lesson #1 (3-6) ratios and proportions - buffalo state...

Lesson #1 (3-6) Ratios and Proportions

Materials: Pen/ Pencil Ratios and Proportions Guided Notes

Measuring CupsClear ContainerWaterRatio and Proportions Worksheet/HomeworkFraction ChartRuler

Overview:

Students will develop a clear understanding of what ratios and proportions are and how to solve proportions. The students will do this by filling out and working through the examples of the guided notes as a whole and independently on the worksheet.

Lesson Objectives:

Students will discover and develop methods on how show two ratios form a proportion. (synthesis)

Solve proportions. (application)

Vocabulary:

Ratio - is a comparison of two numbers by division. The ratio of x to y can be expressed in the following ways:

a) x to yb) x:y

c) yx

Proportions - an equation stating that two ratios are equal is called a proportion. Rate – the ratio of two measurements having different units of measure Scale – a ratio or rate is used when making a model or drawing of something that

is too large or too small to be conveniently drawn at actual size.

Anticipatory Set: (8 to 10 minutes)

Have the students relate:How many of you cook at home for your family? (If students are not participating, share one of your own experiences with them)Did you ever have a hard time, figuring out if you have enough food or ingredients for everyone who is eating it?

Revised copy

**All supplies and copies of work sheets will be taken care of and supplied to you

I have a hard time measuring especially, when it ask for ½ of a cup of something and I only have the ¼ , 1/3, 1/8, and the 1 cup measuring cup to measure with.

What should I do if that problem occurs? Let’s say we are making mac-n-cheese and I need ½ cup of milk, but I don’t have the ½ measuring cup. Can I still make the mac-n-cheese? Why?

Yes, because you can measure two ¼ cups of milk, which is the same as ½ cup of milk.

Have students measure the cups with water, to show they are equivalent. This is what we are working on today, Ratios and Proportions. YAY!

Developmental Activity: (20 to 25 minutes)

1. Pass out Ratio and Proportion Guided Notes.2. Work through the examples and fill in the blanks with students by asking

questions.3. Pass out Ratio and Proportion Worksheet.4. Have students work on the worksheet independently. Walk around helping

students if they are having trouble.5. If students are struggling do a couple problems together as a class, but have them

guide you through the problem. Keep asking, “What do I do next?”, if no one knows or responds, have them refer to their notes.

6. If students do not finish the Ratio and Proportion Worksheet, have them finish it for homework.

Closure: (5 to 8 minutes)

(See review section on Ratio and Proportions Worksheet.) Recap and ask guided questions for students to collaborate on the answers to summarize what they have learned.

Assessments: (7 to 10 minutes)

Students will be given homework that will be collected next class. Their homework is the Ratios and Proportions Worksheet/Homework (what ever they don’t finish in class is homework.) As well as a quiz/ test at the end of the week (Friday) that will consist of similar problems from the note sheet and homework sheet done in class.

Name _____________________________________________ Date _____________

Ratios and Proportions Guided Notes:

Section #1 - Determining Rations and Proportions:

What is a RATIO? ______________________________________________________

It can be expressed in the three following ways:

1) 2) 3)

Ratios are often expressed in _________________________

What is a PROPORTION? ________________________________________________

Example:42

= 21

124

31

=

How can we tell these two proportions are equivalent to each other? To check if two ratios form a proportion is to use________________________. If the cross products are equal, then the ratios form a proportion.

Verify the examples above:

42

= 21

124

31

=

2 (2) = 1(4) 4 = 4 YES!

Practice:

3024

= 54

246

41

=

2÷

2÷

4×

4×

Section #2 – Solving Proportions:

You will come across proportions that involve variables. But have no fear you can solve them using _______________________and the techniques used to solve other equations.

Example: Solve the Proportion:

1624

15=

nStep #1: Write out the original equation

1624

15=

nStep #2: Find the cross products

16n = 24(15) Step #3: ________________

16n = 360 Step #4: Isolate the variable (_________ both sides by _____ )

16360

1616

=n

Step #5: Simplify

n = 22.5 Step #6: _____________________

1624

155.22

=

22.5 (16) = 24 (15) 360 = 360 YES!

Practice: Solve the Proportion:

k6

512

= CHECK:

Section #3 – Using Rates:

RATES - _______________________________________________________________

Example: What are they comparing?

A price of $1.99 per dozen eggs ________________________

A speed of 55 miles per hour ________________________

A salary of $30,000 per year ________________________

Practice:

Alex goes on a 30-mile bike ride every Saturday. He rides the distance in 4 hours. At this rate, how far can he ride in 6 hours? How far can he ride in 1 hour?

