Lesson one. Proportions in the Port of Long Beach 1. Terminal Objective. Lesson 1

Transcription

1 Proportions in the Port of Long Beach Lesson one Terminal Objective Content Standard Reference: Students will solve Port of Long Beach word problems by writing a proportion and using the cross product property to write an equation and solve it. Grade 6 Number Sense 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. Materials 1. Proportions PowerPoint 2. Popsicle stick Time Required 1 class 1

2 Keyword 1. Proportion two equal ratios Introduction of Lesson Anticipatory Set: Students should know how to determine whether two ratios form a proportion and how to solve a proportion in two ways: using the equal fraction approach or crossproducts. Recall #1: Do these two ratios form a proportion? 2 3 Tell you neighbor yes or no Answer: Yes Recall #2: Tell your neighbor the cross product property. Explain how it can be used in the first problem. Answer: , so this must be a proportion. Recall #3: Equal fraction approach: Solve this proportion by thinking of them as two different fractions: 2 3 Answer: x 12 (multiply the denominator by 4 to get 12) Recall #4: Crossproducts: Solve this proportion using cross products to write an equation and solve it. 8 x Solution: 2 x x x 30 5 Student Objective: Students will solve word problems involving the Port of Long Beach that use proportions. Purpose: Proportions are used in measurement, cooking, and enlarging and reducing photos. They are also used in the Port of Long Beach to calculate quantity 2

3 Introduction of Lesson cont d changes, like changes in cargo container sizes. Explain any other real-life use of proportions that come to mind. Lesson A proportion is two equal ratios. Today all of the numbers in the ratios will include a unit or label, such as feet or shoes. Each proportion will have four numbers, each with a label. Students can use an equal fraction approach to determine if the ratios are equal or use cross-products. Show the four numbers, each with a label in this situation. Also show students that the ratios are equal. A twenty-foot container holds 15,000 shoes. A forty-foot container holds 30,000 shoes. 1 TEU 2 TEUs 15, 000 shoes 30, 000 shoes What is the definition of a proportion? Answer: A proportion is two equal ratios. How many numbers will be in the proportion? Answer: 4. Each of the numbers will have a. Answer: unit or label. Is this a proportion? 1 cargo hold 6 cargo holds 10, 000 tons 60, 000 tons 3

4 Lesson cont d The teacher will pull a Popsicle stick to pick a random person to answer. Answer: Yes, multiply both the holds and the tons by 6 or the cross-products are equal. Write a proportion to represent a situation. There are four steps: 1. Read the situation. Identify the four numbers and their labels. 2. Write two fraction bars and the equal sign. 3. Write the labels in the proportion so they match either vertically or horizontally. 4. Fill in the matching numbers. There must be a relationship both vertically and horizontally. Model the four steps to write a proportion for this situation: A crane operator unloads one container in two minutes. The operator can unload 10 containers in 20 minutes. 1. Step 1 is to identify the units. In this case, the units are containers and minutes. 2. Step 2 is to write one unit on top of the other: cont' r 1 10 min Step 3 is to write two fraction bars and the equal sign: cont' r min 4. Step 4 is to fill in the numbers so each ratio has a meaning: cont' r 1 10 min

5 Lesson cont d Show two new proportions for the same problem. Partners discuss if the new proportions make sense. After one minute, select a student at random to say the answer. 1. Does this proportion make sense? min 2 20 cont' r 1 10 Answer: Yes. 2. Does this proportion make sense? cont' r 1 2 min Answer: No, this proportion shows that it takes 10 minutes to unload one container Write a proportion for this situation: One shipping container is 8 feet tall, so 5 containers are 40 feet tall. Answer: cont'r ft Write a proportion to represent a word problem with one number missing. Students will follow the same steps, but a variable will need to be used to represent the missing number. Write a proportion to represent this problem: The Morton Salt Company can package pound bags of salt in a minute. How many bags can it package in an hour? Follow the four steps. Answer: bags 20 x or 20 bags 1min min 1 60 x bags 60 min 5

6 Lesson cont d 1. How do you know which numbers go with each label? Answer: The proportion has to match the situation you were given, both horizontally and vertically. 2. Does it matter if I put bags on the top or bottom? Answer: No. Write a proportion only for this word problem. Do not answer the how many bags? question: One bag sells for $5. How many bags would you get for $75? Answer: bag $ 1 5 x 75 or x 1 bag bags $5 $75 Solve a proportion by using an equal fractions approach. There are four steps: 1. Write the proportion for the problem. 2. Determine the factor you need to multiply by. 3. Multiply to find the missing number. 4. Rewrite the answer so it answers the question. Use the Morton Salt Company example: The Morton Salt Company can package pound bags of salt in a minute. How many bags can it package in an hour? bags x bags 1 min 60 min 6

7 Lesson cont d 2. 1 min x min bags x bags bags are packaged in an hour. When is the equal fraction approach appropriate? Answer: One of the numerators or denominators must be a factor of the other. Here, 1 goes into 60: 20 bags x bags 1 min 60 min Solve the Morton Salt proportion you wrote earlier: 1 bag sells for $5. How many bags would you get for $75? 1. 1 bag x bags $5 $75 2. $5 x 15 $ bag x bags 4. I would get 15 bags for $75. Solve a proportion by using cross-products to write the equation and solving it. 1. Write the proportion for the problem. 2. Use cross-products property to write equation. 3. Divide both sides by the coefficient. 4. Rewrite answer so it answers the question. 7

8 Lesson cont d The Morton Salt Company receives two shipments each year totaling 120,000 tons of salt. How much salt would it receive in a year if it receives three shipments? 1. 2 shipments 3 shipments 120, 000 tons x tons 2. 2x 120, x 120, x 180, It would receive 180,000 tons of salt. Why was cross-multiplying a better way to solve the last proportion? Answer: Because 2 does not go into 3, or 2 is not a factor of 3. Write and solve a proportion to solve this problem: A crane operator unloads 30 containers in 60 minutes. How many containers are unloaded in 90 minutes? cont' r x 60 min x x x 45 cont' r 90 min containers are unloaded in 90 minutes. 8

9 Closure Have students exchange notes with their partner and look for any differences. Students should add to their notes if they see something they would like to include. Lastly, they should write the summary on the bottom of their notes. 9