in it.
3. Answer will vary. Regular hexagons and triangles will also tessellate easily and may be found.
4. The number of bricks added from one ring to the next should stay constant because as you add a new ring, you increase the diameter of the circle the amount of two bricks (one on each side). This increase in diameter causes the circumference to increase by approximately three times that amount (circumference = pi x diameter or approximately 3.14 x diameter). As long as the bricks stay the same size and are laid in the same direction, there should be a relatively constant increase in the number of bricks in each ring.
5. Answers will vary.
6. Answers will vary.
7. Answers will vary.
8. Answers will vary.
9. Answers will vary.
10. Answers will vary.
11. Answers will vary.
12. The shadow factor is very small when the shadow is much longer than the height of the object. This occurs when the sun is low in the sky, which is early in the morning or late in the day.
13. The shadow factor is very large when the shadow is much shorter than the height of the object. This occurs when the sun is high in the sky, which is during the middle of the day.
14. This occurs when the length of the shadow is the same as the height of the object. This occurs when the sun is at a 45-degree angle.

Mathematics of Bicycles I: "Wheel" Figure This Out

1. No.
2. The two factors that influence how far the bike traveled are the different sizes of the wheels and the different gear settings. We will explore these two separately.
3. Answers will vary.
4. Answers will vary.
5. It should be close to 3.
6. Should be true.
7. If you increase the height of the tire by 1 inch, the distance around the tire increases by about 3 inches.

Mathematics of Bicycles II: Gearing Up!

1. One complete turn of the pedal produces one complete turn of the wheel because the pedals are directly attached to the wheel.
2. Observe the tricycle.
3. One turn of the pedal should produce more than one turn of the wheel. This is different from the triangle because of the gears. The number of teeth in the pedal gear is larger than the number of teeth in the wheel gear. So one turn of the pedal produces more than one turn of the wheel.
4. Answers will vary.
5. The number should be very close because the number of revolutions the wheel makes is directly related to the number of times the teeth in the wheel gear would go into the teeth in the pedal gear.
6. If the number of teeth in the wheel gear is larger than the number of teeth in the pedal gear, then the wheel will make less than one complete turn when the pedals make one turn.
7. The smallest pedal to wheel ratio occurs when you are on the small gear on the pedals and the largest gear on the wheel. In this setting, one turn of the pedals produces much less than one turn of the wheel.
8. You would need to find a setting in which the number of teeth in the pedal gear and the wheel gear are equal. It does not matter how many teeth; they just need to be the same.