Explore: Let m =

Plan: What are we comparing? Write a proportion for the problem.

Solve:

Examine:

Section #4 – Using a Scales as Proportions:

A ratio or rate called a ______________is used when making a model or drawing of something that is too large or too small to be conveniently drawn at actual size.

Example: This line (above) is 2 inches long but it can represent 400 meters for a map

Set up a proportion to find out how many meters it would be to move across the map 4.3 inches?

Practice: The scale of a map for Crater Lake Island is 2 inches = 9 miles. The distance between Discovery Point and Phantom Ship Overlook on the map is about 1 ¾ inches. What is the actual distance between these two places?

Phantom Ship OverlookDiscovery Point

Crater Lake Island

Explore:

Plan:

Solve:

Examine:

Name ___________________________________ Date _______________

Ratios and Proportions Worksheet/Homework (SHOW YOUR WORK!!!):

1) Review:

a) Compare and contrast the similarities and differences of ratios and proportions.

_______________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________

b) How can cross products help us solve proportions? Explain your answer.

_______________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________

c) Give two examples of a rate and state the different units of measure.

______________________________________________________________________________________________________________________

2) Determine whether each pair of ratios form a proportion

a) 3312,

114

b) 98,

1716

3) Solve each proportion:

a) x6

43

=

b) 155

45=

a

3) A blueprint for a house states that 2.5 inches equals 10 feet. If the length of a wall is 12 feet, how long is the wall in the blueprint?

4) Rachel earns $152 in 4 days. At that rate, how many days will it take her to earn $532?

5) Lafayette HS is hosting a concert. The math department is in charge of refreshments. One of the items to be served is punch. The school cook has given you four different recipes calling for sparkling water and cranberry juice. (Hint: Answer question #3 first and find out how much punch each recipe makes)

RECIPE A

2 cups cranberry juice3 cups sparkling water

RECIPE B


RECIPE C


RECIPE D

1 cup cranberry juice4 cups sparkling water

1) Which recipe will make punch that has the strongest cranberry flavor? Explain your answer.

2) Which recipe will make punch that has the weakest cranberry flavor? Explain your answer.

3) We need 120 cups of punch. For each recipe, how many cups of cranberry juice and how many cups of sparkling water are needed? Explain your answer. (Hint: find out how much punch each recipe makes)

Name _________ANSWER KEY_________________________ Date _____________

Ratios and Proportions Guided Notes:

Section #1 - Determining Rations and Proportions:

What is a RATIO? A ratio is a comparison of two numbers by division.

It can be expressed in the three following ways: 1) x to y2) x: y

3) yx

Ratios are often expressed in SIMPLIEST FORM.

What is a PROPORTION? An equation stating that two ratios are equal is called a PROPORTION.

Example:42

= 21

124

31

=

How can we tell these two proportions are equivalent to each other? To check if two ratios form a proportion is to use CROSS PRODUCTS. If the cross products are equal, then the ratios form a proportion.

Verify the examples above:

42

= 21

124

31

=

If students are struggling show them the Fraction chart or give them the measuring cups to find out what is equivalent.

2÷

2÷

4×

4×

2 (2) = 1(4) 1(12) = 3 (4) 4 = 4 YES! 12 = 12 YES!

Practice:

3024

= 54

246

41

=

24 (5) = 4(30) 1(24) = 6(4) 120 = 120 YES! 24 = 24 YES!

Section #2 – Solving Proportions:

You will come across proportions that involve variables. But have no fear you can solve them using CROSS PRODUCTS and the techniques used to solve other equations.

Example: Solve the Proportion:

1624

15=

nStep #1: Write out the original equation

1624

15=

nStep #2: Find the cross products

16n = 24(15) Step #3: Simplify

16n = 360 Step #4: Isolate the variable (Divide both sides by 16)

16360

1616

=n

Step #5: Simplify

n = 22.5 Step #6: Check your answer

1624

155.22

=

22.5 (16) = 24 (15) 360 = 360 YES!

Practice: Solve the Proportion:

k6

512

= CHECK: k6

512

=

k6

512

=15.1

65

12+

=

12k = 6(5)5.2

65

12=

12k = 30 12(2.5) = 6(5)

1230

1212

=k

30 = 30 YES!

k = 1.5

Section #3 – Using Rates:

RATES- the ratio of two measurements having different units of measure

Example: What are they comparing?

