2m
Perimeter
and Area
5 Day Unit for the 7th Grade
Graph Paper
Geoboards
Geometer’s Sketchpad
Tangrams
TI-83+ Graphing Calculator
Kimberly Witwer
1.4 m
Table of Contents
Objectives for the Unit
Standards for the Unit
Materials for the Unit
Overview of Unit
Resources for the Unit
Day #1: Graph Paper and the Internet
Area and Perimeter Packet
Day #2: Geoboards
Overhead situations
Overhead chart for perimeter and area
Perimeter and Area With Geoboards worksheet
Geoboard Paper
Homework
Day #3: Geometer’s Sketchpad
Area and Perimeter Using GSP worksheet
Design a House Project paper
Day #4: Tangrams
Area and Tangrams packet
Day #5: TI-83+ Graphing Calculator
Exploring Area packet
Overhead of introduction problem
Final Assessment: Design a House Project
Design a House Project paper
Pages
3
4
5
6
7
8-14
11-14
15-24
17-18
19
20-21
22
23-24
25-29
27-28
29
30-36
33-36
37-47
41-46
47
48-49
49
Witwer – Page 2
Objectives for the Unit:
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
11.
12.
13.
14.
15.
16.
17.
Students will be able to find the area and perimeter of squares and rectangles.
Students will be able to devise a formula for the area of squares and rectangles.
Students will be able to devise a formula for the perimeter of any shape.
Students will be able to make shapes that have a given area.
Students will be able to make shapes that have a given perimeter.
Students will be able to find fractional areas.
Students will be able to find the maximum area of a rectangle given a
perimeter.
Students will be able to design a one-floor house using their knowledge of area
and perimeter.
Students will be able to differentiate between area and perimeter.
Students will be able to use area and perimeter in real life situations.
Students will use appropriate units of measurement.
Students will determine the area and the perimeter of shapes by using
geoboards.
Students will use Geometer’s Sketchpad to determine area and perimeter.
Students will apply knowledge of area and perimeter to answer questions in a
game on the Internet.
Students will compare areas through the use of tangrams.
Students will be able to make a scatterplot on the TI-83+ Graphing Calculator.
Students will find the maximum area of a rectangular shape using the TI-83+
Graphing Calculator.
Witwer – Page 3
Standards for the Unit:
New York State Key Ideas Covered:
1A- Students will use patterns to develop formulas for the area and perimeter
of rectangles and squares.
1B- Students will make and evaluate their conjectures about the formulas for
the area and perimeter of rectangles and squares.
1C- Students will devise formulas for the perimeter and area of rectangles
and squares.
4B- Students will select appropriate units of measure for finding the area and
perimeter of rectangles and squares.
4C- Students will use geoboards to further develop the ideas of area and
perimeter.
4F- Students will model situations when solving the problem with
determining the maximum area of a garden.
5A- Students will use measurement in the everyday situation of a fence
around a garden.
5B- Students will use appropriate units of measure for perimeter and area.
5C- Students will know and apply the formulas for how to find the area and
perimeter of polygons (squares and rectangles).
5E- Students will use graphing calculators to organize and analyze data.
7B- Students will represent geometric patterns using graphing calculators.
NCTM Standards Covered:
Geometry- Students will understand the relationships among side lengths and
area and perimeter.
Measurement- Students will use appropriate units to measure area and
perimeter.
Reasoning and Proof- Students will make and investigate their own
conjectures about how to find the area and perimeter of squares and
rectangles.
Communication- Students will communicate their mathematical thinking
coherently to others as they work with partners, work in groups, write
responses to questions, and speak to the teacher or class as a whole.
Representation- Students will represent area and perimeter by using
geoboards and graphing calculators.
Witwer – Page 4
Materials for the Unit
Student’s Textbooks: The University of Chicago School Mathematics
Project Transition Mathematics (1998)
Pencils
Overhead of graph paper
Overhead markers
Overhead square and rectangle
Overhead projector
Square and rectangular shapes with different areas and perimeters
Graph paper
Area and perimeter packets
Computers with the internet
Geoboards
Rubber bands
Overhead pieces of situations
Overhead chart for perimeter and area
Perimeter and area with geoboards worksheets
Overhead geoboard
Homework worksheets
Computers with Geometer’s Sketchpad (GSP)
Area and perimeter using GSP worksheets
Design a house project papers
Large sheets of graph paper
Tangrams labeled 1-7
Area and tangrams packets
Overhead tangrams
Overhead of area and tangrams packet
TI-83+ Calculator
Teacher’s TI-83+ Calculator with overhead connection
20 Popsicle sticks for each student
Exploring area packet
Overhead of introduction problem
Witwer – Page 5
Perimeter and Area Overview
5 Day Unit for the 7th Grade
Day #1: Graph Paper and the Internet
® Students will find the areas and perimeters of squares and rectangles by counting
squares on graph paper.
® Students will work in groups to discover the relationships between the length and
width of a shape and the shape’s area and perimeter.
® Students will play a game called “Area and Perimeter” on the Internet at
http://www.cut-the-knot.com/games/shtml.
Day #2: Geoboards
® Students will classify real-life situations by determining which are related to finding
the perimeter and which are related to finding the area.
® Students will use geoboards to find rectangles and squares that have a given
perimeter and record the shapes’ areas.
® Students will use geoboards to find rectangles and squares that have a given area
and record the shapes’ perimeters.
® Students will use their findings to devise a formula to find the area of a square or
rectangle and determine whether or not two shapes can have the same area but
different perimeters and whether or not two shapes can have the same perimeter but
different areas.
Day #3: Geometer’s Sketchpad
® Students will use Geometer’s Sketchpad to calculate the perimeter and area of
rectangles and squares they create.
