Exploring Number Sense and Spatial Sense

LEAP UNIT 1

**Standards:**

3.1(4,5,8,9,10)

3.2(2,3,5,6)

3.3(1,2,3,7)

3.5(1,2,3,4)

3.6(1)

**Tools/Technology:**

1. Graph paper

2. Overhead graph paper

3. 5 Rectangles and squares with different areas and perimeters

4. Geoboards and rubberbands for each student

5. Overhead geoboard

6. Tangrams

9. Corresponding worksheets

**OBJECTIVES:**

1. Students will be able to find area and perimeter of squares and rectangles

2. Students will be able to devise formula for area of squares and rectangles

3. Students will be able to devise formula for perimeter of any shape

4. Students will be able to make shapes a given area

5. Students will be able to make shapes a given perimeter

6. Students will be able to find fractional areas

7. Students will be able to find the maximum area of a rectangle given a
perimeter

8. Students will be able to design a 1 story home using their knowledge of area
and

perimeter

**Outline of Schedule(Tentative 15 Days):**

Week 1: *Use graph paper to explore area and perimeter.*

-Students will trace shapes onto graph paper to count the area and perimeter of
the

shapes

-Students will come up with the formula for area and perimeter of a square and

rectangle

**Week 2:** *Use geoboards to explore area and perimeter.*

-Students will be asked to find area of various polygons using the geoboard

-Contest: who can find the most shapes with a given area and or perimeter

**Week 3: ***Use of tangrams to explore fractional areas.*

-Students will use tangrams pieces to see how the areas relate

-Given one piece area students will find the other 6 areas

**Week**** 4:** *Use a graphing calculator to explore maximum area and perimeter.*

-Students will use a graphing calculator to make conjectures and scatter plots
to

find the maximum area given a perimeter

**Week 5:** *Final assessment: Students will design a 1 floor house with various rooms
on
graph paper using area and perimeter.*

-Students will use a large sheet of graph paper to design a one-floor home with

various rooms and label each room with area and perimeter

-Students will find the overall area and perimeter of the home he/she designed

**Week 1:**

**Objectives:**

1. Students will be able to find area of rectangles and squares by counting
squares on

graph paper

2. Students will be able to find perimeter of rectangles and squares by counting
squares

on graph paper

3. Students will devise formulas for area and perimeter of rectangles and
squares

**Materials**:

-Overhead of graph paper

-Square and rectangular shapes with different area and perimeter

-Graph paper

-Area and perimeter worksheet #1

**Outline of Activity:**

Today we are going to be talking about perimeter and area of square and
rectangular

shapes. Show overhead of graph paper and define one square as 1 square unit.
Place a

square shape (transparent) on the overhead so that students can see the units
through the

shape. Explain that perimeter is the distance around the shape, and that area is
how many

square units are in the shape.

Tell students what the perimeter is and pick a volunteer to explain how he/she
found it.

(Answer should be that they counted the boxes around the shape)

Tell students what area of object is and pick a volunteer to explain how he/she
found it.

Do another example with students this time using a rectangle. Instead of telling
students

area and perimeter, see if they can come up with it on their own and explain it.

Have students explore area and perimeter through completing area and perimeter

worksheet #1.

**Homeroom: _________________ Name: __________________
Date: ______________________ #1
**

**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

**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 SHAPE #4: Draw a square
Perimeter of 14 units Perimeter of 12 units
Area of 12 square 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 with of shape.
_____________________________________________________________________
_____________________________________________________________________
5. Does this relationship hold true for all rectangles? What about squares?
Explore this
idea by looking back at the example you have completed.
_____________________________________________________________________
_____________________________________________________________________
6. What is the area of the 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 = ______________
**

**Week**** #2**

**Objectives:**

1. Students will be able to find area and perimeter of various polygons by using
the

geoboards.

2. Students will be able to make polygons with certain areas and perimeters

**Materials:**

Overhead geoboard

Rubberbands

Geoboard for each student

Geoboard paper

**Description of Lesson**

Explain the objective of today’s lesson. Using overhead geoboard, define one
square

unit. Show students other ways to show one square unit. Show _ square unit and
have

students guess what the area of the triangle would be. Once students come up
with _

square unit, have him/her explain how he/she got that answer. Show another
irregular

polygon, have students explore the area and perimeter of it. Explain the use of
“helper

rubberbands” to break up the shape into easier areas and perimeters. Show how to
do this

on the overhead. Do not complete the whole shape, but have students finish on
their

own. Have students explain different ways they came up with the answer.

