PROPORTIONS Judo Math Inc.

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1 PROPORTIONS 2013 Judo Math Inc.

2 7 th grade Grade Yellow Belt Training: Algebra/Proportions Discipline Order of Mastery: Proportional Thinking 1.Intro to proportions 2. Determining proportionality (tables, equivalent ratios) (RP2a) 3. Proportionality constant (RP2b) 4. Proportions in equations (RP2c) 5. Proportionality in graphs (RP2d) 6. Interest, tax, gratuity, fees (RP3) 7. Percent increase and decrease and percent error (RP3) Welcome to the Yellow Belt - Proportions! Though it sounds like a complicated word, proportions are little more than fractions or ratios that equal each other. Every time you look at a map, pay tax or tip at a restaurant, or figure out your percent on a quiz, you are using a proportional relationship! For example, you could solve the problem, Polly ran the 100-yard dash in 15 seconds. How long did it take her to run 10 yards? Now, a word of advice about Proportions. In order for a proportion to be, well, proportionate, the cross products have to be equal. This means that you will often have to do some cross multiplying and dividing to solve them. But don t try cross multiplying in other situations (I ve seen you do it!) It doesn t work for multiplying and dividing fractions it is simply for solving proportions. I know you can keep all of this straight. Standards Included: 7.RP.2 Recognize and represent proportional relationships between quantities. 7.RP.2a Decide whether two quantities are in a proportional relationship, e.g., by testing for equivalent ratios in a table or graphing on a coordinate plane and observing whether the graph is a straight line through the origin. 7.RP. 2b Identify the constant of proportionality (unit rate) in tables, graphs, equations, diagrams, and verbal descriptions of proportional relationships. 7.RP.2c Represent proportional relationships by equations. For example, if total cost t is proportional to the number n of items purchased at a constant price p, the relationship between the total cost and the number of items can be expressed as t = pn. 7.RP.2d Explain what a point (x, y) on the graph of a proportional relationship means in terms of the situation, with special attention to the points (0, 0) and (1,r) where r is the unit rate. 7.RP.3 Use proportional relationships to solve multistep ratio and percent problems. Examples: simple interest, tax, markups and markdowns, gratuities and commissions, fees, percent increase and decrease, percent error Judo Math Inc.

3 1. Intro: Looking at proportions First look at these pictures and write about how proportions could be used to help you learn more about the picture (hint: use words like scale up, scale down, ratio, proportionate, not to scale, etc.) 1

4 1. Intro: A proportions mystery There has been a smuggling of pancakes out of Ms. Wong s classroom and a handprint has been found as the only evidence. It was placed about 3 feet off the ground and the size of the handprint is exactly the same as pictures below. Use proportions and explanations to tell me - How tall the thief is. - How high off the ground the thief s shoulders are. -The arm span of the thief. -How old you think the thief might be! -Who the thief is!!! 2

5 2. Determining proportionality: equivalent ratios Just like with fractions that are equal, proportions are equal if you can multiply the numerator or denominator by the same number Show what each numerator and denominator would be multiplied by in this situation to get an equivalent fraction Example x5 1 = x5 3 = = = Another interesting thing about proportions is their diagonals! Multiply them here 5 = = = = Since we know the product of diagonals are equal, let s try to use that to figure out what X is below. (You will be setting up an equation) = 6 x 2. 3x=48 3. Solve for x Our RULE: When solving proportions, first multiply on the diagonals and set them equal. To solve, divide! 3

6 2. Determining proportionality: tables Sometimes we will look at proportions in tables. Inspect the three tables below. There are three unique relationships to notice with proportion tables. Horizontal: Each value in the "green" column is multiplied by a constant to get the corresponding value in the "red" column. Vertical: If the "green" value is multiplied by a number, then the "red" number is multiplied by the same number. Addition: If two numbers in the "green" column are added together, then the corresponding "red" numbers also are added together. Sometimes they are written this way: Green Red x Complete the following ratio tables: 1. A person who weighs 160 pounds on earth will weigh 416 pounds on Jupiter. If a person weighs 120 pounds on earth, how much would he weigh on Jupiter? 2. There are 12 boys and 16 girls in a class. At that rate, how many boys would there be if there were 28 girls? Boys Girls If five boys can eat 16 slices of pizza, then how many slices can 20 boys eat?(create a table for this) 28 4

