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The first formula can be rewritten as:
height in inches when grown up12(height father height mother) 11 2. a. Show that this formula is equal to the first one.
b. Rewrite the second formula.
c. Can the formula for girls be described as 2 less than the mean of father’s and mother’s height? Explain why or why not.
The higher you stand, the farther you can look! Here is a rule to estimate the relationship between the height (h) in meters and the distance to the horizon (d ) in kilometers. The rule holds only for a clear day.
d
13 h3. a. Write this formula as an arrow string. Use the sign.
b. Hank is at the beach and is 1.80 m tall. How many kilometers away can he still see a ship?
c. Bert is 1.50 m tall. He used the formula and found the distance to be 4.416 km. Comment on Bert’s answer.
d. Frances claims: “If you stand twice as high, you can see twice as far!” Is Frances right? Explain.
e. Mohamed claims: “If you stand four times as high, you can see twice as far!” Is Mohamed right? Explain.
Additional Practice
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1. Yes, they describe the same pattern. Here are some sample strategies you could use:
Strategy 1
A pattern could be:
T P 2 P can be explained as a row of P orange tiles, 2 white tiles, and another row of orange tiles.
T 2P 2 can be explained as 2 rows of P orange tiles and 2 white tiles.
T 2(P 1) can be explained as 2 rows of P orange tiles and a white tile.
Strategy 2
P 2 P is the same as 2P 2, and this can also be written as 2(P 1).
2. a. Sample table:
Section A Patterns
Path 2 Path 3 Path 4
Patio Number Number of Orange Number of White
(P ) Tiles (O) Tiles (W )
1 1 8
2 4 12
3 9 16
4 16 20
5 25 24
6 36 28
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b. Discuss your formula with a classmate. Sample formulas:
The Number of Orange Tiles is the Patio Number times itself.
Number of Orange Tiles Patio Number Patio Number G P P
G P2
c. Discuss your formula with a classmate. You can find many equivalent formulas.
W (P 2) (P 2) P P W 2(P 2) 2P
W 4 4P W 4(P 1) W 4(P 2) 4
W (P 2) (P 2) (P P)
d. You can find the answer by extending the table or by using the formulas you found.
Number of Orange Tiles in Patio Number 10 is 100.
Number of White Tiles in Patio Number 10 is 44.
3. a. Sample table:
b. Yes, both formulas give the same result.
Sample drawing:
The length of the side of the whole square (including orange and white tiles) is P 2, so the total number of tiles is (P 2) (P 2).
Answers to Check Your Work
1 2
4 3
P P F
Path Number of Number of Total Number
Length (L) Orange Tiles (O) White Tiles (W ) of Tiles (T )
1 1 8 9
2 4 12 16
3 9 16 25
4 16 20 36
5 25 24 49
6 36 28 64
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4. a. Terry is building patio number 12.
The side of the square Terry is going to build has a length of 14 since 14 14 196, so the patio number is two less.
b. Use one of the formulas from 2b to find the number of white tiles. For example, W 4(P 1) will give you 4(12 1) 4 13 52 white tiles.
1. a. Length 1U 1D or, since the 1 in 1U is the same as U, Length U D
b. Length 30U 29D
c. If you think very hard, both yes and no can be your answer!
Discuss your explanation with a classmate.
Sample explanations: No, it is not possible, because 30 and 29 are not divisible by the same number; there is no number that goes into 29 and 30 evenly.
But you might also say it is possible if you first rewrite the formula; for instance:
Length(29U U) 29D, so Length 29U 29D U Length 29(U D) U
2. a. Sample response: A row with a basic pattern with two lying bricks and three standing bricks that is shown four times.
b. Length 8L 12S
3. a. Row 4. Sample explanation: A basic pattern cannot be repeated with Row 4 since there is no number that goes into 9 and 13 evenly. All other rows have basic patterns.
b. The formulas for the length of the different basic patterns for Row 5 are:
Length 3L 2S Length 6L 4S Length 9L 6S
Answers to Check Your Work
Section B Brick Patterns
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1. Here is a sample letter. Your letter will be different, but your explanation should be similar to the one shown in the letter.
To Whom It May Concern:
Recently I realized that your company has printed a formula incorrectly. It is located on the notebook cover as follows:
• To convert Celsius temperatures to Fahrenheit temperatures, use this formula:
C95 (F 32).
• To convert Fahrenheit temperatures to Celsius temperatures, use this formula:
F 5 9(C 32).