® Students will verify that the areas and perimeters are correct by adding and
multiplying to find the areas and perimeters.
® Students will modify the rectangles and squares they create by dragging one side of
the shape and determine if the perimeters and areas changed as they expected.
Day #4: Tangrams
® Students will compare areas by using tangrams. For instance, the students are told
that one piece is 1 square unit and then they must determine the area of the other
pieces by relating them to the piece with the given area of 1 unit2.
® Students will find fractional areas by using all the tangrams of the entire square as
one square unit and then finding the areas of each individual piece.
Day #5: TI-83+ Graphing Calculator
® Students will use 20 craft sticks to design a fence for a garden.
® Students will use graphing calculators to enter in the possible widths and determine
the possible lengths of the fence and then make a scatterplot to show the
relationship between the width of the garden and the area of it.
® Students will determine the maximum area of a given perimeter.
Assessment: Design a House Project, Homework, Classwork
® Students will design a one-floor home as their final assessment project.
® Students will complete homework and worksheets in class to serve as further
assessment.
Witwer – Page 6
Resources:
Feldman, Cathy, et. al. (1998) Transition Mathematics. Glenview, IL:
Scott, Foresman and Company.
http://www.cut-the-knot.com/games/shtml
http://www.education.ti.com
Witwer – Page 7
Day #1: Graph Paper and the Internet
NAME: Kimberly Witwer
GRADE LEVEL: Seventh
I.
ACTIVITY:
Seventh grade students will find the area of rectangles and squares by
counting squares on graph paper.
Seventh grade students will find the perimeter of rectangles and squares
by counting squares on graph paper.
Seventh grade students will devise formulas for the area and the perimeter
of rectangles and squares.
Seventh grade students will find areas and perimeters to play a game on
the Internet.
II.
STANDARDS:
New York State Key Ideas Covered: 1A, 1B, 1C, 4B, 4C, 5B, 5C
NCTM Standards Covered: Geometry, Measurement, Reasoning and
Proof, Communication, Representation
III.
MATERIALS:
Pencils
Overhead of graph paper
Overhead markers
Overhead square and rectangle
Overhead Projector
Square and rectangular shapes with different areas and perimeters
Graph paper
Area and perimeter packets
Computers with the Internet
IV.
OPENING ACTIVITY:
The students will be seated in their desks. I will put a square shape on the
overhead and ask the students to describe it to me. Then I will ask them to
tell me the size of the square. I will ask the students to tell me how we
could find the exact size of the square. I will ask them if we could use
graph paper while placing graph paper on the overhead project and what
each square could represent (1 square unit). Then I will place a
transparent square shape on the overhead so the students can see the units
through the shape. I will ask if anyone remembers what the perimeter of
the shape is (the distance around the shape) and if anyone remembers what
the area of the shape is (how many square units are in the shape). Then I
will ask for a volunteer to find the perimeter of the square (12 units) and to
explain how he/she found it (counted the boxes around the shape). I will
tell the class that the area of the square is 9 square units and ask for a
volunteer to explain how I know the area is 9 square units (counted the
boxes inside the shape). I will ask the students to see if they can figure out
why the area is in square units by remembering the square and looking at
the next example of a rectangle. I will put up a transparent rectangle and
ask the students to find the perimeter and area of the rectangle and explain
how they got their answer. (perimeter is 16 units- counted boxes around
the shape, area is 15 square units- counted boxes inside the shape). I will
ask the students to tell me why the area is in square units (describes the
space it fills, not just a line). I will explain that a square is twodimensional.
I will ask the students to work in their groups of four to complete a packet
that will help them discover more about perimeter and area. I will explain
that we are going to look at the areas and perimeters of square and
Witwer – Page 8
rectangular shapes only today. I will explain that when the students have
finished, they are to go to the computers and play the game on the
computers (Area and Perimeter) at http://www.cut-theknot.com/games.shtml. I will explain to the class that the game will be set
up for them to begin and if they need help to please raise their hands.
Also, I will explain that this game will be available to the students when
they have finished their work throughout the next week.
V.
MAIN ACTIVITY:
I will have the students (in their groups of four) work on the packets by
tracing the shapes given to them and finding the perimeters and areas of
each shape. I will ask that each student has a completed packet and piece
of graph paper with the shapes traced on it by the time class ends today.
I will assess the students by hearing the groups’ responses. I will move
about the room and look at each individual student’s paper to see if he/she
understands perimeter and area and I will collect the papers to do an
informal assessment (not for a grade). I will also assess students by
calling on different students to volunteer answers to questions.
VI.
CLOSING ACTIVITY:
The students will be in their seats and I will ask them about the different
squares and rectangles they worked with in their groups. I will ask for a
volunteer to share his/her response to number 4 of Part #3, the relationship
between perimeter and length and width of a shape (perimeter is adding all
the sides of the length and width together). I will ask for a volunteer to
share his/her response to number 7 of Part #3, the relationship between the
area and length and width of the rectangle (area is length times width). I
will ask the students if these relationships are always true (yes).
VII.
HOMEWORK:
I will assign homework in the students’ textbook The University of
Chicago School Mathematics Project Transition Mathematics (1998) as
pages 160-161, problems 1-4 and 9-11.
Witwer – Page 9
ANSWER KEY TO HOMEWORK
The University of Chicago School Mathematics Project Transition Mathematics
(1998)
Pages 160-161
1-4 and 9-11
Please see answers on the Teacher’s Edition of the textbook.
Witwer – Page 10
Name__________________________
Part 1:
In this section you will use the graph paper and the shapes provided to explore the area
and perimeter of each shape. (Shapes are numbered 1-5). Trace each shape onto your
graph paper and compute the area and perimeter for each.
