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Demystify Challenging Problems with Bar Modeling

Demystify Challenging Problems with Bar Modeling. Gregg Velatini Dianna Spence 2012 Georgia Mathematics Conference. 72 inches. 9 x 4 = 36 inches. Simple Ratios and Proportions.

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Demystify Challenging Problems with Bar Modeling

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  1. Demystify Challenging Problems with Bar Modeling Gregg Velatini Dianna Spence 2012 Georgia Mathematics Conference

  2. 72 inches 9 x 4 = 36 inches Simple Ratios and Proportions • The lengths of three rods are in the ratio of 1:3:4. If the total length is 72 inches find the length of the longest rod. 9 Rod 1 9 9 9 Rod 2 72 / 8 = 9 inches Rod 3 9 9 9 9 The length of the longest rod is 36 inches

  3. Ratios and Proportions • A garbage man had 3 times as much money as a teacher. After the teacher earned an extra $200 moonlighting, the garbage man only had twice as much money. How much money did the teacher have at first? Garbage 3 Parts Teacher 1 Part

  4. Ratios and Proportions • A garbage man had 3 times as much money as a teacher. After the teacher earned an extra $200 moonlighting, the garbage man only had twice as much money. How much money did the teacher have at first? Garbage 2 Parts Teacher 200 1 Part The teacher had $400 at first.

  5. Ratios - Practice • Karen’s cat condo boards cute calicos for companionless curmudgeons. In September, the condo boarded cats and the ratio of female to male cats was 3:2. In October, she boarded several more cats, 35 of which were female. After adding the new cats, the ratio of female to male cats was reduced to 1:1 If she wound up with 250 cats, how many of the original cats were female?

  6. Karen’s cat condo boards cute calicos for companionless curmudgeons. In September, the condo boarded cats and the ratio of female to male cats was 3:2. In October, she boarded several more cats, 35 of which were female. After adding the new cats, the ratio of female to male cats was reduced to 1:1 If she wound up with 250 cats, how many of the original cats were female? Before Female Cats 30 30 30 Male Cats After Female Cats 35 30 250 Cats 35 Male Cats 90 of the original cats were female.

  7. Solving Fraction Equations • Mom bought 1 carton of eggs. She used 1/6 of the eggs to make cookies and 1/4 of the eggs to bake a cake. How many eggs did mom have left? 12 Eggs 1 7 eggs Cookies Cake

  8. Solving Fraction Equations-- Practice • Brad spent 1/3 of his money on Beanie Babies and 1/2 of it on Nascar collectables. • What fraction of his money did he spend altogether? • What fraction did he have remaining?

  9. Solving Simple Fraction Problems • Brad spent 1/3 of his money on Beanie Babies and 1/2 of it on Nascar collectables. • What fraction of his money did he spend altogether? • What fraction did he have remaining? 1/3 1/2 Brad’s Money Beanies Nascar 1/6 • Brad spent 5/6 of his money. • Brad had 1/6 of his money remaining.

  10. 15 Solving an Algebraic Equation • Six less than three times a number is fifteen. What is the number? 7 21/3=7 21 6 The number is 7

  11. Solving an Algebraic Equation--Practice • Three more than twice a number is eleven. What is the number?

  12. 8 11 1 1 1 Solving a Simple Algebraic Equation • Three more than twice a number is eleven. What is the number? 2x + 3 = 11 2x = 8 x = 8/2 x = 4 4 The number is 4

  13. 409 - 17 - 32 lbs = 360 lbs 409 lbs • The combined weight of Brad, John and Gregg is 409 lbs. Gregg is 32 lbs heavier than Brad and Brad is 17 lbs lighter than John. Find John’s weight. John John 17 lbs 17 lbs 17 lbs Brad Brad Gregg Gregg 32 lbs 32 lbs 360 lbs / 3 = 120 lbs John weighs 120 + 17 = 137 lbs John 120 lbs 137 lbs

  14. Solving an Algebraic Equation - Practice • The combined IQ’s of Mitt, Gary, and Barack is 397. Barack’s IQ is 7 points higher than Mitt’s, which is 15 points less than Gary’s. Find Barack’s IQ.

