Expressions and Equations 6.EE.9


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1 Expressions and Equations 6.EE.9
2 Teacher Notes Common Core State Standard Expressions and Equations 6.EE Represent and analyze quantitative relationships between dependent and independent variables. 9. Use variables to represent two quantities in a realworld problem that change in relationship to one another; write an equation to express one quantity, thought of as the dependent variable, in terms of the other quantity, thought of as the independent variable. Analyze the relationship between the dependent and independent variables using graphs and tables, and relate these to the equation. For example, in a problem involving motion at constant speed, list and graph ordered pairs of distances and times, and write the equation d = 65t to represent the relationship between distance and time. Copyright National Governors Association Center for Best Practices and Council of Chief State School Officers. All rights reserved. 1 st Activity Notes and Examples A variable could be a shape such as a or a, but by 6 th grade it will be a letter. A quantity is a number of things 25 days, 20 miles, 3 inches, 6 books, etc. As the teacher defines variable, independent variable and dependent variable, students will fill in blanks with the correct word. Guided practice may be used on #5 and #6. Questions 7 10 may be used as independent practice and checking for understanding. By the time students work problems 9 and 10 they should remember to define variables before writing an equation. Throughout all activities question students about identifying which variable is the independent variable and which variable is the dependent variable. 2 nd Activity Tables, Graphs and Equations In this activity students will be given a table of values representing a situation and they will be required to graph the data and write an equation. Students will be given a graph of data and must determine an equation and a table of values using the data in the graph.
3 Students will be given a word problem and must determine all three ways to represent the situation equation, table and graph. Teach the students when the situation requires a scatter plot (graph points and do not draw a line through them). This is when fractional parts of a quantity make no sense. If the independent variable is number of hamburgers to be purchased, fractional parts make no sense. You cannot buy half a hamburger. If the independent variable is number of hours worked, it is possible to work part of an hour, so a line may be drawn through the points to represent that. When students make a graph on their own they must label the axes with the quantities they represent. The dependent variable customarily is on the vertical axis and the independent variable is on the horizontal axis. Students should learn to use a scale to match the data. Make sure students have equal spaces representing the same value.
4 3 rd Activity Group Activity Each group will receive a card with instructions. The group will complete one of the following tasks. Given an equation, construct a table and a graph representing the situation. Given a table of values, construct a graph and write an equation representing the same situation. Given a graph, construct a table and write an equation representing the same situation. Materials: 6 cards provided with this resource Grid Chart Paper and Meter Stick OR Graph Paper and Straight Edge Markers The groups may make their graphs and tables on grid chart paper to be displayed on the wall. Extension Activity Check for Understanding! A single page extension activity is included. The students are presented with a word problem and must represent the problem in three different ways.
5 Writing Equations Name 1. A is a symbol (usually a letter) that represents a number. 2. In the expression 2a 5, is the variable. 3. In the equation 15 = 3d, is the variable. 4. We describe how two quantities are related using. You will identify two kinds of variables: independent and dependent. The variable depends on the variable. Example: The total cost, t, of hamburgers which cost $1.95 each depends on the number of hamburgers, h to be bought. Independent variable: Dependent variable: 5. Coach Tally will buy basketball shoes for each person on the team. Each pair of shoes will cost $ Write an equation showing the relationship between the number of players and the total cost to be spent on the team. Define variables. = number of players = total cost Identify the independent variable and the dependent variable. Independent Variable Dependent Variable Write an equation for this situation. Find the total cost of buying shoes for 15 basketball players.
6 6. Mario is selling magazines for a school fundraiser. The school will receive $2.10 for each magazine he sells. The variable quantities are number of magazines sold and total money for the school. Assign a variable for each quantity. = number of magazines = total money depends on, so is the dependent variable. Write an equation showing the relationship between these two variables. The school made a profit of $ on the magazines that Mario sold. Use the equation to determine the number of magazines Mario sold. 7. Heather plans to read three books each month. Write an equation relating the total number of books Heather reads and the number of months. Define variables. (Tell what each variable represents.) Independent Variable Dependent Variable Write the equation. Use the equation to find the number of books Heather will read in one year.
7 8. Sam will mow lawns for people in his neighborhood. He will charge $12.00 for each lawn. Write an equation to show the relationship between the number of lawns he mows and the total money he will earn. Define variables. Independent Variable Dependent Variable Write the equation. How many lawns must Sam mow to have enough money to buy a video game that costs $64.00? 9. John gets an allowance of $8.00 per week. He plans to save half his allowance each week. Write an equation to show the relationship between the number of weeks he saves and the total amount he will save. (Don t forget to define the variables.) 10. Olivia s mom will bake four cookies for each guest that is invited to Olivia s birthday party. Write an equation relating the total number of cookies her mom will bake to the number of guests invited.
