Number Sense
Developed by Using Geometry
Middle School
By:
Jessica Eng, [email protected]
Kendra Wold, [email protected]
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Executive Summary
This unit will be for two weeks, completing one lesson per
day for ten consecutive days early in the year during fifth
grade. It uses Tangrams to introduce simple geometry and
move into more complex figures the breakdown/construction
of some polygons. The students will first be introduced to
Tangrams and will talk about different shapes. We will
discuss how most polygons are interrelated.
The students will create their own Tangram sets and will
work through a variety of activities and lessons with the
Tangram pieces. Some activities will be virtual and others will
be physical.
Students will recall their prior knowledge of translations,
reflections, and rotations. They will use this knowledge to
manipulate the pieces in their Tangram sets to create simple,
semi-simple, and complex figures. Students are also learning
about position in space, new vocabulary, and properties of
figures all at the same time.
At the end of two weeks students will create their own
bulletin board to display in the hallway. They will each create
an animal for the “Zoometry” Zoo and calculate the area and
perimeter of their animals. They will also be comparing and
contrasting their animal with other animals in the zoo.
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Lesson 1
Objective: Students will use Tangram pieces to form given polygons.
MN Standards:
Geometry & Measurement
4.3.3.1
Apply translations (slides) to figures.
4.3.3.2
Apply reflections (flips) to figures by reflecting over vertical or horizontal lines and relate reflections to lines
of symmetry.
4.3.3.3
Apply rotations (turns) of 90˚ clockwise or counterclockwise.
7.3.2.1
Describe the properties of similarity, compare geometric figures for similarity, and determine scale factors.
For example: Corresponding shapes in similar Tangram sets are the same size.
Launch: Has anyone ever heard of Tangrams? What do you think they are? Who
do you think invented/created them? Why? Tangrams are an Ancient Chinese
moving piece puzzle, consisting of 7 geometric shapes. Show a set of Tangrams.
Here is one Tangram set. What do you think the pieces are for? What do you
think you can create from these pieces?
Explore:
Describing figures and visualizing what they look like when they are transformed
through rotations or flips or are put together or taken apart in different ways are
important aspects of geometry in the lower grades. This two-part Tangram
example demonstrates the potential for high-quality experiences provided by using
hands on shape manipulatives. Write the words translation (slide), reflection (flip),
and rotation (turn) on the board.
Is it possible to complete all these tasks? Try these Tangram challenges with the
virtual Tangrams:
* Make a square using only one Tangram piece.
* Make a square using two, three, four, five, and seven Tangram pieces.
Share: Have students discuss and share everything they discovered with their
pieces for day one.
Summarize: Who can tell me what a slide is? What is a flip? What does it mean
to turn an object? Can anyone show class what they are? How are the squares
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similar or different? How are the Tangram pieces similar or different? Is it
possible to create our own Tangram pieces? What are some other shapes or
polygons that we can make? So to sum it up, today we used seven pieces of
different shapes to create a number of different squares.
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Lesson 2
Objective: Students will create their own Tangram set and explore the options of
making squares by using a different amount of pieces each time.
MN Standard:
Geometry & Measurement
4.3.3.1
Apply translations (slides) to figures.
4.3.3.2
Apply reflections (flips) to figures by reflecting over vertical or horizontal lines and relate reflections to lines
of symmetry.
4.3.3.3
Apply rotations (turns) of 90˚ clockwise or counterclockwise.
4.3.3.4
Recognize that translations, reflections and rotations preserve congruency and use them to show that two
figures are congruent.
5.1.3.4
Solve real-world and mathematical problems requiring addition and subtraction of decimals, fractions and mixed
numbers, including those involving measurement, geometry and data.
For example: Combining different polygons to create one. Each polygon is considered a fraction of the whole.
7.3.2.1
Describe the properties of similarity, compare geometric figures for similarity, and determine scale factors.
For example: Corresponding shapes in similar Tangram sets are the same size.
Launch: Who thinks they can make a square with two different shapes? How about
three shapes? How about more than three shapes shapes? Today we will make our
own set of Tangrams and work with them to see if we can create squares..
Explore: Students will construct their own set of Tangram pieces. Materials
needed are: a piece of rectangular paper for folding, a pair of scissors, envelopes,
and a ruler (optional).
The steps are on page five.