A price of $1.99 per dozen eggs dollars to a dozen eggs

A speed of 55 miles per hour miles to hours

A salary of $30,000 per year dollars to years

Practice:

Alex goes on a 30-mile bike ride every Saturday. He rides the distance in 4 hours. At this rate, how far can he ride in 6 hours? How far can he ride in 1 hour?

Explore: Let m = the number of miles Alex can ride in 6 hours

Plan: What are we comparing? Write a proportion for the problem.

Miles to hours or hoursmiles

6430 m

=

Solve: 64

30 m=

30(6) = 4m 180 = 4m

4

44

180 m=

45 = m

Does this makes sense? Think about it!

In the first hour how many miles did ride? 7.5 miles

In the second hour? 7.5 + 7.5 = 15 miles

In the third hour? 7.5 + 7.5 + 7.5 = 22.5 miles

In the sixth hour? 6(7.5) = 45 miles

Review: Cross product says to do 30(6) = m(4). What property allows me to write it as 30(6) = 4m?

Answer: Commutative Property

Examine: 45 miles per 6 hours

6430 m

=

645

430

=

30(6) = 45(4) 180 = 180 YES!

Section #4 – Using a Scales as Proportions:

A ratio or rate called a SCALE is used when making a model or drawing of something that is too large or too small to be conveniently drawn at actual size.

Example: This line (above) is 2 inches long but it can represent 400 meters for a map (show map scale on board)

Set up a proportion to find out how many meters it would be to move across the map 4.3 inches?

m3.4

4002

=

Practice: The scale of a map for Crater Lake Island is 2 inches = 9 miles. The distance between Discovery Point and Phantom Ship Overlook on the map is about 1 ¾ inches. What is the actual distance between these two places?

Explore: Scale is 2 inches = 9 miles inches to miles

30/ 4 = 7.5 miles per hour

615.7 m

=

7.5(6) = m45 = m

OR

Phantom Ship OverlookDiscovery Point

Crater Lake Island

1 ¾ inches = ?? miles

Let m = the number of miles between Phantom Ship Overlook and Discovery Point

Plan: We are comparing inches to miles OR milesinches

m431

92

= ORm75.1

92

=

Solve:m75.1

92

=

2m = 1.75(9) 2m = 15.75

275.15

22

=m

m = 7.875 miles

Examine: m75.1

92

=

875.775.1

92

=

2(7.875) = 1.75(9) 15.75 = 15.75 YES!

Does this make sense? Before checking or even before they solve the answer have them predict what the answer would be. If 2 inches = 9 miles then, 1 4

3 inches be equal to more than 9 miles or less than 9 miles? Is it going to be close to 9 miles?

Name _______ANSWER KEY_____________________ Date _______________

Ratios and Proportions Worksheet/Homework (SHOW YOUR WORK!!!):

1) Review:

d) Compare and contrast the similarities and differences of ratios and proportions.

Ratios Proportions Compares two numbers or two

measurements Shows us how two ratios are

equal Involves division Involves division In simplest form Cross products are used to solve

proportions Can be written in three ways:x:y, x to y, y

x Is an equation

Rates and scales are examples of proportions

e) How can cross products help us solve proportions? Explain your answer.

Cross Products helps us solve proportions, because it makes the equation much simpler to solve. Cross Products let us get rid of the fractions and work with whole numbers.

f) Give two examples of a rate and state the different units of measure.

speed mph miles and hourssalaries per year dollars and years

2) Determine whether each pair of ratios form a proportion

a) 3312,

114

b) 98,

1716

3312

114

= 98

1716

=

4(33) = 12(11) 16(9) = 8(17) 132 = 132 YES! 144 ≠ 136 NO!

3) Solve each proportion:

**Answers may vary

a) x6

43

= CHECK:x6

43

=

x6

43

=86

43

=

3x = 6(4) 3(8) = 6(4)3x = 24 24 = 24 YES!x = 8

b) 155

45=

aCHECK:

155

45=

a

155

45=

a155

4515

=

15a = 5(45) 15(15) = 5(45) 15a = 225 225 = 225 YES!

15255

1515

=a

a = 15

3) A blueprint for a house states that 2.5 inches equals 10 feet. If the length of a wall is 12 feet, how long is the wall in the blueprint?

2.5 inches = 10 feet ?? inches = 12 feet

Comparing inches to feet OR feetinches

Let i = the length of the wall in the blueprint

12105.2 i

= CHECK:1210

5.2 i=

12105.2 i

=123

105.2

=

2.5(12) = 10i 2.5(12) = 10(3) 30 = 10i 30 = 30 YES!

