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