1. Advanced Bar Modeling Gregg Velatini Dianna Spence 2009 GCTM

2. Solving an Algebraic Equation Three more than twice a number is eleven. What is the number ? 11 8 2x + 3 = 11 2x = 8 x = 8/2 x = 4 4111 The number is 4

3. Solving an Algebraic Equation Mrs. Spence has a bag of treats for her cats. If she gives each cat 7 treats, she will have 17 treats left. If she gives each cat 4 treats, she will have 50 treats left. How many cats does Mrs. Spence own? 7 treats 4 treats 17 nn nn nnn nnnn 50 Let n = Number of Cats 7n + 17 = 4n + 50 ?

4. Solving an Algebraic Equation Mrs. Spence has a bag of treats for her cats. If she gives each cat 7 treats, she will have 17 treats left. If she gives each cat 4 treats, she will have 50 treats left. How many cats does Mrs. Spence own? 7 treats 4 treats 17 nn nn nnn nnnn 33 7n + 17 = 4n + 33 + 17 3n = 33 n=11 17 50 11 Mrs. Spence has 11 cats.

5. Solving an Algebraic Equation The average weight of Peter, Paul and Mary is 80 lbs. Paul is twice as heavy as Mary. Peter is 10 lbs lighter than Paul. Find Paul’s weight. Mary Paul Peter 10 lbs 80 x 3 = 240 lbs Re-Draw Let M = Mary’s weight W = Paul’s weight P = Peter’s weight W = 2M P = W – 10 P+W+M = 240

6. The average weight of Peter, Paul and Mary is 80 lbs. Paul is twice as heavy as Mary. Peter is 10 lbs lighter than Paul. Find Paul’s weight. Mary Paul Peter 10 lbs 80 x 3 = 240 lbs Mary Paul Peter 10 lbs 80 x 3 = 240 lbs + 10 = 250 lbs 250/5 = 50 lbs 50 lbs 100 lbs Let M = Mary’s weight W = Paul’s weight P = Peter’s weight P+W+M = 240 W = 2M P = W – 10 5M -10 =240 5M = 250 M=50 W=2M ? Paul weighs 100 lbs.

7. Solving Fraction Equations Maria won a cash prize. She spent 3/8 of it on washing machine, 2/5 of it on a refrigerator, and $180 on a fan. If she had $270 left, how much was the prize? Maria’s Prize

8. Solving Fraction Equations Maria won a cash prize. She spent 3/8 of it on washing machine, 2/5 of it on a refrigerator, and $180 on a fan. If she had $270 left, how much was the prize? Maria’s Prize 1/8

9. Solving Fraction Equations Maria won a cash prize. She spent 3/8 of it on washing machine, 2/5 of it on a refrigerator, and $180 on a fan. If she had $270 left, how much was the prize? Maria’s Prize 1/5 1/8

10. Solving Fraction Equations Maria won a cash prize. She spent 3/8 of it on washing machine, 2/5 of it on a refrigerator, and $180 on a fan. If she had $270 left, how much was the prize? Maria’s Prize 1/5 1/8 Washing Machine Refrigerator $180 + $270 = $450 $450 $450/9 =$50 $50 Total prize = 40 x $50 = $2000

11. Solving Fraction Equations In a jar filled with beads, 2/5 of the beads are blue, 1/3 of them are red, and the rest are green and yellow. The total number of red, green and yellow beads is 126. There are ¾ as many green beads as there are yellow beads. How many yellow beads are there? Beads 1/3 Red2/5 BlueRest Green and Yellow 126 Red, Green, and Yellow 14 126/9 = 14 “There are ¾ as many green beads as there are yellow beads.” 4 Yellow3 Green 56 There are 4 x 14 = 56 Green and Yellow Beads 8 There are 8 x 4 = 32 Yellow Beads 56/7 = 8

12. Ratios and Proportions The ratio of Clinton’s baseball cards to Jesse’s baseball cards was 3:4. After Clinton bought another 40 baseball cards, he had twice as many baseball cards as Jesse. How many baseball cards did Clinton have at first? Clinton Jesse 3 Parts 4 Parts

13. Ratios and Proportions Set this up as a “Before and After” problem. Clinton Jesse Clinton Jesse After 3 Parts 4 Parts Before 40 Cards 2 Parts 1 Part 8 40/5 = 8 8 x 3 = 24 88 8 Clinton had 24 cards to begin with

14. Ratios and Proportions If you mix 1 gal of 40% acid solution with 2 gal of 60% acid solution, what is the resulting acid concentration?. += 1 gal2 gal3 gal 40 %60 % ? % 16/30 = 53 1 / 3 % The final concentration is 53 1/3 % acid.

15. Ratios and Proportions 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? += 1 gal ? gal 3 gal 60 % ? % 80 % 3 gal -1 gal = 2 gal 2 gal 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.

16. Percentages A toy plane which originally cost $92 was sold at a loss of 25%. For how much was it sold? Plane Cost $92

17. Percentages A toy plane which originally cost $92 was sold at a loss of 25%. For how much was it sold? Plane Cost 25% 75% $92 100%-25% = 75% ? 0.75 x $92 = $69 $69 The plane was sold for $69. Alternative: Tie percentages to fractions. 25% = 1/4 $92 1/4 ? ¼ of $92 = $23 So ¾ of $92 = $69 $69 Plane Cost

18. Time, Rate, Distance Lydia and Nicholas left Atlanta, GA at the same time. Both were headed for Panama City, FL and each drove at a uniform speed for the entire trip. When Lydia reached Panama City, Nicholas was still 150 miles away. After 3 hours, Nicholas reached Panama City. If the two cities are 375 mi. apart, what speed was Lydia traveling? 375 mi. AtlantaPanama City Lydia Nicholas 150 mi 3 hrs ? Speed

19. The bar model here has become more abstract. It is used more as an organizational and conceptual tool, and less as a computational tool. 375 mi. AtlantaPanama City Lydia Nicholas 150 mi, 3 hrs ? Speed Nicholas’s Speed = 150 mi ÷ 3hrs= 50 mph Nicholas’s time = 375 mi ÷ 50 mph = 7.5 hrs Lydia’s time = 7.5 hrs – 3 hrs = 4.5 hrs Lydia’s speed = 375 mi ÷ 4.5 hrs = 83 1/3 mph Lydia’s was traveling at a rate of 83 1/3 mph 83.1/3 mph