SHAPE #1
1. Identify this shape:____________________
2. The perimeter is _________________units
3. The area is ___________________ square units
SHAPE #2
4. Identify this shape:____________________
5. The perimeter is _________________units
6. The area is ___________________ square units
SHAPE #3
7. Identify this shape:____________________
8. The perimeter is _________________units
9. The area is ___________________ square units
SHAPE #4
10. Identify this shape:____________________
11. The perimeter is _________________units
12. The area is ___________________ square units
SHAPE #5
13. Identify this shape:____________________
14. The perimeter is _________________units
15. The area is ___________________ square units
Witwer – Page 11
Part 2:
In this section you will use your graph paper to draw shapes with given areas and
perimeters. For each question, label the shape with the number, the area, and the
perimeter.
SHAPE #1 Draw a square
SHAPE #2: Draw a rectangle
Perimeter of 8 units
Perimeter of 12 units
Area of 4 square units
Area of 8 square units
SHAPE #3:
Draw a rectangle
Perimeter of 14 units
Area of 12 square units
SHAPE #4:
Draw a square
Perimeter of 12 units
Area of 9 square units
Part #3:
In this section, you will answer the question and discover the formula for area and
perimeter.
Look at the rectangle to the right.
1. What are the two lengths? ____________ in. and ____________ in.
2. What are the two widths? ____________ in. and ____________ in.
3. What is the perimeter? ______________ in.
4. What is the relationship between perimeter and length and width of the shape?
______________________________________________________________
______________________________________________________________
5. Does this relationship hold true for all rectangles? What about squares? Explore
this idea by looking back at the axample you have completed.
______________________________________________________________
______________________________________________________________
6. What is the area of above shape? ____________________ square in
7. What is the relationship between area and length and width of the rectangle?
______________________________________________________________
______________________________________________________________
8. What would the perimeter and area of a rectangle be if the length is 82cm and the
width is 5cm? Perimeter = __________________ Area = __________________
Witwer – Page 12
Name
Key
Part 1:
In this section you will use the graph paper and the shapes provided to explore the area
and perimeter of each shape. (Shapes are numbered 1-5). Trace each shape onto your
graph paper and compute the area and perimeter for each.
SHAPE #1
Square (2 x 2)
1. Identify this shape:____________________
2. The perimeter is _________________units
8
3. The area is ___________________
square units
4
SHAPE #2
4. Identify this shape:____________________
Rectangle (10 x 3 )
5. The perimeter is _________________units
26
6. The area is ___________________
square units
30
SHAPE #3
7. Identify this shape:____________________
Rectangle (5 x 6)
8. The perimeter is _________________units
22
9. The area is ___________________
square units
30
SHAPE #4
10. Identify this shape:____________________
Rectangle (15 x 2)
11. The perimeter is _________________units
34
12. The area is ___________________
square units
30
SHAPE #5
13. Identify this shape:____________________
Rectangle (8 x 9)
14. The perimeter is _________________units
34
15. The area is ___________________ square units
72
Witwer – Page 13
Look at the rectangle to the right.
5
5
1. What are the two lengths? ____________
in. and ____________
in.
2
2
2. What are the two widths? ____________
in. and ____________
in.
3. What is the perimeter? ______________
in.
14
4. What is the relationship between perimeter and length and width of the shape?
Answers will vary: Perimeter is adding all the sides together
______________________________________________________________
______________________________________________________________
5. Does this relationship hold true for all rectangles? What about squares? Explore
this idea by looking back at the axample you have completed.
______________________________________________________________
Yes, the perimeter is always found by adding all the sides together
______________________________________________________________
6. What is the area of above shape? ____________________
10
square in
7. What is the relationship between area and length and width of the rectangle?
Answers will vary: Area is length times width
______________________________________________________________
______________________________________________________________
8. What would the perimeter and area of a rectangle be if the length is 82cm and the
174 cm
width is 5cm? Perimeter = __________________
Area = __________________
410 cm2
Witwer – Page 14
Day #2: Geoboards
NAME: Kimberly Witwer
GRADE LEVEL: Seventh
I.
ACTIVITY:
Seventh grade students will classify real-life situations by determining
which are related to finding the perimeter and which are related to finding
the area on the overhead projector.
Seventh grade students will use geoboards to find rectangles and squares
that have a given perimeter or a given area. .
Seventh grade students will be ale to find the area and perimeter of
rectangles and squares given their dimensions.
II.
STANDARDS:
New York State Key Ideas Covered: 1A, 1C, 4F, 5A, 5B
NCTM Standards Covered: Geometry, Measurement, Reasoning and
Proof, Communication, Representation
III.
MATERIALS:
Geoboards
Rubber Bands
Geoboard Paper
Pencils
Overhead pieces of situations
Overhead chart for perimeter and area
Perimeter and Area With Geoboards worksheets
Overhead Geoboard
Homework Worksheets
IV.
OPENING ACTIVITY:
The students will be seated in their desks. I will put the situations below
on the overhead and ask the students to classify them into lists by which
relate to finding perimeter and which relate to finding area:
How much fencing is needed to go around your pool?
How many plants can you fit into your garden?
How many tiles do you need to cover the floor of your patio?
How much wood do you need to make a picture frame?
How long of a gutter do you need to go around the roof of your
house?
How much carpeting do you need to cover your bedroom floor?
I will have the students come up and move the questions into the
appropriate columns on the “T chart” for area and perimeter. Once they
have organized the events, we will discuss why they choose each one to go
into the area or perimeter column. Then, I will explain that each student
will work with his/her partner (assigned earlier in the year- based on
students who work well together) to complete the day’s activities.
V.
MAIN ACTIVITY:
I will give each student a geoboard and some rubber bands. I will show
them by modeling on the overhead what one unit is on the geoboard and
how to find the area and the perimeter of a shape. (This will be review
from Day 1). I will then ask them to work with their partners to complete
the Perimeter and Area With Geoboards worksheet by finding some
squares or rectangles that have a perimeter of the given values.
