MODULE 1 CHANGES IN QUANTITIES, CONSTANT RATE OF CHANGE, AND LINEAR FUNCTIONS Part 2: Investigations #6-10 Investigation #6: Developing a Formula for Linear Functions Investigation #7: Extra Practice with Linear Functions Investigation #8: Slope-Intercept Form of a Linear Function Investigation #9: Proportionality Investigation #10: Extra Practice with Proportionality Copyright 2013 Carlson, OBryan, & Joyner 1 I#6 DEVELOPING A FORMULA FOR LINEAR FUNCTIONS
1. Suppose that y = 3x. Copyright 2013 Carlson, OBryan, & Joyner 2 I#6 1. Suppose that y = 3x. a. x = 11 i. x = 11 4 = 7 Copyright 2013 Carlson, OBryan, & Joyner 3 I#6
1. Suppose that y = 3x. a. x = 11 i. x = 11 4 = 7 ii. y = 3(11 4) = 21 Copyright 2013 Carlson, OBryan, & Joyner 4 I#6 1. Suppose that y = 3x. a. x = 11 i. x = 11 4 = 7 ii. y = 3(11 4) = 21
iii. y = 3(11 4) + 2 = 23 Copyright 2013 Carlson, OBryan, & Joyner 5 I#6 1. Suppose that y = 3x. b. x = 5.75 i. x = 5.75 4 = 1.75 Copyright 2013 Carlson, OBryan, & Joyner 6 I#6
1. Suppose that y = 3x. b. x = 5.75 i. x = 5.75 4 = 1.75 ii. y = 3(5.75 4) = 5.25 Copyright 2013 Carlson, OBryan, & Joyner 7 I#6 1. Suppose that y = 3x. b. x = 5.75 i. x = 5.75 4 = 1.75 ii. y = 3(5.75 4)
= 5.25 iii. y = 3(5.75 4) + 2 = 7.25 Copyright 2013 Carlson, OBryan, & Joyner 8 I#6 1. Suppose that y = 3x. c. x = 1 i. x = 1 4 = 5 Copyright 2013 Carlson, OBryan, & Joyner 9
I#6 1. Suppose that y = 3x. c. x = 1 i. x = 1 4 = 5 ii. y = 3(1 4) = 15 Copyright 2013 Carlson, OBryan, & Joyner 10 I#6 1. Suppose that y = 3x. c. x = 1 i. x = 1 4 = 5
ii. y = 3(1 4) = 15 iii. y = 3(1 4) + 2 = 13 Copyright 2013 Carlson, OBryan, & Joyner 11 I#6 1. Suppose that y = 3x. d. How can we determine the value of y for some unknown x value, such as x = n? We determine the change in x from 4 to n using n 4. Then we determine the change in y using y = mx, so
y = 3(n 4). Then the value of y when x = n is y = 3(n 4) + 2. Copyright 2013 Carlson, OBryan, & Joyner 12 I#6 2. Suppose we want to know the new value of y when x = 10. Answer the questions that follow. y = 3(x 4) + 2 y = 3(10 4) + 2 y = 3(6) + 2 y = 18 + 2 y = 20 a.
b. c. d. What does 10 4 represent? What does 3(6) represent? What does 18 + 2 represent? What does 20 represent? a. the change in x from x = 4 to x = 10 b. the change in y from y = 2 to the new value of y c. the value of y when x = 10 (because it shows the change in y added to the initial value of y) d. the value of y when x = 10 Copyright 2013 Carlson, OBryan, & Joyner 13
I#6 3. Suppose we want to know the new value of y when x = 1.5. Answer the questions that follow. y = 3(x 4) + 2 y = 3(1.5 4) + 2 y = 3(2.5) + 2 y = 7.5 + 2 y = 5.5 a. b. c. d. What does 1.5 4 represent? What does 3(2.5) represent? What does 7.5 + 2 represent?
What does 5.5 represent? a. the change in x from x = 4 to x = 1.5 b. the change in y from y = 2 to the new value of y c. the value of y when x = 1.5 (because it shows the change in y added to the initial value of y) d. the value of y when x = 1.5 Copyright 2013 Carlson, OBryan, & Joyner 14 I#6 4. Suppose we want to know the new value of y for any value of x. Answer the questions that follow. y = 3(x 4) + 2 a. What does x 4 represent?
b. What does 3(x 4) represent? c. What does 3(x 4) + 2 represent? a. the change in x from x = 4 to any give value of x b. the change in y from y = 2 to the new value of y c. the value of y for the given value of x Copyright 2013 Carlson, OBryan, & Joyner 15 I#8 Linear Function Formula If y is a linear function of x with a constant rate of change of m and if (x1, y1) is an ordered pair solution of the function, then the following is a formula for the linear relationship. y = m(x x1) + y1
______________________________________________ Ex: If the constant rate of change of y with respect to x is 3, and if (1, 7) is a point on the graph, then the formula for the linear relationship is y = m(x x1) + y1 y = 3(x (1)) + 7 or y = 3(x + 1) + 7 Copyright 2013 Carlson, OBryan, & Joyner 16 I#6 5. The constant rate of change of y with respect to x is 3.5, and (6, 2) is a point on the graph. a. Write the formula for the linear function.
