p.1 The Broomsticks
You have three broomsticks: The RED broomstick is three feet long The YELLOW broomstick is four feet long The GREEN broomstick is six feet long
1. How much longer is the GREEN broomstick than the RED broomstick?
2. How much longer is the YELLOW broomstick than the RED broomstick?
3. The GREEN broomstick is ______times as long as the YELLOW broomstick.
4. The YELLOW broomstick is ______times as long as the GREEN broomstick.
5. The YELLOW broomstick is ______times as long as the RED broomstick.
6. The RED broomstick is ______times as long as the YELLOW broomstick.
2014 Ted Coe, [email protected] p.2 7. A certain stock started at a value $74. One year later it was valued at $89.54. By what percent did the stock’s value increase?
8. The Willis tower (formerly the Sears tower) is 1730 feet high. The Burj Khalifa (formerly Burj Dubai) is 2717 feet high. a. The Burj is ______times as large as the Willis tower.
b. The Willis tower is ______times as large as the Burj
c. The Burj is ______percent the size of the Willis tower.
d. The Willis tower is ______percent the size of the Burj.
a 9. How can you think about ? b
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a 10. If you are given c , how can you think about c? b
11. On the broomstick questions (#3-#6) we did not need to mention “feet” in the answer. Why is that?
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What is the length of “d”? You may choose the unit.
What is the measure of the angle? You may choose the unit.
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2014 Ted Coe, [email protected] p.10 Geometric Fractions: Area, Length, Sets. Use pattern blocks to answer questions 1-9.
1. If = 1 what is ?
2. If =1 what is ?
3. If = ¼ what is ?
4. If = ¼ what is ?
5. If =2 what is ?
6. If = 2/3 what is 1? Explain
7. If = 5/2 what is 1? What is 3/4? Explain.
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8. If = 7 what is ?
9. If = 1 what is 5/3 ?
10. If = 3/5 what is 1?
5 11. Use your thinking in #9 to explain and finish: 1 3
5 12. Use your thinking in #9 to explain and finish 2 3
3 13. Use your thinking in #10 to explain and finish 1 5
14. If segment AB has a length of 7/4, how could you determine a length of 1?
4 7 15. Use your thinking in #14 to explain and finish 7 4
16. Provide two different ways that you could determine a length of 7/8.
17. If segment CD has a length of 4/7 how could you determine a length of 1?
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18. Provide two different ways that you could determine a length of 8/7.
19. The following set is 3/5 of some original set. Can the original set be represented as a rectangular ar- ray? Explain without using algebra.
20. What is 5/3 of the given set? Explain without using algebra.
21. You are told that the following set is 3/5 of some original set. Can the original set be represented as a rectangular array? Explain without using algebra.
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2014 Ted Coe, [email protected] p.15 Similarity
a. When we say two figures are similar we mean…
b. You are given the similar triangles below. Using only our meaning of similarity (and no formulas) explain how to find x.
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c. A’B’ is ______times as long as 1/7 of AB. d. So A’B’ is ______times as long as AB. e. Since the figures are similar, this means that A’C’ is ______times as long as AC. So A’C’ is ______or ______f. How long is is B’C’? g. Without using formulas, find x and y in the similar figures:
h. Bonus! Find the areas of the two figures.
2014 Ted Coe, [email protected] p.17 9. You are given the three similar, right triangles below:
I. Find the length of the hypotenuse for each of the triangles.
II. Complete each of the following sentences: a. In triangle ABC the length of the side opposite angle C is ______times as large as the hypotenuse. That is, it has a length ______“hypotenuse lengths” long. b. In triangle A’B’C’ the length of the side opposite angle C’ is ______times as large as the hypotenuse. That is, it has a length ______“hypotenuse lengths” long. c. In triangle A’’B’’C’’ the length of the side opposite angle C” is ______times as large as the hypotenuse. That is, it has a length ______“hypotenuse lengths” long. III. Will this constant relationship be true for all triangles similar to ABC? Explain.