  15. 397 - 7 - 15 = 375 397 • The combined IQ’s of Mitt, Gary,and Barack is 397. Barack’s IQ is 7 points higher than Mitt’s, which is 15 points less than Gary’s. Find Barack’s IQ. Barack Barack 7 7 7 Mitt Mitt Gary Gary 15 15 375 lbs / 3 = 125 Barack’s IQ is 125 + 7 = 132 Barack 125 132

  16. Mixture Problems • A “recipe” requires mixing 1 oz of 20% alcohol with 2 oz of 80% alcohol and 5 oz of orange juice. What is the resulting alcohol concentration? 1 oz 2 oz 5 oz 8 oz ? % 20 % + 80 % + 0 % = 18/80 = 22 1/2 % The final concentration is 22 1/2 % alcohol

  17. Mixture Problems -- Practice • 2 liters of 30% acid are mixed with 1 liter of 60% acid. What is the resulting acid concentration?

  18. 2 liters of 30% acid are mixed with 1 liter of 60% acid. What is the resulting acid concentration? 2 liters 1 liter 3 liters ? % 30 % + 60 % = The final concentration is 40% acid

  19. Mixture Problems • What amount and concentration of acid solution must be added to 1 gal of 60% acid solution in order to get 3 gal of 80% acid solution? 3 gal -1 gal = 2 gal 2 gal ? gal 1 gal 3 gal ? % 80 % 60 % + = There are 24 shaded units here. 6 come from the first bucket. 18 must come from the second bucket. Shading each gallon equally to get 18 total shaded units results in each gallon with 9 of 10 shaded units 2 gal of 90% acid solution must be added to 1 gal of 60 % acid solution to yield 3 gal of 80% acid solution.

  20. Mixture Problems -- Practice • What amount and concentration of acid solution must be added to 2 gal of 30% acid solution in order to get 5 gal of 60% acid solution?

  21. What amount and concentration of acid solution must be added to 2 gal of 30% acid solution in order to get 5 gal of 60% acid solution? 2 gallons 3 gallons 5 gallons 60 % 30 % + ? % = 3 gallons of 80% acid must be added.

  22. Mixture Problems • How much $1.20 per pound chocolate must be added to 4 pounds of $0.90 per pound chocolate to get chocolate that averages $1.00 per pound? ? pounds ? pounds 4 pounds + = $1.20 /lb $0.90 /lb $1.00 /lb Each segment represents $0.10 $0.90 2 pounds of $1.20 per pound chocolate must be added to 4 pounds of $0.90 per pound chocolate to get 6 pounds of chocolate that averages $1.00 per pound

  23. Mixture Problem - Practice • How much $1.20 per pound chocolate must be added to 4 pounds of $0.90 per pound chocolate to get chocolate that averages $1.10 per pound?

  24. How much $1.20 per pound chocolate must be added to 4 pounds of $0.90 per pound chocolate to get chocolate that averages $1.10 per pound? ? pounds ? pounds 4 pounds $1.20 /lb + $0.90 /lb = $1.10 /lb 8 pounds of $1.20 per pound chocolate must be added to 4 pounds of $0.90 per pound chocolate to get 12 pounds of chocolate that averages $1.10 per pound

  25. A 32 pound of radioactive material decays to 4 lbs in 3000 years Half Life – Radioactive Decay Half - Life in years Half - Life Half - Life Amount Remaining 4 lbs … 32 lbs 16 lbs 8 lbs 3000 years Definition: The amount of time it takes for a material to decay to ½ of it’s original amount is called the half-life Half life =3000/3 =1000 years

  26. Half Life - Practice • If it takes 2 hrs for a sample to decay from 96 pounds to 12 lbs, how long will it take to decay to 3 lbs?

  27. If it takes 2 hrs for a sample to decay from 96 pounds to 12 lbs, how long will it take to decay to 3 lbs? half life 48 lbs 6 lbs 3 lbs 96 lbs Amount Remaining 24 lbs 12 lbs 2 hrs It will take two more “half lifes” to get from 12 pounds to 3 pounds. The half life is 120/3 min. = 40 min It will take 5 x 40 min. = 200 minutes to decay from 96 pounds to 3 pounds.

