Slide 1 / 65 Slide 2 / 65 6th Grade Dependent & Independent Variables 2015-11-25 www.njctl.org Slide 3 / 65 Slide 4 / 65 Table of Contents Click on a topic Translating to Equations to go to that section. Dependent and Independent Variables Translating to Equations and Tables Equations Graphing Equations Glossary Return to Table of Contents Slide 5 / 65 Slide 6 / 65 Equation Example Mrs. Smith bought pencils for her class and spent $5.50 total. If each pencil costs $0.15, how many students are in the class? This tells us one pencil An equation is a statement that shows that two costs $0.15. This tells us the equation must equal $5.50. mathematical expressions are equal. Do we know how many students are in the class? These are examples of equations: No, so this will be our variable , so let s = students in the class. 12 + a = 15 x / 5 = 10 y = 3 x Now, think about how you would calculate the total price of the pencils. You would multiply the cost of one pencil by the When writing equations, you want to pull the pieces of number of students in the class and get the total, so important information from the problem, and turn them that is how you must set up your equation. into mathematical expressions. $0.15s = $5.50
Slide 7 / 65 Slide 8 / 65 1 The Dolphins scored 9 more points than the Jets. The Try This: Jets scored 34 points. Which equation could be used to find the number of points p that the Dolphins scored during the game? Kristin is three years older than her sister, Sarah. If Kristin is 13 years old, how old is Sarah? A p + 9 = 34 B p + 34 = 9 C p - 9 = 34 D p - 34 = 9 Slide 9 / 65 Slide 10 / 65 2 Frank earns $8.50 per hour for mowing his neighbors' 3 Scott is working at a local restaurant. He earns $45 for yards. Last weekend Frank earned $51. Which each shift plus the tips he receives from his customers. equation can be used to determine the number of hours Last night, he earned $73 for his shift. Which equation h Frank worked last weekend? can be used to determine how much Scott earned in tips, t, last night? A 8.5h = 51 B 51h = 8.5 A 45 + t = 73 C 8.5/h = 51 B 73 + t = 45 D h/51 = 8.5 C 73 - t = 45 D 45 - t = 73 Slide 11 / 65 Slide 12 / 65 4 Gabby was trying to pay off her credit card bill. After her 5 Melinda and her friends went to the theater and payment of $75, her credit card balance was $263. purchased 3 adult tickets and one large popcorn. One Which equation can be used to determine the initial adult ticket costs $9, and they spent $33.75 at the balance, b, on Gabby's credit card? theater. Which equation could be used to determine the cost c of the popcorn? A b - 263 = 75 A 9c - 3 = 33.75 B b - 75 = 263 B 33.75-9 = c C b + 75 = 263 C 3c + 9 = 33.75 D b + 263 = 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 The value of a dependent variable relies on the Table of values of the independent variable. Contents Slide 15 / 65 Slide 16 / 65 Try This: Example Try to guess the missing variable. Frank earns $8 per hour mowing his neighbors' lawns. Independent Dependent Math Practice how far you drive how much gas you use The amount of money he earns, m, depends on how many hours he works, h. weight of a sick child dosage of medicine given your test score how you study for the test The more hours he works, the more money he earns. With your group, try to think of at least three Therefore the depend ent variable is money, m, examples of independent and dependent variables. and the independent variable is hours, h. Independent Dependent The amount of hours he works does not rely on the money he earns. Slide 17 / 65 Slide 18 / 65 Click for Question 6 The number of tickets I can buy depends on how The number of tickets I can buy depends much money I have. on how much money I have. 7 So which value is the independent variable? True A amount of money False B number of tickets
Slide 19 / 65 Slide 20 / 65 8 It costs $4.25 to rent a movie. The amount of money 9 The older I get, the taller I am. I spend depends on how many movies I rent. My height is the... So the dependent variable is the number of movies I rent. A Independent Variable B Dependent Variable True False 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 B Dependent Variable Equations and Tables Return to Table of Contents Slide 23 / 65 Slide 24 / 65 Rule Machine Tables The relationship between dependent The value of the output relies on 1. The value of the input and independent variables can be Input 2. The rule represented with a table. Output Independent Dependent Input Output The independent variable is always in the left column, and the dependent variable is always in the right column. Rule The rule is the relationship between the input and the output. It says what happens to the input inside the machine. The relationship between independent & dependent variables and input & output works like a machine. The value of the output always depends on the value of the input.