y = 3.5(x 6) 2 b. Find the value of y when x = 14. y = 3.5(14 6) 2 = 26 Copyright 2013 Carlson, OBryan, & Joyner 17 I#6 6. The constant rate of change of p with respect to t is 4, and (t, p) = (3, 5) is a point on the graph. a. Write the formula for the linear function. p = 4(t + 3) +5 b. Find the value of p when t = 10. p = 4(10 + 3) +5 = 33 Copyright 2013 Carlson, OBryan, & Joyner
18 Investigation #6b I#6 7. Write the formula that defines the linear relationship represented in each of the following graphs. a. y = 2(x + 2) + 9 or y = 2(x 3) 1 Copyright 2013 Carlson, OBryan, & Joyner
20 I#6 7. Write the formula that defines the linear relationship represented in each of the following graphs. b. r = 1.5(v + 2) +3 or r = 1.5(v 1) + 7.5 Copyright 2013 Carlson, OBryan, & Joyner 21 I#6
8. Write the formula that defines the linear relationship given in each of the following tables. a. b. x y w d 3 23 7
2 1 13 4 5 4 12 2 11
9 37 20 29 y = 5(x 4) + 12 d = (w + 7) 2 Copyright 2013 Carlson, OBryan, & Joyner 22 I#6
9. Recall the burning candle context from Investigation 5. The length of a burning candle changes at a constant rate of 1.6 inches per hour. When the candle had been burning for 3.5 hours it was 8.4 inches long. Let L represent the length of candle remaining (in inches) and let t represent the number of hours the candle has been burning. a. Write the formula that will calculate the length of candle remaining (in inches) given the number of hours the candle has been burning. L = 1.6(t 3.5) + 8.4 Copyright 2013 Carlson, OBryan, & Joyner
23 I#6 b. Describe what the different parts of the formula calculate. L = 1.6(t 3.5) + 8.4 t 3.5 is the change in t from t = 3.5 to some new value of t 1.6 is the constant rate of change of L with respect to t 1.6(t 3.5) is the change in L for the corresponding change in t 8.4 is the value of L when t = 3.5 1.6(t 3.5) + 8.4 (and L) represent the value of L for the given value of t Copyright 2013 Carlson, OBryan, & Joyner 24
I#6 c. Use your formula to find the length of candle remaining for each of the following number of hours burned. i. 6 hours L = 1.6(6 3.5) + 8.4 = 4.4, so the candle is 4.4 inches long after burning for 6 hours ii. 3 hours L = 1.6(3 3.5) + 8.4 = 9.2, so the candle is 9.2 inches long after burning for 3 hours iii. 0 hours (so prior to burning) L = 1.6(0 3.5) + 8.4 = 14, so the candle is 14 inches long prior to burning Copyright 2013 Carlson, OBryan, & Joyner 25
I#6 d. Graph the function. What does the graph represent? The graph represents the length of the candle (in inches) for all possible values of t (the number of hours the candle has burned). Copyright 2013 Carlson, OBryan, & Joyner 26 I#6 In this investigation we developed the general linear formula y = m(x x1) + y1.
This is one version of the point-slope formula for a linear function [named this because we can write the formula if we know the constant rate of change (which is sometimes called the slope) and one point]. Most textbooks give the point-slope formula as y y1= m(x x1). The two formulas provide the same information but one is solved for y. Copyright 2013 Carlson, OBryan, & Joyner 27 I#6 10. Explain why y y1= m(x x1) is just another way of writing y = mx. y y1 represents a change in y from a reference point (y = y1) to the new value of y (which is equivalent to y).
Additionally, x x1 is representing a change in x from a reference point (x = x1) to the chosen value of x (which is equivalent to x). Thus, these formulas are representing the same information given two points (x, y) and (x1, y1). Copyright 2013 Carlson, OBryan, & Joyner 28 I#6 11. Given the formula y 1 = 4(x 9), answer the following questions. a. What is the constant rate of change of the linear relationship? The constant rate of change of y with respect to x is 4. (The change in y is always 4 times as large as the
corresponding change in x.) b. Based on the formula, what point do we know for sure must be on the graph of the function? (9, 1) Copyright 2013 Carlson, OBryan, & Joyner 29 I#6 c. For some ordered pair solution (x, y), explain what each of the following represents. i. x 9 ii. 4(x 9) iii. y 1 i.
x 9 is the change in x from x = 9 to the new value of x ii. 4(x 9) is (rate of change)(change in x), which is the change in y from y = 1 to the new value of y iii. y 1 is the change in y from y = 1 to the new value of y d. Solve for y to rewrite the formula in the form we used earlier in this investigation. y = 4(x 9) + 1 Copyright 2013 Carlson, OBryan, & Joyner 30 I#6 12. In the candle burning context we could have written L 8.4 = 1.6(t 3.5). Explain what each of the following represents.
a. t 3.5 The change in the number of hours the candle has been burning from t = 3.5 to any other value of t. b. 1.6(t 3.5) The change in the candles length (in inches) from L = 8.4. c. L 8.4 The change in the candles length (in inches) from L = 8.4. Copyright 2013 Carlson, OBryan, & Joyner 31 Investigation
#7 I#7 EXTRA PRACTICE WITH LINEAR FUNCTIONS 1. After leaving a rest stop and getting on the highway, you are traveling at a constant speed of 65 mph heading towards Kansas City. Half an hour later you pass a sign that says Kansas City 80 miles. a. Write a formula to define the relationship between the quantities distance from Kansas City (in miles) and time since leaving the rest stop (in hours). Be sure to define your variables. Let d be the distance from Kansas City and t be the elapsed time (in hours) since leaving the rest stop. d = 65(t 0.5) + 80 or d = 80 65(t 0.5) b. How far is the rest stop from Kansas City? In one half hour you travel 0.5(65) = 32.5 miles, so the rest
stop was 80 + 32.5 = 112.5 miles from Kansas City. 33 I#7 1. After leaving a rest stop and getting on the highway, you are traveling at a constant speed of 65 mph heading towards Kansas City. Half an hour later you pass a sign that says Kansas City 80 miles. c. Assuming you continue at the same speed, how long does it take you to reach Kansas City from the time you leave the rest stop? It takes 1/65th of an hour to travel one mile, so it takes you hours to travel 80 miles, and 1.23 + 0.5 = 1.73 hours to travel from the rest stop to Kansas City Copyright 2013 Carlson, OBryan, & Joyner
34 I#7 2. Write the formula that defines each of the following linear functions. a. There is a constant rate of change of y with respect to x of 5 and (x, y) = (2, 9) is a point on the graph. y = 5(x 2) + 9 b. (3, 11) and (6, 5) are two points on the graph of y with respect to x. The constant rate of change of y with respect to x is (so the change in y is always times as large as the change in x), and the formula is y = (x + 3) 11 [or equivalently y = (x 6) 5]. Copyright 2013 Carlson, OBryan, & Joyner
35 I#7 2. Write the formula that defines each of the following linear functions. c. After a recent rain, Johns swimming pool was nearly overflowing. He decided to pump out the extra water. The pool contained 9,250 gallons of water after 20 minutes of pumping and 9,142 gallons of water after 28 minutes pumping. Be sure to define your variables. Let v be the volume of the pool (in gallons) and t be the number of minutes spent pumping. v = 13.5(t 20) + 9,250 [or equivalently v = 13.5(t 28) + 9,142] Copyright 2013 Carlson, OBryan, & Joyner
36 I#7 3a. Given values of x and y in the tables below, which table(s) contain values that could define a linear relationship between quantities? Table 1 x y Table 2 x y Table 3 x
y 2 13 2.5 8.5 1 9.6 1 4
1.5 2.5 1 15 5 8 4.5 2 4
