6th Grade. Dependent & Independent Variables

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1 Slide 1 / 68

2 Slide 2 / 68 6th Grade Dependent & Independent Variables

3 Slide 3 / 68 Table of Contents Translating to Equations Dependent and Independent Variables Click on a topic to go to that section. Equations and Tables Graphing Equations Glossary

4 Slide 4 / 68 Vocabulary words are identified with a dotted underline. 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. How many thirds are in 1 whole? (Click on the dotted underline.) How many fifths are in 1 whole? How many ninths are in 1 whole? The underline is linked to the page in the presentation's glossary containing the vocab chart.

5 1 Vocab Word Slide 5 / 68 The charts have 4 parts. Factor A whole number that can divide into another number with no remainder. A whole number that multiplies with another number to make a third number. 2 Its meaning (As it is used in the lesson.) R is a factor of 15 3 Examples/ Counterexamples 3 x 5 = 15 3 and 5 are factors of 15 3 is not a factor of 16 4 Link to return to the instructional page. Back to Instruction

6 Slide 6 / 68 Translating to Equations Return to Table of Contents

7 Slide 7 / 68 An equation is a statement that shows that two mathematical expressions are equal. These are examples of equations: 12 + a = 15 x / 5 = 10 y = 3x When writing equations, you want to pull the pieces of important information from the problem, and turn them into mathematical expressions.

8 Slide 8 / 68 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 costs $0.15. This tells us the equation must equal $5.50. 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 = $5.50

9 Slide 9 / 68 Try this: Kristin is three years older than her sister, Sarah. If Kristin is 13 years old, how old is Sarah? for answer Pull

10 Slide 10 / 68 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 B p + 34 = 9 C p - 9 = 34 for answer Pull D p - 34 = 9

11 Slide 11 / 68 2 Frank earns $8.50 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 B 51h = 8.5 C 8.5/h = 51 D h/51 = 8.5 for answer Pull

12 Slide 12 / 68 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 B 73 + t = 45 C 73 - t = 45 for answer Pull D 45 - t = 73

13 Slide 13 / 68 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 = 75 B b - 75 = 263 C b + 75 = 263 for answer Pull D b = 75

14 Slide 14 / 68 5Melinda 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 = B = c C 3c + 9 = D 3(9) + c = for answer Pull

15 Slide 15 / 68 Dependent and Independent Variables Return to Table of Contents

16 Slide 16 / 68 Vocabulary An independent variable is the variable that is subject to choice, or one that is not influenced by another variable. The value of a dependent variable relies on the values of the independent variable.

17 Slide 17 / 68 Example 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.

18 Slide 18 / 68 Try to guess the missing variable. 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 examples of independent and dependent variables. Independent Dependent

19 Slide 19 / 68 6The number of tickets I can buy depends on how much money I have. True False for answer Pull

20 Slide 20 / 68 Click for Question The number of tickets I can buy depends on how much money I have. 7So which value is the independent variable? A B amount of money number of tickets for answer Pull

21 Slide 21 / 68 8It 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 for answer Pull

22 Slide 22 / 68 9The older I get, the taller I am. My height is the... A B Independent Variable Dependent Variable for answer Pull

23 Slide 23 / 68 10The more people I have at my party, the more brownies I need to bake. The number of people at my party is the... A B Independent Variable Dependent Variable for answer Pull

24 Slide 24 / 68 Equations and Tables Return to Table of Contents

25 Slide 25 / 68 The relationship between dependent and independent variables can be represented with a table. Independent Input Dependent Output 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.

26 Slide 26 / 68 Input The value of the output relies on 1. The value of the input 2. The rule Output 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.

27 Slide 27 / 68 Let's Practice figuring out the rule. Step 1. Assign a value to the input. The input and output values will show on this table. Step 2. Hit Enter to see the output. Step 3. Once you have enough input/output values to figure out the rule, select + or * and the addend or factor. Step 4. Check Your Rule Click here for online practice.

28 Slide 28 / 68 The value of n is the input. Given the value for n, find the output using the given rule. Input Output n 2n click click click

29 Slide 29 / 68 The manager of the department store raised the price $15 on each video game. Can you find an expression (rule) that will satisfy the total cost of the video game if given the original price? Original price Price after mark up $100 $55 $38 x click click click click $115 $70 $53 x + 15

30 Slide 30 / 68 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 expression (rule) containing a variable which satisfies the difference in grade level of the two boys. older son's grade level younger son's grade level 6 click 4th grade 10 click 8th grade 2 click Kindergarten g click g - 2

31 Slide 31 / 68 Tables can be used to represent equations. The table below represents the equation y = x + 3. x y The output (y) depends on the input (x). So x is the independent variable, on the left, and y is the dependent variable, on the right.