1010

1030 i

=

3 = i

4) Rachel earns $152 in 4 days. At that rate, how many days will it take her to earn $532?

$152 = 4 days$532 = ?? days

Comparing money and days OR daysmoney

Let d = the number of days it takes Rachel to earn $532

d532

4152

= CHECK:d

5324

152=

d532

4152

=14532

4152

=

152d = 532(4) 152(14) = 532(4) 152d = 2128 2128 = 2128 YES!

1522128

152152

=d

d = 14 days

5) Lafayette HS is hosting a concert. The math department is in charge of refreshments. One of the items to be served is punch. The school cook has given you four different recipes calling for sparkling water and cranberry juice. (Hint: Answer question #3 first and find out how much punch each recipe makes)

RECIPE A


RECIPE B


RECIPE C


RECIPE D

1 cup cranberry juice4 cups sparkling water

4) Which recipe will make punch that has the strongest cranberry flavor? Explain your answer.

5) Which recipe will make punch that has the weakest cranberry flavor? Explain your answer.

I compare each ratio and put them in order from least to greatest. I did this by using their proportions, for example 8

3 < 21 = 8

4 , therefore D < B.

6) We need 120 cups of punch. For each recipe, how many cups of cranberry juice and how many cups of sparkling water are needed? Explain your answer. (Hint: find out how much punch each recipe makes)

We are comparing cranberry juice to punch AND sparkling water to punch

punchuicecranberryj

punchatersparklingw

A) 2 cups of cranberry juice + 3 cups of sparkling water = 5 cups of punch

Let c = the number of cups of cranberry juice in the punch

2 cups of cranberry juice = 5 cups of punch?? cups of cranberry juice = 120 cups of punch

12052 c

= CHECK:12048

52

=

2(120) = 5c 2(120) = 48(5)240 = 5c 240 = 240 YES!

55

5240 c

=

48 = c

Let s = the number of cups of cranberry sparkling water in the punch

3 cups of sparkling water = 5 cups of punch?? cups of sparkling water = 120 cups of punch

12053 s

= CHECK:12072

53

=

3(120) = 5s 3(120) = 72(5)360 = 5s 360 = 360 YES!72 = s

Weakest Strongest D B C A

If students struggle with this use the fraction worksheet.

ANS: 48 cups of cranberry juice72 cups of sparkling water

B) 4 cups of cranberry juice + 8 cups of sparkling water = 12 cups of punch



120124 c

= CHECK:12040

124

=

4(120) = 12c 4(120) = 40(12)480 = 12c 480 = 480 YES!

1212

12480 c

=

40 = c



120128 s

= CHECK: 12080

128

=

8(120) = 12s 8(120) = 80(12)960 = 12s 960 = 960 YES!80 = s


C) 3 cups of cranberry juice + 5 cups of sparkling water = 8 cups of punch



12083 c

= CHECK:12045

83

=

3(120) = 8c 3(120) = 45(8)360 = 8c 360 = 360 YES!

88

8360 c

=

45 = c



12085 s

= CHECK:12075

85

=

5(120) = 8s 5(120) = 75(8)600 = 8s 600 = 600 YES!75 = s


D) 1 cup of cranberry juice + 4 cups of sparkling water = 5 cups of punchLet c = the number of cups of cranberry juice in the punch

1 cup of cranberry juice = 5 cups of punch?? cups of cranberry juice = 120 cups of punch

12051 c

= CHECK:12060

51

=

120 = 5c 120 = 60(5)120 = 120 YES!

55

5120 c

=

60 = c



12054 s

= CHECK:12096

54

=

4(120) = 5s 4(120) = 96(5)

480 = 5s 480 = 480 YES!96 = s


Shannon TrimperMED308Dr. Cushman

Lesson Plan

Topic: Percent of Change

Grade Level: 9th grade

Materials: pencils, paper, worksheets

Lesson Overview: The students will find percents of increase and decrease and solve problems involving percents of change. Using mostly real life situations, the students will compute percentages – using sales tax and discounts and compare what these percents do to prices. They will also find the percent of change between two numbers and determine if it is a percent of increase or a percent of decrease.

Objectives: The students will compare the differences between a

percentage increase and decrease. The students will solve problems and determine the

percentage increase or the percentage decrease between two numbers.

New York State Standards: A.CN.6 – Recognize and apply mathematics to situations in

the outside world. A.N.5 – Solve algebraic problems arising from situations that

involvefractions, decimals, percents (decrease/increase and discount),and proportionality/direct variation.