Fold a rectangular piece of paper so that you form a square.
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Cut off the extra flap.
1. Cut the square into two triangles.
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2. Take one triangle and fold it in half. Cut the triangle along the fold into two
smaller triangles.
3. Take the other triangle and crease it in the middle. Fold the corner of the
triangle opposite the crease and cut.
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4. Fold the trapezoid in half and fold again. Cut along both folds.
5. Fold the remaining small trapezoid and cut it in two.
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Now that students have their 7 pieces cut apart and all together, prompt them to
create a list of all their pieces. They could even record this list on the outside of
their storage envelope. Begin writing a list on the board by having five slots
bulleted for the following:
¾
¾
¾
¾
¾
a small square
two small congruent triangles
two large congruent triangles
a medium triangle
a parallelogram
Ask students to try to make a square using all 7 pieces.
Share: Students will discuss and show how they found a square using all 7 pieces, if
they did, and they will also share more if any other discoveries were found. These
can be shown on an overhead or an Elmo (document camera). What are the three
terms we use when manipulating or moving the Tangram pieces? Can anyone give an
example of a rotation? Reflection? Translation? How did we categorize the
different pieces? What do we create when we add pieces together? A triangle is
a fraction of the whole square. Can anyone find any other fractions of the whole
square?
Summarize: The three terms we use when manipulating or moving Tangram pieces
are rotation, reflection, and translation. We hear more of flips and turns. We now
know the more mathematical terms. We also categorized our Tangram pieces by
creating a list of pieces in one set.
Store all pieces in the envelope given at the beginning of the lesson. Pose
questions about what other types of shapes students could create with their
Tangrams. Ask whether students think they could create animals or other shapes
using pieces for the next lesson. Come back to the launch questions and find out if
they were able to create that square and how they feel about it now.
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Lessons 3 and 4
Objective: Students will use their Tangram pieces to create other polygons and
practice their addition skills in adding pieces together.
MN Standards:
Geometry & Measurement
5.1.3.4
Solve real-world and mathematical problems requiring addition and subtraction of decimals, fractions and mixed
numbers, including those involving measurement, geometry and data.
For example: Combining different polygons to create one. Each polygon is considered a fraction of the whole.
7.3.2.1
Describe the properties of similarity, compare geometric figures for similarity, and determine scale factors.
For example: Corresponding shapes in similar Tangram sets are the same size.
Launch: If we gave each Tangram piece a point value, do you think you could create
a polygon with the most points possible? Is there more then one way to add
numbers up to equal 15 or even 20?
Explore: Students will need a copy of the point labeled Tangram sheet, scissors to
cut the pieces out, a notebook to keep track of points and shapes. They will also
receive a copy of the “Daring Dozen Polygons”.
Pass out the point labeled Tangram sheet to each student. Each student may work
with another, alone, or in a group of three. One Daring Dozen Polygon page (front
and back) is needed per group. If students would like a copy of their own they
may. They will record their scores and the way they used pieces to create specific
polygons by tracing their findings next to the one they figures out.
For example: Draw the triangle worth 2 points connected to the triangle worth 8
points at their right angles. Show example. Here you have a triangle worth 10
points. If you place the triangle on a different side you will not create a triangle.
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Daring Dozen Polygons
Chosen Polygon
1.
Use 2 pieces to create a triangle
worth 10 points.
2.
Use 2 pieces to make a square of 9
points.
3.
Use 2 pieces to create a trapezoid
worth 10 points.
4.
Use 3 pieces to make a 12-point
rectangle.
5.
Use 3 pieces to make a 16-point
square.
6.
Use 3 pieces to create a 15-point
parallelogram.
Illustrations
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Points
7.
Use 4 pieces to create a 14-point
square.
8.
Use 4 pieces to create a rectangle
that equals 18 points.
9.
Use 4 pieces to create a square
that equals 20 points.
10.
Use 5 pieces to create a 25-point
parallelogram.
11.
Use 5 pieces to make a trapezoid
worth 25 points.
12.
Use 5 pieces to make a 22-point
rectangle.
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Share: Have students discuss how many were able to find possible solutions for all
12. Also discuss how many found other polygons and points that weren’t listed.
Have some students share on the overhead what they created.