  28. Fact: A 3 dB increase is equivalent to a doubling in sound volume.* Decibels Intensity of original sound = V Volume V 0 dB 2V 3 dB 6 dB 4V 9 dB 8V 16V 12 dB

  29. Fact: A 3 dB increase is equivalent to a doubling in sound volume.* A sound engineer finds that adjusting the volume on his console results in an increase of 15 decibels. By what factor has the volume increased? Decibels V 0 dB 2V 3 dB 6 dB 4V 9 dB 8V 16V 12 dB 15 dB 32V 15/3=5, so the volume will be doubled 5 times. A 15 dB increase results in the volume increasing by a factor of 32

  30. Axel hears his favorite song on his fancy stereo which and he turns up the volume such that the volume is increased by a factor of 64. How many decibels did the sound level increase? Decibels

  31. Axel hears his favorite song on his fancy stereo which and he turns up the volume such that the volume is increased by a factor of 128. How many decibels did the sound level increase? V 0 dB 2V 3 dB 6 dB 4V 9 dB 8V 16V 12 dB 15 dB 32V 64V 18 dB 128V 21 dB 256V 24 dB • The sound level increased by 21 dB.

  32. System of Equations Solve y x 1 x x y y y 13 x x x x 1 x 1 x x x x x 1 • Remove the three “1’s” y=x+1 10 2 2 2 2 2 y=x+1 x=2,y=3 10

  33. Systems of Equations - Practice • A local bake sale sells brownies for $2 each and cakes for $6 each. At the end of the day 60 more cakes were sold than brownies and the total revenues were $600. How many brownies and cakes were sold?

  34. A local bake sale sells brownies for $2 each and cakes for $6 each. At the end of the day 60 more cakes were sold than brownies and the total revenues were $600. How many brownies and cakes were sold? B B C C C C C C 600 B B B B B B B B 60 60 60 60 60 60 B B B B B B B B 30 There were 30 Brownies and 90 cakes sold. 240

  35. 500 ft Geometry – • A path up the side of a 500 foot tall hill is 1000 ft. long A hiker travels 800 feet up the path. What was his change in elevation? 800 ft

  36. y 500 ft x Geometry – • A path up the side of a 500 foot tall hill is 1000 ft. long A hiker travels of 800 feet up the path. What was his change in elevation? 200 ft • The ratio of the line segments on both sides must be the same. 200 800 ft 800 y 500 ft 500/5 = 100 x 100 100 100 100 His change in elevation was 400 feet.

  37. Geometry – Practice • The triangles shown are similar. Find z. 6 4 z 9

  38. Geometry – • The triangle ABC has angles such that angle B is 3 times the measure of angle C and ½ the measure of angle A. Find the measures of angles A,B, and C. 18 C 108 54 B 18 A C 18 18 180 degrees 18 B 18 18 18 18 18 180/10 = 18 18 A

  39. Geometry – Practice • Angles A and B are complementary. Angle A is 2/3 the measure of angle B. Find the measure of angles A and B A B

  40. Sue can paint a mailbox in 2 hours. It takes Bill 3 hours to paint the same mailbox. How long will it take them to paint three of the mailboxes working together? Rate of Work Problems 1/2 Mailbox per hour Bar represents one mailbox Sue Bill 1/3 Mailbox per hour Sue and Bill can paint 5/6 of a mailbox in one hour if they work together. Both 5/6 Mailbox per hour

  41. Sue can paint a mailbox in 2 hours. It takes Bill 3 hours to paint the same mailbox. How long will it take them to paint three of the mailboxes working together? Rate of Work Problems 1 hour Sue and Bill can paint 5/6 of a mailbox in one hour if they work together. Both 12 Min 1 mailbox 12 5/6 Mailbox per hour Second Hour Third Hour First Hour 36 min

  42. Rate of Work Problems -- Practice A pro cyclist can complete a race in 2 hours. A teacher takes 4 hours to complete the same race. If they share a tandem bike, how long will it take them to complete the race pedaling together?

  43. A pro cyclist can complete a race in 2 hours. A teacher takes 4 hours to complete the same race. If they share a tandem bike, how long will it take them to complete the race pedaling together? Rate of Work Problems 1/2 race per hour Bar represents one race Pro Teacher 1/4 race per hour They can complete 3/4 of the race in one hour if they work together. Both 3/4 race per hour

  44. A pro cyclist can complete a race in 2 hours. A teacher takes 4 hours to complete the same race. If they share a tandem bike, how long will it take them to complete the race pedaling together? Rate of Work Problems -- Practice 1 hour They can complete 3/4 of the race in one hour if they work together. Both 20 Min 1 race It will take them 1 hour and 20 minutes working together. 20 ¾ race per hour One hour

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