Key Ideas:o 3 – Operationso 5 - Measurement

Anticipatory Set: (approx 3-5 minutes)

Begin by passing out the Percents of Change: Practice worksheet and asking the students the following questions. Have them write their responses on the worksheet provided:

What happens to the price of something when the sales tax is added?o The price increases

What happens to the price of a product when there is a sale? o The price decreases

As you ask the questions and the students are writing their responses, take time to walk around and glance at their answer to make sure they are following along. Next ask them to share their answers.Developmental Activity: (approx 20-25 minutes)

Hand out the worksheet provided to each student, read or ask a student to read the first problem to the group. As you or a student is doing this, write the work shown on the board as you go through the problem.

Sara bought a pair of jeans for $29.99. If the sales tax was 8.25%, find the total price of the jeans that Sara paid.

29.99 x .0825 = 2.4729.99 + 2.47 = 32.46

Write the numbers on the board and see if the students can figure out or tell you how to go about solving this problem. The students will first need to know that in order to find the tax they must move the decimal point over two places to the left. Ask if they know why that is.

The students should at least be able to tell you that percents are out of 100. If not, ask a question like: what is the highest percentage you can score on a test? Once you have found the tax, ask the students what they are to do with that amount – add or subtract it to the original price. If they can not figure this out, go back to the questions in the beginning – they should have said that sales tax increases the price.

Next ask the students to find what the price of the jeans would be if they were on sale for 20% off. The students should begin the same way as the previous problem. Have the students do this on their own and then go over it once they have finished it or at least give them time to think about it. (No more than 5 minutes.) Ask the students what they notice about the prices, when tax is added and also when a product is marked at a discounted price. They should write these responses on their paper, to make sure they are following along walk around and check their work and responses, then ask someone to share.

29.99 x .20 = 6.0029.99 – 6 = 23.99

Next, again write the following problem on the board (the students can follow along on their worksheet):

Original: 25New: 28

Find whether the percent of change is a percent of increase or a percent of decrease. First you will need to find the amount of change. Since the new amount is greater than the original, the percent of change is a percent of increase.

28 – 23 = 3

100253 r

= In order to solve this proportion the students may see that 25

must be multiplied by 4 go get 100. If not make it obvious, tell them to look at it like money: 25 cents and a dollar, ask how many quarters do you need to get a dollar?

i.e. 24(4) = 100 and therefore 3(4) = 12

Another way to solve this proportion is to multiply each side by the common denominator.

)25100)(100

()253)(25100( xrx =

On the left side of the equation the 25’s would cancel out and on the right side the 100’s would cancel each other out. Leaving the following:

100(3) = 25(r) 300 = 25r

2525

25300 r

=

12 = rThe percent of increase is 12%.

This problem should be done as a class, ask the students for help as you go through the algebra part of the problem. The students should be able to solve this and walk you through it.

The next problem is an example of a percent of decrease. Again, write this on the board but this time see if the students can solve this on their own, assist them where needed. Once you have given them a chance to get started, go over it and have them explain their process.

Original: 30New: 12

The percent of change is a percent of decrease because the new amount is less than the original. Find the change.

30 – 12 = 18

10030

18 r=

18(100) = 30(r) 1800 = 30r

3030

301800 r

=

60 = rThe percent of decrease is 60%.

Once you have finished this problem, give the students the worksheet to complete. They can begin if there is time left at the end of class or it is to be completed for homework.

Closure: (3-5 minutes)Before the students begin the homework assignment, hand out

the Exit Ticket. This will consist of questions based on today’s lesson to see how well the students followed along. Collect the ticket before they leave class.

Assessment:The homework and the exit tickets will be used as the

assessment. Also, as the students are working on examples 2 and 4 use this time to walk around and see if they were paying attention to the previous examples and they should be using those as a guide to assist them in completing them. Give the students about 3-5 minutes to work on each of these problems before you go over them.

