Factor. 7th Grade Math. Ratios & Proportions. Writing Ratios. 3 Examples/ Counterexamples. Vocab Word. Slide 2 / 184. Slide 1 / 184.

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1 Slide / Slide / New Jersey Center for Teaching and Learning Progressive Mathematics Initiative This material is made freely available at and is intended for the non-commercial use of students and teachers. These materials may not be used for any commercial purpose without the written permission of the owners. NJCTL maintains its website for the convenience of teachers who wish to make their work available to other teachers, participate in a virtual professional learning community, and/or provide access to course materials to parents, students and others. th Grade Math Ratios & Proportions -- Click to go to website: Slide / Slide / Table of Contents Vocabulary words are identified with a dotted underline. Click on the topic to go to that section Writing Ratios Equivalent Ratios Rates Proportions Direct & Indirect Relationships in Tables & Graphs Constant of Proportionality Writing Equations for Proportions Understanding Graphs of Proportions Problem Solving Scale Drawings Similar Figures Glossary Sometimes when you subtract the fractions, you find that you can't because the first numerator is smaller than the second! When this happens, you need to regroup from the whole number. (Click on the dotted underline.) How many thirds are in whole? How many fifths are in whole? How many ninths are in whole? The underline is linked to the glossary at the end of the Notebook. It can also be printed for a word wall. Common Core:.RP.,.RP.,.G. Slide / Slide / The charts have parts. Factor Vocab Word Its meaning A whole number that can divide into another number with no remainder. Eamples/ Countereamples (As it is used in the lesson.) Writing Ratios R. is a factor of A whole number that multiplies with another number to make a third number. = is not a factor of Back to and are factors of Instruction Link to return to the instructional page. Return to Table of Contents

2 Slide / Slide / Ratios Ratios Ratio - A comparison of two numbers by division What do you know about ratios? Ratios can be written three different ways: a to b a:b b a Each is read, "the ratio of a to b." Each ratio should be in simplest form. When have you seen or used ratios? Find the ratio of boys to girls in this class Slide / Slide / Remember to write your ratios in simplest form! Write the ratio in three ways: a. The number of cows to the number of horses There are cupcakes. Nine are chocolate, are vanilla and the rest are strawberry. What is the ratio of vanilla cupcakes to strawberry cupcakes? There are animals in the field. Twenty are cows and the rest are horses. A : b. The number of horses to the number of animals in the field B C D : Remember to write your ratios in simplest form! Slide / Slide / A There are cupcakes. Nine are chocolate, are vanilla and the rest are strawberry. What is the ratio of chocolate & strawberry cupcakes to vanilla & chocolate cupcakes? There are cupcakes. Nine are chocolate, are vanilla and the rest are strawberry. What is the ratio of chocolate cupcakes to total cupcakes? A B B C C D D Remember to write your ratios in simplest form! Remember to write your ratios in simplest form!

3 Remember to write your ratios in simplest form! There are cupcakes. Nine are chocolate, are vanilla and the rest are strawberry. What is the ratio of total cupcakes to vanilla cupcakes? A to Slide / Slide / Equivalent Ratios B to C to Return to Table of Contents D to Slide / Equivalent ratios have the same value Slide / There are two ways to determine if ratios are equivalent.. Common Factor : is equivalent to : to is equivalent to to Since the numerator and denominator were multiplied by the same value, the ratios are equivalent Slide / Slide /. Cross Products is equivalent to True Since the cross products are equal, the ratios are equivalent. = = False is equivalent to

4 Slide / is equivalent to True False : is equivalent to, which is equivalent to Slide / True False Slide / True False : is equivalent to, which is equivalent to to is equivalent to, which is equivalent to Slide / True False Slide / Slide / Rates Rate: a ratio of two quantities measured in different units Rates Eamples of rates: participants/ teams gallons/ rooms Return to Table of Contents burgers/ tomatoes

