Port Proportions

Proportions

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Before we start We need to learn a little bit about the Port of Long Beach.

If you haven’t seen the introductory lesson, do that first! 2

Before we begin… You already know how to: • Determine if 2 ratios form a proportion • Solve a proportion using an equalfraction approach • Solve a proportion using crossproducts and an equation

Refresh your memory with a little practice…

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Refresher #1

Do these two ratios form a proportion?

2 ? 3 = 8 12 Tell your neighbor yes or no. If you disagree, discuss until you agree. Yes 4

Refresher #2

Explain to your neighbor how the cross product property is used to determine if these two ratios form a proportion.

2 ? 3 = 8 12 2 • 12 = 3 • 8, so this must be a proportion 5

Refresher #3

Solve this proportion by thinking of the two ratios as equivalent fractions. •4=

2 8 = 3 x •4=

x = 12 6

Refresher #4

Solve this proportion by using crossproducts to write an equation and solve it.

2 x = 6 15

6x = 2 • 15

6x = 30 6 6

x=5

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Today you will solve word problems which involve proportions. All the problems will involve situations from the Port of Long Beach.

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By the end of the lesson, you will: • • •

Write proportions to represent situations at the Port of Long Beach Write proportions with one variable Use cross-products to solve the proportions

CA Math Content Standard 6.1.3 Use proportions to solve problems. Use crossmultiplication as a method for solving such problems, understanding it as the multiplication of both sides of an equation by a multiplicative inverse. 9

Why learn about proportions?

• • • • •

Proportions are used in many real life situations Converting one unit of measure to another Making a recipe larger or smaller Enlarging or reducing photographs Just about anywhere! 10

A proportion is two equal ratios

1 2 = 2 4 11

An example: • •

A twenty-foot container (1 TEU) holds 15,000 tennis shoes. A forty-foot container (2 TEUs) holds 30,000 tennis shoes.

1 TEU 2 TEUs = 15,000 shoes 30,000 shoes 12

The proportion:

double

1 TEU 2 TEUs = 15,000 shoes 30,000 shoes double is two equal ratios. 13

In pairs: •

Determine which partner will be person A and which is person B.

•

Person A: Tell your partner the definition of a proportion.

•

Person B: Tell your partner he or she is correct, or tell your partner the correct definition.

A proportion is two equal ratios. 14

The ships that carry salt have 6 cargo holds. Each can hold 10,000 tons of salt.

Is1this a hold proportion?6 cargo holds cargo = 10,000 tons of salt 60,000 tons of salt Don’t say the answer out loud, wait for the count of three. Yes Tell your partner how you know it is a proportion. I will pick a person at random to explain. 15

Next, you will write a proportion for a situation. There are four steps:

this 1. Read the situation. Identify the Write in your notes two units. 2. Write one unit on top of the other. 3. Write two fraction bars with an equal sign between. 4. Fill in the numbers so each ratio has meaning. 16

Watch me use the 4 steps Write a proportion to represent this situation:

A crane operator unloads one container every two minutes. The operator can unload 10 containers in 20 minutes.

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Step 1:

Identify the two units. A crane operator unloads 1 container every 2 minutes. The operator can unload 10 containers in 20 Hint: minutes. The units come after the numbers. 18

Step 2:

Write 1 unit on top of the other. A crane operator unloads 1 container every 2 minutes. The operator can unload 10 containers in 20 minutes. containers minutes 19

Step 3:

Write the fraction bars and = sign. A crane operator unloads 1 container every 2 minutes. The operator can unload 10 containers in 20 minutes. containers = minutes 20

Step 4:

Write numbers in ratios. A crane operator unloads 1 container every 2 minutes. The operator can unload 10 containers in 20 minutes. containers 1 10 = minutes 2 20 We wrote a proportion to represent the problem. 21

Would this proportion also make sense? I’ll show you another proportion for the SAME problem. Discuss with your partner if this proportion also makes sense. After one minute, I will call one student at random to answer.

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Would this proportion also make sense?

1m

inu A crane operator unloads o te ve i 1 container every 2 r s minutes. The operator can unload 10 containers in 20 20 minutes 2 minutes. = containers 1 10

Yes 23

One more… I’ll show you another proportion for the SAME problem. Discuss with your partner if this proportion also makes sense. After one minute, I will call one student at random to answer.

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Does this proportion also make sense? A crane operator unloads 1 container every 2 minutes.

1m

inu ov te is er

The operator can unload 10 containers in 20 2 containers 1 minutes. = 10 20 minutes

No

This proportion shows that the crane operator takes 10 minutes to unload 1 container. 25

Time to practice •

• • •

•

I will give you a situation and you will write a proportion to represent it. First you will write it on your own. Then you will check your proportion with your group members. Then I will select one person at random to teach the class how to write the proportion. Be sure to refer to your notes for the 4 steps. 26

Time to practice

Write a proportion for this situation.

1 container is 8 feet wide, so 5 containers are 40 feet wide. 27

Teach anyone who needs help.

1 container is 8 feet wide, so 5 containers are 40 feet wide. 28

Time to practice

I will select one person to teach the class.