29.3 Tables 1 and 2 can represent linear relationships, but Table 3 cannot. b. For each linear relationship you identified, write the formula that defines the relationship. (answers may Table 1: y = 3(x + 2) 13 Table 2: y = 1.5(x + 2.5) + 8.5 Copyright 2013 Carlson, OBryan, & Joyner vary depending on the reference point chosen) 37 I#7 4. Kim is a freelance writer for a major magazine. The
given table shows the amount of money Kim charged the magazine for varying numbers of words in her articles. Number words in Kims article Total amount charged, in dollars 602 884.70 1,245 1,752.75 2,450
3,379.50 3,600 4,932.00 a. There is a constant rate of change of the total amount she charges (in dollars) with respect to the number of words in her article. Determine the constant rate and explain what it represents in this context. Copyright 2013 Carlson, OBryan, & Joyner 38 I#7 Number words in Kims article
Total amount charged, in dollars 602 884.70 1,245 1,752.75 2,450 3,379.50 3,600
4,932.00 a. There is a constant rate of change of the total amount she charges (in dollars) with respect to the number of words in her article. Determine the constant rate and explain what it represents in this context. The constant rate of change is $1.35 per word. The change in the total amount she charges (in dollars) is 1.35 times as large as the change in the number of words for the article. Copyright 2013 Carlson, OBryan, & Joyner 39 I#7 b. Suppose she writes an article, and at the last minute the magazine asks her to increase the articles length by 450
words. By how much will her fee change? If F is the total fee she charges (in dollars) and w is the number of words in the final article, then F = 1.35w = 1.35(450) = 607.50. Her fee increases by $607.50. c. Write a formula to define the relationship between the total amount she charges (in dollars) in terms of the number of words in an article. Define any necessary variables. F = 1.35(w 602) + 884.70 Copyright 2013 Carlson, OBryan, & Joyner 40 I#7 5. According to the American Heart Association, people should maintain a heart rate between 50% and 85% of
their maximum heart rate during exercise. The following graph shows the high and low target heart rates during exercise (in beats per minute) for people of different ages. a. Is a 40-year-old who maintains a heart rate of 100 bpm while exercising following the AHA recommendations? Yes, the point (40, 100) falls between the lines. Copyright 2013 Carlson, OBryan, & Joyner 41 I#7
b. What does the point (30, 95) on the line marked low represent in this context? A 30-year old person should try to have their heart rate at least 95 beats per minute while exercising. c. Define variables and write formulas to define the high and low target exercise heart rates for people of different ages. Let h be the target exercise heart rate (in bpm) and n be a persons age in years. High: h = 0.85(n 30) + 162 or h = 0.85(n 50) + 145 Low: h = 0.5(n 30) + 92 or h = 0.5(n 50) + 85 Copyright 2013 Carlson, OBryan, & Joyner 42 I#7
d. Find the vertical intercept for the low target exercise heart rate if we were to extend the line. Does this value have any meaning in our context? Explain. The low target exercise heart rate line would cross the vertical axis at h = 110 (use the reasoning weve developed in this module to determine this value). However, this answer doesnt really mean anything in this context since the American Heart Association is not going to provide recommended exercise heart rates for newborns. Copyright 2013 Carlson, OBryan, & Joyner 43 I#7
6. Sketch a graph for each of the following linear functions. a. y = 2.5(x 4) 1 b. w = 0.8(r + 1) + 5 Copyright 2013 Carlson, OBryan, & Joyner 44 I#7 6. Sketch a graph for each of the following linear functions. a. y = 2.5(x 4) 1 b. w = 0.8(r + 1) + 5 Copyright 2013 Carlson, OBryan, & Joyner
45 I#7 6. Sketch a graph for each of the following linear functions. a. y = 2.5(x 4) 1 b. w = 0.8(r + 1) + 5 Copyright 2013 Carlson, OBryan, & Joyner 46 I#7 7. Write the formula that defines the linear relationship shown in each graph. a.
b. y = 1.4(x + 1) + 7 or y = 1.4(x 2.5) + 2.1 y = 3.2(x 2) + 5.9 or y = 3.2(x 4.6) + 14.22 Copyright 2013 Carlson, OBryan, & Joyner 47 I#7 8. The points (0, 5), (1, 5), (2.5, 5), (4, 5), and (12, 5) lie on a line. a. Plot these points and construct the line that passes
through all of these points. Copyright 2013 Carlson, OBryan, & Joyner 48 I#7 b. As the value of x varies, what can you say about the value of y? Write a formula to represent all points that lie on this line. The values of y remain constant. y = 5 c. Linear functions of the form y = c, where c is a constant, are horizontal lines that are c units away from the x-axis (or horizontal axis). Describe the lines y = 2.5 and y = 0. The first is a horizontal line 2.5 units above the horizontal axis. The second is a horizontal line that lies on the horizontal axis.
Copyright 2013 Carlson, OBryan, & Joyner 49 I#7 d. What is the slope of y = 2.5? Will all horizontal lines have this slope? Explain. The slope (constant rate of change of y with respect to x) is 0. Yes, all horizontal lines will have a slope of 0 because for any change in x, the change in y is zero. Zero (0) is the only value of m that makes the statement y = 0 = mx true for any value of x. Copyright 2013 Carlson, OBryan, & Joyner
50 I#7 9. The points (3, 6), (3, 5), (3, 2), (3, 0), (3, 6) and (3, 10) lie on a line. a. Plot these points and construct the line that passes through all of these points. Copyright 2013 Carlson, OBryan, & Joyner 51 I#7 b. How do the changes in x and y compare? Write a formula to represent all points that lie on this line.
The values of x will remain constant at a value of 3. x=3 c. Relationships of the form x = c, where c is a constant, are vertical lines that are c units away from the y-axis (or vertical axis). Describe the lines x = 4 and x = 0. The first is a vertical line that is 4 units to the left of the vertical axis. The second is a vertical line that lies on the vertical axis. Copyright 2013 Carlson, OBryan, & Joyner 52 I#7 d. What is the slope of x = 4.5? What can you say about the slope of vertical lines? The slopes (constant rate of change of y with respect to x)
are undefined. e. Why does it make sense to say that the slope of a vertical line is undefined based on the meaning of constant rate of change? If the constant rate of change is a number m such that y = mx, and the change in x is always 0 for any change in y , there is no value of m that will make the statement y = m0 true for y 0. Copyright 2013 Carlson, OBryan, & Joyner 53 Investigation #8
I#8 SLOPE-INTERCEPT FORM OF A LINEAR FUNCTION In previous investigations we learned how to write the formula for a linear relationship if we know the rate of change of one quantity with respect to the other quantity (m) and one ordered pair for the relationship (x1, y1): y = m(x x1) + y1, or alternatively y y1 = m(x x1). In this investigation we will explore the special case where x1 = 0. Copyright 2013 Carlson, OBryan, & Joyner 55 I#8
1. The given graph shows the relationship where y changes at a constant rate of 2 with respect to x. a. What is the change in x from the given reference point to x = 7? Represent this on the graph. b. What is the value of y when x = 7? Represent this on the graph. Copyright 2013 Carlson, OBryan, & Joyner 56 I#8 1. The given graph shows the relationship where y changes at a constant rate of 2 with respect to x.