Witwer – Page 15
I will assess the students by hearing the groups’ responses. I will move
about the room and look at each individual student’s paper to see if he/she
understands the perimeters and areas and the relationship between the area
and the lengths and widths of the shapes. I will collect the papers to do an
informal assessment (not for a grade). I will also assess students by
calling on different students to volunteer answers to questions.
VI.
CLOSING ACTIVITY:
The students will be in their seats and I will ask them about the different
perimeters on the worksheet. I will ask for volunteers to give us examples
of the areas they came up with from the given perimeters. Then I will ask
if the students found a relationship between the length and width of a
square or rectangle and its area. (A=L*W). I will also ask for someone to
give an answer to the question about two shapes having the same area but
different perimeters and the question about two shapes having the same
perimeter but different areas.
VII.
HOMEWORK:
I will assign homework as a worksheet with three shapes to find the area
of and a word problem involving area. (Please see attached worksheet.)
Witwer – Page 16
How much fencing is
needed to go around
your pool?
How many plants
can you fit into
your garden?
How many tiles
do you need to
cover the floor
of your patio?
How much wood
do you need
to make a
picture frame?
How long of
a gutter do you need
to go around the
roof of your house?
Witwer – Page 17
How much carpeting
do you need to
cover your
bedroom floor?
Witwer – Page 18
Area
Perimeter
Witwer – Page 19
Name ___________________
Date ____________________
Perimeter and Area
With Geoboards
Activity:
Using the geoboard, design different squares or rectangles with a perimeter
of 10.
Record these on the dot paper.
Now use the geoboard again to design more squares andrectangles with the
following perimeters: 16, 20, and 24. Find the areas for each shape. If
you finish earlier, try other perimeters. Record the perimeters and the
areas below.
Perimeter of 16
Perimeter of 20
Perimeter of 24
Area: ______
Area: ______
Area: ______
Area: ______
Area: ______
Area: _______
Area: _______
Area: _______
Area: _______
Area: _______
Area: ______
Area: ______
Area: ______
Area: ______
Area: ______
What patterns did you notice when the perimeters stayed the same and the areas changed?
Can you devise a formula from the information above to find the area of a square and
rectangle?
Is it possible for a shape to have the same area and perimeter?
Can to shapes have the same area but differenet perimeters?
Can two shapes have the same perimeter but different areas?
Witwer – Page 20
Key
Name ___________________
Date ____________________
Perimeter and Area
With Geoboards
Activity:
Using the geoboard, design different squares or rectangles with a perimeter
of 10.
Record these on the dot paper.
Now use the geoboard again to design more squares andrectangles with the
following perimeters: 16, 20, and 24. Find the areas for each shape. If
you finish earlier, try other perimeters. Record the perimeters and the
areas below.
* ANSWERS WILL VARY IN THE CHART*
Perimeter of 16
Perimeter of 20
Perimeter of 24
Area: ______
Area: ______
Area: ______
Area: ______
Area: ______
Area: _______
Area: _______
Area: _______
Area: _______
Area: _______
Area: ______
Area: ______
Area: ______
Area: ______
Area: ______
What patterns did you notice when the perimeters stayed the same and the areas changed?
You can multiply the length and the width to get the area
Can you devise a formula from the information above to find the area of a square and
rectangle?
A=L*W
Is it possible for a shape to have the same area and perimeter?
Yes – The numbers can be the same (a square with a length of 4 has an
area of 16 units2 and a perimeter of 16 units2)
No – The units are always different
Can to shapes have the same area but differenet perimeters?
Yes – for example two rectangles with area of 20 could be 4 x 5 with
perimeter of 18, or 10 x 2 with perimeter 24
Can two shapes have the same perimeter but different areas?
Yes – for example two rectangles with a perimeter of 30 could be 10 x 5
with area of 50, or 12 x 3 with area 36.
Witwer – Page 21
Witwer – Page 22
Name___________________________
Find the area. Show your work and answers in the space provided.
1.
15 mm
15 mm
2.
2m
1.4 m
3.
5 in
2 in
4 in
2 in
Solve the following problem. Draw diagrams if necessary. Show your work
and answer in the space provided.
4. Richard is inputting a page for the school newspaper on his computer.
The useable page area measures 7 inches by 9 inches. He has art that
measures 3 inches by 5 inches. How much of the page does he have left
for text?
Witwer – Page 23
Answer Key
1.
Area = length x width
A = 15 mm x 15 mm
A = 225 mm2
2.
Area = length x width
A = 2 m x 1.4 m
A = 2.8 m2
3.
Area = length x width
A1 = 3 in x 2 in
A1 = 6 in2
A2 = 2 in x 4 in
A2 = 8 in2
Atotal = A1 + A2
Atotal = 6 in2 + 8 in2
Atotal = 14 in2
4.
Area = length x width
Atyping = 7 in x 9 in
Atyping = 63 in2
Aart = 3 in x 5 in
Aart = 15 in2
Atotal left = Atyping - Aart
Atotal left = 63 in2 – 15 in2
Atotal left = 48 in2
Witwer – Page 24
Day #3: Geometer’s Sketchpad
NAME: Kimberly Witwer
GRADE LEVEL: Seventh
I.
ACTIVITY:
Seventh grade students will find the perimeter and area of a square given
the length of one side.
Seventh grade students will find the perimeter and area of a rectangle
given its dimensions.
Seventh grade students will use Geometer’s Sketchpad to determine the
area and perimeter of squares and rectangles.
II.
STANDARDS:
New York State Key Ideas Covered: 1A, 4B, 4C, 5C
NCTM Standards Covered: Geometry, Measurement, Reasoning and
Proof, Representation
III.
