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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
1
2
6
5
10
11 11
3
4
24
6
7
8
9
24
24
24
24
24
20
27
32
35
36
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
1
2 12
24 24
3 228
28
50
24
24
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 46 + 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 6 212 cubes + 12 cubes = 24 cubes
2 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
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.
Proportional Reasoning
Determine 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 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 Riddle
Create 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 Information
The power point will be available on our website http://tidewaterteam.wm.edu.
Eric Shippee: [email protected] Jernigan [email protected]
Thank you and have a great day!