a. What is the change in x from the given reference point to x = 7? Represent this on the graph. x = 7 0 = 7. Note that the change in x from x = 0 is the same as the value of x. b. What is the value of y when x = 7? Represent this on the graph. Copyright 2013 Carlson, OBryan, & Joyner 57 I#8 1. The given graph shows the relationship where y changes at a constant rate of 2 with respect to x.
a. What is the change in x from the given reference point to x = 7? Represent this on the graph. x = 7 0 = 7. Note that the change in x from x = 0 is the same as the value of x. b. What is the value of y when x = 7? Represent this on the graph. y = 2(7) + 5 = 9 Copyright 2013 Carlson, OBryan, & Joyner 58 I#8 1. The given graph shows the relationship where y changes at a constant rate of 2 with respect to x.
c. What is the change in x from the given reference point to x = 1? Represent this on the graph. d. What is the value of y when x = 1? Represent this on the graph. Copyright 2013 Carlson, OBryan, & Joyner 59 I#8 1. The given graph shows the relationship where y changes at a constant rate of 2 with respect to x. c. What is the change in x from
the given reference point to x = 1? Represent this on the graph. x = 1 0 = 1 d. What is the value of y when x = 1? Represent this on the graph. Copyright 2013 Carlson, OBryan, & Joyner 60 I#8 1. The given graph shows the relationship where y changes at a constant rate of 2 with respect to x. c. What is the change in x from the given reference point to
x = 1? Represent this on the graph. x = 1 0 = 1 d. What is the value of y when x = 1? Represent this on the graph. y = 2(1) + 5 = 7 Copyright 2013 Carlson, OBryan, & Joyner 61 I#8 1. The given graph shows the relationship where y changes at a constant rate of 2 with respect to x. e. Write a formula to calculate the value of y for any possible value of x.
y = 2(x 0) + 5 = 2x + 5 f. Is there a benefit to using the vertical intercept as our reference point? Explain. Yes. The change in x from x = 0 is the same as the value of x, so the formula becomes simpler. Copyright 2013 Carlson, OBryan, & Joyner 62 I#8 Slope-Intercept Form of a Linear Function If y is a linear function of x with a constant rate of change of m and if we know that (0, b) is a point on the graph, then the following is a useful formula for the function relationship. y = m(x x1) + y1
y = m(x 0) + b y = mx + b In this form we rely on the fact that the value of x is also the change in x from x = 0. This is just a special case of y = m(x x1) + y1 its not a new formula to memorize! Ex: If y changes at a constant rate of 4.2 with respect to x, and the graph of the function passes through the y = 4.2x 6 point (0, 6), then . Copyright 2013 Carlson, OBryan, & Joyner 63 I#8 2. Consider the formula y = 3.5x 9. Suppose we want to find the value of y when x = 6.5. Explain how the
formula determines the corresponding value of y using the meaning of constant rate of change. (You may sketch a graph or draw a diagram if it helps you explain.) The formula determines the value of y for a given value of x using a reference point of (0, 9). We change x from x = 0 to x = 6.5, so x = 6.5 0 = 6.5, and thus the value of x is also the change in x from x = 0. The formula then determines the change in y from y = 9 using m(x) = 3.5(6.5). We add this change to y = 9 to find the value of y when x = 6.5. Copyright 2013 Carlson, OBryan, & Joyner 64 I#8 2. Consider the formula y = 3.5x 9. Suppose we want to
find the value of y when x = 6.5. Explain how the formula determines the corresponding value of y using the meaning of constant rate of change. (You may sketch a graph or draw a diagram if it helps you explain.) Copyright 2013 Carlson, OBryan, & Joyner 65 I#8 3. Savanna and Zoe met for their daily run. Savanna arrived late so Zoe started running before Savanna arrived. When Savanna started running, Zoe had already run 3 laps. Savanna joined Zoe at the beginning of the 4th lap and they ran together for 5 more laps before they both stopped. a. Complete the table that represents the relationship between
the number of laps run by Savanna x and the number of laps run by Zoe y. Number of Laps Run by Savanna, x 0 1 2 3 4 5 Number of Laps Run by Zoe, y Copyright 2013 Carlson, OBryan, & Joyner 66
I#8 3. Savanna and Zoe met for their daily run. Savanna arrived late so Zoe started running before Savanna arrived. When Savanna started running, Zoe had already run 3 laps. Savanna joined Zoe at the beginning of the 4th lap and they ran together for 5 more laps before they both stopped. a. Complete the table that represents the relationship between the number of laps run by Savanna x and the number of laps run by Zoe y. Number of Laps Run by Savanna, x 0 1 2 3 4
5 Number of Laps Run by Zoe, y 3 4 5 6 7 8 Copyright 2013 Carlson, OBryan, & Joyner 67 I#8 b. What is the constant rate of change in this context? What
does it tell us? 1 lap run by Zoe per lap run by Savanna. For any change in the number of laps run by Savanna is always the same as the change in the number of laps run by Zoe. c. Write a formula to express the number of laps run by Zoe y in terms of the number of laps run by Savanna x. y=x+3 Copyright 2013 Carlson, OBryan, & Joyner 68 I#8 d. Construct a graph that represents the number of laps run by Zoe in terms of the number of laps run by Savanna. Be sure to label your axes.
