# Day 5 - Bemidji State University ```Geometry &amp;
Measurement
Travis Whittington
twhittington@clbs.k12.mn.us
Executive Summary
This Geometry and Measurement unit will take about fifteen class days.
The unit is composed of two parts: the first part focuses on increasing student
background with area, surface area and nets. Part two develops student
understanding of volume and it’s application. Both parts of the unit were
developed to meet and exceed Minnesota state standards along the way.
Minnesota Standards Covered
Standards utilized in the unit are highlighted
Describe and classify three-dimensional figures including cubes,
Describe, classify,
5.3.1.1 prisms and pyramids by the number of edges, faces or vertices
and draw
Geometry &amp;
as well as the types of faces.
representations of
Measurement
three-dimensional
5.3.1.2 Recognize and draw a net for a three-dimensional figure.
figures.
Develop and use formulas to determine the area of triangles,
5.3.2.1 parallelograms and figures that can be decomposed into
triangles.
Use various tools and strategies to measure the volume and
surface area of objects that are shaped like rectangular prisms.
5.3.2.2
Determine the area
of triangles and
Geometry &amp; determine the
Measurement surface area and
volume of
rectangular prisms
in various contexts.
For example: Use a net or decompose the surface into
rectangles.
Another example: Measure the volume of a cereal box by using a
ruler to measure its height, width and length, or by filling it with
cereal and then emptying the cereal into containers of known
volume.
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
5.3.2.3 volume measurements.
For example: Use cubes to find the volume of a small box.
Develop and use the formulas V = ℓwh and V = Bh to determine
the volume of rectangular prisms. Justify why base area B and
5.3.2.4
height h are multiplied to find the volume of a rectangular prism
by breaking the prism into layers of unit cubes.
Pretest
By the end of this unit, students should be able to answer the following questions correctly. MCA III
sampler questions relating to the unit are included.
1)
2)
3)
4)
5)
6)
Part 1
Part One
Day 1 – Pre-test.
Day 2 – Area Introduction
Day 3 – Extending Area
Day 4 – Nets
Day 5 – Exploring Geosolids
Day 6 – Surface Area
Part Two
Day 7 – Which Solid Is Bigger
Day 8 – Which Solid Is Bigger pt. 2
Day 9 – Volume Unknown
Day 10 – Deconstructed Volume
Day 11– Finding Volume
Day 12 - Peas in a room
Day 13 – Peas in a room
Day 14 – Review Day
Day 15 – Post-test.
Note: All materials are ordered and attached as referenced in the plan.
Day 1
(Pre-test)
Standards Covered:
5.3.1.1 - Describe and classify three-dimensional figures including cubes, prisms and pyramids by the number of
edges, faces or vertices as well as the types of faces.
5.3.1.2 – Recognize and draw a net for a three-dimensional figure.
5.3.2.1 – Develop and use formulas to determine the area of triangles, parallelograms and figures that can be
decomposed into triangles.
5.3.2.2 – Use various tools and strategies to measure the volume and surface area of objects that are shaped like
rectangular prisms.
5.3.2.3 – 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.
5.3.2.4 - Develop and use the formulas V = ℓwh and V = Bh to determine the volume of rectangular prisms. Justify
why base area B and height h are multiplied to find the volume of a rectangular prism by breaking the prism into
layers of unit cubes.
Materials: MCA Pre-test, Wrap It Up activity.
Launch: Question of the day activity, (students are given five minutes to gather their notebooks, pencils,
and answer the following question on the smart board daily before class starts, at the end, we discuss and
talk about a few answers/solutions.) “Make a list of shapes you know of.”
“Today we’re going to be taking a pre-test to see how much you all know about our next unit. Remember
this helps us measure where we are starting at and where we are ending up, so be honest with yourself
Explore: Hand out MCA Pre-Test. Once students complete the pretest they can work on the Wrap It Up
activity.
Share: Save five minutes at the end for students to share their wrap it up surveys with a neighbor.
Summarize: Use information turned in from Wrap It Up survey to pinpoint on areas that need work.
Day 2
(Area)
Standards Covered:
5.3.2.1 – Develop and use formulas to determine the area of triangles, parallelograms and figures that can be
decomposed into triangles.
5.3.2.2 – Use various tools and strategies to measure the volume and surface area of objects that are shaped like
rectangular prisms.