MATERIALS:
Computers with Geometer’s Sketchpad (GSP)
Area and Perimeter Using GSP worksheets
Pencils
Design a House Project papers
Large sheets of graph paper
IV.
OPENING ACTIVITY:
The students will be seated at computers in the computer lab. I will ask
them to remember the definition of perimeter and how it is determined
(the distance around an object determined by adding the lengths of the
sides of the object). Students will also be asked what area is (the amount
of space covered by a figure or object). I will ask the students how you
can find the area of a rectangle (use the formula A=L*W).
V.
MAIN ACTIVITY:
The students will be at their own computer with GSP on it. I will ask
them to follow the directions on the Area and Perimeter Using GSP
worksheets.
I will assess the students by collecting the homework from the night
before (worksheet with three shapes and word problem) for a grade. I will
move about the room and look at each individual student’s paper and
computer screen to see if he/she understands area and perimeter. I will
collect the papers to do an informal assessment (not for a grade) and I will
call on different students to volunteer answers to questions.
VI.
CLOSING ACTIVITY:
The students will be seated at their computers and I will ask for volunteers
to share whether the things they found on GSP were consistent with what
they expected.
I will explain to the students that they will be designing a one-floor house
and I will hand out the project description (Design a House Project
papers). I will explain that it will be due the following Friday so they have
a week and a half to complete it. We will read it aloud together and I will
answer any questions the students have. If time allows, the students will
begin working on their project.
Witwer – Page 25
VII.
HOMEWORK:
Work on Design a House Project- come with any questions about it
tomorrow.
Witwer – Page 26
Name
Date
Area and Perimeter
Using GSP
Follow the directions below to answer the questions about area and perimeter by using
GSP.
1. Create a rectangle and measure the length and width. Calculate the perimeter
using the calculator feature and through two methods, adding all sides and
confirming the 2 times length plus 2 times width method. List your
measurements below.
Length _____________
Width ______________
Perimeter _____________
Try dragging any one side of the rectangle to see if the perimeter formula and the
addition of all sides still agree.
2. Now create a square and show that the perimeter is 4 times the length of any one
side. List your measurements below.
Length of one side ______________
Perimeter _____________
3. Draw a rectangle on the screen. Find the area by measuring the length and width.
Then multiply the measures to find the area. Record your information below.
Length _____________
Width ______________
Area ________________
At this point I will demonstrate to the students how line segments can be selected and the
rectangle can be filled and area determined in this method. It should be shown that both
methods reveal the same answer. Which method do you prefer and why?
4. Now you can create a rectangle and confirm the area by calculation and by filling
the polygon. List your data below.
Length _____________
Width ______________
Calculated Area __________
Filling Area _____________
Witwer – Page 27
Name
Date
Key
Area and Perimeter
Using GSP
Follow the directions below to answer the questions about area and perimeter by using
GSP.
1. Create a rectangle and measure the length and width. Calculate the perimeter
using the calculator feature and through two methods, adding all sides and
confirming the 2 times length plus 2 times width method. List your
measurements below.
Length _____________
Width ______________
Perimeter _____________
Try dragging any one side of the rectangle to see if the perimeter formula and the
addition of all sides still agree.
2. Now create a square and show that the perimeter is 4 times the length of any one
side. List your measurements below.
Length of one side ______________
Perimeter _____________
3. Draw a rectangle on the screen. Find the area by measuring the length and width.
Then multiply the measures to find the area. Record your information below.
Length _____________
Width ______________
Area ________________
At this point I will demonstrate to the students how line segments can be selected and the
rectangle can be filled and area determined in this method. It should be shown that both
methods reveal the same answer. Which method do you prefer and why?
4. Now you can create a rectangle and confirm the area by calculation and by filling
the polygon. List your data below.
Length _____________
Width ______________
Calculated Area __________
Filling Area _____________
* All answers will vary depending on what size shape the student creates. *
Witwer – Page 28
Name _________________
Due Date______________
Design a House Project
You are to design a one-floor house plan that includes the following:
2 bedrooms, 1 bathroom, 1 kitchen, 1 living room, 2 exits
Use the large sheets of graph paper provided to design your house.
Choose your scale and write it on the design.
The total area for your house must be between 150 - 200 square meters.
The following information should be attached to your house plans:
___ Total area of house (15 points)
___ Outside perimeter of house (15 points)
___ Area of bedroom one ( 5 points)
___ Perimeter of bedroom one ( 5 points)
___ Area of bedroom two ( 5 points)
___ Perimeter of bedroom two ( 5 points)
___ Area of bathroom ( 5 points)
___ Perimeter of bathroom ( 5 points)
___ Area of kitchen ( 5 points)
___ Perimeter of kitchen ( 5 points)
___ Area of living room ( 5 points)
___ Perimeter of living room ( 5 points)
___ 2 exits are clearly marked ( 10 points)
___ Neatness ( 10 points)
You may find the following websites helpful as you complete your plan:
www.coolhouseplans.com
www.dreamhomesource.com
www.houseplanguys.com
HAVE FUN!!!
Witwer – Page 29
Day #4: Tangrams
NAME: Kimberly Witwer
GRADE LEVEL: Seventh
I.
ACTIVITY:
Seventh grade students will compare areas through the use of tangrams.
Seventh grade students will find fractional areas by using tangrams.
Seventh grade students will find the area of 6 tangram pieces given one
piece’s area.
II.
STANDARDS:
New York State Key Ideas Covered: 1A, 4B, 4C, 5C
NCTM Standards Covered: Geometry, Measurement, Reasoning and
Proof, Communication, Representation
III.