Copyright 2013 Carlson, OBryan, & Joyner 69 I#8 e. Illustrate on your graph in part (d) a change from 2 to 5 laps run by Savanna and the corresponding change in the number of laps run by Zoe. Copyright 2013 Carlson, OBryan, & Joyner 70 I#8 4. After pulling the plug from a bathtub, the water started draining. In the first 20 seconds 5 gallons drained from
the tub. Suppose that the tub originally contained 56 gallons of water. a. Assuming the water is draining at a constant rate, what is the change in the volume of water remaining after 10 seconds? After 1 second? How much water remains in each case? After 10 seconds the change in volume is 2.5 gallons (and the volume remaining is 2.5 + 56 = 53.5 gallons). After 1 second the change in volume is 0.25 gallons (and the volume remaining is 0.25 + 56 = 55.75 gallons). Copyright 2013 Carlson, OBryan, & Joyner 71 I#8 b. Write the formula that calculates the number of gallons
of water remaining in the tub in terms of the number of seconds that have elapsed since pulling the plug. Be sure to define any necessary variables. Let v represent the volume of water remaining in the tub (in gallons) and t represent the time elapsed (in seconds) since pulling the plug. Then v = 0.25t + 56 which might also be written v = 56 0.25t. Copyright 2013 Carlson, OBryan, & Joyner 72 I#8 c. Sketch a graph that relates the number of gallons of water remaining in the tub to the amount of time elapsed since the tub began to drain. Be sure to label your axes.
Copyright 2013 Carlson, OBryan, & Joyner 73 I#8 d. State the vertical intercept of the graph and explain what it represents in this situation. The vertical intercept is 56 [the graph intersects the vertical axis at (0, 56)]. The tub contains 56 gallons of water before the plug is pulled. e. The horizontal intercept in this context is the value of t where the graph intersects the horizontal axis. Find the horizontal intercept and explain what it represents in this situation. The horizontal intercept is 224 [the graph crosses the horizontal axis at (224, 0)]. The tub is empty 224
seconds (about 3.7 minutes) after the plug is pulled. We can find this value by substituting v = 0 and solving for t. If we do, what does each step of the process represent? Copyright 2013 Carlson, OBryan, & Joyner 74 I#8 If we do, what does each step of the process represent? The first step is to subtract 56 from 0, which calculates the change in the output. Then we divide by the constant rate of change to find the change in the input from 0, which is also the value of the input were looking for. Copyright 2013 Carlson, OBryan, & Joyner
75 I#8 5. Write the formula for each linear relationship described or represented in the following exercises. a. The constant rate of change of y with respect to x is 5 and the graph passes through the point (0, 3.6). y = 5x + 3.6 b. Carlos left his house and started driving. When Carlos is 12 miles from his house he passes Exit 201. If he continues to drive at a constant speed of 58 miles per hour, write a formula that calculates the total distance hes traveled since leaving his house as a function of the number of hours since passing Exit 201. Let d represent the distance (in miles) Carlos has traveled since leaving his house and t represent the number of hours since he passed Exit 201. Then d = 58t + 12.
Copyright 2013 Carlson, OBryan, & Joyner 76 I#8 5. Write the formula for each linear relationship described or represented in the following exercises. c. Write the formula for y in terms of x. x 2 0 6 9 y 9 4
43 62.5 y = 6.5x + 4 Copyright 2013 Carlson, OBryan, & Joyner 77 I#8 6. Sometimes we want to write a linear relationship using the slope-intercept form (y = mx + b) but we dont know the vertical intercept. Lets revisit the bathtub draining context. Suppose we pull the plug on a different bathtub and allow it to drain. The volume is changing at a constant rate of 18 gallons per minute, and after 2 minutes there are 16 gallons of water remaining in the
tub. Let v be the number of gallons of water remaining in the tub and let t be the number of minutes that have elapsed since pulling the plug. Copyright 2013 Carlson, OBryan, & Joyner 78 I#8 a. How many gallons were in the tub before it started draining? From the reference point we change t by 2 minutes, so v changes by (18)(2) = 36 gallons. The original volume of water in the tub was 16 + 36 = 52 gallons. Copyright 2013 Carlson, OBryan, & Joyner
79 I#8 b. Write the relationship between v and t using the slopeintercept form (v = mt + b). v = 18t + 52 c. In a previous investigation we would have written the formula as v = 18(t 2) + 16. Use the distributive property and combine like terms to simplify v = 18(t 2) + 16. What did you find? Copyright 2013 Carlson, OBryan, & Joyner 80 I#8 d. (optional extension) How do the calculations you
performed in part (c) relate to the work you did in part (a)? When we distribute, we are changing the reference point from t = 2 to t = 0 for the formula, and the term (18)(2) appears, which is the amount v changes when t goes from t = 2 to t = 0. Then the process adds 36 + 12, which generates the value of v when t = 0. 81 I#8 7. Write a formula for each of the following linear relationships in i) the form y = m(x x1) + y1. ii) the form y = mx + b. a.
y changes at a constant rate of 4 with respect to x, and (6, 1) is a point on the line. i) y = 4(x + 6) + 1 ii) y = 4x +25 Copyright 2013 Carlson, OBryan, & Joyner 82 I#8 7. Write a formula for each of the following linear relationships in i) the form y = m(x x1) + y1. ii) the form y = mx + b. b.
i) y = 1.25(x 2) 7.5 or y = 1.25(x + 4) ii) y = 1.25x 5 Copyright 2013 Carlson, OBryan, & Joyner 83 I#8 8. Sketch the graph for each of the following functions. a. y = 2x + 7 b. y = 1.5x
Copyright 2013 Carlson, OBryan, & Joyner 84 I#8 8. Sketch the graph for each of the following functions. a. y = 2x + 7 b. y = 1.5x Copyright 2013 Carlson, OBryan, & Joyner 85 I#8 8. Sketch the graph for each of the following functions.
a. y = 2x + 7 b. y = 1.5x Copyright 2013 Carlson, OBryan, & Joyner 86 Investigation #9 I#9 PROPORTIONALITY For Exercises #1-4, use the following context. In computer word processing or layout programs you have the option to insert images, such as a photo. Then you can
change the size of this image by clicking on it and dragging the edges. Suppose you insert a photo into a program that is a rectangle 5 inches wide by 4 inches high. Copyright 2013 Carlson, OBryan, & Joyner 88 I#9 1a. Suppose you know how wide the photo needs to be. What must you pay attention to so that the photo is not distorted from the original? The ratio of the height of the photo compared to the width must remain constant. That is, the height of the photo must always be times as large as the width of the photo. b. The first step in correctly resizing the photo is to determine the height of the photo based on the width
that you need. If you need the photo to be 13 inches wide, what should you make its height? Explain how you are thinking about this problem to come up with your answer. The height should be times as large as the width of the photo, so 13() = 10.4 inches. Copyright 2013 Carlson, OBryan, & Joyner 89 I#9 c. Complete the table that relates the heights h of the photo (in inches) to the widths w (in inches). How do we know that these measurements will produce a photo that is not distorted? Width of photo, in inches (w)
Height of photo, in inches (h) 1 2 6 13 Copyright 2013 Carlson, OBryan, & Joyner 90 I#9 c. Complete the table that relates the heights h of the photo (in inches) to the widths w (in inches). How do we know that these measurements will produce a photo that is not
distorted? Width of photo, in inches (w) 1 2 6 13 Height of photo, in inches (h) 1() = = 0.8 2() = 8/5 = 1.6 6() = 24/5 = 4.8 13() = 52/5 = 10.4 In each case the height is times as large as the width, which ensures that the photo will not be distorted.
