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Do Now What is the pattern?

Do Now What is the pattern?. Materials Needed:. Notebook Calculator Communicator Marker Eraser. Direct Variation. What is it and how do I know when I see it?. Bill is happy. He is thinking about that bag of candy that his mom just bought! One entire bag of candy just for him!!!

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Do Now What is the pattern?

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  1. Do NowWhat is the pattern?

  2. Materials Needed: • Notebook • Calculator • Communicator • Marker • Eraser

  3. Direct Variation What is it and how do I know when I see it?

  4. Bill is happy. He is thinking about that bag of candy that his mom just bought! One entire bag of candy just for him!!! That’s 25 pieces of candy all to himself!!! How does it work?

  5. Oh look! His friend Gina is over. She even brought a bag of candy with her!! Her bag has 25 pieces too!! Now, two people and 50 pieces of candy! It’s cool - Bill likes Gina. Now what?

  6. Bill looks out the window… Robert is running to his house! What’s that he’s holding? He has a bag of candy too! His bag also has 25 pieces! Now there are 3 people to share 75 pieces of candy! OH MY!! And…

  7. Word is out on the street…. Bill is having a party?! Who invited Spike? It’s okay – Spike is bringing a bag of candy to share with the group. Another 25 pieces to share! What does that have to do with a direct variation? Oh well…

  8. Let’s look at what happened… • Bill had one bag of candy - 25 pieces • Gina arrives with a bag of candy – 50 pieces • Robert runs to Bill’s house with a bag of candy – 75 pieces • Spike is headed to Bill’s with a bag of candy – 100 pieces • As each person enters, the number of pieces of candy increases • Lets make a table and graph it!

  9. And… • An algebraic form of this would be… • Y = the amount of candy each one gets • X = the number of people • K = the constant (amount of candy in each bag) • Y = KX • As the number of people increased, the amount of candy each one got increased. • That’s easy!

  10. What is a direct variation? • A direct variation is described by an equation of the form y = kx, where • K 0 • As x increases y increases, as x decreases y decreases • The graph is a straight line that always passes through the origin.

  11. Examples of Direct Variation: Note: X increases, And Y increases. What is the constant of variation of the table above? Since y = kx we can say Therefore: 12/6=k or k = 2 14/7=k or k = 2 16/8=k or k =2 Note k stays constant. y = 2x is the equation!

  12. Examples of Direct Variation: Note: X decreases, And Y decreases. What is the constant of variation of the table above? Since y = kx we can say Therefore: 30/10=k or k = 3 15/5=k or k = 3 9/3=k or k =3 Note k stays constant. y = 3x is the equation!

  13. Examples of Direct Variation: Note: X decreases, -4, -16, -40 And Y decreases. -1,-4,-10 What is the constant of variation of the table above? Since y = kx we can say Therefore: -1/-4=k or k = ¼ -4/-16=k or k = ¼ -10/-40=k or k = ¼ Note k stays constant. y = ¼ x is the equation!

  14. Example of a Direct Variation Problem • If 4 pounds of peanuts cost $7.50, how much will 2.5 pounds cost. • This is a Direct Variation problem because: • as the weight of the peanuts increases the cost also increases.

  15. Answer Now What is the constant of variation for the following direct variation? • 2 • -2 • -½ • ½

  16. Is this a direct variation? If yes, give the constant of variation (k) and the equation. Yes! k = 6/4 or 3/2 Equation? y = 3/2 x

  17. Is this a direct variation? If yes, give the constant of variation (k) and the equation. Yes! k = 25/10 or 5/2 k = 10/4 or 5/2 Equation? y = 5/2 x

  18. Is this a direct variation? If yes, give the constant of variation (k) and the equation. No! The k values are different!

  19. Answer Now Which of the following is a direct variation? • A • B • C • D

  20. Answer Now Which is the equation that describes the following table of values? • y = -2x • y = 2x • y = ½ x • xy = 200

  21. Using Direct Variation to find unknowns (y = kx) Given that y varies directly with x, and y = 28 when x=7, Find x when y = 52. HOW??? 2 step process 1. Find the constant variation k = y/x or k = 28/7 = 4 k=4 2. Use y = kx. Find the unknown (x). 52= 4x or 52/4 = x x= 13 Therefore: X =13 when Y=52

  22. Using Direct Variation to find unknowns (y = kx) Given that y varies directly with x, and y = 3 when x=9, Find y when x = 40.5. HOW??? 2 step process 1. Find the constant variation. k = y/x or k = 3/9 = 1/3 K = 1/3 2. Use y = kx. Find the unknown (x). y= (1/3)40.5 y= 13.5 Therefore: X =40.5 when Y=13.5

  23. Using Direct Variation to find unknowns (y = kx) Given that y varies directly with x, and y = 6 when x=-5, Find y when x = -8. HOW??? 2 step process 1. Find the constant variation. k = y/x or k = 6/-5 = -1.2 k = -1.2 2. Use y = kx. Find the unknown (x). y= -1.2(-8) x= 9.6 Therefore: X =-8 when Y=9.6

  24. Using Direct Variation to solve word problems Problem: A car uses 8 gallons of gasoline to travel 290 miles. How much gasoline will the car use to travel 400 miles? Step One: Find points in table Step Three: Use the equation to find the unknown. 400 =36.25x 400 =36.25x 36.25 36.25 or x = 11.03 Step Two: Find the constant variation and equation: k = y/x or k = 290/8 or 36.25 y = 36.25 x

  25. Using Direct Variation to solve word problems Problem: Julio wages vary directly as the number of hours that he works. If his wages for 5 hours are $29.75, how much will they be for 30 hours Step One: Find points in table. Step Three: Use the equation to find the unknown. y=kx y=5.95(30) or Y=178.50 Step Two: Find the constant variation. k = y/x or k = 29.75/5 = 5.95

  26. Direct Variation and its graph y = mx +b, m = slope and b = y-intercept With direction variation the equation is y = kx Note: m = k or the constant and b = 0 therefore the graph will always go through…

  27. the ORIGIN!!!!!

  28. Tell if the following graph is a Direct Variation or not. Yes No No No

  29. Tell if the following graph is a Direct Variation or not. Yes No No Yes

  30. Hooray! • You can do Direct Variation Problems!!!!

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