Geometry
Laura Dahl
Clearbrook-Gonvick Elementary School, Grade 5
ldahl@clearbrook-gonvick.k12.mn.us
Lexi Geisler
Robert J. Elkington Middle School, Grade 6
ageisler@isd318.org
1
This unit addresses the following Minnesota Math Standards:
Develop and use formulas to determine the area of
5.3.2.1 triangles, parallelograms and figures that can be
decomposed into triangles.
Use various tools and strategies to measure the
Determine the
volume and surface area of objects that are
area of
shaped like rectangular prisms.
triangles and
quadrilaterals;
For example: Use a net or decompose the surface into
5.3.2.2
rectangles.
determine the
Geometry &
surface area
5
Another example: Measure the volume of a cereal box by using
Measurement
a ruler to measure its height, width and length, or by filling it
and volume of
with cereal and then emptying the cereal into containers of
rectangular
known volume.
prisms in
Understand that the volume of a three-dimensional
various
figure can be found by counting the total number
contexts.
of same-sized cubic units that fill a shape without
5.3.2.3 gaps or overlaps. Use cubic units to label volume
measurements.
For example: Use cubes to find the volume of a small box.
Calculate the surface area and volume of prisms
and use appropriate units, such as cm2 and cm3.
Justify the formulas used. Justification may
6.3.1.1 involve decomposition, nets or other models.
Calculate
For example: The surface area of a triangular prism can be
perimeter,
found by decomposing the surface into two triangles and three
rectangles.
area, surface
area and volume
Calculate the area of quadrilaterals. Quadrilaterals
of two- and
include squares, rectangles, rhombuses,
Geometry &
three6
parallelograms, trapezoids and kites. When
Measurement
dimensional
formulas are used, be able to explain why they are
6.3.1.2
figures to solve
valid.
real-world and
For example: The area of a kite is one-half the product of the
mathematical
lengths of the diagonals, and this can be justified by
problems.
decomposing the kite into two triangles.
Estimate the perimeter and area of irregular
figures on a grid when they cannot be decomposed
6.3.1.3
into common figures and use correct units, such as
cm and cm2.
2
Geometry Outline
Outline Length - 10 periods
1 period
Pretest…………………………………………………….p. 4-6
Scavenger Hunt
Activity 1…………………………..………………………… p. 7
2 periods
Metric Olympics
Activity 2………………….…………..……………………..p. 8-9
1 period
Geoboard
Activity 3………………….………….………………………p. 10-12
1 period
Shape Explorer
Activity 4..……………………….………………………….p. 13
1 period
Cereal Box
Activity 5……………………………….………………..…p. 14-15
2 periods
Make a Polyhedral City
Activity 6………….……….……….………………………p. 16
2 periods
Design a Fish Aquarium
Activity 7……………….………….……………………….p. 17-18
Posttest………………….…..……………………....p. 19-21
3
Name ________________
Period ___
Geometry Pretest
Show all your work. Do your best!
1. Find the perimeter of the shapes below.
5 ft
a.
3 cm
2.5 in
b.
4.5 ft
5 in
7 cm
10 ft
_______________
_______________
_______________
2. Find the area of the shapes below.
b.
2 cm
b.
h = 2 in
c.
5 ft
5 cm
_______________
4 in
6 ft
_______________
_______________
3. Find the length of the lines below in centimeters? (use a ruler)
a.
a. _______
b.
b. _______
4. Create the following shapes on the geoboard below.
a. 5 x 5 square
b. triangle with a perimeter of 10
c. rectangle with an area of 24
4
Find the perimeter and area of the irregular shapes below.
Note:
Add lengths to shapes.
5.
Perimeter
Area
_________
_________
6.
Perimeter
_________
Area
_________
7. What is the formula for area of a triangle? _______________
8. Show how to find the area of a triangle using the rectangle below. Show
your work and explain what you did in a short paragraph.
__________________________________________________
__________________________________________________
__________________________________________________
__________________________________________________
5
9. Find the volume of the rectangular prism. Show all your work. Units are in
inches.
Volume = ________
10. Find the surface area of the cube below. Use the net to help you find the
area of each face. Show ALL your work.
5 cm
Surface Area = _______
6
Activity 1: Scavenger Hunt
For Scavenger Hunt Worksheet go to:
http://cmase.uark.edu/teacher/workshops/AIMS-lessons/mini-metrics.pdf
1 class period
Objective: Students will practice estimating lengths of objects and then practice
measuring objects to prepare them for Metric Olympics tomorrow.