Copyright 2013 Carlson, OBryan, & Joyner 91 I#9 d. Complete the following statement: As the width of the photo changes, the height of the photo changes so that the height is always times as large as the width of the photo. (Other answers are possible.) e. Write a formula that determines the height of the photo given its width such that the photo will not be distorted from the original. h = ()w Copyright 2013 Carlson, OBryan, & Joyner
92 I#9 2a. Draw a graph of the height vs. the width for undistorted photos with a width up to 14 inches. Be sure to label your axes with the appropriate quantities and units of measure. Represent the photo widths on the horizontal axis. Copyright 2013 Carlson, OBryan, & Joyner 93 I#9 b. Explain what a point on your graph represents. It represents the width and height of a photo that is not
distorted from the original. c. What is the constant rate of change for the relationship you graphed? What does this tell us? ; for any change in the width of the photo (in inches), the height of the photo (in inches) must change by times as much so that the photo will not become distorted. Copyright 2013 Carlson, OBryan, & Joyner 94 I#9 d. Where does your graph cross the vertical axis? What does this point represent? (0, 0); If the width of the photo is 0 inches, the height must be 0 inches as well.
e. Write a formula that determines the width of the photo given its height such that the photo will not be distorted from the original. w = (5/4)h Copyright 2013 Carlson, OBryan, & Joyner 95 I#9 3a. What is the ratio equal to? What does this tell us? ; the height of the photo is always times as large as the width of the photo (assuming the photo is not distorted) b. Plot the points (0, 0) and (5, 4) on the graph in Exercise #2, then calculate w and h from the first point to the second point and represent these changes on the graph.
w = 5 0 = 5 and h = 4 0 = 4 Copyright 2013 Carlson, OBryan, & Joyner 96 I#9 c. Talk about the link between and when we change away from (0, 0). Why did this happen? h = h and w = w, so ; this happens because we are changing away from (0, 0), so h = h
0 = h and w = w 0 = w d. Is the result you found in part (c) true for all linear functions? If so, why does this happen. If not, what makes this context different? You may want to look back at exercises in previous investigations. Its not generally true. What makes this context special is that (0, 0) is a solution, and the value of a variable and its change away from 0 are equal Copyright 2013 Carlson, OBryan, & Joyner 97 I#8 Proportionality If y is a linear function of x, and if y = 0 when x = 0, then we say y is proportional to x and
y = mx where m is the constant rate of change of y with respect to x. Note 1: When y is proportional to x the constant rate of change m is sometimes called the constant of proportionality because it tells us that y is always m times as large as x. You might also see m replaced with another letter, like k. Note 2: The relationship y = mx is also sometimes called direct variation and we might say y varies directly with x. Copyright 2013 Carlson, OBryan, & Joyner 98 I#9 4. In earlier investigations we used the general formula y = m(x x1) + y1 to represent a linear function. a. If we know y is proportional to x, what point must be
on the graph of the relationship? (0, 0) b. Use your answer from part (a) as the reference point (x1, y1) to show that y = mx is the formula when y is proportional to x. y = m(x x1) + y1 y = m(x 0) + 0 y = mx Copyright 2013 Carlson, OBryan, & Joyner 99 I#9 In the photograph context (Exercises #1-4), we say that the height of the photo is proportional to the width of the photo and the constant of proportionality is . This creates some interesting properties.
As the two quantities change together, I) the height of the photo will always be times as large as the width of the photo. II) the ratio of the height of the photo to the width of the photo will always be . However, its also true that III) if one quantity is scaled by some factor m, then the other quantity will also be scaled by the same factor m. Copyright 2013 Carlson, OBryan, & Joyner 100 I#9 In Exercises #5-7, use the following context. When making pancakes from a powdered mix, a recipe is given for the amounts of each ingredient to be used. One popular brand calls for 20 tablespoons of mix for every 6
ounces of water. 5a. Suppose you want to use 10 tablespoons of powdered mix. How much water do you need? Explain how you are thinking about this problem to come up with your answer. Since 10 tablespoons is half of 20 tablespoons of mix, you will need half of the original amount of water, or 3 ounces of water. Copyright 2013 Carlson, OBryan, & Joyner 101 I#9 b. Suppose you want to use 5 tablespoons of powdered mix. How much water do you need? What about for 15 tablespoons of mix? Explain the thinking you used to determine your answers.
Since 5 tablespoons is half of 10 tablespoons of mix, you will need half of the amount of water determined in part (a), or 1.5 ounces of water. 15 tablespoons of mix is 3 times this amount, so you will need 3 times as much water, or 4.5 ounces. (These could also be scaled from the original 20 tablespoons of mix.) Copyright 2013 Carlson, OBryan, & Joyner 102 I#9 c. How many tablespoons of powdered mix should be used with 12 ounces of water? 15 ounces of water? 29 ounces of water? 12 ounces of water is two times as much as the original
amount of water, so you will need two times as much mix, or 40 tablespoons. 15 ounces is 15/16 times as much water (or 2.5 times) so you will need 15/16 (or 2.5) times as much mix, or 50 tablespoons of mix. 29 ounces of water is 29/6 times as much water as the original amount, so you will need 29/6 times as much mix as the original, or 96 tablespoons of mix. Copyright 2013 Carlson, OBryan, & Joyner 103 I#9 d. How many ounces of water should be used if you only have 8 tablespoons of powdered mix left in the box? 8 tablespoons of mix is 8/20 times as much as (or 0.4 times) the original amount of mix, so you will need 8/20 (or 0.4)
times as much water as originally called for, or 2.4 ounces of water. e. How much water should be used with one tablespoon of powdered mix? How much powdered mix should be used with one ounce of water? One tablespoon of mix is 1/20 as much as called for, so you will need 1/20 as much water, or 0.3 ounces of water for one tablespoon of mix. For one ounce of water, you will need 20/6 tablespoons of mix, or 3 tablespoons. Copyright 2013 Carlson, OBryan, & Joyner 104 I#9 6. Some people vary the ratio of powder to water based on how thick they like their pancakes. Some people like their batter thicker while others prefer thinner batter.
