Slide 1 / 65 Slide 2 / 65 6th Grade Dependent & Independent Variables 15-11-25 www.njctl.org Slide 3 / 65 Slide 4 / 65 Table of Contents Translating to Equations Dependent and Independent Variables Equations and Tables Graphing Equations Glossary Click on a topic to go to that section. Translating to Equations Return to Table of Contents Slide 5 / 65 Equation An equation is a statement that shows that two mathematical epressions are equal. These are eamples of equations: 12 + a = 15 / 5 = 10 y = 3 When writing equations, you want to pull the pieces of important information from the problem, and turn them into mathematical epressions. Slide 6 / 65 Eample Mrs. Smith bought pencils for her class and spent $5. total. If each pencil costs $0.15, how many students are in the class? This tells us one pencil costs $0.15. This tells us the equation must equal $5.. Do we know how many students are in the class? No, so this will be our variable, so let s = students in the class. Now, think about how you would calculate the total price of the pencils. You would multiply the cost of one pencil by the number of students in the class and get the total, so that is how you must set up your equation. $0.15s = $5.
Slide 7 / 65 Slide 8 / 65 Try This: Kristin is three years older than her sister, Sarah. If Kristin is 13 years old, how old is Sarah? 1 The Dolphins scored 9 more points than the Jets. The Jets scored 34 points. Which equation could be used to find the number of points p that the Dolphins scored during the game? A p + 9 = 34 p + 34 = 9 C p - 9 = 34 D p - 34 = 9 Slide 9 / 65 2 Frank earns $8. per hour for mowing his neighbors' yards. Last weekend Frank earned $51. Which equation can be used to determine the number of hours h Frank worked last weekend? A 8.5h = 51 51h = 8.5 C 8.5/h = 51 D h/51 = 8.5 Slide 10 / 65 3 Scott is working at a local restaurant. He earns $45 for each shift plus the tips he receives from his customers. Last night, he earned $73 for his shift. Which equation can be used to determine how much Scott earned in tips, t, last night? A 45 + t = 73 73 + t = 45 C 73 - t = 45 D 45 - t = 73 Slide 11 / 65 4 Gabby was trying to pay off her credit card bill. After her payment of $75, her credit card balance was $263. Which equation can be used to determine the initial balance, b, on Gabby's credit card? A b - 263 = 75 b - 75 = 263 C b + 75 = 263 D b + 263 = 75 Slide 12 / 65 5 Melinda and her friends went to the theater and purchased 3 adult tickets and one large popcorn. One adult ticket costs $9, and they spent $33.75 at the theater. Which equation could be used to determine the cost c of the popcorn? A 9c - 3 = 33.75 33.75-9 = c C 3c + 9 = 33.75 D 3(9) + c = 33.75
Slide 13 / 65 Slide 14 / 65 Vocabulary Dependent and Independent Variables An independent variable is the variable that is subject to choice, or one that is not influenced by another variable. Return to Table of Contents The value of a dependent variable relies on the values of the independent variable. Slide 15 / 65 Eample Slide 16 / 65 Try This: Try to guess the missing variable. Frank earns $8 per hour mowing his neighbors' lawns. The amount of money he earns, m, depends on how many hours he works, h. The more hours he works, the more money he earns. Therefore the dependent variable is money, m, and the independent variable is hours, h. The amount of hours he works does not rely on the money he earns. Independent how far you drive Dependent how much gas you use weight of a sick child dosage of medicine given your test score how you study for the test With your group, try to think of at least three eamples of independent and dependent variables. Independent Dependent Math Slide 17 / 65 6 The number of tickets I can buy depends on how much money I have. True False Slide 18 / 65 Click for Question The number of tickets I can buy depends on how much money I have. 7 So which value is the independent variable? A amount of money number of tickets
Slide 19 / 65 8 It costs $4.25 to rent a movie. The amount of money I spend depends on how many movies I rent. So the dependent variable is the number of movies I rent. True False Slide / 65 9 The older I get, the taller I am. My height is the... A Independent Variable Dependent Variable Slide 21 / 65 Slide 22 / 65 10 The more people I have at my party, the more brownies I need to bake. The number of people at my party is the... A Independent Variable Dependent Variable Equations and Tables Return to Table of Contents Slide 23 / 65 Tables Slide 24 / 65 Rule Machine The relationship between dependent and independent variables can be represented with a table. The value of the output relies on 1. The value of the input 2. The rule Independent Dependent The independent variable is always in the left column, and the dependent variable is always in the right column. The relationship between independent & dependent variables and input & output works like a machine. Rule The rule is the relationship between the input and the output. It says what happens to the input inside the machine. The value of the output always depends on the value of the input.