7 4. Martin can read 45 pages in 30 minutes. At this rate, how long will it take him to read a 300- page book? (create a table for this) 5. The ratio of boys to girls at this school is 4 to 5. Complete the following table for possible quantities of boys and girls: Boys Girls Explain how you got your answers above: 6. For every 5 teachers in the village, 3 of them drive a Honda. Complete the following ratio table for possible quantities of teachers and Hondas: Teachers Hondas Search around your house or around this school for something you can write a ratio about. Write your topic and then complete a ratio table for your information: Topic: 5

8 8. Sam is planning to drive from New York to San Francisco in his Prius. Sam Started to fill out a table (Shown here) showing how far in miles he can travel for each gallon of gas he buys: Gallons Miles Use the information in the table to answer these questions: a. Please complete the table and assume that the relationship is proportional. b. How many miles per gallon is Sam getting, based on the table? c. If you can, write an equation that Sam can use to find the distance (d) he can drive on any number of gallons of gas. 6

9 3. Constant of proportionality A Proportionality constant is a number that is used to convert a measurement in one system to the equivalent measurement in another system. For instance, people who are familiar with the traditional system of units used in the United States, pounds, feet, inches, etc., may need to find out the metric equivalent for these measures in grams and meters. To make these calculations they would need some proportionality constants. For instance, people may know that they have 100 eggs and want to know how many dozen eggs they have. The proportionality constant K is then 1 dozen/ 12 eggs. 100 eggs * 1 dozen / 12 eggs = 8 dozen eggs + 4 eggs. This is also known as the UNIT RATE Determine the proportionality constant in each of these situations (all of your answers should say k= ) 1. At Vons, canned Chicken Noodle Soup costs $1.29 for a 12 ounce can. How much is this per ounce? 2. An accountant earn $1200 for every 45 hour work week. What is the unit rate in dollars per hour? 3. My car s odometer told me that I drove 155 miles in 3 hours. What was my unit rate in miles per hour? 4. A 32 gram serving of Cinnamon Life contains 9 grams of sugar. A 55 gram serving of Raisin Nut Bran contains 16 grams of sugar. Which cereal has less sugar per gram of sugar? (hint: convert each rate to unit rates and compare!) 5. Red Delicious apples are $1.25 for 2 pounds and Granny Smith apples are $2.75 for 5 pounds. Which ones are cheaper? (hint: convert each rate to unit rates and compare!) 7

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11 4. Proportions in equations (direct variation) One way to write a formula showing how to use a proportionality constant (let us call it "K") is: Y=K*X When two variables are related in such a way that the ratio of their values always remains the same, the two variables are said to be in direct variation. In simpler terms, that means if A is always twice as much as B, then they directly vary. If a gallon of milk costs $2, and I buy 1 gallon, the total cost is $2. If I buy 10 gallons, the price is $20. In this example the total cost of milk and the number of gallons purchased are subject to direct variation -- the ratio of the cost to the number of gallons is always 2. Example: Given that the variables vary directly, write an equation relating x and y. x = 6, y = 18 Solution: Now you try! 1. Given that the variables vary directly, write an equation relating x and y. x = 5, y = Given that the variables vary directly, write an equation relating x and y. x = 21, y = Given that the variables vary directly, write an equation relating x and y. x = 6.9, y =

12 4. Given that the variables vary directly, write an equation relating x and y. x = 4, y = Given that the variables vary directly, write an equation relating x and y. x = 16, y = A paycheck varies directly with the number of hours worked. Suppose the pay for 20 hours of work is $ What is the pay for 500 hours of work? 10

13 5. Proportionality in graphs If y varies directly as x, then the graph of all points that describe this relationship is a line going through the origin (0, 0) whose slope is the proportionality constant. That's because each of the variables is a constant multiple of the other, like in the graph shown below: Check out the following graphs and answer the questions about proportionality

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15 6. Interest, tax, gratuity, fees (RP3) Percents are a ratio out of 100 and can be solved very effectively using proportions % = part 100 whole OR % = is 100 of Example: What percent of 30 is 12? x 100 =12 30 Practice: 1. What is 45% of 12? 2. What is 20% of 15? 3. What percent of 20 is 2? 4. What percent of 50 is 10? 5. 25% of what is 60? 6. 40% of what is 50? 13