5 Slide / Slide / Finding a Unit Rate Unit Rates Si friends have pizza together. The bill is $. What is the cost per person? Unit rate: Rate with a denominator of one Often epressed with the word "per" Hint: Since the question asks for cost per person, the cost should be first, or in the numerator. $ people Eamples of unit rates: miles/gallon Since unit rates always have a denominator of one, rewrite the rate so that the denominator is one. cookies per person words/minute to reveal $ people $. person Slide / The cost of pizza is $. per person Slide / Click for practice. Sity cupcakes are at a party for twenty children. How many cupcakes per person? John's car can travel. miles on gallons of gas. How many miles per gallon can the car travel? Slide / The snake can slither feet in half a day. How many feet can the snake move in an hour? Slide /

6 Slide / There are five chaperones at the dance of students. How many students per chaperone are there? The recipe calls for cups of flour for every four eggs. How many cups of flour are needed for one egg? Slide / Slide / Sarah rode her bike miles in Slide / We often use unit rates to easily compare rates. hour. What is Sarah's unit rate in miles per hour? Eample: Sebastian and Aleandra both work during the summer. Sebastian worked hours one week and earned $. before taes. Aleandra worked hours and earned $. before taes. Who earns more per hour at their job? Sebastian Aleandra Sebastian earned more per hour Slide / Tara traveled miles on a full tank of gas. Her gas tank holds gallons. Tahira and Brendan going running at the track. Tahira runs. miles in minutes and Brendan runs miles in minutes. Who runs at a faster pace (miles per hour)? Show your work! Which person's car gets better gas mileage? Jim Tara A Tahira B Brendan Jim traveled miles on a full tank of gas. His gas tank holds gallons. Slide /

7 Show your work! Fruity Oats is $. for a oz. bo. Snappy Rice is $. for a oz. bo. Which cereal is cheaper per ounce? Show your work! A Red apples A Fruity Oats B Green apples B Snappy Rice Slide / Two families drive to their vacation spot. The Jones family drives miles and used gallons of gas. The Alverez family drives miles and uses gallons of gas. Which family got more miles per gallon of gas? Slide / Mariella typed words in minutes. Enrique typed words in minutes. Who typed more words per minute? Show your work! Show your work! A Jones Family A Mariella B Alverez Family B Enrique Slide / Population Density Population Density: A unit rate of people per square mile This data is compiled by the US Census Bureau every years and is used when determining the number of Representatives each state gets in the House of Representatives. Slide / Population Density Red apples cost $. for ten. Green apples cost $. for si. Which type of apple is cheaper per apple? Slide / Slide /

8 Slide / Slide / Click for National Geographic Web Site To calculate population density: Find the population of the state. NJ =,, people Find the area of the state. NJ =, square miles Divide Population,, =, =, people per square mile Area Slide / We know that New Jersey has a population density of, people per square mile. Use the links below to compare this data with two other states. Population Density = Population Area Click here for population data Click here for area data The population of Newark, NJ is, people in. square miles. What is its population density? Newark, NJ The population of Moorestown, NJ is, people in square miles. What is its population density? Slide / The population of Waco, TX is, people in. square miles. What is its population density? Waco Moorestown, NJ Slide / Slide /

9 The population of Argentina is,, people and Argentina is,, square miles. What is the population density? The population of San Luis, Argentina is, people and the Provence is, square miles. What is the population density? Slide / San Luis, Argentina Slide / Slide / Proportions A proportion is an equation that states that two ratios are equivalent. Eample: Proportions Return to Table of Contents Slide / Slide / If one of the numbers in a proportion is unknown, mental math can be used to find an equivalent ratio. Eample : Hint: To find the value of, multiply by also. Slide /