Step 1. Assign a value to the input. Step 2. Hit Enter to see the output. Slide 25 / 65 Let's figuring out the rule. The input and output values will show on this table. Slide 26 / 65 The value of n is the input. Given the value for n, find the output using the given rule. n 2n Step 4. Check Your Rule Step 3. Once you have enough input/output values to figure out the rule, select + or * and the addend or factor. Click here for online practice. 80 0 Slide 27 / 65 The manager of the department store raised the price $15 on each video game. Can you find an epression (rule) that will satisfy the total cost of the video game if given the original price? Original price $ Price after mark up $115 Slide 28 / 65 A parent wants to figure out the differences in grade level of her two sons. The younger son is two years behind the older one in terms of grade level. Write an epression (rule) containing a variable whic satisfies the difference in grade level of the two boys. older son's grade level 6 younger son's grade level 4th grade $55 $70 10 8th grade $38 $53 2 Kindergarten + 15 Slide 29 / 65 Tables can be used to represent equations. The table below represents the equation y = + 3. Tables 3 6 4 7 6 9 The output (y) depends on the input. So is the independent variable, on the left, and y is the dependent variable, on the right. y Math g n 10 28 g - 2 Slide 30 / 65 This table represents the equation t = n + 11 Find the values for t, given the values for n. t 21 39 51
Slide 31 / 65 Slide 32 / 65 80 This table represents the equation y = - 60 Find the values for y, given the values for n. y 1 180 60 1 Slide 33 / 65 This table represents the equation y = 2 Find the values for y, given the values for. y Slide 34 / 65 Real Life Eample Equations and tables can also be used to represent real-life information mathematically. Natalie is going ice skating. The skating rink charges $6.25 per hour of skating. We will let h represent the number of hours of skating and c represent the total cost. Equation: c = 6.25 h Teacher Notes & Math 80 0 hours (h) cost (c) 1 $6.25 2 $12. 3 $18.75 Slide 35 / 65 Mary's age is twice the age of Jack. Can you think of an algebraic equation which determines Mary's age (m), given Jack's age (j)? Jack's Age 12 Real Life Eample Mary's Age 24 Slide 36 / 65 11 Henry downloads songs into itunes. The amount of time it takes him to download a song depends on the song's file size. Which is the independent variable? A Download time File size 14 28 24 j 48 2j = m
Slide 37 / 65 Slide 38 / 65 12 If it takes seconds to download one megabyte, which equation represents this scenario? Use the variable t for download time and s for file size. A s = t C t = s - s = t 13 Which table represents the equation t = s? A s t s t 0 4 4 0 2 5 2 1 6 Answer C D s t 0 1 1 s t 1 51 2 52 3 53 Slide 39 / 65 Slide / 65 14 Find the missing value in this table. 15 Use the equation y = 5 to complete the table. It takes Jonathan 6 minutes to run a mile. Let t represent the number of minutes and d represent the number of miles. y? (t) (d) 6 1 Answer? 1? Answer 18? Slide 41 / 65 A C D?= 4?= 30?= 2?= 4?= 30?= 10?=?= 30?= 2?=?= 3?= 10 Slide 42 / 65 Tables to Graphs You have learned that equations and tables are two ways to represent real-life scenarios. Graphing Equations Equations and tables can also be graphed to represent a real-life scenario. Return to Table of Contents Eample: A cafeteria has an automatic waffle-making machine. The table shows the relationship between the time in hours and the number of waffles the machine can make (y). (y) 1 2 3 1 4 0 5 2
The equation for this scenario is y = Slide 43 / 65 Each hour, waffles are made. The value of y depends on the value of. y is dependent Tables to Equation is independent (y) 1 2 3 1 4 0 5 2 Slide 44 / 65 This scenario can be represented with a graph. When graphing: The independent variable is always across the ais. The dependent variable is always up the y ais. y ais (dependent) 0 3 2 0 1 Variables on Graphs ais (independent) Slide 45 / 65 Slide 46 / 65 Once you have represented the equation in a function table, you can utilize the independent and dependent variable values as coordinates. Plot the coordinates from the table below to graph the scenario. Time Waffles Made (y) Function Table Coordinat e (,y) 0 3 Math A bookstore is running a special and is charging $5 for any childrens' book. 1. Write an equation to represent the scenario. 2. Complete the table to represent the scenario. 3. Graph the function. Number of ooks Total Cost (y) Tables to Graphs Coordinat e (,y) 35 30 Answer & Math 1 (1, ) 2 1 5 (1, 5) 25 2 (2, ) 3 1 (3, 1) 4 0 (4, 0) 5 2 (5, 2) 0 1 2 10 (2, 10) 3 15 (3, 15) 4 (4, ) 5 25 (5, 25) 15 10 5 Slide 47 / 65 16 A plane descends at a rate of feet per minute. Which graph represents this scenario? A 0 3 2 0 1 0 3 2 0 1 Slide 48 / 65 17 Which scenario does the graph represent? A Mia earns $12 per hour. The river rose at a steady rate of 15 feet per hour. C The stock value decreased by $. per minute D The plane traveled miles per hour. 80 70 60 0 C 3 2 0 1 30 10