Name: ____________________

Percent of Change: Practice

1. What happens to the price of something when the sales tax is added?

2. What happens to the price of a product when there is a sale?

3. Sara bought a pair of jeans for $29.99. If the sales tax was 8.25%, find the total price of the jeans that Sara paid.

4. How much would Sara pay for the jeans if they were marked 20% off?

5. Find whether the percent of change is a percent of increase or a percent of decrease.

a. Original: 25New: 28

b. Original: 30New: 12

Exit Ticket

1. If the new number is greater than the original number, is this a percent increase or percent decrease?

2. If the new number is less than the original number, is this a percent increase or percent decrease?

Name: ____________________

Percents of Change: Homework

State whether each percent of change is a percent of increase or a percent of decrease. Then find each percent of change. Round to the nearest whole percent.1. Original: 85 2. Original: 32.5

New: 30 New: 90

Find the total price of each item.3. backpack: $35.00 4. hat: $18.50 tax: 7% tax: 6.25%

Find the discounted price of each item.5. shirt: $45.00 6. watch: $37.55 discount: 40% discount: 35%

Find the final price of each item.7. dress: $70.00 8. camera: $58.00 discount: 30% discount: 25% tax: 7% tax: 6.5%

Percents of Change: Homework Key

1. 65% decrease

2. 177% increase

3. $37.45

4. $19.66

5. $27

6. $24.41

7. $52.43

8. $46.33

Lesson #3 (4-4) Equations as Relations

Materials: Pen/PencilNotebook Paper

Equation as Relations Guided NotesEquation as Relations Worksheet/ HomeworkGraph Paper

Lesson Overview: Students will develop a clear understanding of what equations and relations are,

their components (domain, range, independent variable, dependent variable, and ordered pair) and how to they are used. The students will do this by filling out and working through the examples of the guided notes through class discussion and independently on the worksheet.

Lesson Objectives:

Students will be able to generate the range and domain of a given equation. (analysis)

Students will graph the solution set for a given domain. (application)

Vocabulary:

Ordered Pair – is the solution of an equation with two variables that results in a true statement when substituted into the equation.

Domain – (all input values) contains all values of the independent variable Range – (all out put values) contains all values of the dependent variable

Anticipatory Set: (8 to 10 minutes)

Check list: writing equations (algebraically and verbally)undoing operationsworking backwards with word problemssolving multi-step equationssolving equations with the variable on each sidesolving equations and formulasusing ratios and proportionspercent of change

So far everything we have been doing, we are working with only one variable. Now we are given an equation with two variables. This type of equations is called Equations of Relations.

Have the students relate:How many of you have been to Canada or maybe a different country? Did any of you buy something while you were there?

Souvenirs, T-shirt, clothes, food … etcDid you have a problem with the exchange rate at all?

The price tag of item they bought did not match how much the item rang up for. The products they bought were cheaper or more expensive than normal.

Problem: Let’s say you and your friends are going to be taking a road trip to Toronto. So far you have saved $500 for your trip, and you want to find out how much that will be worth in Toronto, Canada. The exchange rate today is 1 US dollar = 1.10 Canadian.

Have the students reflect:

Can you set up an equation to be able to convert US dollars to Canadian dollars? YES! The equation to convert dollars (d) to Canadian dollars (c): c = 1.10d.

Optional: If students are having trouble coming up with an equation, make a table:

Note: Today the exchange rate is pretty much even, but you could talk about other places. I thought this would be more beneficial to students because some of them have actually been to Canada and might be able to relate. You could make the connection that in England the exchange rate is 1 US dollar = 0.69 pounds.

US Money (d = dollars)

Canadian Money (c = Canadian dollars)

Input value (independent variable)

Output value(dependent variable)

1.00 1.102.00 2.20

3.00 3.304.00 4.405.00 5.50

500.00 550.00d 1.10 * d

Why are Equations of Relations important in traveling? An equation of Relations shows us relationships between the exchange rate of two different countries’ currency. So we know how much money we have. (The equation describes the relationship)

Developmental Activity: (40 minutes 20 minutes on Section #1, 20 minutes on Section #2 and 3)

1. Pass out Equations of Relations Guided Notes. 2. Have students take out a sheet of notebook paper and have them write their name

and date across the top. 3. Work through the examples and fill in the blanks with students by asking

questions.4. When asking a closure question, have students write their responses on the

separate sheet of paper.5. Pass out Equations of Relations Worksheet.

Ask students how they converted $2 into $1.10? If they still cannot explain it ask them how they converted $3.00 into $3.30? If they still cannot explain themselves, ask if they see a pattern on the right column.

6. Have students work on the worksheet independently. Walk around helping students if they are having trouble.

7. If students are struggling do a couple problems together as a class, but have them guide you through the problem. Keep asking, “What do I do next?”, if no one knows or responds, have them refer to their notes.

8. If students do not finish the Equations of Relations Worksheet, have them finish it for homework.

Closure: (5 minutes) After each practice example of each section, I have them answer a question on a

separate sheet of paper to check their understanding. Collect each paper before class ends. (See notes on Answer Key)

Students will work independently on Equations of Relations Worksheet. Teachers should circulate around as students are working on the handout to make sure they are staying on task and doing the work.

