Advanced Bar Modeling
Gregg Velatini
Dianna Spence2009 GCTM
Solving an Algebraic Equation
Three more than twice a number is eleven. What is the number ?
11
82x + 3 = 11
2x = 8
x = 8/2
x = 4
4 1 1 1
The number is 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
17n n n n n n n
n n n n 50
Let n = Number of Cats
7n + 17 = 4n + 50
?
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
17n n n n n n n
n n n n 33
7n + 17 = 4n + 33 + 17
3n = 33
n=11
17
5011
Mrs. Spence has 11 cats.
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
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.
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
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
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
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
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 Blue Rest 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 Yellow 3 Green
56
There are 4 x 14 = 56Green and Yellow Beads
8There are 8 x 4 = 32Yellow Beads
56/7 = 8
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
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 = 248 8 8
Clinton had 24 cards to begin with
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 gal 2 gal 3 gal
40 % 60 % ? %
16/30 = 53 1/3 %
The final concentration is 53 1/3 % acid.
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 gal2 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.
Percentages
A toy plane which originally cost $92 was sold at a loss of 25%. For how much was it sold?
Plane Cost
$92
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%
$92100%-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 = $23So¾ of $92 = $69
$69
Plane Cost
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.
Atlanta Panama City
Lydia
Nicholas
150 mi3 hrs
? Speed
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.
Atlanta Panama 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
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