LESSONS - TypePadcicobb.typepad.com/files/ghsgt-obj.36-dir… · Web view · 2008-12-12LESSONS. FOR GHSGT. MATH OBJECTIVE 36. Direct and Inverse Proportions. Direct and Inverse

GHSGT-Math-Obj. 36-Direct & Inverse Proportions

INTERVENTIONLESSONS

FOR GHSGTMATH

OBJECTIVE 36

Direct and Inverse Proportions

1


Direct and Inverse ProportionsLesson I

GHSGT Objective#36 Identifies and applies mathematics to practical problems requiring direct and inverse proportions. EQ: How do I identify and solve problems involving direct proportions? Warm-Up/Activator Decide whether you believe the following statements are True or False. Before After A direct variation can be written as a

proportion.

In y=kx, x varies directly as y varies.

A direct variation has a constant of proportionality.

A direct variation is nonlinear

The origin is not a possible solution in a direct variation.

Mini Lesson

Have students in collaborative pairs make a list of things they believe have a direct relationship. (Give an example . . . the more cans of coke you buy, the higher the cost.) Share items on lists. Eliminate those items that are not direct proportions. Develop definition of direct proportion (variation) with students.

DIRECT PROPORTION – An equation of the form y=kx, where k is the constant of proportionality (variation); k≠0.

2


Have students draw a T-chart for y=3x and find 5 values for x and y.

Ex. x y y = 3x0 0 y = 3(0)2 6 y = 3(2)3 9 y = 3(3)5 15 y = 3(5)6 18 y = 3(6)

In Pairs – answer the following questions in relation to the above example:

Are the values for y directly proportional to the values for x? If so, what is the constant of proportionality? If I know that an x with a value of 6 produces a y with a value of 18, how can I set up a proportion to find the y value if x = 10? Explain.

Ans. 6 = 10

18 y 6y = 180 y = 30

When the 10 is used as a value for x in y = 3x, do you get the same answer? Explain. Graph your ordered pairs. What do you notice about the graph? (linear) Which of the following are direct proportions and how do you know?

1. Sam pays 80¢ for 3 cookies and $1.00 for 4.2. Two books cost $44 and six books cost $132.3. A hotdog costs $2.00 for the 1st one and $1.50 for each additional hot dog.

What we want students to know about direct proportions:

Straight when graphed (can be continuous or discreet) Contains the origin Can be put in the form of y = kx You can find the constant of proportionality from given data You can find missing data by using a proportion.

3


Model how to find the constant of proportionality (variation) usinga. y = 10xb. -3a = bc. If y = 12 when x = 3, find y when x = 7. d. If 6 m of copper wire weights 2 kg. how much will 30 m of wire weigh?

Model how to find the missing value.

a. Six pounds of sugar costs $2.00; at that rate, how much will 40 pounds of sugar cost?b. If y = 12 when x = 15, find x when y = 21.

Pairs Checking Have students circle #3 and #7. Students must work individually until they get to the circled problem, then check answers with partner before continuing. Find the constant of proportionality (variation).

1. When Sam works 5 hours he gets paid $29.75, and when he works 20 hours he gets paid $119.

2. y = x 7

3. If y = 3 when x = 15, then y = -5 when x = -25.

4. If y varies directly as x, and y = 27 when x = 6, then y = 45 when

x = 10. Solve for the missing value.

5. A car uses 5 gallons of gasoline to travel 140 miles. At that rate, how much gasoline will the car use to travel 196 miles?

6. If y = 15 when x = 5, find y when x = 7.

True or False ? Explain your reasoning

7. A person’s height varies directly as his or her age.

8. The formula d = rt is a direct proportion if the rate is constant.

4


Answers to Work Session1. $5.952. 73. 54. 4.55. 7 gallons6. 217. False, because you don’t grow at a constant rate and at some point you

stop growing.8. True, because the distance will increase at a constant rate as the time

increases.

Summarizer

List 3 things you know about direct proportions.

Write 2 problem situations that are direct proportions.

Use one of the above problems to explain how you know it is a direct proportion.

Go back to the activator and allow students to change answer if need be.

5


Direct and Inverse ProportionsLesson 2

GHSGT Objective#36 Identifies and applies mathematics to practical problems requiring direct and inverse proportions.