10 Slide / Slide / If one of the numbers in a proportion is unknown, mental math can be used to find an equivalent ratio. Solve the proportion using equivalent ratios Eample: Hint: To find the value of, divide by also. Slide / Solve the proportion using equivalent ratios Solve the proportion using equivalent ratios Slide / Solve the proportion using equivalent ratios Slide / Solve the proportion using equivalent ratios Slide /

11 Slide / Slide / Proportions can also be solved using cross products. In a proportion, the cross products are equal. Cross multiply = = Solve for = Eample Cross multiply = = Solve for = Slide / Use cross products to solve the proportion Use cross products to solve the proportion Slide / Use cross products to solve the proportion Slide / Use cross products to solve the proportion Slide /

12 Slide / Use cross products to solve the proportion Direct & Indirect Relationships in Tables & Graphs Slide / Return to Table of Contents Slide / Slide / You can determine if a relationship is proportional by looking at a table of values or the graph. How? Table If all the ratios of numbers in the table are equivalent, the relationship is proportional. Graph If the graph of the numbers forms a straight line through the origin (,), the relationship is proportional. Eample. On a field trip, every chaperone is assigned students. Is the student to chaperone ratio proportional? If you use a table to demonstrate, you would need several ratios to start. Chaperones Students Net, find the simplified ratios and compare them. Are they the same? to reveal The relationship is proportional. Slide / Slide / The local pizza place sells a plain pie for $. Each topping costs an additional $.. Is the cost of pizza proportional to the number of toppings purchased? Toppings Cost ($).... to reveal cost toppings Ratios: Since the ratios are not equivalent, the relationship is not proportional. Is the relationship shown in the table proportional? Try this: No Year Income $, $, $, $,

13 Slide / Is the relationship shown in the table proportional? Is the relationship shown in the table proportional? No Slide / No y.. y Slide / Is the relationship shown in the table proportional? Is the relationship shown in the table proportional? No Slide / No y y Slide / Remember: Table If all the ratios of numbers in the table are equivalent, the relationship is proportional. Slide / Eample. On a field trip, every chaperone is assigned students. Is the student to chaperone ratio proportional? Chaperones Students Graph If the graph of the numbers forms a straight line through the origin (,), the relationship is proportional. Connected points form a straight line Students Since the graph is a straight line through the origin, the relationship is proportional. Line crosses through the origin Chaperones

14 Slide / Slide / Draw a graph to represent the relationship. Is the relationship proportional? Y No the relationship is not proportional, it does not go through the origin... No Salary ($) X Click for answer Is the relationship shown in the graph proportional? Eample. No Seconds Cost ($)... Feet Slide / Is the relationship shown in the graph proportional? No Is the relationship shown in the graph proportional? Students Cost ($). Slide / No. Toppings Is the relationship shown in the graph proportional? Slide / No Is the relationship shown in the graph proportional? Hours Slide / Tet Messages Teachers

15 Slide / Slide / The constant of proportionality is a constant ratio (unit rate) in any proportional relationship. We use the letterk to represent the constant of proportionality. Constant of Proportionality Equations: y = k or k=y Return to Table of Contents Slide / Slide / Find the constant of proportionality: We can find the constant of proportionality from a table of values, equation and a graph. Apples (lbs).. Cost ($)..... In a table, simplify any one of the ratios. Chaperones Students Click Slide / X Find the constant of proportionality. Y X Y Click Find the constant of proportionality: Slide /

16 Slide / X Y... Find the constant of proportionality. X Y.. Slide / Find the constant of proportionality. Slide / Slide / Find the constant of proportionality: In an equation, write the equation in the form y = k. ( to reveal) Eamples: Click Click Click Find the constant of proportionality. Slide / Find the constant of proportionality. Slide /

17 Slide / Slide / In a graph, choose a point (, y) to find and simplify the ratio. Find the constant of proportionality. (, ) Students Chaperones Slide / Slide / Find the constant of proportionality. Find the constant of proportionality. Click... Slide / Find the constant of proportionality Find the constant of proportionality. Slide /