32 Slide 32 / 68 This table represents the equation t = n + 11 Find the values for t, given the values for n. n t click click click

33 Slide 33 / 68 This table represents the equation y = x - 60 Find the values for y, given the values for n. x y click click click

34 Slide 34 / 68 This table represents the equation Find the values for t, given the values for n. n t click click click

35 Slide 35 / 68 This table represents the equation y = 2x Find the values for y, given the values for x. x y

36 Slide 36 / 68 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 hours (h) cost (c) 1 $ $ $18.75

37 Slide 37 / 68 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 Mary's Age j 2j = m

38 Slide 38 / 68 11Henry 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 B Download time File size for answer Pull

39 Slide 39 / 68 12If it takes 50 seconds to download one megabyte, which equation represents this scenario? Use the variable t for download time and s for file size. A B C s = 50t t = 50s 50 - s = t for answer Pull

40 Slide 40 / 68 13Which table represents the equation t = 50s? A B s t s t for answer Pull C D s t s t

41 Slide 41 / 68 14Find the missing value in this table. It takes Jonathan 6 minutes to run a mile. Let t represent the number of minutes and d represent the number of miles. (t) (d) ? for answer Pull

42 Slide 42 / 68 15Use the equation y = 5x to complete the table. x y 20?? ? for answer Pull A B C D?= 4?= 30?= 250?= 4?= 30?= 10?= 100?= 30?= 250?= 100?= 3?= 10

43 Slide 43 / 68 Graphing Equations Return to Table of Contents

44 Slide 44 / 68 You have learned that equations and tables are two ways to represent real-life scenarios. Equations and tables can also be graphed to represent a real-life scenario. Example: A cafeteria has an automatic waffle-making machine. The table shows the relationship between the time in hours (x) and the number of waffles the machine can make (y). (x) (y)

45 Slide 45 / 68 The equation for this scenario is y = 50x Each hour, 50 waffles are made. The value of y depends on the value of x. y is dependent x is independent (x) (y)

46 When graphing: Slide 46 / 68 This scenario can be represented with a graph. The independent variable is always across the x axis. The dependent variable is always up the y axis y axis (dependent) x axis (independent)

47 Slide 47 / 68 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 (x) Waffles Made (y) Coordinate (x,y) (1, 50) (2, 100) (3, 150) (4, 200) (5, 250)

48 Slide 48 / 68 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. for answer Pull 40 Number of Books (x) Total Cost (y) Coordinate (x,y) (1, 5) (2, 10) 3 15 (3, 15) 4 20 (4, 20) (5, 25)

49 Slide 49 / 68 16A plane descends at a rate of 50 feet per minute. Which graph represents this scenario? A B C for answer Pull

50 Slide 50 / 68 17Which scenario does the graph represent? A Mia earns $12 per hour. B C D The river rose at a steady rate of 15 feet per hour. The stock value decreased by $.50 per minute The plane traveled 300 miles per hour for answer Pull

51 Slide 51 / 68 18Which scenario does the graph represent? A Tia starts out with $30. Every hour Tia earns an additional $20. B Tia starts out with $50. Every minute she spends $5. C Tia runs a mile every 20 minutes for answer Pull

52 Slide 52 / 68 19A 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 B t h for answer Pull C t h D t h

53 Slide 53 / 68 20The table and graph represent which equation? A B y = 50x 50y = x minutes (x) # of words typed (y) for answer Pull 400 C y = x D y = x

54 Slide 54 / 68 Glossary Return to Table of Contents

55 Slide 55 / 68 Coordinates A pair of values that show an exact position on a coordinate plane. (x,y) y (2,3) (0,0) x Back to Instruction

56 Slide 56 / 68 Dependent Variable The variable whose value varies based on the value of another variable. Equations y = x 2 The value of y is determined by the value of x. Function Tables The (x) (y) dependent 1 1 variable is always the 2 4 output (y). 3 9 The dependent variable is always along the y axis Graphing Back to Instruction

57 Slide 57 / 68 Equation Two expressions that are equivalent to each other. Equivalence is shown with an equal sign. 4x=8 equivalent expressions 4 =x 3 equivalent expressions x 3 no equivalence Back to Instruction

58 Slide 58 / 68 Expression Numbers, symbols and operations grouped together that show the value of something An expression is one side of an equation. 2 x 3 = 6 Expressions DO NOT have equal signs. Back to Instruction

59 Slide 59 / 68 Function Table Organizes the special relationship between x and y values. Each of its input values gives back exactly one output value. (x) (y) The output (y) depends on the input (x). x is the independent variable, on the left, and y is the dependent variable, on the right. Back to Instruction

60 Slide 60 / 68 Independent Variable The variable that holds its value and is not influenced by another variable. Equations y = x 2 x holds its value, regardless of what y is. Function Tables The independent variable is always the input (x). (x) (y) The independent variable is always across the x axis Graphing Back to Instruction

61 Slide 61 / 68 Input The input is the independent variable in a function (y). The output (y) depends on the input (x). (x) (y) y = x 2 Input The value of y is determined by the value of x. 3 Input Rule y = x 2 9 Output Back to Instruction

62 Slide 62 / 68 Output The output is the dependent variable in a function (y). The output (y) depends on the input (x). (x) (y) y = x 2 Output The value of y is determined by the value of x. 3 Input Rule y = x 2 9 Output Back to Instruction

63 Slide 63 / 68 Rule Defines the relationship between the input and the output, using an algebraic equation. It says what happens to the input in a function. 3 Input 9 Rule: y = 3x Rule: y = x 2 (x) (y) 1 3 Output Rule: y = 4x+2 (x) (y) Back to Instruction

64 Slide 64 / 68 Variable A letter or symbol that represents a changeable or unknown value. 4x + 2 variable x x x =? 2x = 6 Back to Instruction

65 Slide 65 / 68 Back to Instruction

66 Slide 66 / 68 Back to Instruction

67 Slide 67 / 68 Back to Instruction

68 Slide 68 / 68 Back to Instruction

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