Minnesota State Mathematics Standards: Geometry and Measurement
Benchmarks covered in these lessons:
5.3.2.1
Develop and use formulas to determine the area of triangles,
parallelograms and figures that can be decomposed into triangles.
6.3.1.3
Estimate the perimeter and area of irregular figures on a grid when
they cannot be decomposed into common figures and use correct
units, such as cm and cm2.
Launch: “How many of you like to build things or sew? What skills do you need for
both of those things? You need to be able to effectively and accurate measure in
order for your project to turn out. Today, we will be review how to use rulers and
other skills to measure objects and estimate what objects we believe are a given
length.”
Explore: Students will each receive a ruler and practice finding a given length.
After they are ready to move on, practice estimating the length of your desk,
pencil, etc in cm. Then provide students a Metric Scavenger Hunt worksheet (1 for
each group of 2) and have them find objects of the given length, estimating first.
The winner is the group with the lowest total difference.
Share: Students will share what was the easiest and hardest part of their
scavenger hunt. We will then share several objects that they found that matched
the given lengths.
Summarize: “Today we learned that estimating and measuring and very important
and needed in several life skills, such as building and sewing. Tomorrow we will use
our measuring skills to play Metric Olympics. Come ready to estimate and measure!”
Extension: Provide students with a length and area and see if they can find a 2 or
3 dimensional shape.
7
Activity 2: Mini Metric Olympics
For Mini Metric Worksheets go to:
http://cmase.uark.edu/teacher/workshops/AIMS-lessons/mini-metrics.pdf
2 class periods
Objective: Students will estimate and measure objects in metric units playing Mini
Metric Olympics.
Minnesota State Mathematics Standards: Geometry and Measurement
Benchmarks covered in these lessons:
5.3.2.1
Develop and use formulas to determine the area of triangles,
parallelograms and figures that can be decomposed into triangles.
6.3.1.3
Estimate the perimeter and area of irregular figures on a grid when
they cannot be decomposed into common figures and use correct
units, such as cm and cm2.
Launch: “Have you ever watched the Olympics? What is your favorite event? Today
you will all be competing in Mini Metric Olympics. Yes, there will be medals handed
out at the end of our competition so get ready to have some fun and practice
measuring like we did yesterday. ”
Explore: Go through the Olympic History on pages 3-5 of Mini Metrics and discuss
how the Olympics have changed and how they are similar to current events. Then
discuss what events they each will be competing in and decide as a class what
parameters and rules we will all follow. Everyone needs to do the same thing! After
that, have students work in pairs and decide who will be the team captain and
competitor of each group. Next, model each of the events with your students and
have them complete the estimation part of their worksheet and estimate to the
nearest whole unit. Lastly, assign each group a station and allow them about 2-4
minutes to complete their event and then have them switch. The winner (s) is the
individual and group with the lowest score.
Share: Students will share their individual and group score on the board and we
will discuss what the groups with the lowest score used for their strategy. If time
allows, talk about outliers and how one bad estimate or measurement can throw off
all of the data. Discuss the easiest and hardest part of the Olympics. Was it
estimating, measuring, etc? What strategies could you use next time to do better?
8
Summarize: “Today we applied our understanding of estimating and measuring and
discovered that they are equally important. We estimate and measure all the time
in our daily lives and this activity increased your knowledge of metrics so that you
will do better on the next Olympic series and be prepared to find the area of a
variety of shapes.”
Extension: Do the Mini Metric Olympic #2 and talk about outliers and how they
affect your overall score. Talk about 2 and 3 dimensional objects and how to
estimate the area and/or volume of an object.
9
Activity 3: Geoboard
1 class period
Objective: Students will use the geoboards to represent an understanding of area
and perimeter by calculating and creating different perimeters and areas on the
geoboard.
Minnesota State Mathematics Standards: Geometry and Measurement
Benchmarks covered in these lessons:
5.3.2.1
Develop and use formulas to determine the area of triangles,
parallelograms and figures that can be decomposed into triangles.
6.3.1.2
Calculate the area of quadrilaterals. Quadrilaterals include squares,
rectangles, rhombuses, parallelograms, trapezoids and kites. When
formulas are used, be able to explain why they are valid.
6.3.1.3
Estimate the perimeter and area of irregular figures on a grid when
they cannot be decomposed into common figures and use correct
units, such as cm and cm2.
Launch: “How much fencing would you need to go around your garden? How many
yards of carpet are in your room? How many gallons of paint would you need to buy
to paint your kitchen or your house? Today we are going to use the geoboards and
explore area and perimeter of several polygons.”