MATERIALS:
Tangrams labeled 1-7
Area and Tangrams packets
Pencils
Overhead tangrams
Overhead of Area and Tangrams packet
IV.
OPENING ACTIVITY:
The students will be seated in their desks. I will hand out the tangram
pieces and allow students some time to try to form a square with the seven
pieces. Then I will have a volunteer come to the overhead and show how
to make the square using all the pieces.
V.
MAIN ACTIVITY:
The class will work together to solve the following area problem:
The tangram piece labeled #5 is 1sqaure unit. Find the area of each
of the other pieces.
I will encourage students to lay pieces on other pieces to see how they are
related. For example two #3 pieces will make up the #4 pieces, therefore,
the area of piece #3 is 1/2 square unit. We will continue working together
until the chart is completed:
I will then hand out the Area and Tangrams packets to each student. I will
ask the students to work on the packet alone and then if they need to talk
with their partner, they will have a chance a bit later in the lesson.
Witwer – Page 30
I will assess the students by moving about the room and looking at each
individual student’s paper to see if he/she understands the areas of the
tangrams and I will collect the papers to do an informal assessment (not
for a grade). Also, I will call on different students to volunteer answers to
questions.
VI.
CLOSING ACTIVITY:
The students will review their packets with their partners. Then we will
come together as a group and I will ask for volunteers to share their
responses to the packet. We will go over the entire packet together as a
group while using the overhead projector.
I will ask if there are any questions on the Design a House Project and
how the students are doing with the project.
VII.
HOMEWORK:
I will assign homework in the students’ textbook The University of
Chicago School Mathematics Project Transition Mathematics (1998) as
page 210, problems 1-8.
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ANSWER KEY TO HOMEWORK
The University of Chicago School Mathematics Project Transition Mathematics
(1998)
Page 210
1-8
Please see answers on the Teacher’s Edition of the textbook.
Witwer – Page 32
Name _____________________________
Area and Tangrams
Task: Below is one set of tangrams, you also have one set punched out for you. Notice
that the pieces are numbered. Complete the following table using your knowledge of
area and tangrams. One row has been completed for you as an example.
Number of Pieces
Name of Piece
1
2
3
4
5
6
7
Right Isosceles Triangle
Area of Piece
(square units)
4
Part II:
Explain a strategy to determine how you found the area for piece #4.
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Part III:
Suppose that the area of the entire square is one square unit, fill in the chart below so that
each piece has the correct area. Notice again that one has been done for you.
Number of piece
1
2
3
4
5
6
7
Cost of piece
1/16
Witwer – Page 34
Key
Name _____________________________
Area and Tangrams
Task: Below is one set of tangrams, you also have one set punched out for you. Notice
that the pieces are numbered. Complete the following table using your knowledge of
area and tangrams. One row has been completed for you as an example.
Number of Pieces
Name of Piece
1
2
3
4
5
6
7
Right Isosceles Triangle
Right Isosceles Triangle
Right Isosceles Triangle
Square
Right Isosceles Triangle
Parallelogram
Area of Piece
(square units)
4
4
1
2
1
2
2
Part II:
Explain a strategy to determine how you found the area for piece #4.
Answers will vary: I found that piece #5 was one square unit, so I took that piece (#5)
and saw how many fit onto piece #4. Two piece #5 fit, so the area of piece #4 is 2 square
units.
Witwer – Page 35
Part III:
Suppose that the area of the entire square is one square unit, fill in the chart below so that
each piece has the correct area. Notice again that one has been done for you.
Number of piece
1
2
3
4
5
6
7
Cost of piece
4/16 = _
4/16 = _
1/16
2/16 = 1/8
1/16
2/16 = 1/8
2/16 =1/8
Witwer – Page 36
Day #5: TI-83+ Graphing Calculator
NAME: Kimberly Witwer
GRADE LEVEL: Seventh
I.
ACTIVITY:
Seventh grade students will find the maximum area of a rectangular shape
using the TI-83+ Graphing Calculator.
Seventh grade students will be able to compute the maximum area given a
perimeter.
Seventh grade students will be able to make a scatterplot on the TI-83+.
II.
STANDARDS:
New York State Key Ideas Covered: 1B, 1C, 5C, 5E, 7B
NCTM Standards Covered: Geometry, Measurement, Reasoning and
Proof, Communication, Representation
III.
MATERIALS:
TI-83+ Calculators
Teacher’s TI-83+ Calculator with Overhead Connection
20 popsicle sticks for each student
Exploring Area packet
Pencils
Overhead of introduction problem
IV.
OPENING ACTIVITY:
The students will be seated in their desks. I will put the introduction
problem on the overhead projector and read it to the students. I will ask a
student to explain the problem in his or her own words:
You need to design a rectangular garden along an existing wall in
your backyard. The garden needs to be surrounded by a small
fence to protect it from animals. You have 20 sections of 1-meter
long fence pieces. You cannot cut the fence pieces because it is too
expensive. Your task is to design a garden to maximize the
amount of vegetables you can grow. How many sections would
you use along the width and the length of the garden?
I will explain that there are many different rectangular gardens that can be
made using the 20 sections of fencing as I pass out 20 popsicle sticks to
each student. I will ask the students to use the manipulatives (popsicle
sticks) to explore possible widths and lengths of the garden while I pass
out the Exploring Area packet. Then, I will ask a few volunteers to come
to overhead to show how they arranged the 20 popsicle sticks to enclose
the garden. I will have the students record the tries in the chart on the
worksheet in the packet. Then I will ask the students to get into their
groups of four with their calculators, packets, and pencils. I will ask them
to use the graphing calculators to explore the question further using their
packets.
V.
