7th and 8th Grade Mathematics

Curriculum Supports Eric Shippee

College of William and Mary

Alfreda Jernigan Norfolk Public Schools

Perimeter, Area, Volume, and Surface

Area

NCTM Focal Points In grade 6 students should:•solve problems involving area and volume

to extend grade 5 work and provide a context for equations

NCTM Focal Points In grade 7 students should:• develop an understanding of and using

formulas to determine surface areas and volumes of three-dimensional shapes

•connect their work on proportionality with their work on area and volume by investigating similar objects.

NCTM Focal Points In grade 7 students should:• develop an understanding of and using formulas to

determine surface areas and volumes of three-dimensional shapes

• Students connect their work on proportionality with their work on area and volume by investigating similar objects. They understand that if a scale factor describes how corresponding lengths in two similar objects are related, then the square of the scale factor describes how corresponding areas are related, and the cube of the scale factor describes how corresponding volumes are related.

The Perimeter is 24 Inches. What’s the Area?

Working in groups of two or three, cut off a strip of adding machine paper that is at least 24 inches long. Glue it together so that it forms a paper collar that is about 24.1 inches around. Count out 38 one-inch cubes. Using as many of these cubes as needed, construct rectangular arrays that fill the paper collar you just made. Record the dimensions of each array and record its area and perimeter. Note any patterns you find. 

The Perimeter is 24 Inches. What’s the Area?

Length Width Perimeter Area

12

65

1011 11

34

24

6789

2424

242424

2027323536

The Area is 24 Inches. What’s the Perimeter?

Working in groups of two or three, count out 24 one-inch cubes. Form these 24 cubes into a rectangular array, using all 24 cubes. Record the dimensions of each array and record its area and perimeter. Note any patterns you find.

The Area is 24 Square Inches. What’s the Perimeter?

 

Length Width Perimeter Area

12 12

24 24

3 2282850

2424

4 6 20 24

Let’s try surface area.

Scale Factor Length Width Height Surface Area in one-inch squares

Starting prism

Doubling the width

Tripling the width

Doubling the height

Triplingthe height

2 3 2 Counted 32 one-inch squares

2 6 2

2

 12 + 12 + 4 + 4 + 12 + 12= 56 one-inch squares

9 22(lw+lh+wh)

= 2(2*9+2*2+9*2) = 80 one-inch squares 

2 3 4 6 + 6 + 8 + 8 + 12 + 12= 52 one-inch squares

2(lw+lh+wh)= 2(2*3+2*6+3*6)

= 72 one-inch squares

2 3 6

Conclusion•When we doubled or tripled one measurement, it

increases the total surface area, but we did not see any constant change.

•There is no direct relationship to changing one measured attribute to its changing surface area.

Now that we have calculated surface area,let’s try volume.

Scale Factor Length Width Height Volume in one-inch cubes

Starting prism

Doubling the height

Tripling the height

Doubling the width

Triplingthe width

2 3 2 Counted 12 one-inch cubes

2 3 4

2

12 cubes + 12 cubes = 24 cubes2 layers = 2 * 12 = 24 cubes

3 612 + 12 + 12 = 36 cubes

3 layers = 3 * 12 = 36 cubes

2 6 2 12 cubes + 12 cubes = 24 cubes2 vert. layers = 2 * 12 = 24 cubes

12 + 12 + 12 = 36 cubes 3 vert. layers = 3 * 12 = 36 cubes

2 9 2

Conclusion•When we doubled or tripled one measurement it

doubled, or tripled the volume. So if we know the volume of the first prism we can multiply its volume by whatever scale factor we are increasing one measurement by to find out how much the volume of a new prism will be.

•There is a direct relationship to changing one measured attribute to its change in Volume.

What about other shapes?

    Volume = (8*6*5) divided by 3 = 80if we double any of the measurements we double the volumeVolume (16*6*5) divided by 3 = 160Volume (8*12*5) divided by 3 = 160Volume (8*6*10) divided by 3 = 160

    Surface Area = 4(1/2bh) + lw SA = 2( ½ * 6 * 5.83 ) + 2 ( ½ * 8 * 6.40 ) + 8 * 4 = 118.18If we double the length SA = 2( ½ *6* 5.83 ) + 2 ( ½ * 16* 9.43 ) + 16 * 6 = 281.86If we double the widthSA = 2( ½ * 12 * 7.8 ) + 2 ( ½ * 8 * 6.40 ) + 8 * 12 = 240.8