Slide 25 / 65 Slide 26 / 65 Practice Practice Let's Practice figuring out the rule. The value of n is the input. Step 1. Given the value for n, find the output using the given rule. Assign a value to the input. The input and output values will show Input Output on this table. n 2n Step 2. Hit Enter to see the output. 20 40 click Step 3. Once you have enough input/output values to figure out the rule, select + or * and the addend or factor. 40 80 Click here for online click Step 4. practice. 100 Check Your Rule 200 click Slide 27 / 65 Slide 28 / 65 Practice Practice A parent wants to figure out the differences in grade level of her The manager of the department store raised the price $15 on each two sons. The younger son is two years behind the older one in video game. Can you find an expression (rule) that will satisfy the total terms of grade level. Write an expression (rule) containing a variable which cost of the video game if given the original price? satisfies the difference in grade level of the two boys. Original price Price after mark up younger son's older son's grade level grade level click 6 4th grade $100 click $115 click $55 10 click 8th grade $70 click $38 click 2 Kindergarten $53 click x click g x + 15 g - 2 Slide 29 / 65 Slide 30 / 65 Tables Practice This table represents the equation Tables can be used to represent equations. t = n + 11 The table below represents the equation Find the values for t, given the values for n. y = x + 3. Math Practice n t x y 3 6 21 10 4 7 click 6 9 28 39 click The output ( y) depends on the input ( x) . 51 40 So x is the independent variable, on the left, click and y is the dependent variable, on the right.
Slide 31 / 65 Slide 32 / 65 Practice This table represents the equation y = x - 60 Find the values for y, given the values for n. y x 80 20 click 120 60 click 180 120 click Slide 33 / 65 Slide 34 / 65 Practice Real Life Example Equations and tables can also be used to represent This table represents the equation real-life information mathematically. y = 2 x Find the values for y, given the values for x. Natalie is going ice skating. Teacher Notes & Math Practice The skating rink charges $6.25 per hour of skating. x y We will let h represent the number of hours of skating and c represent the total cost. 20 40 Equation: c = 6.25 h click hours (h) cost (c) 40 80 1 $6.25 click 2 $12.50 100 200 3 $18.75 click Slide 35 / 65 Slide 36 / 65 Real Life Example 11 Henry downloads songs into iTunes. The amount of Mary's age is twice the age of Jack. time it takes him to download a song depends on Can you think of an algebraic equation which determines Mary's age (m), given Jack's age (j)? the song's file size. Which is the independent variable? Jack's Age Mary's Age A Download time B File size 12 24 click 14 28 click 24 48 click j 2j = m click
Slide 37 / 65 Slide 38 / 65 12 If it takes 50 seconds to download one megabyte, 13 Which table represents the equation t = 50s ? which equation represents this scenario ? Use the variable t for download time and s for file size. A B s t s t A s = 50t 200 4 4 200 Answer B t = 50s 100 2 5 250 C 50 - s = t 50 1 6 300 D C s t s t 1 51 200 150 2 52 100 50 3 53 50 1 Slide 39 / 65 Slide 40 / 65 14 Find the missing value in this table. 15 Use the equation y = 5x to complete the table. It takes Jonathan 6 minutes to run a mile. Let t x y represent the number of minutes and d represent the number of miles. 20 ? Answer Answer ? 150 (t) (d) 50 ? 6 1 18 ? A ? = 4 ? = 30 ? = 250 B ? = 4 ? = 30 ? = 10 C ? = 100 ? = 30 ? = 250 D ? = 100 ? = 3 ? = 10 Slide 41 / 65 Slide 42 / 65 Tables to Graphs You have learned that equations and tables are two ways to represent real-life scenarios. Equations and tables can also be graphed to Graphing Equations represent a real-life scenario. Example: (x) (y) A cafeteria has an automatic 1 50 waffle-making machine. The table 2 100 shows the relationship between the time in hours (x) and the 3 150 Return to number of waffles the machine Table of 4 200 can make (y). Contents 5 250
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