16 Percents come in very handy in lots of situations in life Tax: A fee charged by a government on a product, income, or activity. When you purchase items at the store, you are charged tax on your payment. Tip: A tip (also called a gratuity) is a sum of money given to a person for performing a service for you (like a waitress, or a valet) Try solving these with your knowledge of proportions and percents! 1. A bicycle is on sale for $ The sales tax rate is 5%. What is the total price? 2. Rosa and her friends are eating out for dinner. The bill was $ They want to leave a 20% tip. How much should they leave for tip? 3. Rick bought 3 shirts for $18 each, 2 pair of socks for $3.99 a pair, and a pair of slacks for $ The sales tax rate is 8.5%. How much did he pay? 4. The Rodriguez family went to dinner at Pasta Palace. Mr. Rodriguez ordered a meal for $6.25; Mrs. R ordered a meal for $7.50; the 2 children ordered individual pizzas for $4.99 each. The sales tax rate was 7.25% and they left a 15% tip. How much was the total bill? 14

17 5. Megan bought 50 beads for $9.95. She received a 10% discount, and she paid a sales tax of 6%. How much did she pay for the beads? 6. Eric bought a canoe for $ The sales tax was 5.25%. What was his total cost? 7. Shinya went to the salon. The cost of her haircut was $ He tipped the barber 15%. What was the total cost? 8. A snowmobile priced at $2,800 is on sale for 20% off. The sales tax rate is 8.25%. What is the sale price including tax? 9. Lisa's store collects 5% sales tax on every item sold. If she collected $22.00 in sales tax, what was the cost of the items her store sold? 15

18 10. Bill and his friends went to Red Robin. Their order consisted of 2 hamburgers t $5.49 each, a chicken sandwich for $6.25, an order of ribs for $12.99, and 4 sodas at $1.75 each. They paid a 7% sales tax, and a 20% tip. How much money did lunch cost them? 11. At a restaurant you only have $20 to spend on dinner. In addition to the cost of the meal you must pay a 5% sales tax and leave a 15% tip. What is the most expensive item you can order? 12. A bicycle shop is selling $ bikes at 10% off. If the sales tax is 7% how much will the bike cost? 13. A magazine advertises that a subscription price of $29.99 (for 12 issues) represents a savings of 70% from the newsstand price. What is the price of one issue at the newsstand? 14. After the 15% sales tax the jeans were $ How much was the jeans before the sales price? 16

19 7. Percent increase and decrease and percent error (RP3) When a beginning and ending amount are given and you are asked to find the percent of increase or decrease from the beginning amount to the ending amount, you need to use the formula shown below: In other words, find out how much the increase or decrease was. Then divide that by the original amount given in the problem. Once you have this number as a fraction or a decimal, change it to a percent and you are done! Try these 1. Suppose the average attendance at a local high school's football games went down from 2000 people in 2003 to 1500 people in What was the percent decrease in attendance at the football games? 2. The enrollment at a university increased from 14,000 students to 16,000 students over a period of 5 years. What is the percent increase in enrollment? 3. The selling price of a home was dropped from $200,000 to $190,000. By what percent did the price drop? 17

20 4. What is the percent decrease on a DVD recorder that is marked down from $400 to $350? 5. My real estate agent told me that my house had appreciated in value over the last three years. In other words, it has gone from being worth $102,500 to being worth $111,000. What is the percent increase in the value of my house? 6. A pair of jeans is marked down 30% and then reduced at the cash register another 10%. Is this a total reduction of 40%? Why or why not? Explain using examples. 7. Describe the pattern using a percent increase. Find the next three numbers in the sequence. 1, 3, 9, 27,?,?,?. 18

21 Percent Error is similar to percent increase or decrease. When calculating percent error, take the ratio of the amount of error to the accepted value or true value, or real value. Then, convert the ratio to a percent. We can express the percent error with the following formula shown below: IMPORTANT! URGENT! Keep in mind that when computing the amount of error, you are always looking for a positive value. Therefore, always subtract the smaller value from the bigger. In other words, amount of error = bigger smaller Example: A man measured his height and found 6 feet. However, after he carefully measured his height a second time, he found his real height to be 5 feet. What is the percent error the man made the first time he measured his height? *Percent error = (amount of error)/accepted value *Amount of error = 6-5 = 1 *The value he found after he carefully measured his height, or 5 *So, percent error = 1/5 = 20% You Try! A student made a mistake when measuring the volume of a big container. He found the volume to be 65 liters. However, the real value for the volume is 50 liters. What is the percent error? 19

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23 NOW, for the TRUE proportions masters 21

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