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IV. What is true about the angles at C, C’, and C’’?
V. If I were to increase the angle at C, C’, and C’’ while keeping the triangle a right triangle, what would happen to the opposite side in terms of hypotenuse-lengths?
VI. If the only length-units we use are “hypotenuse-lengths”, what is the absolute longest the opposite side could ever measure in a right triangle?
VII. What is the absolute shortest?
VIII. In a right triangle, this relationship (the opposite side as “how many times as large” as the hypotenuse) is known as “Sine.” Each “Sine” value corresponds to a given acute angle in a right triangle. According to the sine chart on the next page, what should be the approximate measure of angle C?
IX. Measure the angle to confirm.
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10. Use the sine chart to find the value of x in each of the following:
t (degrees) Radians Sin(t) t (degrees) Radians Sin(t) 0 0.000 0.00000 45 0.785 0.70711 1 0.017 0.01745 46 0.803 0.71934 2 0.035 0.03490 47 0.820 0.73135 3 0.052 0.05234 48 0.838 0.74314 4 0.070 0.06976 49 0.855 0.75471 5 0.087 0.08716 50 0.873 0.76604 6 0.105 0.10453 51 0.890 0.77715 7 0.122 0.12187 52 0.908 0.78801 8 0.140 0.13917 53 0.925 0.79864 9 0.157 0.15643 54 0.942 0.80902 10 0.175 0.17365 55 0.960 0.81915 11 0.192 0.19081 56 0.977 0.82904 12 0.209 0.20791 57 0.995 0.83867 13 0.227 0.22495 58 1.012 0.84805 14 0.244 0.24192 59 1.030 0.85717 15 0.262 0.25882 60 1.047 0.86603 16 0.279 0.27564 61 1.065 0.87462 17 0.297 0.29237 62 1.082 0.88295 18 0.314 0.30902 63 1.100 0.89101 19 0.332 0.32557 64 1.117 0.89879 20 0.349 0.34202 65 1.134 0.90631 21 0.367 0.35837 66 1.152 0.91355 22 0.384 0.37461 67 1.169 0.92050 23 0.401 0.39073 68 1.187 0.92718 24 0.419 0.40674 69 1.204 0.93358 25 0.436 0.42262 70 1.222 0.93969 26 0.454 0.43837 71 1.239 0.94552 27 0.471 0.45399 72 1.257 0.95106 28 0.489 0.46947 73 1.274 0.95630 29 0.506 0.48481 74 1.292 0.96126 30 0.524 0.50000 75 1.309 0.96593 31 0.541 0.51504 76 1.326 0.97030 32 0.559 0.52992 77 1.344 0.97437 33 0.576 0.54464 78 1.361 0.97815 34 0.593 0.55919 79 1.379 0.98163 35 0.611 0.57358 80 1.396 0.98481 36 0.628 0.58779 81 1.414 0.98769 37 0.646 0.60182 82 1.431 0.99027 38 0.663 0.61566 83 1.449 0.99255 39 0.681 0.62932 84 1.466 0.99452 40 0.698 0.64279 85 1.484 0.99619 41 0.716 0.65606 86 1.501 0.99756 42 0.733 0.66913 87 1.518 0.99863 2014 Ted Coe, [email protected] 43 0.750 0.68200 88 1.536 0.99939 44 0.768 0.69466 89 1.553 0.99985 45 0.785 0.70711 90 1.571 1.00000 p.20
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2014 Ted Coe, [email protected] p.22 You have an investment account that grows from $60 to $103.68 over three years.
2014 Ted Coe, [email protected] p.23 We can compare two numbers additively or multiplicatively.
1.. Consider a bank account that grows from $98 to $114 over 3 years.
a) Additive comparison b) Multiplicative comparison*
* so ______is ______times as large as ______
2. We can determine how the amount changes on a single-year basis:
a) Additively b) Multiplicatively
3. We can write functions that describe the growth if the growth was:
a) Linear (additive) b) Exponential (multiplicative)
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