18 Slide / Slide / The constant of proportionality and the unit rate are equivalent. Writing Equations For Proportions Return to Table of Contents We can use the constant of proportionality to help write equations using proportional relationships. By transforming the equation from: to y =k, we can write an equation that can be applied to various situations. *Remember: is the independent variable and y is the dependent variable. This means that a change in will effect y. Slide / Slide / EXAMPLE TRY THIS: You are buying Jersey Tomatoes for a cost of pounds for $.. Write an equation to represent the proportional relationship. At the candy store, you purchase lbs for $.. Write an equation to represent the proportional relationship. Let c = cost p = pounds Let c = cost p = pounds Determine the unit rate: Determine the unit rate: k = $. per pound k = $. per pound Write an equation to relate the two quantities: c = kp c =.p Write an equation to relate the two quantities: c = kp c =.p Slide / Slide / Write an equation to represent the proportional relationship shown in the table. Gallons Miles.. Write an equation that represents the proportional relationship. The total cost (c) of grapes for $. per pound(p) A c =.p B p =.c Let g = gallons m = miles m =.g TRY THIS:

19 Slide / Cost $. $. $. $. Write an equation that represents the proportional relationship.. A B. C A s =.c. D B c =.s. C c =.s D s =.c. Slide / Slide / Write an equation that represents the proportional relationship. You are ordering new menus for your restaurant. You pay $. for menus. Write an equation that represents the proportional relationship. Days, d Hours, h.. A c =.m A B m =.c B C m =.c C D c =.m D Slide / Slide / Remember, you can use a graph to determine if a relationship is proportional. How? If the graph is a straight line going through the origin (, ). Understanding Graphs of Proportions Return to Table of Contents Once you determine that the relationship is proportional, you can calculate k, the constant of proportionality. Then, write an equation to represent the relationship. What do these equations mean? Once we have determined the equation, we can understand what the graph was showing us visually. Shirts Write an equation that represents the proportional relationship. Slide /

20 Slide / Slide / EXAMPLE The jitneys in Atlantic City charge passengers for rides. What amount do they charge per ride? Dollars Find a point on the graph (,.) Use the point to find the unit rate What does the unit rate represent? The jitneys charge $. per ride. Passengers Does the line run through the unit rate? What does the unit rate represent? Mark drives miles per gallon on average. Gallons Slide / Does the line run through the unit rate? What coordinate pair represents the unit rate? (, ) Slide / TRY THIS TRY THIS Mary drives the bus. Her rate is shown in the graph. What is the unit rate? What does it represent? Dollars Find a point on the graph (, ) Use the point to find the unit rate Use the point to find the unit rate What does the unit rate represent? She drives people per hour What does the unit rate represent? She earns $. per dog What coordinate pair represents the unit rate? (,.) Dogs Does the line run through the unit rate? Find a point on the graph (, ) People Jasmine gets paid for every dog that she walks according to the graph at the right. What does she earn per dog? Use the point to find the unit rate What coordinate pair represents the unit rate? (,.) Mark drives to work each day. His gas mileage is shown in the graph. What is the unit rate? What does it represent? Find a point on the graph (, ) Miles EXAMPLE What coordinate pair represents the unit rate? (, ) Hours Does the line run through the unit rate? Slide / Slide / Chocolates at the candy store cost $. per dozen. How much does one candy cost? Round your answer to the nearest cent. Solution: $ candy Problem Solving. =. () =. = Return to Table of Contents $. per candy (Use equivalent rates to set up a proportions)

21 Slide / Slide / Eample : The ratio of boys to girls is to. There are people on a team. How many are girls? Eample : There are books per student. There are students. How many books are there? Set up the proportion: Girls People How did we determine this ratio? Set up the proportion: Books Students Where does the go? = = = =, books girls Slide / Cereal costs $. for a one pound bo. What is the price per ounce? Round your answer to the nearest penny. Slide / Slide / A Brand A B Brand B Slide / There are girls for every boys at the party. There are girls at the party. How many boys are there? Which is the better buy? Brand A: $. for ounces Brand B: $. for ounces Where does the go? The farmer has cows and chickens. He owns chickens for every cow. He has a total of animals. How many cows does he own? =