MAIN ACTIVITY:
I will have the students use their TI-83+ calculators in their groups to
complete the Exploring Area packet:
1. Enter possible widths into L1 by typing each number and pressing enter
Witwer – Page 37
(Note: Numbers will range from 1-9 because the width can not be negative
and cannot exceed the number of fence pieces being used…keep in mind
that there are 2 widths).
2. Enter the lengths into L2:
Through the opening activity, students should have noticed that the length
will be 20-2*width. In this case, the width is L1.
Therefore, the formula students will enter into L2 will be 20-2*L1.
(Go to top of L2, so that L2 is highlighted and enter in desired formula.
Press enter and the desired lengths will appear).
3. Answer Question #4 to help you figure out the formula to enter into L3.
(Enter this formula just as you did for L2).
Witwer – Page 38
4. Use L1 and L3 to make a scatterplot. This will visually display the
relationship between the width of the garden and the area of it.
5. From the information in the lists, come up with a "friendly" window for
your graph.
6. Once your graph is displayed, the trace button can be used to explore
the data and display each of the coordinates. The left and right arrows can
be used to change which coordinate the cursor is on, so further
investigation can be done.
Witwer – Page 39
7. Complete the rest of the corresponding worksheet.
I will assess the students by hearing the groups’ responses. I will move
about the room and look at each individual student’s paper to see if he/she
understands the use of the graphing calculators and I will collect the
papers to do an informal assessment (not for a grade). I will call on
different students to volunteer answers to questions.
VI.
CLOSING ACTIVITY:
I will bring the class together as a whole and we will discuss the different
strategies and problems faced in the activity. I will ask for volunteers to
explain how they came up with their final answer and answers to other
questions on the worksheet that they would like to share with the class. I
will ask that the other students are comparing what they got through their
investigation.
VII.
HOMEWORK:
Finish Design a House Project
Witwer – Page 40
Name: ______________________________
Exploring Area
Using the Graphing Calculator
Within your groups, your job is to complete this activity following the directions and
suggestions very carefully. Once you have finished and discussed this activity, you will
have a better understanding of maximizing area using your graphing calculators.
The Problem: You need to design a rectangular garden along an existing wall in your
backyard. The garden needs to be surrounded by a small fence to protect it from animals.
You have 20 sections of 1-meter long fence pieces to work with. Your task is to design a
garden to maximize the amount of vegetables that can be grown. How many sections
would you use along the width? Along the length?
From the opening activity just completed, you should have some possible widths and
lengths for the garden. But, did you find the largest possible area for the garden? Follow
the instructions and questions below to make sure you have found the largest area.
Suggestion: Start by drawing a picture
1. If you were to use 2 sections of the fencing along each width, how many would
remain for the length? _______________________________
What would be the area of the garden? ________________________________
(Remember to label your answer with the correct units)
2. Fill in the table with other possible widths and length
Width (m)
Length (m)
Area (m2)
2
Witwer – Page 41
Enter possible widths into L1 of your calculator…remember there are two widths so
your values will not go up to 19.
3. If you know the width, how can you find the length? Write an equation that shows
how the width and length are related. (Keep in mind that the total amount of fencing
being used is 20 meters)
________________________________________________
4. Now that you have that formula, enter it into L2. (Make sure that you go to the top of
L2 so that it is highlighted, enter the formula, and hit enter. Once you hit enter, the
values for length will appear.
5. Would it make sense to enter 0 in for the width of your garden? Why / Why not?
________________________________________________________________________
________________________________________________________________________
6. What is the largest number of pieces that you can use for the width? (Keep in mind
that there are 20 pieces of fencing and 2 widths)
________________________________________________________________________
________________________________________________________________________
7. Now that you have the values for the length and width, you need to find a formula for
area using L1 and L2. (Remember that L1 is your widths, and L2 is the areas). Write
a formula for area in terms of L1 and L2.
______________________________________________
Enter this formula into L3 just as you entered the formula in L2.
8. Scroll through L3. Are the values you found in the opening activity in your list?
Describe a pattern you see when looking at the data in L3.
________________________________________________________________________
________________________________________________________________________
9. Use L1 and L3 to make a scatterplot. This will visually display the relationship
between the width of the garden and its area. Make sure you find a “friendly”
window. Sketch the graph below and be sure to include your viewing window.
Witwer – Page 42
10. Use the trace button on your calculator to complete the following statement.
The largest possible area of ____________________ m2 is found by using
_______________ pieces for the width, and ____________________ pieces
for the length.
11. How does the pattern you observed in question #8 show up in the scatter plot?
_____________________________________________________________________
_____________________________________________________________________
_____________________________________________________________________
12. Suppose you had to plant a garden in the corner of your yard where there are 2
existing walls. How would you expect the values for the length and width to change?
Would this make the garden larger or smaller? Explain your reasoning.
_____________________________________________________________________
_____________________________________________________________________
_____________________________________________________________________
13. If you had the choice between putting the garden along the one existing wall or in the
corner, what would you choose? Keep in mind that there are only two back corners
in you yard, would this effect your decision? Explain your answer.
_____________________________________________________________________
_____________________________________________________________________
_____________________________________________________________________
14. Did you find this activity useful in helping you understand maximum area? Why or
why not? What did you like most about this activity? What did you like least?
Would you suggest that this activity be used again?
_____________________________________________________________________
_____________________________________________________________________
_____________________________________________________________________
Witwer – Page 43
Key
Name: ______________________________
Exploring Area
Using the Graphing Calculator
Within your groups, your job is to complete this activity following the directions and
suggestions very carefully. Once you have finished and discussed this activity, you will
have a better understanding of maximizing area using your graphing calculators.
The Problem: You need to design a rectangular garden along an existing wall in your
backyard. The garden needs to be surrounded by a small fence to protect it from animals.
You have 20 sections of 1-meter long fence pieces to work with. Your task is to design a
garden to maximize the amount of vegetables that can be grown. How many sections
would you use along the width? Along the length?