Slide 49 / 65 18 Which scenario does the graph represent? A Tia starts out with $30. Every hour Tia earns an additional $ Tia starts out with $. Every minute she spends $5. C Tia runs a mile every minutes. 80 70 60 Slide / 65 19 A dogwood tree grew at the rate of 4 feet per year. Which table represents the relationship between the height of the tree (h) and the number of years (t)? A t h t h 1 4 1 4 2 5 3 12 3 6 5 Answer 30 10 C t h 5 1 10 2 15 3 D t h 4 1 8 2 12 3 Slide 51 / 65 Slide 52 / 65 The table and graph represent which equation? A y = y = C y = - D y = + minutes # of words typed (y) 0 3 2 0 1 1 3 4 7 1 0 3 Answer The graph shows the number of teaspoons of water, y, that have dripped from a leaky faucet at the end of minutes. 21 Which equation represents the relationship between and y shown in the graph? A y = 3 y = - 3 C y = 1/3 D y = + 3 From PARCC PA sample test calculator #5 Slide 53 / 65 The graph shows the number of teaspoons of water, y, that have dripped from a leaky faucet at the end of minutes. 22 ased on the relationship in the graph, how many teaspoons of water will have dripped from the faucet at the end of 21 minutes? Slide 54 / 65 Glossary Return to Table of Contents From PARCC PA sample test calculator #5
Slide 55 / 65 Coordinates A pair of values that show an eact position on a coordinate plane. Slide 56 / 65 Dependent Variable The variable whose value varies based on the value of another variable. (,y) y (0,0) (2,3) Equations The value of y is determined by the value of. Function Tables The (y) dependent variable is 1 1 always the 2 4 output (y). 3 9 Graphing The dependent variable is always along the y ais. 8 7 6 5 4 3 2 1 ack to ack to Slide 57 / 65 Equation Two epressions that are equivalent to each other. Equivalence is shown with an equal sign. Slide 58 / 65 Epression Numbers, symbols and operations grouped together that show the value of something. 4=8 equivalent epressions 4 = 3 equivalent epressions 3 no equivalence 2 3 2 + 1 An epression is one side of an equation. 2 3 = 6 Epressions DO NOT have equal signs. ack to ack to Slide 59 / 65 Function Table Organizes the special relationship between and y values. Each of its input values gives back eactly one output value. Slide 60 / 65 Independent Variable The variable that holds its value and is not influenced by another variable. (y) 1 1 2 4 3 9 The output (y) depends on the input. is the independent variable, on the left, and y is the dependent variable, on the right. Equations holds its value, regardless of what y is. Function Tables The independent variable is always the input. (y) 1 1 2 4 3 9 Graphing The independent variable is always across the ais. 8 7 6 5 4 3 2 1 ack to ack to
Slide 61 / 65 The input is the independent variable in a function (y). Slide 62 / 65 The output is the dependent variable in a function (y). The output (y) depends on the input. (y) 1 1 2 4 3 9 The value of y is determined by the value of. 3 Rule 9 The output (y) depends on the input. (y) 1 1 2 4 3 9 The value of y is determined by the value of. 3 Rule 9 ack to ack to Slide 63 / 65 Rule Defines the relationship between the input and the output, using an algebraic equation. It says what happens to the input in a function. Slide 64 / 65 Variable A letter or symbol that represents a changeable or unknown value. 3 9 Rule: Rule: y = 3 (y) 1 3 2 6 3 9 Rule: y = 4+2 (y) 1 6 2 10 3 14 4 + 2 =? variable 2 = 6 ack to ack to Slide 65 / 65 Standards for Mathematical MP1: Making sense of problems & persevere in solving them. MP2: Reason abstractly & quantitatively. MP3: Construct viable arguments and critique the reasoning of others. MP4: Model with mathematics. MP5: Use appropriate tools strategically. MP6: Attend to precision. MP7: Look for & make use of structure. MP8: Look for & epress regularity in repeated reasoning. Additional questions are included on the slides using the "Math " Pull-tabs (e.g. a blank one is shown to the right on this slide) with a reference to the standards used. If questions already eist on a slide, then the specific MPs that the questions address are listed in the Pull-tab.