Algebra
Chapter 7 Answers
Lines and Distance
Let's explore lines and distance!
Part A, Absolute Value
1. Browse through the BART Downtown San Francisco Map Web site.
a. Locate the Grace Cathedral on the map. Which stations are within 4 blocks of the Grace Cathedral? Justify your answer.
[Cable Car Museum, 3 blocks north of the Cathedral; Chinese Recreation Center, 4 blocks northeast of the Cathedral; Masonic Auditorium, 1 block south of the Cathedral. ]b. If x represents a point on the map, write an absolute value expression to indicate all points that are less than or equal to a distance of four blocks from x.
[x ≤ |4|]
Part B, Graphing Absolute Value Functions
2. Graph your function from b (above) on a graphing calculator or a piece of graph paper.
a. Write an equation if x is the Grace Cathedral and y is a point10 blocks away. Explain why an absolute value equation is appropriate. [ y = | x + 10 | ]
b. Graph your function from a.
[Graphs will vary.]
Part C, Absolute Value Equations
3. If x represents the distance from the Chinese Hospital to the Montgomery Station and y represents the total distance, write an equation for the distance from the San Francisco Art Institute to the Montgomery Station. [y = 2 | x |]
4. Graph your equation from 3. How is it alike and how is it different
from the graph y = | x |?
[The graph of y = 2 |x| is steeper.
The "v" is smaller, or narrower. The slopes of the arms of the
graph are twice the slope of y = | x |.]
Part D, Making Connections
5. Using the BART map, name two places that are less than one half block
from the Montgomery Street Station.
[Golden Gate University and Academy of Art]
6. Use an absolute value inequality to describe the situation from number
5.
[y ≤ | x + .5 |, where x is the Montgomery Street Station. ]
Let's explore lines, distance, and a very interesting mathematician, Pythagoras!
Part A, Square Roots
1. Read through the Pythagoras Web site.
a. Write two ideas you have gleaned from the reading about Pythagoras' life, teachings, or discoveries.
[ Answers will vary. ]
Part B, Square Root Functions and Equations
2. Open the Circle
and Square Web site. Using a compass, a straight edge, and the circular
model as a guide, reproduce the circular model for the square root of 2
on a piece of 8 1/2 " x 11" paper. Make the radius of your circle
between 10 and 20 centimeters.
[Check the students' work.]
a. Measure the length of your square to the nearest centimeter.
[Anwers will vary.]
b. Measure the diagonal of your square to the nearest centimeter.
[Answers will vary. ]c. Using your calculator, find the approximate ratio of the length of the diagonal of your square to the length of the side of your square?
[The approximate ratio is 1 : 1.414.]d. Compare your answers to c with those obtained by other students. What do you notice?
[Possible answers include: The lengths are different but the ratio is the same. The ratio is the square root of 2. The length of the diagonal of a square is equal to the length of the side of the square times the square root of two.]
Part C, Making Connections
3. Explain why the square root of two is not rational.
[The square root of two represents a ratio that
cannot be expressed as the quotient of two integers.]
4. Explain why the square root of four is rational.
[The square root of four can be evaluated to the
integer 2. It can be expressed as the quotient of two integers, for example,
12 ÷ 3.]
5. Is the square root of 12.96 rational or irrational? Explain.
[The square root of 12.96 is rational since it
equals 3.6, a terminating decimal.]
How are taxis and math related? How about taxis, math, and Pythagoras? Let's investigate!
Part A, Taxi Distance
1. The state capital of California is Sacramento. Open the Maps Web site and type Sacramento, CA 95814 in the blanks provided.
a. The star on the map is the Governor's Mansion State Historic Park. Find the taxi distance between the Governor's Mansion State Historic Park and the California State Capitol Museum.
[The horizontal distance is 4 blocks and the vertical distance is 5 blocks. The total distance is 4 + 5, or 9 blocks.]b. Find the taxi distance between the Governor's Mansion State Historic Park and the Towe Ford Museum, a wonderful museum to see old cars.