1 container is 8 feet wide, so 5 containers are 40 feet wide. 29

One Solution 1 container is 8 feet wide, so 5 containers are 40 feet wide. containers feet

1 5 = 8 40

Next we will write proportions to represent word problems.

• • •

One of the four numbers will be missing. Use a variable in its place. Follow the same 4 steps.

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I’ll do the first one • The Morton Salt Company can package 20 bags of salt in 1 minute. • How many bags can it package in 1 hour (60 min)? : 1 p s e t t i n S u y f i t n e Id 32

I’ll do the first one • The Morton Salt Company can package 20 bags of salt in 1 minute. • How many bags can it package in 1 hour (60 min)? : 2 p ts e t S ni u ite r W

bags minutes 33

I’ll do the first one • The Morton Salt Company can package 20 bags of salt in 1 minute. • How many bags can it package in 1 hour (60 min)? : 3 p e t S empty ite os r W ati r

bags minutes

=

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I’ll do the first one • The Morton Salt Company can package 20 bags of salt in 1 minute. • How many bags can it package in 1 hour (60 min)? : 4 p e t S in bags x l 20 l i F ers = b minutes 60 1 num We wrote a proportion to represent the problem. 35

Think about it… • •

I’m going to ask you a question about the Morton Salt problem. I’ll give you a minute to think about it, then ask for a thumbs up for Yes, or a thumbs down for No.

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Your Question • The Morton Salt Company can package 20 bags of salt in 1 minute. • How many bags can it package in 1 hour (60 min)? bags x 20 = minutes 60 1 Could you write a proportion for this problem with “bags” on the bottom? Think time… 10 1 minute 50 40 30 20 Time’s seconds Up 37

Your Question • The Morton Salt Company can package 20 bags of salt in 1 minute. • How many bags can it package in 1 hour (60 min)? bags minutes

x 20 = 60 1

Yes, just flip the numbers from top to bottom 38

Time to practice •

• • •

•

I will give you a problem and you will write a proportion to represent it. First you will write it on your own. Then you will check your proportion with your group members. Then I will select one person at random to teach the class how to write the proportion. Be sure to refer to your notes for the 4 steps. 39

On your own… Write a proportion only for this problem. (Do not answer the “how many bags” question!) 1 bag sells for $5. How many bags would you get for $75?

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Check with your group members

Teach anyone who needs help.

1 bag sells for $5. How many bags would you get for $75?

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One person teaches I will select one person to teach the class.

1 bag sells for $5. How many bags would you get for $75?

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One Solution 1 bag sells for $5. How many bags would you get for $75? bags $

1 x = 5 75

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Finally we will solve these proportions There are four steps: Write this 1. Write the proportion for the in your notes problem. 2. Use the cross-products property to write an equation. 3. Divide both sides by the coefficient to solve. 4. Rewrite your answer so it answers the question. 44

Watch me use the four steps Use a proportion to solve the Morton Salt problem:

• 1 bag sells for $5. • How many bags would you get for $75?

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Watch me use the four steps • 1 bag sells for $5. • How many bags would you get for $75? : 1 p e e t S h t ite ion r W ort p o r p

bags $

1 x = 5 75

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Watch me use the four steps • 1 bag sells for $5. • How many bags would you get for $75? : 2 p e e t S h t ite on r W ati u q e

bags $

1 x = 5 75

5x = 75 • 1

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Watch me use the four steps • 1 bag sells for $5. • How many bags would you get for $75? : 3 p e h t t S o b e id s v i D ide s

bags $

1 x = 5 75

5x = 75 • 1 5 5 x = 15 48

Watch me use the four steps • 1 bag sells for $5. • How many bags would you get for $75? bags 1 x = $ 5 75 : 4 5x = 75 • 1 p e t S ite to r 5 5 w r e R swe x = 15 an tion s e You would get 15 bags. qu 49

In pairs: •

Determine which partner will be person A and which is person B.

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Person A: Tell your partner how to write an equation from a proportion.

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Multiply diagonally, and write an equal sign in between

Person B: Tell your partner how to solve the equation.

Divide both sides by the coefficient 50

Time to practice • • •

•

I will give you a problem and you will write a proportion and solve it. First you will work on your own. Then you will check your proportion and answer with your group members. I will select one person at random to teach the class how to solve the problem. 51

On your own… •

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Crane operator Mike Metri said in an interview that his record number of containers unloaded in an 8-hour shift is 330. On average, about how many containers were unloaded in each hour?

Write the proportion first, then solve it. 52

Check with your group members Teach anyone who needs help. •

•

Crane operator Mike Metri said in an interview that his record number of containers unloaded in an 8-hour shift is 330. On average, about how many containers were unloaded in each hour? 53

One person teaches I will select one person to teach the class. •

•

Crane operator Mike Metri said in an interview that his record number of containers unloaded in an 8-hour shift is 330. On average, about how many containers were unloaded in each hour? 54

One Solution • •

330 containers in 8 hours. On average, about how many containers were unloaded in each hour?

containers hours

330 8

x = 1

8x = 330 • 1 8 8 x = 41.25 About 41 containers each hour 55

Check Your Notes and Summarize • • •

Exchange notes with your partner and look for any differences. Add anything to your notes that you feel is missing. Write a summary of how to write a proportion from a word problem and how to solve the proportion at the bottom of your notes.

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