8 Writing Equations Name ANSWER KEY 1. A variable is a symbol (usually a letter) that represents a number. 2. In the expression 2a 5, a is the variable. 3. In the equation 15 = 3d, d is the variable. 4. We describe how two quantities are related using _variables. You will identify two kinds of variables: independent and dependent. The dependent variable depends on the independent variable. Example: The total cost, t, of hamburgers which cost $1.95 each depends on the number of hamburgers, h, to be bought. Independent variable: _h or number of hamburgers Dependent variable: t or total cost 5. Coach Tally will buy basketball shoes for each person on the team. Each pair of shoes will cost $ Write an equation showing the relationship between the number of players and the total cost to be spent on the team. Define variables. n_(or any letter) = number of players _c (or any letter) = total cost Identify the independent variable and the dependent variable. Ind. Variable n or no. of players Dep. Variable c or total cost Write an equation for this situation. c = 98n Find the total cost of buying shoes for 15 basketball players. c = 98n c = 98(15) c = 1470 Total cost = $1470
9 6. Mario is selling magazines for a school fundraiser. The school will receive $2.10 for each magazine he sells. The variable quantities are number of magazines sold and total money for the school. Assign a variable for each quantity. n = number of magazines t = total money Total money depends on number of magazines sold, so total money is the dependent variable. Write an equation showing the relationship between these two variables. t = 2.10n The school made a profit of $ on the magazines that Mario sold. Use the equation to determine the number of magazines Mario sold. t = 2.10n 105 = 2.10n Divide both sides of equation by = n Mario sold 50 magazines. 7. Heather plans to read three books each month. Write an equation relating the total number of books Heather reads and the number of months. Define variables. (Tell what each variable represents.) b = number of books m = number of months Independent Variable m Dependent Variable b The number of books read depends on the number of months she reads books. Write the equation. b = 3m Use the equation to find the number of books Heather will read in one year. 1 year = 12 months b = 3m b = 3(12) b = 36 books
10 8. Sam will mow lawns for people in his neighborhood. He will charge $12.00 for each lawn. Write an equation to show the relationship between the number of lawns he mows and the total money he will earn. Define variables. n = number of lawns m = total money earned Independent Variable n Dependent Variable _m Write the equation. m = 12n How many lawns must Sam mow to have enough money to buy a video game that costs $64.00? m = 12n 64 = 12n He cannot mow lawns. 64 = 12n He must mow 6 lawns to have enough money = n 9. John gets an allowance of $8.00 per week. He plans to save half his allowance each week. Write an equation to show the relationship between the number of weeks he saves and the total amount he will save. (Don t forget to define the variables.) w = no. of weeks t = total amount t = 4w 10. Olivia s mom will bake four cookies for each guest that is invited to Olivia s birthday party. Write an equation relating the total number of cookies her mom will bake to the number of guests invited. g = number of guests c = total no. of cookies c = 4g
11 Tables, Graphs and Equations Name 1. The Hamburger Hut charges $2.50 for each hamburger. Mrs. Shannon s 6 th grade class will eat lunch at the Hamburger Hut next week. Fill in the table to show the total cost for the number of hamburgers in the table. Number of Total Cost Hamburgers h c 1 $ $ $ n Write an equation showing the relationship between the total cost and the number of hamburgers. 16 Equation: On the grid, graph the first six entries in the table above. The independent variable is graphed on the horizontal axis. The dependent variable is graphed on the vertical axis. Total Cost No. of Hamburgers
12 2. The graph shows the distance Marla walked in a certain amount of time Distance (feet) Time (minutes) Make a table of values for the data in the graph. Time t Distance d Write an equation describing the relationship of time and distance. What distance could Marla walk in 12 minutes? What distance could she walk in 4 ½ minutes?
13 3. Callie charges $5.00 per hour to babysit. Fill in the table to show the relationship of hours worked and total earnings. Hours h 1 Earnings e Graph the data in the table on the grid. 25 Write an equation relating hours worked to total earnings. 20 How much money will Callie earn if she babysits 3 ½ hours? Earnings How many hours must Callie work to buy a computer which costs $487.00? Hours
14 Tables, Graphs and Equations Name ANSWER KEY 1. The Hamburger Hut charges $2.50 for each hamburger. Mrs. Shannon s 6 th grade class will eat lunch at the Hamburger Hut next week. Fill in the table to show the total cost for the number of hamburgers in the table. Number of Hamburgers h Total Cost c 1 $ $ $ $ $ $ $50.00 n $2.50n Write an equation showing the relationship between the total cost and the number of hamburgers. 16 Equation: 6 14 c = 2.50n 12 On the grid, graph the first six entries in the table above. The independent variable is graphed on the horizontal axis. The dependent variable is graphed on the vertical axis. Total Cost No. of Hamburgers
15 2. The graph shows the distance Marla walked in a certain amount of time Distance (feet) Time (minutes) Make a table of values for the data in the graph. Time Distance t d Write an equation describing the relationship of time and distance. d = 200t What distance could Marla walk in 12 minutes? d = 200(12) d = 2400 feet What distance could she walk in 4 ½ minutes? d = 200 (4 ½) d = 900 feet
16 3. Callie charges $5.00 per hour to babysit. Fill in the table to show the relationship of hours worked and total earnings. Hours h Earnings e 1 $ $ $ $ $25.00 Graph the data in the table on the grid. 25 Write an equation relating hours worked to total earnings. e = 5h How much money will Callie earn if she babysits 3 ½ hours? e = 5(3 ½) e = $ Earnings 10 5 How many hours must Callie work to buy a computer which costs $487.00? e = 5h 487 = 5h 487 = 5h = h She must work 97.4 hours to earn $ Hours
17 Card A Mr. Jones drives on the interstate at a constant rate of 70 miles per hour. The distance he travels for a given time may be represented by the equation d = 70 t. d = distance (miles) t = time (hours) Make a table with at least 6 entries to represent this situation. Graph the points from your table on a grid. Use a reasonable scale and label the horizontal and vertical axes with the words time and distance. Card B Jolie wants to download different ringtones in her cell phone. The table below shows the total cost of downloading a particular number of ringtones. Number of Total Cost Ringtones r c 1 $ $ $ $6.00 Represent this data as a graph. Show the scale and label the horizontal and the vertical axes with Number of Ringtones and Total Cost. Write an equation to represent the data using r and c.