EQ: How do I identify and solve problems involving inverse proportions (variations)?

Warm-up/ActivatorUsing the letters in the word DIRECT, write a phrase or sentence beginning with each letter to express what you know about direct proportions (variations).

Mini LessonHave students discuss in pairs what the word inverse means. Share your thoughts with the class.

Discuss the following example: Area of three rectangles.Have students sketch what each would look like.

Rectangle 1 Rectangle 2 Rectangle 3Length 2 4 8Width 2 1 ½ Area 4 4 4Option: Use area of 24 and let students graph the relationship between width and length.

Go over inverse proportion (variation) definition. *An inverse variation is described by an equation of the form xy = k or y = k , k ≠ 0. X Small Groups Have students brainstorm to make a list of three things they think might be inversely proportional. Share out.

6


Discuss in each case whether the product of the two factors will be a constant. Model: The pitch of a musical tone varies inversely as its wavelength. If one tone has a pitch of 440 vibrations per second and a wavelength of 2.4 feet, find the wavelength of a tone that has a pitch of 660 vibrations per second.

Let p = pitch and w = wavelength. Find the value of k. pw = k(440) (2.4) = k Substitute 440 for p and 2.4 for w. 1056 = k The constant of variation is 1056. Next, find the wavelength of the second tone. w = k Divide each side of pw = k by p. pw = 1056 Substitute 1056 for k and 660 for p. 660w = 1.6 The wavelength is 1.6 feet.

Think, Pair, Share! If (x1, y1) is a solution of an inverse variation, xy = k and (x2, y2) is a second solution of xy = k, how can we find k? How can we find a missing number if the other three are given?Students may not understand this notation. If not, use specific numbers to model like below. Model: If y varies inversely as x, and y = 3 when x = 12, find x when y = 4. Let x1 = 12, y1 = 3, y2 = 4. Solve for x2.

Method 1 Method 2Use the product rule. Use the proportion. x1y1 = x2y2 x1 = y2

(12)3 = (x2)4 x2 y1

36 = 4 x2 12 = 4 36 = 4 x2 x2 3

7


4 4 36 = 4x2 9 = x2 9 = x2

Distributed Practice If you have observed people on a seesaw, you may have noticed that the heavier person must sit closer to the fulcrum (pivot point) to balance the seesaw. A seesaw is a type of lever, and all lever problems involve inverse variation. Solve. 1. An 8-ounce weight is placed at one end of a yardstick. A 10- ounce weight is placed at the other end. Where should the fulcrum be placed to balance the yardstick? (The fulcrum should be placed 20 inches from the 8-ounce weight.) 2. Suppose two people are on a seesaw. For the seesaw to balance, which person must sit closer to the fulcrum?

a. Jorge, 168 pounds

Emilio, 220 pounds* b. Shawn, 114 pounds* Shannon, 97 pounds Have students compare answers. Share with the whole class. Summarizer. Give each student one of the following equations and ask him/her to determine if it is a direct or inverse proportion and ask him/her to find the constant of proportionality.

● ab = 6 ● c = 3.14d ● 50 = x

y● 1a = d ● s = 3t ● xy = 1 5 ● a = 7 ● 14 = ab ● 2x = y b

8


Have students find someone else who has the same equation and compare answers.

Direct and Inverse Proportion

Lesson 3


EQ: How do I differentiate between direct and inverse proportions (variations)? Warm-up/ActivatorStudents will be given 5 small pieces of paper.

You will designate a place in the room to represent DIRECT and INVERSE. Showing one real world problem at a time have students put a D or I on the scrap paper. At the signal, students will go to the selected location (taking the paper to show their choice). Students discuss why they made the decision and share with the whole class. Collect papers before allowing students to sit and move on to the next problem.

Examples:

(I) I. It takes 45 minutes for 2 copiers to finish a printing job. If 5 copiers work together to print the same job, how long would it

take to finish?

(D) 2. At the local grocery store, 2 pineapples cost $ 2.78. How much 5 pineapples cost?

(D) 3. If 4 inches represents 5 feet on a scale drawing, how many feet does 6 inches represent? (D) 4. Aunt Bess uses 3 cups of oatmeal to bake 6 dozen oatmeal cookies. How many cups of oatmeal would she need to bake 15 dozen cookies?