5.3.2.4 - Develop and use the formulas V = ℓwh and V = Bh to determine the volume of rectangular prisms. Justify
why base area B and height h are multiplied to find the volume of a rectangular prism by breaking the prism into
layers of unit cubes.
Materials: Smartboard, graph paper, meter sticks.
Launch: Question of the day activity, “How would you find the area of this room?)” Discuss results and student
“Area is a way to know how large something is in two dimensions. Like a piece of paper has two dimensions, length
and width.” Explain area &amp; concepts to class, answer questions.
Explore: Give the students meter sticks to measure the room. Give them time to measure the room and determine
the size of the room. (Room has 12 inch tiles). If they don’t catch onto the 12 inch tiles, hint. Ask, “how many tiles
are in the room?”
Once students come up with answers, have them grab a piece of graph paper and create an array, (5 x 5, 10 x 7,
etc.). Have students come up with an area of their shape they created as an answer key.
Share: Once complete, they will trade with a partner and solve each other’s arrays. Challenge them to come up with
a creative way to quickly find the area. Discuss different methods of solving the arrays using student examples.
Summarize: “What is area?” “What was the shortcut for solving arrays (rectangles, squares).”
Day 3
(Extending Area)
Standards Covered:
5.3.2.1 – Develop and use formulas to determine the area of triangles, parallelograms and figures that can be
decomposed into triangles.
5.3.2.2 – Use various tools and strategies to measure the volume and surface area of objects that are shaped like
rectangular prisms.
5.3.2.4 - Develop and use the formulas V = ℓwh and V = Bh to determine the volume of rectangular prisms. Justify
why base area B and height h are multiplied to find the volume of a rectangular prism by breaking the prism into
layers of unit cubes.
Materials: Smartboard, graph paper, scissors.
Launch: Question of the day activity, “What was the quick way to find the area, make me rectangle or a square and
show me how to find the area.” Discuss student answers and results.
Hand out graph paper, “Simon’s Square Factory is trying to figure out a quick way to find the area of their 6 x 6
square.”
Explore: In pairs have students come up with a method to find the area of the 6 x 6 square.
Share: Have student pairs share their solution with another pair. Then pick a few groups to explain their ideas to the
class. Go over base x height for squares and rectangles.
Explore: Tami’s triangle factory has heard about your fantastic work, and now they would like your help. They start
with a 6 x 6 square from Simon’s factory, then make their triangle from that. The base of the triangle is six, and the
point goes to the top of the square, and the point splits the top of the square in half. (Model this for students) What’s
the area of her triangle?
Share: Have student pairs share their solution with another pair. Then pick a few groups to explain their ideas to the
class.
Explore: Use your scissors to cut out the square, then cut out the triangle (just like Tami’s factory does), is there
Share: Have students report their findings; introduce &frac12; base x height to find the area of a triangle.
Summarize: “What were the quick way to find the area of a rectangle or square, how about a triangle?”
Day 4
(Nets)
Standards Covered:
5.3.1.1 - Describe and classify three-dimensional figures including cubes, prisms and pyramids by the number of
edges, faces or vertices as well as the types of faces.
5.3.1.2 – Recognize and draw a net for a three-dimensional figure.
5.3.2.2 – Use various tools and strategies to measure the volume and surface area of objects that are shaped like
rectangular prisms.
Materials: Smartboard, scissors, NCTM Nets (see attachments), geosolids. glue. large grid paper, cubic nets
handout.
Launch: Question of the day activity, (Have a cube on hand) “If I had a cube, and was able to take it apart, what
would it look like? Draw me a picture and explain.” Discuss results and student answers.
Draw an example of a net of a cube. Ask students “If we were to cut it out and fold it together if it would make a
cube?” Introduce the concept of a net. Hand out the cube net in the NCTM Net packet and scissors.
Explore: Let students assemble the nets, pose the question “could we change the net around and still get a cube?
Give students large grid paper and allow them to create more nets for a cube. “How many can we find?”
Share: When students find a net that works, have them draw it up on the board. Once they have them all, or are
stuck, hand out the cubic nets worksheet and have them either confirm or check/add to the class list.
Summarize: “How many ways can we find the net of a cube? Do you think we can make nets of other solids?”
Day 5
(Exploring Geosolids)
Standards Covered:
5.3.1.1 - Describe and classify three-dimensional figures including cubes, prisms and pyramids by the number of
edges, faces or vertices as well as the types of faces.