22 Slide / The auditorium can hold person for every square feet. It is square feet. How many people can the auditorium hold? The recipe for one serving calls for oz of beef and oz of bread crumbs. people will be attending the dinner. How many ounces of bread crumbs should be purchased? Mary received votes for every vote that Jane received. people voted. How many votes did Jane receive? Slide / Slide / Making Sense of Your s Sometimes your answer will be a decimal or fraction that may not make sense as an answer. Double check: - Reread the problem - Does your answer make sense? - Do you need to round your answer? - If so, which way should you round your answer? To make the desired shade of pink paint, Brandy uses oz. of red paint for each oz. of white paint. She needs one quart of pink paint. How many oz. of red paint will she need? ( quart = ounces) Slide / Slide / Cole earned a total of $ by selling cups of lemonade. How many cups of lemonade does Cole need to sell in all to earn $? Assume the relationship is directly proportional. Slide /

23 Slide / Hayley learned a total of appetizer recipes over the course of weeks of culinary school. How many weeks does she need to complete to have learned appetizers? Assume the relationship is directly proportional. Kailyn took a total of quizzes over the course of days. After attending days of school this quarter, how many quizzes will Kailyn have taken in total? Assume the relationship is directly proportional. Slide / Slide / Brittany baked cookies with cup of flour. How many cups of flour does Brittany need in order to bake cookies? Assume the relationship is directly proportional. Shane caught a total of fish over the course of days on a family fishing trip. At the end of what day will Shane have caught his fish? Assume the relationship is directly proportional. Slide / Slide / In a sample of randomly selected students at a school, students eat breakfast every morning. There are students in the school. Using these results, predict the number of students that eat breakfast. A B Slide / Scale Drawings C D Return to Table of Contents Question from ADP Algebra I End-of-Course Practice Test

24 Slide / Slide / A scale is always provided with a scale drawing. Scale drawings are used to represent objects that are either too large or too small for a life size drawing to be useful. The scale is the ratio: Eamples: drawing real life (actual) When solving a problem involving scale drawings you should: A life size drawing of an ant or an atom would be too small to be useful. Write the scale as a ratio Write the second ratio by putting the provided information in the correct location (drawing on top & real life on the bottom) Solve the proportion A life size drawing of the state of New Jersey or the Solar System would be too large to be useful. Slide / Slide / Eample: Eample: This drawing has a scale of ":", so anything drawn with the size of "" would have a size of "" in the real world, so a measurement of mm on the drawing would be mm on the real horse. The distance between Philadelphia and San Francisco is, miles. You look on a map and see the scale is inch : miles. What is the distance between the two cities on the map? drawing = actual Write the scale as a ratio = = =.. inches on the map Slide / Slide / Try This: On a map with a scale of inch = miles, the distance A hrs min. B hrs min. C hrs D hrs min. between two cities is. inches. If a car travels miles per hour, about how long will it take to get from one city to the other. On a map, the distance between your town and Washington DC is. inches. The scale is inch : miles. What is the distance between the two cities?