Assessment: (5 to 10 minutes) Students will be given homework that will be collected next class. As well as a

quiz/ test at the end of the week (Friday) that will consist of similar problems from the note sheet and homework sheet done in class.

Name: _______________________________________ Date: __________________

Equation as Relations Guided Notes:

The equation c = .90d is an example of an equation with TWO VARIABLES. A solution of an equation with two variables is an ORDERED PAIR that results in a true statement when substituted into the equation. Written as (independent variable, dependent variable) or (x, y).

Section #1 - Solve Using a Replacement Set:

Example: Find the solution set for y = 2x + 3, {(-2, -1), (-1, 3), (0, 4), (3, 9), (x, y)}.

Step #1: Make a table. Step #2: Substitute each ordered pair into the equation.Step #3: Find out which ordered pairs belong to this equation.Step #4: State the solution set.

This is called the REPLACEMENT SET – a set of possible solutions, some might be right and some might be wrong, so check ALL the ordered pairs!

The ordered pairs (-2, 1) and (3, 9) result in true statements. The solution set to this equation is: {(-2, 1), (3, 9)}

Practice: Find the solution set for 3x – 2y = 5, given the replacement set:{(3, 4), (5, 5), (-1, -4), (0, 2)}.

x y 3x-2y = 5 Solution?3 4 3(3) - 2(4) = 5

9 – 8 = 5 1 = 5

NO

5 5 3(5) - 2(5) = 5 15 – 10 = 5 5 = 5

YES!

-1 -4 3(-1) - 2(-4) = 5 -3 – (-8) = 5 -3 + 8 = 5 5 = 5

YES!

4 2 3(0) - 2(2) = 5 0 – 4 = 5 -4 = 5

NO!

x y y = 2x + 3 Solution?-2 -1 -1 = 2 (-2) + 3

-1 = -4 + 3 -1 = -1

YES!

-1 3 3 = 2 (-1) + 3 3 = -2 + 3 3 = 1

NO

0 4 4 = 2 (0) +3 4 = 0 + 3 4 = 3

NO

3 9 9 = 2 (3) + 3 9 = 6 + 3 9 = 9

YES!

Have students pick their own ordered pair to see if it is a solution.

Question #1: Why is the ordered pair (4, 2) not in the solution set for this equation?

Answer: When you substitute or plug in (4, 2) into the equation, both sides of the equation do not equal each other.

The ordered pairs (5, 5) and (-1, -4) result in true statements. The solution set is {(5, 5), (-1, -4)}.

Section #2 - Solve Using a Given Domain:

DOMAIN – (all input values) contains all values of the independent variableRANGE - (all output values) contains all values of the dependent variable

What is the domain and range to our solution set in the previous example?Domain: {5, -1}Range: {5, -4}

Example: Solve b = a +5 if the domain is {-3, -1, 0, 2, 4}.

Step #1: Make a table.Step #2: Substitute each value of a into the equation to determine the values of b in the RANGE.

a a + 5 b (a, b)-3 -3 + 5 2 ( -3, 2)-1 -1 + 5 4 (-1, 4)0 0 + 5 5 (0, 5)2 2 + 5 7 (2, 7)4 4 + 5 9 (4, 9)

The solution set is {(-3, 2), (-1, 4), (0, 5), (2, 7)}.

What is the range to this equation? {2, 4, 5, 7, 9}.

Practice: (1) Solve y = 3x + 1 if the domain is {-4, -2, 0, 2, 4}.

What is the range to this equation? {-11, -5, 1, 7, 13}

Section #3 – Solve and Graph the Solution Set:

Example: Solve 4x + 2y = 10 if the domain is {-1, 0, 2, 4}. Graph the solution set.

x 3x + 1 y (x, y)-4 3(-4) +1 -11 ( -4, -11)-2 3(-2) +1 -5 (-2, -5)0 3(0) +1 1 (0, 1)2 3(2) + 1 7 (2, 7)4 3(4) +1 13 (4, 13)

Question #2: When we substitute all the independent variables (the x values) into the equation, we get dependent variables (the y values). What is the set that contains all the dependent variables called?

Answer: the Range

On the test, they might give you the range and you have to find the domain. This is the same process except instead of substitute in the domain into all the x values you would substitute the range into the y values.

