Demystify Challenging Problems with Bar Modeling

Gregg Velatini

Dianna Spence

2012 Georgia Mathematics Conference

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.

Rod 2

9 x 4 = 36 inches

Rod 1

72 inches

72 / 8 = 9 inchesRod 3

9

9 9 9 9

The length of the longest rod is 36 inches

9 9 9

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

Teacher

3 Parts

1 Part

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

Teacher

2 Parts

1 Part

200

The teacher had $400 at first.

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?

Female Cats

After

Before

90 of the original cats were female.

Male Cats

Female Cats 35

Male Cats 35

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?

250 Cats

30

30

30 30

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

Cookies

Cake

1

7 eggs

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?

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?

1/6

1/3

Brad spent 5/6 of his money.

Brad’s Money

1/2

Beanies Nascar

Brad had 1/6 of his money remaining.

Solving an Algebraic Equation

Six less than three times a number is fifteen. What is the number ?

15

21

The number is 7

6

7 21/3=7

Solving an Algebraic Equation--Practice

Three more than twice a number is eleven. What is the number ?

Solving a Simple Algebraic Equation

Three more than twice a number is eleven. What is the number ?

11

82x + 3 = 11

2x = 8

x = 8/2

x = 44 1 1 1

The number is 4

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.

409 lbs

32 lbs

17 lbs

Brad

John

Gregg

409 - 17 - 32 lbs = 360 lbs

32 lbs

17 lbs

Brad

John

Gregg 360 lbs / 3 = 120 lbs

John 17 lbs120 lbs John weighs 120 + 17 = 137 lbs

137 lbs

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.

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.

397

15

7

Mitt

Barack

Gary

397 - 7 - 15 = 375

15

7

Mitt

Barack

Gary 375 lbs / 3 = 125

Barack 7 125 Barack’s IQ is 125 + 7 = 132

132

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 8 oz

20 % 80 % ? %

18/80 = 22 1/2 %

The final concentration is 22 1/2 % alcohol

+

5 oz

0 %

Mixture Problems -- Practice

2 liters of 30% acid are mixed with 1 liter of 60% acid. What is the resulting acid concentration?

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

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?

+ =

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.

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?

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

+30 % ? % 60 %

3 gallons of 80% acid must be added.

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

$1.20 /lb $1.00 /lb

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

4 pounds

$0.90 /lbEach

segment represents

$0.10

$0.90

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?

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?

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

+ =

? pounds 4 pounds ? pounds

$1.20 /lb $1.10 /lb$0.90 /lb

Half Life – Radioactive Decay

A 32 pound of radioactive material decays to 4 lbs in 3000 years

Half life =3000/3 =1000 years

3000 years

Amount Remaining

Half - Life in 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

32 lbs 16 lbs 8 lbs

4 lbs …

Half - Life Half - Life

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?

half life

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?

2 hrs

Amount Remaining

96 lbs 48 lbs

24 lbs 12 lbs

3 lbs6 lbs

The half life is 120/3 min. = 40 min

It will take two more “half lifes” to get from 12 pounds to 3 pounds.

It will take 5 x 40 min. = 200 minutes to decay from 96 pounds to 3 pounds.

Decibels

Volume

atioIntensityRL log10* dBdBL 30103.32log10

Fact: A 3 dB increase is equivalent to a doubling in sound volume.*

Intensity of original sound = V

V

2V

4V

8V

16V

0 dB

3 dB

6 dB

9 dB

12 dB

Decibels

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?

Fact: A 3 dB increase is equivalent to a doubling in sound volume.*

V

2V

4V

8V

16V

0 dB

3 dB

6 dB

9 dB

12 dB

15 dB32V

15/3=5, so the volume will be doubled 5 times. 3225

A 15 dB increase results in the volume increasing by a factor of 32

Decibels

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?

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

2V

4V

8V

16V

0 dB

3 dB

6 dB

9 dB

12 dB

15 dB32V

18 dB

21 dB

24 dB

64V

128V

256V

The sound level increased by 21 dB.

System of Equations

Solve

1332

1

yx

xy

13

x x y y y

y=x+1

x x 11 x xx 1

x x x xx

10

2 2 2 22

10x=2,y=3

y

x 1y=x+1

Remove the three “1’s”

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?

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?

600

B

60

60062

BC

CB

B C C C

240 There were 30 Brownies and 90 cakes sold.

C C C

B B B B B B B B60 60 60 60 60 60

B B B B B B B B 30

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?

500 ft

800

ft

Geometry –

500 ft

x

y

800

ft

200

ft

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?

The ratio of the line segments on both sides must be the same.

800

200

x

y500 ft 500/5 = 100

100 100

His change in elevation was 400 feet.

100 100

Geometry – Practice

The triangles shown are similar. Find z.

4

z

6

9

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.

A B

C

C

B

A

180 degrees

180/10 = 18 18 18 18 18 18 18

18 18 18

18

18

54 108

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

Rate of Work Problems

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?

Sue

Bill

Both

1/3 Mailbox per hour

1/2 Mailbox per hour

Bar represents one mailbox

5/6 Mailbox per hour

Sue and Bill can paint 5/6 of a mailbox in one hour if they work together.

Rate of Work Problems

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?

Both

5/6 Mailbox per hour

Sue and Bill can paint 5/6 of a mailbox in one hour if they work together.

12 Min

1 hour

12

1 mailbox

First Hour Second Hour Third Hour 36 min

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?

Rate of Work Problems

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?

Pro

Both

1/4 race per hour

1/2 race per hour

Bar represents one race

3/4 race per hour

They can complete 3/4 of the race in one hour if they work together.

Teacher

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?

Both

¾ race per hour

They can complete 3/4 of the race in one hour if they work together.

20 Min

1 hour

1 race

One hour

It will take them 1 hour and 20 minutes working together.

20