5.3.1.2 – Recognize and draw a net for a three-dimensional figure.
5.3.2.2 – Use various tools and strategies to measure the volume and surface area of objects that are shaped like
rectangular prisms.
5.3.2.4 - Develop and use the formulas V = ℓwh and V = Bh to determine the volume of rectangular prisms. Justify
why base area B and height h are multiplied to find the volume of a rectangular prism by breaking the prism into
layers of unit cubes.
Materials: Smartboard, paper, NCTM Net packet, scissors, glue.
Launch: Question of the day activity, “What other types of solids besides cubes do you think we could make with a
net? What would they look like?” Discuss results and student answers.
Explore: Discuss the concept of faces, vertices, and edges from yesterday’s cube. Hand out NCTM Net packet,
work through cutting out each net and labeling them as a group. Give them time to assemble. Hand out NCTM
Exploring Geosolids worksheet for students to complete.
Share: Have students share with a neighbor their results and findings. As a pair see if they can find any patterns or
similarities between the solids.
Summarize: “What is the face?” “What are the vertices?” “What is the edge?”
Day 6
(Surface Area)
Standards Covered:
5.3.1.2 – Recognize and draw a net for a three-dimensional figure.
5.3.2.2 – Use various tools and strategies to measure the volume and surface area of objects that are shaped like
rectangular prisms.
5.3.2.4 - Develop and use the formulas V = ℓwh and V = Bh to determine the volume of rectangular prisms. Justify
why base area B and height h are multiplied to find the volume of a rectangular prism by breaking the prism into
layers of unit cubes.
Materials: Smartboard, paper, CM grid paper, geosolids, scissors.
Launch: Question of the day activity, “If I had a present I wanted to wrap that a cube, how could I find out how much
wrapping paper I need?” Discuss student results and answers.
Pair students and give each pair, grid paper, scissors, and their cube from previous day.
Explore: Give students time to develop a method to find how much paper is needed to wrap the present.
Share: Select a few students to show their different surface area models on the Smartboard. Encourage and lead
the discussion/activity to come up with a strategy. Work over a few examples as a group.
Explore: Now bring out the geosolids and give each pair a rectangular prism or triangular prism. When they come
up with an answer they can switch with another group. Encourage them to come with some kind of formula.
Share: Have students share their answers for their methods. “Does our method from before work?” “What about for
triangular prisms?
Summarize: “What is the surface area?”
Day 7
(Which Solid Is Bigger)
Standards Covered:
5.3.1.1 - Describe and classify three-dimensional figures including cubes, prisms and pyramids by the number of
edges, faces or vertices as well as the types of faces.
5.3.2.2 – Use various tools and strategies to measure the volume and surface area of objects that are shaped like
rectangular prisms.
5.3.2.3 – 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.
5.3.2.4 - Develop and use the formulas V = ℓwh and V = Bh to determine the volume of rectangular prisms. Justify
why base area B and height h are multiplied to find the volume of a rectangular prism by breaking the prism into
layers of unit cubes.
Materials: Smartboard, paper, geosolids, rice.
Launch: Question of the day activity, Draw two containers on the Smartboard, “Which one is bigger? Why?” Discuss
Introduce all of the geosolids. Using the Smartboard have students copy down the names of the various geosolids
as they are introduced one by one.
Explore: In groups of 3-4, have students use a geosolid kit and sort out the geosolids from biggest to smallest.
Survey the different groups on how they’re classifying.
Share: Have student groups compare their lineup of solids to another group. Focus on differences. “Is there a way
we could know for sure which is bigger?” Introduce rice.
Break for Day 2
Summarize: “How will the rice help us figure out with solid is bigger? How could we use it?”
Day 8
(Which Solid is Bigger pt.2)
Standards Covered:
5.3.1.1 - Describe and classify three-dimensional figures including cubes, prisms and pyramids by the number of
edges, faces or vertices as well as the types of faces.
5.3.2.2 – Use various tools and strategies to measure the volume and surface area of objects that are shaped like
rectangular prisms.
5.3.2.3 – 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.
5.3.2.4 - Develop and use the formulas V = ℓwh and V = Bh to determine the volume of rectangular prisms. Justify
why base area B and height h are multiplied to find the volume of a rectangular prism by breaking the prism into
layers of unit cubes.