Step #1: Solve the equation for y in terms of x. (This makes creating a table of values much easier to work with)

4x + 2y = 10 2y = 10 – 4xy = 5 – 2x

Step #2: Make a table to find the range and the solution set.

Step #3: Graph the solution set {(-1, 7), (0, 5), (2, 1), (4, -3)}.

Practice: Solve 2x – y = 3 if the domain is {-5, -2, 0, 1, 3}. Graph the solution set.

x 5 - 2x y (x, y)-1 5 – 2(-1) 7 ( -1, 7)0 5 – 2(0) 5 (0, 5)2 5 – 2(2) 1 (2, 1)4 5 – 2(4) -3 (4, -3)

Isolate the y variable: Subtract 4x from both sides and simplify Divide both sides by 2 and simplify

1 2 3 4 5 6 7 8 9 10-10 -9 -8 -7 -6 -5 -4 -3 -2 -1 0

10987654321

-1-2-3-4-5-6-7-8-9-10

x-axis

y - axis

Step #1: 2x – y = 3-y = 3 – 2xy = - (3 – 2x)y = -3 + 2x

Step #2:

x -3 + 2x y (x, y)-5 -3 + 2(-5) -13 (-5, -13)-2 -3 + 2(-2) -7 (-2, -7)0 -3 + 2(0) -3 (0, -3)1 -3 + 2(1) -1 (1, -1,)3 -3 + 2(3) 3 (3, 3)

Step #3: (see graph paper)

Practice: (refer to anticipatory set)

You and your friend just won a trip to England!! The exchange rate is 1 US dollar = 0.69 pounds (£) [p = .69d]. Your friend made a list of the expenses he plans on spending while you are in England. Use the conversion rate to find the equivalent US dollar for these amounts given in pounds (£) and graph the ordered pairs. (Round to the nearest dollar)

Explore: Let d = the amount of US dollars p = the amount of pounds (£)

p = .69d

domain: {40, 30, 15, 6}

Plan: We are looking for dollars (our dependent variable)Pounds is our independent variable

Write our equation to find d in terms of p

Daily Expenses:

Hotel 40 £Meals 30 £Transportation 15 £Entertainment 6 £

Solve: p = .69d

69.

69.69.

dp=

69.p

= d

(see graph paper)

Examine: Look at the values in the range the cost in dollars is higher than the cost in pounds. Do the results make sense?

Name: _______________________________________ Date: __________________

Equation as Relations Worksheet:

1) Find the error: Malena says that (5, 1) is a solution of y = 2x + 3. Bryan says it’s not a solution.

Malena: y = 2x + 3 Bran: y = 2x + 3 5 = 2(1) + 3 1 = 2(5) + 3 5 = 5 1 = 13

Who is correct? Explain your answer. ____________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________

2) Find the solution set for each equation, given the replacement set. (Use graph paper)

a) y = 3x + 4; {(-1, 1), (2, 10), (3, 12), (7,1)}

b) 2x – 5y = 1; {( -7, -3), (7, 3, (2, 1), (-2, -1)}

3) Solve each equation for the given domain. Graph the solution set. (Use graph paper) a) 5x + 4y = 8 for x = {-4, -1, 0, 2, 4, 6}

b) 1/2x +y = 2 for x = {-4, -1, 1, 4, 7, 8}

p69.p d (p, d)

4069.40 58.00 (40, 58)

3069.30 43.00 (30, 43)

1569.15 22.00 (15, 22)

669.6 9.00 (6, 9)

c) y = 3x -1 for x = {-5, -2, 1, 3, 4}

4) The domain for y = 8 – 3x is {-1, 2, 5, 8}. Find the range. (Use graph paper)

5) The range for 2y – x = 6 is {-4, -3, 1, 6, 7}. Find the domain. (Use graph paper)

6) Henry and his brother live in Germany. They are taking a trip to the United States. They are unfamiliar with the Fahrenheit scale, so they would like to convert US temperatures to Celsius. The equation F = 1.8C + 32 relates the temperature in degrees Celsius C to degrees Fahrenheit F

City Temperature (degrees F)NY 34Chicago 23San Francisco 55Miami 72Washington DC 40

a) Solve the equation for C.

b) Find the temperatures in degrees Celsius for each city.

7) The equation for the perimeter of a rectangle is P = 2L + 2W. Suppose the perimeter of rectangle ABCD is 24 centimeters.

a) Solve the equation for L.

b) State the independent and dependent variables.

c) Choose five values for W and find the corresponding values of L.

lesson #1 (3-6) ratios and proportions - buffalo state...

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