9


(I) 5. Grant, who weighs 150 pounds, is seated 8 feet from the fulcrum of a season. Mary is seated 10 feet from the fulcrum. If the seesaw is balanced, how much does Mariel weight?

Mini Lesson

Review what the graph of a direct proportion (variation) looks like by posing questions. Ex: If I have a graph, how do I know that it is a graph of a direct proportion?

Give students graph paper. Ask students to graph the following information and determine whether the situation is a direct or inverse variation.

Surprise Birthday Party

Sue decided to give Maria a surprise birthday party. She ordered a large ice cream cake for the party, but is not sure how many people to invite. How will the number of people attending the party affect the portion of cake each person gets? Does this represent a direct or inverse proportion? Explain your answer and illustrate with a graph. What is the constant of proportionality and what does it represent?

● Pieces of cake: 12 Number of People Number of pieces of cake for each person

2 63 44 36 2

12 1

*This is an inverse variation with a constant of proportionality of 12. When graphed you will get the 1st quadrant half of a hyperbola.

10


Have students complete a compare/contrast graphic organizer on Direct and Inverse Proportion.

Inverse proportion (variation) should include:

½ of a hyperbola for graph Will not contain origin Will never intersect x or y axis y = k or xy = k for equations Has constant of proportionality

Students will now work in collaborative pairs to graph the given 6 equations and answer the 4 questions. After a given amount of time have students pair-square to compare answers. Share out with whole class. Summarizer.Students answer the essential question and turn it in before they leave.

What’s My Graph

11


Create a graphic representation to show at least three solutions for the following equations: y = 2x y = ½ x

y = 2/x y = -2x

y = 2x + 1 y = 2x - 1

1. Which graph(s) are directly proportional? How do you know?

12


2. Which graph(s) are inversely proportional? How do you know? 3. Which graph(s) are neither? Why? 4. What is a real-world situation for?

a. A proportional relationship? (Draw from the equations on the front of the sheet for your example)b. An inverse relationship? (Draw from the equations on the front of the sheet for your example)


13


Lesson 4


EQ: How do I differentiate between direct proportion and nonproportion problems? Warm-up/ActivatorIn pairs have students discuss the terms direct proportion, inverse proportion and nonproportion. Allow students to use notes, dictionaries, the internet, etc. Share results with whole class. Mini LessonIn groups of four have students work the problems entitled Proportions and Graphs. Share answers with whole class.

Proportions and Graphs

Solve the following problems. Record your solution strategy and then answer the questions that follow the problem set. (non) Problem 1: Exhausted Examiners: Elke and Faye corrected final exams at the same rate but Elke got a head start. When Elke had completed 12 exams Faye had finished only 4. When Elke had finished 60 exams, how many exams had Faye completed? (direct) Problem 2: A Metric Conversion: If 6 inches is 15.24 cm, 9 inches is how many centimeters? (direct) Problem 3: An Exchange Rate: If 5 Canadian dollars can be exchanged for 4 US dollars, what is 35 Canadian dollars worth in US dollars?

(non) Problem 4: Taken for a Ride: A taxicab charged $1 plus

14


50 cents a mile. If it costs $3 to go four miles, how much would it cost to go 6 miles? Of Problems 1-4, which are proportion problems and which are not? Briefly justify your answers. To distinguish between problems having a direct proportion and nonproportion problems, it can be helpful to record the data in a table and graph it. Draw a line connecting the dots of the graph and if necessary, extend the line so that it intersects the left or bottom side of the graph. Graph Problems 1-4. Graphs of directly proportional relationships have what characteristics? Why do these graphs have these characteristics? In what way are they different from graphs of nonproportional situations? Summarizer.Students will work in pairs to create a real world problem that exemplifies a direct proportion, and inverse proportion and a nonproportional problem. Students will trade with another pair. Students will verify the types of problems and then solve the problems.


