Bemidji State Summer Math Institute Geometry for Elementary and Middle Level Pat Greendahl Horace May Elementary [email protected] Zach McDermott Horace May Elementary [email protected] Overview
We are attempting to incorporate the additional discovery experiences for
our students into our Everyday Math curriculum.
We tried to keep the needs of our fifth grade students in mind as we planned
our two-week unit. All materials used will be available in our Appendix or
directly within the Everyday Math teacher materials for fifth grade.
Standard
We will address the following standard and the associated benchmarks.
Assessment
In additional to our observations we will be giving a pretest and a posttest.
Both are available in the Appendix.
Standard
Determine the area of triangles and
quadrilaterals; determine the surface
area and volume of rectangular prisms
in various contexts.
Benchmarks
1. Develop and use formulas to determine the area of triangles,
parallelograms and figures that can be decomposed into
triangles.
2. Use various tools and strategies to measure the volume and
surface area of objects that are shaped like rectangular prisms.
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.
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.
UNIT OUTLINE
Day
Description
1
Pre-test
2
Rectangular Surface Area
3-4
Triangular Surface Area
5-6
Parallelograms Surface Area & Perimeter
7-9
Volume and Surface Area of Rectangular Prisms
10-11
Volume and Surface Area of Right Prisms
12
Post-test
Area of Rectangle:
Benchmark 1:
Launch:
Develop and use formulas to determine the area of
triangles, parallelograms and figures that can be
decomposed into triangles.
How many of you would like to make a lot of money today?
Your job will be to design a patio for my house so I can enjoy
barbeques outside. My patio will be made of square bricks (3
different colors) and at last count I had 480(or less) bricks that
you need to you use to build my patio. Your job right now will
be to design my patio as many ways you can.
Rules:
1. Students must design more than one patio using squares or
rectangles
2. Students must work in groups of 2 or 3
3. Students must use all the bricks
4. Students must share their design in class
5. Students must figure out a way to find the area of the patio without
counting each square.
Explore:
Students will work in groups of 2 or 3 and design their patio.
Some will be urged to be more creative by adding extras to the
patio to get the job or will be asked to break up the patio into
smaller parts to make it look more appealing.
Sharing:
Students will come to the board to share their patio design with
the class and to their teacher what their plan was.
Summarizing:
Can anyone tell me a way of figuring out if you have
used all 480 bricks without counting each individual
brick. Basically we will be working on coming up with a
formula for the area of squares, rectangles, or multiple
rectangles.
Area of a Right Triangle
Benchmark 1:
Develop and use formulas to determine the area of
triangles, parallelograms and figures that can be
decomposed into triangles.
Materials: Graph Paper, Problem Sheet
Launch:
Explore:
In groups of 3-4 students, the students will use the handout and
graph paper to solve the provided problem. Some students will
arrive at the conclusion that a triangle is half a rectangle on
their own. After some exploration, a hint may be required to
arrive at a conclusion for triangles.
Share:
Students will come to the board to model the different ways that
they broke up the area to solve for the total area.
Summarize:
As a group, we will try to arrive at a rule for solving the
area of right triangles. Tomorrow we will explore more
examples of why this is true.
Area of Triangles
Benchmark 1:
Develop and use formulas to determine the area of
triangles, parallelograms and figures that can be
decomposed into triangles.
Materials: Geoboards
Launch:
“Last night I couldn’t sleep. I had an image that continued to
pop into my head.
Yesterday, we created a formula that found the area of a right
triangle, but how do we find the area of a triangle that isn’t a
right triangle?”
Explore:
“Create a triangle on your geoboard that isn’t a right triangle.
Exchange geoboards with your neighbor and find the area of
the triangle she created.”
Share:
Students will share examples on the board of the strategies they
used to solve the areas of their triangles.
Summarize:
“Using the triangles that we have looked at today, did
anyone find any connections to our triangles from
yesterday?” Guide the discussion to using the formula
that we arrived at yesterday. “Does it work?”
Area of Parallelograms
Benchmark 1:
Develop and use formulas to determine the area of
triangles, parallelograms and figures that can be
decomposed into triangles.
Launch:
Farmer Francine has been contacted by the state. They want to
build a highway across part of her land. They have offered her
a generous price, but she is looking to renegotiate. Each square
of the diagram represents one acre (43, 560 ft2). Francine wants
to have $1,000,000 to retire. How much should she ask per
acre?
Explore:
In groups of 3-4 students will attempt to find the area required
by the state to build the highway. They will also work on
Francine’s retirement plan.
Share:
Groups will share how they arrived at the surface area of the
highway and Francine’s asking price.
Summarize:
Review and list the strategies that students used to
calculate the total area of the land.
Area and Perimeter of Parallelograms
Benchmark 1:
Develop and use formulas to determine the area of
triangles, parallelograms and figures that can be
decomposed into triangles.
Launch:
“Today you need to use what you know about area and
perimeter to score more points than your partner.”
Explore:
In pairs the kids are going to play the Everyday Math
game rugs and fences. Students need to decide to use the
perimeter or the surface area of a polygon drawn from
the deck for their score each round. The game is
described in the Masters section of the Everday Math
teacher materials.