Materials: Smartboard, paper, geosolids, rice.
Launch: Question of the day activity, “Yesterday when we stopped we introduced the rice as a way to measure, how
do you think your biggest to smallest lineup will change?”
Explore: Hand out geosolids and rice to each group of four, continuing yesterday’s lesson.
Share: Once groups have an answer, compare the whole class’s biggest to smallest lineup. Rule out any arguments
as a class with measuring.
Summarize: “What were we measuring?” “Measuring how much space something takes up is volume, is that what
we did?”
Day 9
(Volume Unknown)
Standards Covered:
5.3.1.1 - Describe and classify three-dimensional figures including cubes, prisms and pyramids by the number of
edges, faces or vertices as well as the types of faces.
5.3.2.2 – Use various tools and strategies to measure the volume and surface area of objects that are shaped like
rectangular prisms.
5.3.2.3 – 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.
5.3.2.4 - Develop and use the formulas V = ℓwh and V = Bh to determine the volume of rectangular prisms. Justify
why base area B and height h are multiplied to find the volume of a rectangular prism by breaking the prism into
layers of unit cubes.
Materials: Graduated cylinders/beakers, objects that fit into cylinders/beakers, paper.
Launch: Question of the day activity, “How would I find the volume of something like an apple?” Discuss student
Show how to find the volume of an apple, and work through the concept as a group.
Explore: Send students to the various stations with cylinders/beakers/objects. Once they have completed the
volume of one object they can move to the next station.
Share: Share answers as a group for the objects on the Smartboard.
Summarize: “Would this work for something like a school bus?”
Day 10
(Deconstructed Volume)
Standards Covered:
5.3.2.2 – Use various tools and strategies to measure the volume and surface area of objects that are shaped like
rectangular prisms.
5.3.2.3 – 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.
5.3.2.4 - Develop and use the formulas V = ℓwh and V = Bh to determine the volume of rectangular prisms. Justify
why base area B and height h are multiplied to find the volume of a rectangular prism by breaking the prism into
layers of unit cubes.
Materials: Smartboard, base ten blocks.
Launch: Question of the day activity, draw a rectangular prism on the Smartboard constructed of cubes. ”How many
cubes would be in this solid.” Discuss student results and answers. Discuss how many cubes would be inside of
what we see.
Explore: Have students construct rectangular prisms out of the base ten blocks, and then find the volume of their
solid they created. Switch with a neighbor and find the volume of their shape.
Share: Share with a partner your method of solving their solid.
Explore: Combine your blocks with your partner and build a new prism using both sets of blocks. Then find the
volume.
Share: Find another pair and switch. Find the area of their solid.
Summarize: “Did anyone have a method of finding the area that worked quicker then counting the unit cubes one by
one?”
Day 11
(Finding Volume)
Standards Covered:
5.3.1.1 - Describe and classify three-dimensional figures including cubes, prisms and pyramids by the number of
edges, faces or vertices as well as the types of faces.
5.3.2.2 – Use various tools and strategies to measure the volume and surface area of objects that are shaped like
rectangular prisms.
5.3.2.3 – 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.
5.3.2.4 - Develop and use the formulas V = ℓwh and V = Bh to determine the volume of rectangular prisms. Justify
why base area B and height h are multiplied to find the volume of a rectangular prism by breaking the prism into
layers of unit cubes.
Materials: Smartboard, paper, base ten blocks.
Launch: Question of the day activity, ”What was your method for quickly finding the area yesterday, explain how it
worked.” Discuss results and student answers.
Introduce finding the area of a rectangular prism using student examples. L x W x H. Use a class example with base
ten blocks to test the formula.
Explore: Have students create rectangular prisms to test the formula.
Share: When students make something composed of more then one prism use it as an example for the class.
Explore: Send students back to workstations to make shapes composed of more then one prism.
Share: Have students walk around and look at other student’s shapes they’ve built.
Summarize: “Can we still find the volume when we have two prisms combined?” “What do you do to find the
Day 12
(Peas in a Room)
Standards Covered:
5.3.2.2 – Use various tools and strategies to measure the volume and surface area of objects that are shaped like
rectangular prisms.
5.3.2.3 – 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.
5.3.2.4 - Develop and use the formulas V = ℓwh and V = Bh to determine the volume of rectangular prisms. Justify
why base area B and height h are multiplied to find the volume of a rectangular prism by breaking the prism into
layers of unit cubes.
