Transcript Getting Geometric Vocabulary To Stick

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 threedimensional 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 threedimensional 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
11
24
11
2
3
10
24
20
9
27
4
5
8
7
24
24
24
32
6
6
24
36
35
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
24
50
24
2
12
28
3
8
22
24
24
4
6
20
24
Let’s try surface area.
SOL 7.5c and 8.7b Describe how changing one measured attribute of a rectangular prism affects its volume and surface area.
Scale Factor
Starting prism
Doubling
Tripling
Quadrupling
Length Width
Height Surface Area in 1cm squares
Scale Factor
Starting prism
Doubling
Tripling
Quadrupling
Length Width
Height Surface Area in 1cm squares
Scale Factor
Starting prism
Length Width
Height Surface Area in 1cm squares
Scale Factor
Length
Width
Height
Surface Area
in one-inch squares
Starting prism
2
3
2
Counted 32 one-inch squares
Doubling
the width
2
6
2
12 + 12 + 4 + 4 + 12 + 12
= 56 one-inch squares
Tripling
the width
2
9
2
2(lw+lh+wh)
= 2(2*9+2*2+9*2)
= 80 one-inch squares
Doubling
the height
2
3
4
6 + 6 + 8 + 8 + 12 + 12
= 52 one-inch squares
6
2(lw+lh+wh)
= 2(2*3+2*6+3*6)
= 72 one-inch squares
Tripling
the height
2
3
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,
SOL 7.5c and 8.7b Describe how changing one measured attribute of a rectangular prism affects its volume and surface area.
let’s try volume.
Scale Factor
Starting prism
Doubling
Tripling
Quadrupling
Length Width
Height Volume in 1cm cubes
Scale Factor
Starting prism
Doubling
Tripling
Quadrupling
Length Width
Height Volume in 1cm cubes
Scale Factor
Starting prism
Length Width
Height Volume in 1cm cubes
Scale Factor
Length Width Height
Volume in one-inch cubes
Starting prism
2
3
2
Counted 12 one-inch cubes
Doubling
the height
2
3
4
12 cubes + 12 cubes = 24 cubes
2 layers = 2 * 12 = 24 cubes
Tripling
the height
2
3
6
12 + 12 + 12 = 36 cubes
3 layers = 3 * 12 = 36 cubes
Doubling
the width
2
6
2
12 cubes + 12 cubes = 24 cubes
2 vert. layers = 2 * 12 = 24 cubes
Tripling
the width
2
9
2
12 + 12 + 12 = 36 cubes
3 vert. layers = 3 * 12 = 36 cubes
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 = 80
if we double any of the measurements we double the
volume
Volume (16*6*5) divided by 3 = 160
Volume (8*12*5) divided by 3 = 160
Volume (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 = 118.18
If we double the length
SA = 2( ½ *6* 5.83 ) + 2 ( ½ * 16* 9.43 ) + 16 * 6 = 281.86
If we double the width
SA = 2( ½ * 12 * 7.8 ) + 2 ( ½ * 8 * 6.40 ) + 8 * 12 = 240.8
Volume =p r2 h
Surface Area = 2(p r2) + 2 p r h
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 = p r2 h
Surface Area = 2(p r2) + 2 p r h
If you double the diameter
Height = 2 in Diameter = 2 in
Volume = 6.28 cu in
Surface Area = 18.84 sq in
Area of Circles
Area of Circles
Area of Circles
Shifting Gears
SOL 7.4
The 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 a
proportional 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?
Model Drawing
$1.60
100%
Model Drawing
$1.60
50%
100%
Model Drawing
.80
$1.60
50%
100%
Model Drawing
.40
$1.60
25%
50%
100%
Model Drawing
$2.00
$1.60
100%
.40
25%
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:
Alfreda Jernigan
[email protected]
[email protected]
Thank you and have a great day!