Sharing:
Individuals will be asked to share strategies that were
successful for them or strategies that did not work.
Summarize:
We will review rules for surface area and perimeter that
we have covered to this point.
Volume of Rectangular Prism
Benchmark 2:
-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.
Launch:
Hold up a model of a rectangular prism to the class, “What is
this? What makes a rectangular prism different than a
rectangle?” I will use this time to write the definition of a
rectangular prism and introduce to them base and height. Give
me some examples in the classroom of things you would
consider rectangular prisms. Remember when we measured
area we were measuring a flat surface or a two dimensional
object. What would I do to measure the inside of this
rectangular prism? Would we use squares? Why? 2
dimensional. What would we use instead of squares? Cubes!
Why? They are three-dimensional.
Vocabulary: Rectangular prism: is formed by six flat surfaces or
faces that are rectangles
Explore:
Students will be given Everyday Math Journal page 321 which
is a group of 3 rectangular prism nets in which they will cut out
and construct. It will be their job to estimate the volume or
number of cm cubes that will fit into each. They will then be
given cubes to find the real answer. I will direct them back to
our list of rectangular prisms on the board and ask them how
many cm cubes would fit into each.
Share:
Students will share how they arrived at their estimates before
being given cubes to check. They will also share individually
with small groups how their answers compared to their
estimates.
Summarize:
“Using what we found today, is there a way to do this
more quickly than counting cubes? How could we find
out more quickly how many cubes would fit inside the
podium or your desk without counting them all?”
Volume and Surface Are of Rectangular Prisms
Benchmarks 2-4:
-Use various tools and strategies to measure the volume and surface area of
objects that are shaped like rectangular prisms.
-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.
-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.
Launch:
“Henry wants to build a castle for his kingdom. It is not a
wealth kingdom, so he has to build his castle using unifix
cubes. Henry is only six years old, so this isn’t all bad.”
Explore:
Henry wants his castle to be six blocks wide and four blocks
deep. He has 408 blocks. How tall can he build his castle?
This castle is 3 blocks wide, 3 blocks deep and 2 blocks tall.
How many blocks will Henry need to build the castles shown
on p. 322 of your Everyday Math Journal?
In groups of 2-3, students will calculate the volumes of the
shapes on p. 322.
Share:
Groups will share one example of how they arrived at their
solution for a problem on p. 322.
Summarize:
Is there a faster way to find how many cubes he needs?
Do we have to count them every time? Together let’s
write a rule that will help us solve these more quickly.
Volume and Surface Are of Rectangular Prisms
Benchmarks 2-4:
-Use various tools and strategies to measure the volume and surface area of
objects that are shaped like rectangular prisms.
-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.
-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.
Launch:
“Henry is not satisfied. He doesn’t think the blocks look nice
enough. He has found gold stickers that are 1 cm2.. He is going
to cover all sides of his castle (including the bottom) with
stickers. How many stickers will he need for the castle designs
that we looked at yesterday?”
Explore:
In groups of 2 -3, students will work through covering the
various designs from Everyday Math Journal p. 322 with
stickers.
Share:
Groups will each share a solution and the process they used
with the class.
Summarize:
“Did any groups use a strategy other than counting
squares. Can we turn this into a rule?”
Volume and Surface Area Right Prisms
Benchmarks 2-4:
-Use various tools and strategies to measure the volume and surface area of
objects that are shaped like rectangular prisms.
-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.
-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.
Launch:
Farmer Francine is back! She is searching for a shed design
that will have the greatest capacity for all the things she wants to buy with
her new-found wealth. She doesn’t want to have a normal shed. She thinks
someone with her wealth should have something fancier. Let’s look at one
together.
We will draw in the layers together to show a height of 50 m.
Explore:
In groups of 2-3, students will solve the volumes for the prisms
found in the Masters on pg. 284 of Everyday Math using the
strategy that we used to work through the triangular prism.
Share:
Groups will be asked to share a strategy that they used for one
of the models on pg. 284.
Summarize:
“Did anyone have a strategy where they didn’t draw in
the layers?” As a group we will try to construct a rule for
finding the volume of the prisms.
Volume and Surface Area Right Prisms (Day 2)
Launch:
Carpenter Casey is no dummy. He knows that Francine has
money to burn, so he is eager to get the job of building her
shed. He is going to charge her according to the surface area of
the building. He is asking $50 per m2. Of the buildings that
Francine considered yesterday, how much will the most
expensive shed be? How much will the least expensive shed
be?
Explore:
In groups of 2-3, students will calculate the surface areas of the
sheds we looked at yesterday. Using those calculations they
need to determine how much Casey will charge her for each
shed.
Extension: How much money could she save if Bob the
Builder wanted to charge her $3 less per m2?
Share:
Groups will be asked to share their process for finding the
surface area of an individual shed and the cost of construction.
Summarize:
“Did anyone recognize any patterns that appeared as the
groups shared their work?” We will create some general
rules for finding surface area of some right prisms.
Appendix
ASSESSMENT
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