Explore: Pass out to each student a geoboard and rubbers bands. Give students
time to explore and play with their board before starting. Have students find the
perimeter of several polygons, and then have students construct polygons with a
given perimeter. Make squares of various sizes and construct a table comparing
the area and perimeter. Do the same with rectangles, kites, trapezoids, and
rhombus.
Share: Have a number of students share their observations from the table on how
perimeter and area are related by shapes with the same, different or similar
length and widths.
10
Summarize: “Today we learned that finding the perimeter and area is something
that you will need to do in everyday life. We also discovered that the same shape
does not share the same perimeter and area.”
Extension: Students could do the above activity including triangles on their table.
11
Geoboards- Area & Perimeter
Shape Length Width Perimeter Area
12
Activity 4: Shape Explorer
1 class period
Objective: Students will find relationships between length, width, perimeter and
area of quadrilaterals and irregular shapes.
Minnesota State Mathematics Standards: Geometry and Measurement
Benchmarks covered in these lessons:
5.3.2.1
Develop and use formulas to determine the area of triangles,
parallelograms and figures that can be decomposed into triangles.
6.3.1.3
Estimate the perimeter and area of irregular figures on a grid when
they cannot be decomposed into common figures and use correct
units, such as cm and cm2.
Launch: “Yesterday you learned how to measure shapes accurately using the metric
system. Is there another way to accurately measure shapes? Today we will learn
how to find the area and perimeter of regular and irregular shapes using skills that
we learned from Geoboards yesterday. Today we will explore in the computer lab.”
Explore: Bring students to the computer lab and have them open up the website
listed below. Do an example or two with them and have them discover patterns with
a partner. They can also keep score so you can assess how well they are doing. If
time allows, have them switch to the Shape Builder game on the same website.
Share: As a whole class, share what patterns you found and what relationships
perimeter and area have with one another. What tools helped you? Did the “show
outline” tool help you? What strategies did you use to find the perimeter and area?
Summarize: “Today we learned that the computer can be a great tool for seeing
relationships between perimeter and area. We also learned that we can find the
area and perimeter in multiple ways (not just adding all the units) and that
irregular shapes are a challenge for us. Tomorrow, we will apply the skills we used
today to find the perimeter, area, and surface area of a cereal box.”
Website for activity found at:
http://www.shodor.org/interactivate/activities/ShapeExplorer/?version=1.6.0_11&
browser=MSIE&vendor=Sun_Microsystems_Inc
13
Activity 5: Cereal Box
http://www.digitallesson.com/PreviewPages/CerealBoxSurfaceAreaPreviewPages.
pdf
1-2 class periods
Objective: Students will use cereal boxes to find the area and/or surface area of
a rectangular prism.
Minnesota State Mathematics Standards: Geometry and Measurement
Benchmarks covered in these lessons:
5.3.2.2
Use various tools and strategies to measure the volume and surface
area of objects that are shaped like rectangular prisms.
6.3.1.1
Calculate the surface area and volume of prisms and use appropriate
units, such as cm2 and cm3. Justify the formulas used. Justification
may involve decomposition, nets or other models.
Launch: “If you could have any cereal that you want, what would you pick? Why?
Today we are going to be using the cereal boxes that you brought to class and find
relationships between the side lengths, area, and surface area, and volume of your
shapes.”
Explore: Students will measure the length, width and height of their cereal boxes
and record their results on the worksheet provided to find the area and volume of
their shape. They will then create a net of their cereal box but cutting it apart and
find the surface area of their box.
Share: Students will share what relationships they found from the side lengths of
their boxes. We will then share our volume and surface area results as a class and
make conclusions on the size of the box versus the size of the area and volume.
Summarize: “Today we learned that cereal boxes can be used to show us how to
find the surface area of a rectangular prism. We also found that the area is found
by multiplying the length by the width and the volume is found by multiplying the
length, width, and height.”
Extension: Discuss that cereal is not sold by volume but by weight. Find the ratio
of weight to volume and discuss what factors might affect the ratio
14
Class Comparison of Boxes
Box Type
Surface Area
Volume
Ratio
(Weight to Volume)
15
Activity 6: Make a Polyhedral City
(Taken from Impact Mathematics, The McGraw-Hill Companies)
2-3 class periods
Objective: Students are going to be creating a polyhedral city made up of a
variety of shapes. Students will use plain paper or graph paper to make nets of
polyhedra that can be folded to make buildings.
Minnesota State Mathematics Standards: Geometry and Measurement
Benchmarks covered in these lessons:
5.3.2.2
Use various tools and strategies to measure the volume and surface
area of objects that are shaped like rectangular prisms.
6.3.1.1
Calculate the surface area and volume of prisms and use appropriate
units, such as cm2 and cm3. Justify the formulas used. Justification
may involve decomposition, nets or other models.