15


Assessment

Place your answer to the left of the question. _____ 1. If y varies directly as x, and y = 6 when x = 3, what is the value of y when x = 5?

a. 2.5b. 18c. 3.6d. 10

_____ 2. If y varies inversely as x, and y = 12 when x = 6, what is the

value of y when x = 8?

a. 9b. 4c. 16d. 20

_____ 3. Find the constant of variation if y varies directly as x, and y = 95 when x = 19.

a. 5b. 1/5c. 19d. 95

_____ 4. Which of these graphs represents a direct proportion?

A. B. C.

_____ 5. (x1, y1) and (x2, y2) are ordered pairs of the same inverse

16


variation. Find the missing value.x1 = 20, y1 = 5x2 = ?, y2 = 50

a. 200b. 8c. 2d. 12.5

_____ 6. (x1, y1) and (x2, y2) are ordered pairs of the same direct variation. Find the missing value.

a. 8b. 48c. 24d. 16

_____ 7. Jim must help his father carry bags of soil to the backyard.

There are 45 bags of soil. Normally this would take Jim 30 minutes to do by himself, but today three of his friends stopped by and offered to help. How long will it take the four boys to carry all of those bags to the backyard?

a. 3.5 min.b. 15 min.c. 10 min.d. 7.5 min.

_____ 8. In a function, y varies inversely as x varies. If y = 18 when x =

12, what is the value of y when x = 6?

a. 36b. 9c. 48d. 12

_____ 9. If it takes Nathan and his friends 6.5 hours to travel 300 miles

17


to Atlanta for a concert which of the following proportions would help him figure out how long it would take to go 650 miles to the group’s next concert?

a. 6.5 = 650 T 300

b. 650 = 300 6.5 T

c. 6.5 = T 300 650

d. 300 = T 650 6.5

_____ 10. Brice can type 22 pages in 3 hours. At this rate approximately how long would it take to type one page?

a. 7.3 minutes

b. 8.2 minutes c. 12.2 minutes d. 13.6 minutes

_____ 11. A fifteen-centimeter pulley runs at 250 r/min (revolutions per minute). How fast does the five-centimeter pulley it drives revolve, if the number of revolutions per minute varies inversely as the diameter?

a. 750 r/min b. 170 r/min c. 1250 r/min d. 83.3 r/min

_____ 12. On a map drawn to scale, 5 centimeters represents 50 miles.

18


A line segment connecting two cities is 9 centimeters long. Which proportion can be used to find the actual distance between the two cities? a. 5 = 50

x 9 b. 5 = 9 50 x

c. x = 9 5 50 d. 5 = x 9 50 _____ 13. Find the missing value if (75, 30) and (__?__, 18) are ordered

pairs of the same inverse variation.

a. 125 b. 75 c. 45 d. 7.2

_____ 14. It takes an average person 60 minutes to type 8 pages on the computer. If three average typists work together to type up an 8-page paper, how long will it take them?

a. 45 min.b. 15 min.c. 30 min.d. 20 min.

_____ 15. One end of a pry bar is under a 350 kg boulder. The fulcrum

19


of the bar is 15 cm from the boulder and 175 cm from the other end of the bar. What mass at that end of the bar will balance the boulder? a. 30 kg b. 150 kg c. 400 kg d. 75 kg

_____ 16. If 3 loaves of bread cost $2.43, what is the cost of 2 loaves? a. $3.49 b. $2 c. $1.62 d. $1.15 _____ 17. Farmer MacDonald’s hen lays eggs at the rate of approximately 50 eggs in three months. If the hen continues at this rate, approximately how many eggs will she lay in 4 years?

a. 67 b. 200 c. 600 d. 800

_____ 18. For a given distance, the speed at which a car travels varies

inversely as the time it travels. If it takes 1.5 h to travel a distance at 84 km/h, how long would it take to travel the same distance at 90 km/h?

a. 2 h b. 1.2 h c. 1 h d. 1.4 h

_____ 19. The number of units manufactured varies directly with the

20


number of hours worked. If 10 units are manufactured in 4 h, how many units are manufactured in 14 h? a. 40 units b. 2.8 units c. 35 units d. 140 units

_____ 20. Doughnuts sell for $1.50 a dozen. How much will 30 doughnuts cost? a. $3 b. $3.75 c. $4 d. $4.50

21

Documents

LESSONS - TypePadcicobb.typepad.com/files/ghsgt-obj.36-dir… · Web view · 2008-12-12LESSONS. FOR GHSGT. MATH OBJECTIVE 36. Direct and Inverse Proportions. Direct and Inverse