Height = 2 in Diameter = 1 in Volume = 1.57 cu in

Surface Area = 7.85 sq in

 If you double the height Height = 4 in Diameter = 1 in Volume = 3.14 cu in Surface Area =  14.13 sq in

Volume = r2 hSurface Area = 2( r2) + 2 r h

Height = 2 in Diameter = 2 in

Volume = 6.28 cu in

Surface Area = 18.84 sq in

If you double the diameter

Volume = r2 hSurface Area = 2( r2) + 2 r h

Area of Circles

Area of Circles

Area of Circles

Shifting Gears

7th and 8th Grade Mathematics

Curriculum Supports Eric Shippee

College of William and Mary

Alfreda Jernigan Norfolk Public Schools

SOL 7.4The student will solve single-step and multistep practical

problems, using proportional reasoning.

• Write proportions that represent equivalent relationships between two sets.

• Solve a proportion to find a missing term.

• Apply proportions to convert units of measurement between the U.S. Customary System and the metric system. Calculators may be used.

• Apply proportions to solve practical problems, including scale drawings. Scale factors shall have denominators no greater than 12 and decimals no less than tenths. Calculators may be used.

• Using 10% as a benchmark, mentally compute 5%,10%, 15%, or 20% in a practical situation such as tips, tax and discounts.

• Solve problems involving tips, tax, and discounts.

SOL 8.3• The student will solve practical problems involving rational

numbers, percents, ratios, and proportions; and determine the percent of increase or decrease for a given situation.

Proportional ReasoningDetermine whether two quantities are in aproportional relationship, e.g, by testing for

an equivalent ratios in a table or graphing in a coordinate plane (does the graph go the

through the origin) •Identify the constant proportionality (unit

rate) in table, graphs, equations, diagrams, and verbal description of proportional relationships

Grab A Handful Reach into the bag or box of color tiles and

grab one handful of color tiles!

Describe your handful of color tiles using ratios.

Determine how many of each color you would have if you grabbed another handful (exact).

What about four handfuls? What if you had …..?

Grab A Handful•How did you organize your thinking?

▫Organize your thinking in table format? ▫What do you notice? ▫How could the table help you determine

the number of each color of 8 handfuls?

Fraction Riddle Create a quadrilateral using the following

clues: ▫½ blue▫¼ yellow▫1/12 green 1/6 red

▫Remainder a color of your choice!

Fraction Riddle Create a quadrilateral using the following

clues: ▫½ green ▫⅓ red ▫Remainder a color of your choice!

Fraction Riddle Create a quadrilateral using the following

clues: •40% yellow • 2/5 green • 1/10 red•10% blue

What if…?

Fraction RiddleCreate your own riddle •Use a minimum of 16 tiles•Use at least three colors•Describe your riddle using percents and

fractions

What do we know about student thinking?

Fraction Strips How can we use fraction strips to make

sense of percents?

•Identify the fraction pieces•Identify percents•What represents 100%? •What about…….?

Transitioning to Models •Represent 100% •Show me…..10%, 20%, 5%, 1%, 50%•Using your models or fraction strips find

25% of 78

Method Drawings and Percents•Find 20% of 25.

•Find 10% of 120

•12 is what percent of 36?

•45 is what percent of 180?

Method Drawings and Percents•21 is 75% of what number?

•12 is 66. 6…% of what number?

Where is the context? Create a scenario for each of the following•Find 5% of 25. •Find 10% of 120 •12 is what percent of 36? •45 is what percent of 180?• 21 is 75% of what number? •12 is 25% of what number?

Percent of Change and Model Drawings

The price of gas rose from $1.60 per gallon in January to $2.00 per gallon in April. What was the percent increase?

$1.60

100%

Model Drawing

$1.60

100%

50%

Model Drawing

$1.60

100%

50%

Model Drawing

.80

$1.60

100%

50%

Model Drawing

.40

25%

$1.60

100%

Model Drawing$2.00

25%

.40

On your own!

Justin wants to buy a wristwatch that normally sells for $60.00. If the wristwatch goes on sale for $45.00, what is the percent decrease in the price of the wristwatch?

Making Model Drawings a Reality•What supports are needed?

•What resources are needed?

•Where do you find the questions?

Contact InformationThe power point will be available on our website http://tidewaterteam.wm.edu.

Eric Shippee: ewship@wm.eduAlfreda Jernigan ajerniga@nps.k12.va.us

Thank you and have a great day!