25 Slide / Slide / On a map, the scale is / inch= miles. Find the actual The figure is a scale of the east side of a house. In the miles A A drawing, the side of each square represents feet. Find the width and height of the door., miles B ft by ft C, miles C ft by ft D, miles D ft by ft The distance between Philadelphia and Las Vegas is. inches on a map with a scale. in : miles. What is the distance in miles? The distance between Moorestown, NJ and Duck, NC is miles. What is the distance on a map with a scale of inch to miles? Slide / ft by ft B Slide / distance between two stores that are / inches apart on the map. You are building a room that is. m long and. m wide. The scale on the architect's drawing is cm :. m. What is the length of the room on the drawing? Slide / You are building a room that is. m long and. m wide. The scale on the architect's drawing is cm :. m. What is the width of the room on the drawing? Slide /

26 Slide / Find the length of a inch wide wall on a scale drawing with a scale inch : feet. You recently purchased a scale model of a car. The scale is cm : m. What is the length of the model car if the real car is m? Slide / You recently purchased a scale model of a car. The scale is cm : m. The length of the model's steering wheel is. cm. What is the actual length of the steering wheel? Slide / Slide / Similar Figures Return to Table of Contents Slide / Two objects are similar if they are the same shape but different sizes. In similar objects: corresponding angles are congruent (the same) corresponding sides are proportional Slide / To check for similarity: Check to see that corresponding angles are congruent Check to see that corresponding sides are proportional (Cross products are equal)

27 Slide / Slide / Eample: Eample: Is the pair of polygons similar? Eplain your answer. Is the pair of polygons similar? Eplain your answer. m m. yd yd =. (.) = () = YES OR m m yd yd =. (.) = () = YES = () = () = NO Slide / No Are the polygons similar? You must be able to justify your answer. (Shapes not drawn to scale.) Are the polygons similar? You must be able to justify your answer. (Shapes not drawn to scale.) No ft OR Slide / = () = () = NO m ft. m m m ft ft Slide / Are the polygons similar? You must be able to justify your answer. (Shapes not drawn to scale.) No Eample: Find the value of in the pair of similar polygons. yd. yd Slide / yd cm cm cm cm yd = () = = cm = OR = () = = cm =

28 Slide / Slide / Find the value of y in the pair of similar polygons. Find the measure of the missing value in the pair of similar polygons. (Shapes not drawn to scale.) in. in y in y Slide / Slide / Find the measure of the missing value in the pair of similar polygons. (Shapes not drawn to scale.) ft Find the measure of the missing value in the pair of similar polygons. (Shapes not drawn to scale.). ft ft Try This: w m m. m ft Slide / mm mm y. mm Find the measure of the missing value in the pair of similar polygons. (Shapes not drawn to scale.) Find the measure of the missing value in the pair of similar polygons. (Shapes not drawn to scale.) Slide / m m m m?

29 Slide / m m m? m Find the measure of the missing value in the pair of similar polygons. (Shapes not drawn to scale.) Find the measure of the missing value in the pair of similar polygons. (Shapes not drawn to scale.) Slide / mm mm m. mm Slide / Slide / Constant of Proportionality A constant ratio (unit rate) in any proportional relationship Glossary Equations: (, ) y = k or Return to Table of Contents k= y = Back to Population Density A unit rate of people per square mile Ratios that have the same value k= Slide / Equivalent Ratios = y = k = k k = y= Instruction Slide / y NJ =,, people NJ =, square miles Population Area Population Area =,,, =, people per square mile Back to Back to Instruction Instruction

30 Slide / Slide / Rate Proportion A ratio of two quantities measured in different units An equation that states that two ratios are equivalent = = = = participants/ teams gallons/ rooms Back to Back to Instruction Instruction Slide / Slide / Ratio Scale A comparison of two numbers by division a to b a b The ratio of a drawing to the real life measurement to different ways: "the ratio of a to b" a:b burgers/ tomatoes There are animals in the field. Twenty are cows and the rest are horses. What is the number of cows to the total number of animals? drawing real life (actual) : S ecal : Drawn Horse mm high Real Horse mm high Back to Instruction Slide / Back to Instruction Slide / Similar Unit Rate Two shapes that are the same shape but different sizes Rate with a denominator of one congruent words/minute miles/gallon not similar! cookies per person Back to Back to Instruction Instruction

31 Slide / Slide / Back to Back to Instruction Instruction Slide / Slide / Back to Back to Instruction Instruction