a. In terms of the recipe, what does it mean for the batter to be thicker or thinner than the recommended mixture? For thicker: It means more tablespoons of mix for the same amount of water, or less water for the same amount of tablespoons of mix. The ratio of mix to water will be larger than the original ratio. For thinner, it means less tablespoons of mix for the same amount of water, or more water for the same amount of tablespoons of mix as called for in the recipe. The ratio of mix to water will be smaller than the original ratio. Copyright 2013 Carlson, OBryan, & Joyner 105 I#9 6. Some people vary the ratio of powder to water based on
how thick they like their pancakes. Some people like their batter thicker while others prefer thinner batter. b. If a person combines 3 ounces of water using 8 tablespoons of powdered mix, is her batter thicker or thinner than the recommended mixture? Explain your reasoning. The ratio of mix to water here is 8/3 (or 2 to 1). So for every ounce of water, you use 2 tablespoons of mix. This is less than what we determined the recipe to be for part (e), which called for 3 tablespoons of mix for each ounce of water. Her batter will be thinner than the recommended mixture. Copyright 2013 Carlson, OBryan, & Joyner 106 I#9
7a. Construct a graph of the number of tablespoons of mix compared to the number of ounces of water based on the recommendations on the given recipe. Be sure to label your axes with the appropriate quantities and units, with the amount of water represented on the horizontal axis. Copyright 2013 Carlson, OBryan, & Joyner 107 I#9 b. Plot the point (4, 10) on your graph and explain what this point represents. This point represents a batter mixture of 4 ounces of water and 10 tablespoons of mix. c. Is the point you plotted in part (b) on the line? What does this
indicate about the batter? This point is not on the line. This tells us that this mixture does not match the recommended ratio it will not have the same consistency as batter made using the recommended ratio. d. Does this ordered pair represent batter that is thicker or thinner than batter made according to the recipe? The batter is thinner. There are 2.5 tablespoons of mix for every ounce of water (as compared with 3 tablespoons for every ounce of water in the original recipe). Copyright 2013 Carlson, OBryan, & Joyner 108 Investigation #10 I#10
EXTRA PRACTICE WITH PROPORTIONALITY 1. Determine if the quantities described in each situation are proportional. Justify your response. (Hint: Think about how the quantities are related as they change together.) a. An online video rental company charges $1.20 for each video rented. Determine if the total cost of the videos rented and the number of videos rented are proportional. The quantities are proportional (they are related by a constant multiple). The total cost of videos rented will always be 1.2 times as large as the number of videos rented. The ratio of the cost (in dollars) to the number of videos is always 1.20 to 1. Copyright 2013 Carlson, OBryan, & Joyner 110
I#10 1. Determine if the quantities described in each situation are proportional. Justify your response. (Hint: Think about how the quantities are related as they change together.) b. A Pilates gym charges a $55 registration fee, then $19 for each 50-minute class a person attends. Determine if the total cost of belonging to the gym and the number of classes a person has attended are proportional. The quantities are not proportional. The quantities total cost of belonging to the gym and the number of classes a person has attended do not remain in a constant ratio. Copyright 2013 Carlson, OBryan, & Joyner 111
I#10 1. Determine if the quantities described in each situation are proportional. Justify your response. (Hint: Think about how the quantities are related as they change together.) c. A recipe for homemade brownies calls for cup of cocoa powder for every cup of flour. Determine if the number of cups of cocoa powder is proportional to the number of cups of flour for different sized batches of brownies. The quantities are proportional. The number of cups of cocoa powder will always be times the number of cups of flour. The ratio of the number of cups of flour to the number of cups of cocoa is always 3:2. Copyright 2013 Carlson, OBryan, & Joyner
112 I#10 2. Determine if the following statements are true or false and explain your reasoning. a. T or F: If Quantity A is proportional to Quantity B, the graph that relates Quantity A and Quantity B passes through the origin. True. We know that when two quantities are proportional the value of one quantity is always a constant multiple times as large as the other quantity. Thus if the value of one quantity is 0, then no matter the constant multiple, the value of the other quantity must be zero (because 0 times any number is 0). So the coordinate point (0, 0) always exists on the graph of proportional quantities. Copyright 2013 Carlson, OBryan, & Joyner
113 I#10 2. Determine if the following statements are true or false and explain your reasoning. b. T or F: If Quantity A is proportional to Quantity B, and Quantity A is increased by some amount, then Quantity B is increased by the same amount. False. Proportionality is not about adding by the same amount. You can easily create a counterexample just choose a proportional relationship y = kx where k 1 and explore what happens if you try to change x and y by the same amount. Copyright 2013 Carlson, OBryan, & Joyner
114 I#10 2. Determine if the following statements are true or false and explain your reasoning. c. T or F: Every graph that passes through the origin represents a proportional relationship between two quantities. False. Its also necessary that there be a constant rate of change. Copyright 2013 Carlson, OBryan, & Joyner 115 I#10
3. Complete the statement. If Quantity A is proportional to Quantity B, then as the value of Quantity A changes, the value of Quantity B changes so that The ratio of the values of Quantity A and Quantity B is always constant. Or they change in such a way that the value of Quantity A is always a constant multiple of the value of Quantity B. Or if the value of Quantity A is scaled by some factor, the value of Quantity B must be scaled by the same factor. Copyright 2013 Carlson, OBryan, & Joyner 116 I#10
4. Use these tables of inputs and outputs to answer the following questions. Table 1 x 3 7 10 y 4 9 12.75 Table 2 k 5 2 12
r 11 4.4 26.4 Table 3 m 4 2.1 6 p 12 2.85 3 a. Which table(s) gives values of quantities that could be related
proportionally? Explain your reasoning. Table 2 is the only one that can represent a proportional relationship. b. For each table that could define a proportional relationship between two quantities, determine a possible formula to relate the values of the two quantities. Table 2: r = 2.2k Copyright 2013 Carlson, OBryan, & Joyner 117 I#10 5. Rebecca has a jar of red jellybeans. She adds 100 blue jellybeans to the jar and shakes them up so the red and blue jellybeans are mixed well. She then takes out a scoop of 50 jellybeans and finds that 25 of them are blue. She pours the 50 jellybeans back into the jar.