The first formula is correct, but the second is not. The second formula should be the reverse of the first, but it is not.
When you use numbers to check out the formulas, you can see that the second one does not work. I know that 0°C is equal to 32°F, so if I put F = 32 in the first formula, I should get C = 0, and when I try it, that happens. When I put C = 0 in the second formula, I should get F =32, but I get 57.6, which is incorrect.
The correct second formula can be found using arrow language to reverse the first formula:
Formula I F 32→ _____ 59
→ C
Formula II C 59
→ _____ 32→ F
Dividing by 95 is the same as multiplying by 5 9. So the correct second formula is F 5 9C 32
It was a mistake to use the parentheses.
Please correct this error.
Sincerely,
Answers to Check Your Work
Section C Using Formulas
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Speed (in mi/h)
Automobile Stopping Distances
Stopping Distance ( in ft)
10 100
200 300 400
20 30 40 50 60 70 80
2. a. Speed (mi/h) Stopping Distance (ft)
20 66
40 206
60 346
b.
c. Yes. Sample explanation:
Restrictions depend on where you are driving. For example, the maximum speed is often 65 mi/h on the interstate and 40 or 35 mi/h in the city. Another restriction is that the formula will not make sense if you use speeds that are too small.
If you are traveling at 10 mi/h, the formula would show that it would take 4 seconds to stop.
Answers to Check Your Work
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3. a. The ramp in part i fits the rule.
0.5 101 5, which is less than18 5
ramp ii. 0.75 38 2, which is more than 18 2
ramp iii. 0.2 102 1 or 15 1, which is more than18 1 b. vertical distance ≤ 18 horizontal distance
4. Designs will vary.
Sample design that has 20 steps.
Make sure you have noted the rise and tread in your design and that they fit the rules.
In the sample design, the measurements were:
rise is 15 cm tread is 33 cm
1. a. 16 4 b. 12 314 12
c. 48 ≈ 6.9 d. 1000 ≈ 31.6
2. a. A circle with a diameter of 30 cm has a radius of 12 30 15 Area
π
15 15 706.85834…Rounded to one decimal place, area ≈ 706.9 cm2
b. You can use a reverse arrow string for the area formula.
area ⎯ π⎯→ ... ⎯⎯→ radius
When you use your calculator, do not round results before the end of your calculations.
10,000⎯ π⎯→ 3183.09886… ⎯⎯→ 56.418958…
The radius of the circle is 56 cm, rounded to the nearest
Answers to Check Your Work
300 cm
660 cm
Section D Formulas and Geometry
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3. a. 6 → 4 2 square→ 4 → 3 12 → 7 19 b. First make the reverse arrow string:
answer→7 …. →:3 …. → …. →4 number 55 →7 48 →:3 16 → 4 →4 8
4. Yes, there is a difference. Discuss your explanation with a
classmate. Sample explanation: In one case you square the 4 and in the other you square the 6.
If you compute the volume of both pyramids, you would find volume (I) (13) 4 4 6 and
volume (II) (13) 6 6 4
1. a. 200 seconds. Strategies will vary. Some students may substitute the given values into the formula:
T 20 L
T 20 100 T 20 10 200 seconds b. 2 seconds
2. The pendulum needs to be 9 cm long. Strategies will vary.
Sample strategy:
T 20 L 60 20 L 3 L 9 L
3. a. Answers will vary. Sample response:
Answers to Check Your Work
Section E Problem Solving
Pendulum Length L Time
4 cm 2 40 sec.
9 cm 3 60 sec.
16 cm 4 80 sec.
25 cm 5 100 sec.
36 cm 6 120 sec.
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b. Graphs will vary. Sample graph:
c. i. True. The time increases as the pendulum length increases.
ii. False. You might use your table or graph to think about your answer.
iii. False. One way to show this is to choose a length from your table, add 1 cm and compute the new time to see if the number of seconds is increased by 20.
4. Greg should make the pendulum 25 cm long.
5. Yes, Caroline’s formula is correct. Explanations will vary. You may reason that the time needed for 1 swing is–—1
100of the time needed for 100 swings, so you need to divide the numbers in the
formula by 100. So, T 0.2 L .
Answers to Check Your Work
Time (in sec.) Pendulum Movement
Length ( in cm)
10 2
4 6 8 10 12 14 16 18 20 22 24 26 28 30
20 30 40 50 60 70 80 90 100
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