Put another polygon the overhead and have students emulate it on their own geoboard

and figure out the area and perimeter. Have volunteer come to overhead and show
how

he/she found the answer. Ask if anyone else found it a different way, and have
him or

her come to overhead and show how he/she got the answer.

Continue this for at least 2 more examples.

Break students into pairs.

Have students find different shapes that have the same area.

For example find as many different shapes with the area of 4 square units.

Have them copy each one they found onto the geoboard paper.

Use the second geoboard paper and have students explore perimeter. Have students
find

different shapes with the same perimeters. For example find all possible shapes
with the

perimeter of 8 units. Copy each one found onto the geoboard paper.

Bring students back together as a whole group to discuss their findings.

Ask students how many of each they found.

Have some students show one way they found the given area/perimeter.

Discuss differences in shapes with the same area/ the same perimeter.

Week 3:

**Objectives:**

1. Students will explore area through use of tangrams

2. Students will find fractional areas

3. Students will find area of 6 pieces given one piece’s area

**Description of Lesson:**

Hand out tangram pieces and allow students some time to try to form a square
with the

seven pieces.

Have a volunteer come to front to show how to make the square using all the
pieces.

This will allow students to become familiar with the pieces and peak their
interest.

Complete an area problem using the tangrams as a whole class.

Ex. The tangram piece labeled #5 is 1sqaure unit. Find the area of each of the

other pieces. 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. Continue until chart is completed.

Tangram piece Area (square unit)

1. 1 1/2

2. 1 1/2

3. 1/2

4. 1

5. 1/2

6. 1

7. 1

Hand out tangram worksheet and have students complete it individually.

**Homeroom: ________________________ Name: __________________
Date: ______________________________ Area and Tangram
**

**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 done for you as an example.
**

**Number of piece Name of piece Area of piece
(square units)
1 Right isosceles triangle 4**

2

3

4

5

6

7

**Part II:
Explain a strategy to determine how you found the area for piece #4.
**

**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
Area of piece
1**

2

3

4

5 1/16

6

7

**Week**** #4**

**Objectives:**

-In small groups, students will use their basic understanding of perimeter and
area to

explore and investigate finding the maximum area of a rectangular shape.

-Students will be able to compute maximum area given a perimeter

-Students will be able to use a graphing calculator to make a scatter plot

**Materials:**

Overhead of introduction problem

20 popsicle sticks each

exploring area worksheet

graphing calculator (TI-73 or TI-83+)

**Description of Lesson**

Pose 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. 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?

There are many different rectangular gardens that can be made using the 20
sections

of fencing. Using the manipulatives (popsicle sticks) explore possible widths
and lengths

or the garden.

Choose a few volunteers to come to overhead to show how they arranged the 20
popsicle

sticks to enclose the garden

Have students record the tries in the chart on the worksheet.

Use the graphing calculator to explore the question further. (Follows along
with

questions on exploring area worksheet).

1. Enter possible widths into L1 by typing each number and pressing enter

(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)

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.

7. Complete the rest of the corresponding worksheet.

Bring class together as a whole and discuss different strategies and problems
faced in the

activity and have various people explain how they came up with their final
answer and

answers to certain questions on the worksheet.

**Name: _____________________________
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 that 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
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 the 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.
**

**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 on 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 with 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 your yard, would this effect your decision? Explain your answer.
_____________________________________________________________________
_____________________________________________________________________
_____________________________________________________________________
**

**14. Did you find this activity useful in helping you understand maximum area?
Why/Why not? What did you like most about this activity? What did you like
least?
Would you suggest that this activity be used again?
_____________________________________________________________________
_____________________________________________________________________
_____________________________________________________________________
_____________________________________________________________________
_____________________________________________________________________
_____________________________________________________________________
**

**Week**** 5:**

**Objectives:**

1. Students will be able to design a one-floor home using their knowledge of
area and

perimeter.

**Materials:**

-large sheets of graph paper

-final assessment worksheet

**Description of Lesson:**

Explain what students will be required to do to earn full points on the
assessment.(See

worksheet) Hand out worksheets and graph paper.

**Homeroom: _____________________ Name: _____________________________
**

**Date: __________________________ Final Assessment
**

**Your Task: Use your knowledge of area and perimeter to design a one-floor house.
Make sure that the house contains at least the following rooms:
**

**1. Living room/Den
2. Bathroom
3. Kitchen
4. 2 bedrooms
**

**If you wish to make more rooms, you may. Make sure that each room has the
dimensions, area, and perimeter labeled. Also, include a key for each box on the
graph paper. You may want to use 1 box = 1 sq. foot
Note: The maximum area for your house is 1200 sq. feet and the minimum is 800sq
feet.**