a. Estimate the total number of jellybeans in the jar and the number of red jellybeans in the jar. Explain your reasoning. Half of the jellybeans in the scoop are blue, so we estimate that half of the jellybeans in the jar are blue. So we estimate there are 200 total jellybeans in the jar and 100 red jellybeans. Copyright 2013 Carlson, OBryan, & Joyner 118 I#10 5. Rebecca has a jar of red jellybeans. She adds 100 blue jellybeans to the jar and shakes them up so the red and blue jellybeans are mixed well. She then takes out a scoop of 50 jellybeans and finds that 25 of them are
blue. She pours the 50 jellybeans back into the jar. b. Suppose she scoops out 150 jellybeans from the jar. How many of each color might she expect to get? Make sure you can explain your reasoning. We expect about half to be blue and half to be red, so 75 of each color in the scoop. Copyright 2013 Carlson, OBryan, & Joyner 119 I#10 6. This time Rebecca adds 58 purple jellybeans to a jar of yellow jellybeans and shakes them up. She then takes out a scoop of 90 jellybeans and finds that exactly 15 of them are purple. She pours the 90 jellybeans back into the jar.
a. Estimate the total number of jellybeans in the jar and the number of yellow jellybeans in the jar. Explain your reasoning. We observe that of the jellybeans in the scoop are purple. So we expect that of the jellybeans in the jar are purple, which means that we expect there to be 6(58) = 348 total jellybeans in the jar, and 348 58 = 290 yellow jellybeans. Copyright 2013 Carlson, OBryan, & Joyner 120 I#10 6. This time Rebecca adds 58 purple jellybeans to a jar of yellow jellybeans and shakes them up. She then takes out a scoop of 90 jellybeans and finds that exactly 15 of
them are purple. She pours the 90 jellybeans back into the jar. b. If Rebecca used a smaller scoop to retrieve jellybeans and finds that exactly 5 of them are purple, how many yellow jellybeans do you expect are in the scoop? Make sure you can explain your reasoning. We expect of the jellybeans in the scoop to be purple, so we expect there to be 5(6) = 30 total jellybeans in the scoop and 30 5 = 25 of them to be yellow. Copyright 2013 Carlson, OBryan, & Joyner 121 I#10 7. Next, Rebecca adds 72 green jellybeans to a jar of orange jellybeans and shakes them up. She then takes
out a scoop of 42 jellybeans and finds that 36 of them are green. She pours the scoop of jellybeans back into the jar. a. Estimate the total number of jellybeans in the jar and the number of orange jellybeans in the jar. Explain your reasoning. In the scoop of the jellybeans are green. So we expect of the jellybeans in the jar to be green, and thus (total jellybeans) = 72 or total jellybeans , so we expect there to be 84 total jellybeans in the jar and 84 72 = 12 orange jellybeans in the jar. Copyright 2013 Carlson, OBryan, & Joyner 122 I#10
7. Next, Rebecca adds 72 green jellybeans to a jar of orange jellybeans and shakes them up. She then takes out a scoop of 42 jellybeans and finds that 36 of them are green. She pours the scoop of jellybeans back into the jar. b. Suppose she scoops 33 jellybeans from the jar. How many of each color might she expect to get? Make sure you can explain your reasoning. We expect of the 33 jellybeans to be green, so (28 or 29) green jellybeans in the scoop, and thus 4 or 5 orange jellybeans. Copyright 2013 Carlson, OBryan, & Joyner
123 I#10 7. Next, Rebecca adds 72 green jellybeans to a jar of orange jellybeans and shakes them up. She then takes out a scoop of 42 jellybeans and finds that 36 of them are green. She pours the scoop of jellybeans back into the jar. c. Suppose that she scoops some jellybeans from the jar and finds that 4 of them are orange. How many total jellybeans do you expect to be in the entire scoop? We expect of the jellybeans in the scoop to be orange, so we expect there to be 7(4) = 28 total jellybeans in the scoop. Copyright 2013 Carlson, OBryan, & Joyner
124 I#10 8. Student Council wants to know how many students plan to attend the pep rally before the Homecoming football game. They randomly select 70 students (from the total school population of 1,820) and 9 of those students say they plan to attend. Based on this survey, about how many students to they expect to attend? Answers may vary since we are asking for an estimate. About of the students surveyed plan to attend, so we should expect about students. Another possible approach is to estimate that students plan to attend.
Copyright 2013 Carlson, OBryan, & Joyner 125 I#10 9. Jamie is practicing his typing skills. On his first trial, he typed 72 words in 3 minutes. On the second trial, he typed 132 words in 5.5 minutes. Assume he types at a constant rate. a. Determine the constant rate of change of the number of words typed with respect to the number of minutes spent typing for each trial. What do you observe? It appears that he typed at a constant rate of 24 words per minute in both trials. [Note that we must first realize that in both trials we have an implied second ordered pair of (0, 0)]. The number of words he can type is 24 times as large as the number of minutes he has spent typing.
Copyright 2013 Carlson, OBryan, & Joyner 126 I#10 b. Represent the two trials as points on the graph. Be sure to label your axes. Copyright 2013 Carlson, OBryan, & Joyner 127 I#10 c. Assume that Jamie always types at the same constant rate (in words per minute). Determine the change in the number of
words Jamie types from 0 to 2.7 minutes. The number of words he types will increase by 64.8 (approximately 65) words. d. When 2.7 minutes had elapsed since starting, how many words had Jamie typed? Jamie will have typed 64.8 (approximately 65) words. e. Your answers to parts (c) and (d) should be the same. Explain why this is the case. In this situation, the change in the time elapsed is the same as the amount of time elapsed because the change is from zero, and the change in the number of words typed is the same as the total number of words typed because the change is from zero. Copyright 2013 Carlson, OBryan, & Joyner 128 I#10
10. Describe three additional pairs of real-world quantities that are related proportionally. Justify your choices. Copyright 2013 Carlson, OBryan, & Joyner 129
Therapeutic Exercise: Foundations and Techniques, 6th Edition
Title: Therapeutic Exercise: Foundations and Techniques, 6th Edition Subject: Chapter 5: Peripheral Joint Mobilization/Manipulation Author: [email protected]Lipschitz functions, proper colorings and cutsets in high ...
Lipschitz functions, proper colorings and cutsets in high dimensions. 2.6.2011 BIRS. Ron Peled (Tel Aviv University) Portions joint with Ohad N. Feldheim(Tel Aviv University)TEST WRITING STRATEGIES Opportunities to Improve Our Educational
Bloom's Taxonomy. One recurring recommendation in the resources is to consider Bloom's Taxonomy. It was created as a method for classifying educational goals for student performance evaluations and has been revised over the years. Bloom's applies to acquiring knowledge, which...