[The horizontal distance is 14 blocks and the vertical distance is 14 blocks. The total distance is 14 + 14, or 28 blocks.]
Part B, The Pythagorean Theorem
2. Continue looking at the site of downtown Sacramento.
a. Find the distance from the Governor's Mansion State Historic Park to the California State Capitol Museum as "the crow flies." If necessary, adjust the map so that you can see both locations.
[Since the horizontal distance is 4 blocks and the vertical distance is 5 blocks, using the Pythagorean Theorem we find the distance to be the square root of 5 squared + 4 squared, or approximately 6.4 blocks.]b. Find the distance from the Governor's Mansion State Historic Park to the Towe Ford Museum as "the crow flies."
[Since the horizontal distance is 14 blocks and the vertical distance is 14 blocks, the total distance is the square root of 14 squared + 14 squared, or approximately 19.8 blocks.]
Part C, Finding Coordinate Distances
3. Use the same Maps
Web site to find a map of your home town. Working with a partner, look at
the map of your town. Assign coordinates to the outside of your map, with
the lower left hand corner being (0,0). Make sure to label your axis.
[Coordinates will vary.]
4. Write five distance questions that pertain to coordinates on your
city map. Put each question on a file card. Exchange cards with other students.
Be sure to show absolute value and square root symbols when appropriate.
[Answers will vary.]
Part D, Making Connections
5. Continue using a map of your town. Choose two intersections on your
map. Explain how to find the distance between the two intersections.
[Possible answers include: One method is to use
the distance formula to assign coordinates and calculate the distance. Some
students will count the horizontal and vertical blocks and then use the
Pythagorean Theorem.]
6. Choose two different intersections on the map of your town. Find the
halfway point between the two intersections. Explain your choice.
[Answers will vary. Notice if students' thinking
and reasoning is appropriate for the problem. It is unlikely, however, that
students will use the midpoint formula.]
Television and the news play a large part in all of our lives. In this section we will explore proportional reasoning though the use of fire trucks...something we see in the news too often!
Part A, Similar Figures and Proportions
1. Open the Fire trucks Web site.
a. Assume that the ladder shown on the truck is 40 feet long from the edge of the platform to the base of the truck. Using proportional reasoning, how long is the fire truck?
[Approximately 42 feet in length]b. Using the same proportions, how high from the ground is the hood of the truck?
[At the highest point, the truck is approximately 16 feet tall.]c. Construct a drawing of the fire truck that is five times the size of the one on your computer screen. You may want to use a grid and enlarge the truck in small sections. Explain why your drawing is similar to the one on your computer screen.
[Similar figures have the same shape, and all lengths are in the same proportion.]
Part C, Right Triangle Trigonometry
2. Continue examining the Fire trucks site.
a. Browse through this site to find the range of degrees for the ladder adjustment.
[-5 degrees to 80 degrees]b. If the ladder is fully extended to 95 feet, how far away from the building will the base of the ladder be if it reaches a point on the side of the building 30 feet from the ground?
[Students will probably use the Pythagorean Theorem to calculate the distance of approximately 90 feet.]
3. What angle would the ladder make with the truck if extended as in
2b?
[Using inverse sine, students can calculate the
angle to be approximately 18 degrees.]
Part D, Making Connections
4. Continue using the fire truck site. Suppose that firemen are using
a 75- foot ladder extended to 70 feet. They encounter a fire, but since
the wind is blowing, they are reluctant to increase the angle of the ladder
with the truck beyond 60 degrees. How many vertical feet will the ladder
extend from the top of the fire truck?
[Approximately 60.6 feet]
5. If the firemen extend the 70-foot ladder at a 45 degree angle, how
many vertical feet will the ladder extend from the top of the fire truck?
Find your answer in two different ways.
[The ladder will be approximately 49.5 feet up
from the top of the fire truck. Most students will use either the Pythagorean
Theorem or the sine function to answer this question.]