From the opening activity just completed, you should have some possible widths and
lengths for the garden. But, did you find the largest possible area for the garden? Follow
the instructions and questions below to make sure you have found the largest area.
Suggestion: Start by drawing a picture
1. If you were to use 2 sections of the fencing along each width, how many would
16 units
remain for the length? _______________________________
What would be the area of the garden? ________________________________
32 units2
(Remember to label your answer with the correct units)
2. Fill in the table with other possible widths and length
Width (m)
Length (m)
Area (m2)
2
16
32
3
14
42
7
6
42
Witwer – Page 44
Enter possible widths into L1 of your calculator…remember there are two widths so
your values will not go up to 19.
3. If you know the width, how can you find the length? Write an equation that shows
how the width and length are related. (Keep in mind that the total amount of fencing
being used is 20 meters)
20 – 2(width) = 2(Length)
________________________________________________
4. Now that you have that formula, enter it into L2. (Make sure that you go to the top of
L2 so that it is highlighted, enter the formula, and hit enter. Once you hit enter, the
values for length will appear.
5. Would it make sense to enter 0 in for the width of your garden? Why / Why not?
Answers will vary. Ex: No, because it would be a straight line with no area
________________________________________________________________________
________________________________________________________________________
6. What is the largest number of pieces that you can use for the width? (Keep in mind
that there are 20 pieces of fencing and 2 widths)
9 because you have 2 widths and 2 lengths, so if the length is 1 that leaves 18 more
________________________________________________________________________
for the width and it is a rectangle so each width has to be the same (i.e. 9)
________________________________________________________________________
7. Now that you have the values for the length and width, you need to find a formula for
area using L1 and L2. (Remember that L1 is your widths, and L2 is the areas). Write
a formula for area in terms of L1 and L2.
L1*L2
______________________________________________
Enter this formula into L3 just as you entered the formula in L2.
8. Scroll through L3. Are the values you found in the opening activity in your list?
Describe a pattern you see when looking at the data in L3.
Yes, the values are in L3. The pattern I see is that the area goes up, then it starts to
________________________________________________________________________
come down
________________________________________________________________________
9. Use L1 and L3 to make a scatterplot. This will visually display the relationship
between the width of the garden and its area. Make sure you find a “friendly”
window. Sketch the graph below and be sure to include your viewing window.
Witwer – Page 45
10. Use the trace button on your calculator to complete the following statement.
50
The largest possible area of ____________________
m2 is found by using
5
10
_______________
pieces for the width, and ____________________
pieces
for the length.
11. How does the pattern you observed in question #8 show up in the scatter plot?
The pattern shows up in the scatterplot because the graph goes up and then it
_____________________________________________________________________
_____________________________________________________________________
comes down just like the numbers in the table
_____________________________________________________________________
12. Suppose you had to plant a garden in the corner of your yard where there are 2
existing walls. How would you expect the values for the length and width to change?
Would this make the garden larger or smaller? Explain your reasoning.
Answers will vary: I would expect the length and width to be larger because you
_____________________________________________________________________
_____________________________________________________________________
only need one of each now. Therefore, thearea of the garden would be larger.
_____________________________________________________________________
13. If you had the choice between putting the garden along the one existing wall or in the
corner, what would you choose? Keep in mind that there are only two back corners
in you yard, would this effect your decision? Explain your answer.
Answers will vary: I would choose to have it in the corner because I could make
_____________________________________________________________________
_____________________________________________________________________
the garden larger with less fence.
_____________________________________________________________________
14. Did you find this activity useful in helping you understand maximum area? Why or
why not? What did you like most about this activity? What did you like least?
Would you suggest that this activity be used again?
Answers will vary: there is no right or wrong answer.
_____________________________________________________________________
_____________________________________________________________________
_____________________________________________________________________
Witwer – Page 46
You need to design a rectangular garden along
an existing wall in your backyard. The garden
needs to be surrounded by a small fence to
protect it from animals. You have 20 sections
of 1-meter long fence pieces. You cannot cut
the fence pieces because it is too expensive.
Your task is to design a garden to maximize
the amount of vegetables you can grow. How
many sections would you use along the width
and the length of the garden?
Witwer – Page 47
Final Assessment: Design a House Project
NAME: Kimberly Witwer
GRADE LEVEL: Seventh
The students’ final assessment will be the following project entitled “Design a House.”
The students will be given the project on Day #3 of the unit and it will be due a week and
a half later (the following Friday).
The students will be given large sheets of graph paper to complete the project.
Witwer – Page 48
Name _________________
Due Date______________
Design a House Project
You are to design a one-floor house plan that includes the following:
2 bedrooms, 1 bathroom, 1 kitchen, 1 living room, 2 exits
Use the large sheets of graph paper provided to design your house.
Choose your scale and write it on the design.
The total area for your house must be between 150 - 200 square meters.
The following information should be attached to your house plans:
___ Total area of house (15 points)
___ Outside perimeter of house (15 points)
___ Area of bedroom one ( 5 points)
___ Perimeter of bedroom one ( 5 points)
___ Area of bedroom two ( 5 points)
___ Perimeter of bedroom two ( 5 points)
___ Area of bathroom ( 5 points)
___ Perimeter of bathroom ( 5 points)
___ Area of kitchen ( 5 points)
___ Perimeter of kitchen ( 5 points)
___ Area of living room ( 5 points)
___ Perimeter of living room ( 5 points)
___ 2 exits are clearly marked ( 10 points)
___ Neatness ( 10 points)
You may find the following websites helpful as you complete your plan:
www.coolhouseplans.com
www.dreamhomesource.com
www.houseplanguys.com
HAVE FUN!!!
Witwer – Page 49