Launch: “If you could create a city what kind of city would you create?” Tell
students to think about the buildings, signs, people, name, and population of their
city.
Explore: Assign groups of 2-4 students and give time for groups to discuss and
brainstorm ideas for creating their cities. Students are to form at least five
buildings and label each solid’s dimensions. Ask students “If you use graph paper,
how can you find the dimensions? If you use another kind of paper, how can you
find the dimensions?” Have students find the surface area of each building. Next,
assemble your city, decorate the buildings, make signs, and add people and other
things found in a city including the name and population.
Share: Have students write and present a short report about their city. Students
should include some of the following details in the report: Describe your city, the
name, location, population, and so on. Make a chart that includes the name, a
sketch, and a brief description of each kind of solid you used. Explain how you
found the dimensions and surface areas. List some of the reasons why you might
need to know the surface areas of the building in a city.
Summarize: “Today we learned that cities are made up of a variety of buildings for
many different purpose. We created our own buildings using surface area to decide
of the size and use of the buildings.”
Extensions: Students could create another builds; such as a circular water tower,
houses with triangle roofs.
16
Activity 7: Design a Fish Aquarium
(Taken from Impact Mathematics, The McGraw-Hill Companies)
2-3 class periods
Objective: Students are going to be creating a fish aquarium. Students will be
using what they learned about finding the volume of prisms to complete this task.
Minnesota State Mathematics Standards: Geometry and Measurement
Benchmarks covered in these lessons:
5.3.2.1
Understand that the volume of a three-dimensional figure can be
found by counting the total number of same-sized cubic units that fill
a shape without gaps or overlaps. Use cubic units to label volume
measurements.
6.3.1.1
Calculate the surface area and volume of prisms and use appropriate
units, such as cm2 and cm3. Justify the formulas used. Justification
may involve decomposition, nets or other models.
Launch: “You won a free fish aquarium in a drawing at school but you have to
personalize the aquarium in order for you to keep it. Today you are going to have
to decide how big you need your aquarium to be, how many fish and what kind you
are going to have in your aquarium.” Pet stores recommend that for each inch of
fish, an aquarium should have 1 gallon of water. One gallon of water is about 231
cubic inches.
Explore: Assign groups of 2-4 students and give time for groups to discuss and
brainstorm ideas for creating their aquariums. Students can use the table, books,
Internet to find what kind of fish they want in the aquarium. They need to
remember to decide on a size and number of fish they are going to be placing in
their aquarium. Students need to have at least 5 different fish in their project.
Students use the size of the fish to determine how many gallons of water the
aquarium needs. Then decide how many cubic inches the aquarium tank needs to be.
Decide on the dimensions of the tank. Use estimation to write the volume and
dimension in centimeters.
Share: Have students write a report about their aquarium. Have students include
some of the following details: An explanation of how they decide on the number of
gallons of water the aquarium needed and how many cubic inches the aquarium
should be. How estimations was used to convert the measurements. Is the aquarium
17
something that is possible to have in their house? Where would they put the new
fish tank?
Summarize: “Today we learned that the volume of prisms depend on the
dimensions that are used to create it. We also found that the volume is found by
multiplying the length, width, and height.”
Extensions: Students could construct their aquarium or construct one for the
classroom
18
Name ________________
Period ___
Geometry Posttest
Show all your work. Do your best!
1. Find the perimeter of the shapes below.
5 ft
c.
3 cm
2.5 in
b.
4.5 ft
5 in
7 cm
10 ft
_______________
_______________
_______________
2. Find the area of the shapes below.
d.
2 cm
b.
h = 2 in
c.
5 ft
5 cm
_______________
4 in
6 ft
_______________
_______________
3. Find the length of the lines below in centimeters? (use a ruler)
a.
a. _______
b.
b. _______
4. Create the following shapes on the geoboard below.
b. 5 x 5 square
b. triangle with a perimeter of 10
c. rectangle with an area of 24
19
Find the perimeter and area of the irregular shapes below.
Note:
Add lengths to shapes.
5.
Perimeter
Area
_________
_________
6.
Perimeter
_________
Area
_________
7. What is the formula for area of a triangle? _______________
8. Show how to find the area of a triangle using the rectangle below. Show
your work and explain what you did in a short paragraph.
__________________________________________________
__________________________________________________
__________________________________________________
__________________________________________________
20
9. Find the volume of the rectangular prism. Show all your work. Units are in
inches.
Volume = ________
10. Find the surface area of the cube below. Use the net to help you find the
area of each face. Show ALL your work.
5 cm
Surface Area = _______
21