18 Card C The graph below shows the number of pounds of meat needed to feed a given number of lions in one day Pounds Make a table to represent the data. Write an equation to represent this situation Lions Card D Cathy will decorate cupcakes for a party. She will place 6 jelly beans on the top of each cupcake. The number of jelly beans she needs may be represented by the equation y = 6x. x = number of cupcakes y = total number of jelly beans Make a table with at least 6 entries to represent this situation. Graph the points from your table on a grid. Use a reasonable scale and label the horizontal and vertical axes with the words cupcakes and total number of jelly beans.
19 Card E The Corner Bookstore sells used books and makes a profit of $2.00 per book. The table below shows the profit on a given number of books. Number of Books b Total Profit t 1 $ $ $ $8.00 Represent this data as a graph. Show the scale and label the horizontal and the vertical axes with Number of Books and Total Profit. Write an equation to represent the data using b and t. Card F The graph below shows the total cost of renting a car for a given number of days. Cost Make a table to represent the data (4, 168) (3, 126) (2, 84) (1, 42) Write an equation to represent this situation Days
20 Group Activity Answer Key Card A Time (hrs) Distance (miles) Distance (miles) Time (hours) Card B c = 1.5r 6 5 Total Cost Ringtones
21 Card C p = pounds of meat n = number of lions p = 30 n No. of lions Pounds of meat Card D No. of No. of jelly cupcakes beans Jelly Beans Cupcakes
22 Card E b = number of books t = total profit t = 2b 8 Total Profit Number of books Card F c = 42d Days d Cost c
23 Check for Understanding! Name Mike is a brick mason. He averages laying 175 bricks each hour. Fill in the table below to show how many total bricks Mike can lay for a given number of hours. Number of Hours h Total Bricks b Define variables. h = b = Independent Variable Dependent Variable Write an equation representing this situation. Graph the points from the table on the grid. Since Mike may work a fractional part of an hour, draw a line through the points. Label the axes with names of the quantities. Mark the scale with equal values for each space. Use a scale appropriate for the numbers in the table.
24 Check for Understanding! Name ANSWER KEY Mike is a brick mason. He averages laying 175 bricks each hour. Fill in the table below to show how many total bricks Mike can lay for a given number of hours. Number Total of Hours Bricks h b Define variables. h = _number of hours_ b = _total bricks Independent Variable h Dependent Variable b The number of bricks he lays depends on the number of hours he works. Write an equation representing this situation. b = 175h Graph the points from the table on the grid. Since Mike may work a fractional part of an hour, draw a line through the points. Label the axes with names of the quantities. Mark the scale with equal values for each space. Use a scale appropriate for the numbers in the table Bricks (5, 875) (4, 700) (3, 525) (2, 350) Graph is a sample answer. 200 (1, 175) Hours
25 Thank you for purchasing a license for this product to be used in your classroom. Additional licenses for colleagues may be purchased for half price. Go to My Purchases and click on Additional Licenses. If you have a question about this product please contact me in Product Q & A. Visit my TpT store for math resources for 3 rd grade and above. Additional Resources by Hilda Ratliff Equations, Tables, Graphs  Math Game Math Board Games Bundle  6th Grade  (6.RP) (6.NS) (6.EE) (6.G) (6.SP) Common Core Math 6th Grade  Solve Equations  Graph Inequalities (6.EE.7,8) Common Core Math 6th Grade Ratios and Proportional Relationships 6.RP.1.2.3ab Common Core Math 6th Grade Ratios and Proportional Relationships 6.RP.3cd Common Core Math 6th Grade  Statistics and Probability  CCSS 6.SP Title page: Graphics by Ashley Hughes
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