Materials: Smartboard, paper, grid paper, geosolids, peas, yard sticks, calculators, Counting on Frank by Rod
Clement.
Launch: Question of the day activity, “Read Counting on Frank by Rod Clement,” pose the question “How many
peas would fit in this room? What would you need to find the answer?” Discuss results and student answers.
Explore: In groups of 3-4, pick team names and have each team come up with an estimate. Define what will stay in
the room, and what we can take out. Answer any questions about the activity. Hand out materials and allow
students to derive a way to find the amount of peas to fill the room.
Share: Towards the end of the hour, have groups come together as a class to explain their method of finding how
many peas would fit in the room.
Summarize: “This room is what kind of a solid we have been learning about?” “What about the windows?” “Door?”
Day 13
(Peas in a room pt 2)
Standards Covered:
5.3.2.2 – Use various tools and strategies to measure the volume and surface area of objects that are shaped like
rectangular prisms.
5.3.2.3 – 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.
5.3.2.4 - Develop and use the formulas V = ℓwh and V = Bh to determine the volume of rectangular prisms. Justify
why base area B and height h are multiplied to find the volume of a rectangular prism by breaking the prism into
layers of unit cubes.
Materials: Smartboard, paper, grid paper, geosolids, peas, yard sticks, calculators, Counting on Frank by Rod
Clement.
Launch: Question of the day activity, “Do you think your estimate is too big or to small now that you’ve done some
measurement, why?”
Explore: Hand out materials to let students finish their activity.
Share: Once all groups have an answer, have them write it on the board and then find out how far off their estimate
was. Discuss differences and similarities. Talk about how groups measured and determined how many peas would
fit.
Summarize: “What was the main shape of this room?” “By investigating how many peas fir in the room, what math
concepts were you working on?”
Day 14
(Review Day)
Standards Covered:
5.3.1.1 - Describe and classify three-dimensional figures including cubes, prisms and pyramids by the number of
edges, faces or vertices as well as the types of faces.
5.3.1.2 – Recognize and draw a net for a three-dimensional figure.
5.3.2.1 – Develop and use formulas to determine the area of triangles, parallelograms and figures that can be
decomposed into triangles.
5.3.2.2 – Use various tools and strategies to measure the volume and surface area of objects that are shaped like
rectangular prisms.
5.3.2.3 – 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.
5.3.2.4 - Develop and use the formulas V = ℓwh and V = Bh to determine the volume of rectangular prisms. Justify
why base area B and height h are multiplied to find the volume of a rectangular prism by breaking the prism into
layers of unit cubes.
Materials: Smartboard, Wrap It Up activity.
Launch: Question of the day activity, “What have you learned over the past three weeks?”
Explore: Review area strategies, area formulas, nets, surface area strategies, surface area formulas,
faces/vertices/edges, volume strategies, volume formulas.
Summarize: Wrap it Up Activity
Day 15
(Post-Test)
Standards Covered:
5.3.1.1 - Describe and classify three-dimensional figures including cubes, prisms and pyramids by the number of
edges, faces or vertices as well as the types of faces.
5.3.1.2 – Recognize and draw a net for a three-dimensional figure.
5.3.2.1 – Develop and use formulas to determine the area of triangles, parallelograms and figures that can be
decomposed into triangles.
5.3.2.2 – Use various tools and strategies to measure the volume and surface area of objects that are shaped like
rectangular prisms.
5.3.2.3 – 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.
5.3.2.4 - Develop and use the formulas V = ℓwh and V = Bh to determine the volume of rectangular prisms. Justify
why base area B and height h are multiplied to find the volume of a rectangular prism by breaking the prism into
layers of unit cubes.
Materials: MCA Post Test
Launch: “We’re going to see how much we’ve learned over the past three weeks by taking a post-test. It’s
very similar to the one you took at the beginning of the unit, but take your time and relate the activities
we’ve done to the questions.”
Explore: Hand out MCA Post Test.
Summarize: “What was an activity you enjoyed from this unit?” “What was an activity you didn’t like?”
Use this sheet to wrap up your ideas, thoughts and questions about the last assignment/test.
One thing I remembered was…
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One thing I liked was…
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One thing I wasn’t sure of was…
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One thing I still have questions about is…
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Do These Make a Cube?
NCTM Worksheet Nets
Prism Nets
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