Summarize: Discuss about the importance of how many different ways polygons
can be created and why that might be. Explain how specific polygons, when
combined in different ways, create other polygons. Discuss how adding point values
gives the created polygon a value. By adding points it is clear that each piece is a
fraction of the whole. Can the same point value be manipulated to create different
shapes?
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Lesson 5
Objective: Students will choose a picture and use Tangram pieces to fill in an
outline and use a website and online game to increase shape awareness.
MN Standards:
Geometry & Measurement
4.3.3.1
Apply translations (slides) to figures.
4.3.3.2
Apply reflections (flips) to figures by reflecting over vertical or horizontal lines and relate reflections to lines
of symmetry.
4.3.3.3
Apply rotations (turns) of 90˚ clockwise or counterclockwise.
7.3.2.1
Describe the properties of similarity, compare geometric figures for similarity, and determine scale factors.
For example: Corresponding shapes in similar Tangram sets are the same size.
Launch: Today we are going to the computer lab to practice our Tangram
skills? If I were to give you an outline of a rabbit, could you fill in the
correct Tangram pieces to make it look like a rabbit and fit? Let’s try it out
today with all sorts of different pictures and figures.
Explore: Materials needed are the computer and internet connection, using
www.kidscom.com/games/Tangram.html
http://pbskids.org/sagwa/games/Tangrams/index.html
Choose a picture and use all seven pieces to fill in the outline. Use the "Hint"
button if you need help. You can choose any picture and once you get it
correct, try another one.
Young students' experiences with puzzles provide a background for
undertaking this activity. Because similar puzzles are available for use with
plastic or paper Tangrams, students can move back and forth between
concrete materials and the computer environment.
Share: What do you do when you cannot figure out a puzzle? Can some
Tangram pieces substitute for others? Did anyone flip or rotate some
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pieces to really make them fit? Would anyone like to share what they
discovered? Can you fill the outlines in more than one way? How many
different ways are there to fill in each shape? Did anyone have to rotate,
flip, or slide their shapes to fit into the outline?
Summarize: This is a great way for us to manipulate the pieces without
using our hands and seeing it a different way. Moving the pieces in different
ways helps us see things in a new perspective. Today we used rotations and
reflections by manipulating virtual Tangrams. We also discovered that some
pieces do have properties of similarity.
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Lessons 6 and 7
Objective: Students will use Tangram pieces to fill in non-outlined shapes and
figures to discover how many different ways the polygons can be used.
MN Standards:
Geometry & Measurement
4.3.3.1
Apply translations (slides) to figures.
4.3.3.2
Apply reflections (flips) to figures by reflecting over vertical or horizontal lines and relate reflections to lines
of symmetry.
4.3.3.3
Apply rotations (turns) of 90˚ clockwise or counterclockwise.
5.1.3.4
Solve real-world and mathematical problems requiring addition and subtraction of decimals, fractions and mixed
numbers, including those involving measurement, geometry and data.
7.3.2.1
Describe the properties of similarity, compare geometric figures for similarity, and determine scale factors.
For example: Corresponding shapes in similar Tangram sets are the same size.
Launch: On Friday we worked with Tangram pieces on the computer and filling in
outlined shapes, today we will work with filling in full shapes and figures. Who
thinks they could create a number or a dinosaur? Take out your Tangram pieces
and let’s see.
Explore: Students will each get one of the four design pages. They will need their
own set of Tangrams to create the designs. Once a student is finished creating all
designs on one sheet they are to move onto another sheet by turning in the first
one for a different one. Students will record their findings in their math
notebooks. They will show how they created each design by tracing the finished
design and adding lines to show each separate piece that they used to create the
design.
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Share: At the end of each day, students will share how they made certain designs
and if there were different ways to come up with the same design. Do you think
we could find the area of each shape? Could some shapes have the same area as
another? Could some shapes look different from each other and still have the
same area? How could we find out? How many of you had to rotate your pieces to
fit in the pictures? How many of you used translations? What about reflections?
Summarize: Today we moved our Tangram pieces in a variety of ways to create
different letter and number designs. What other types of designs do you think we
could create?
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Lesson 8
Objective: Students will have a better understanding of area without formulas,
compute the area of polygons by decomposition, become familiar with names of
certain polygons, learn the term similar and develop geometric intuition.
MN Standards:
Geometry & Measurement
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
Determine the surface area of a rectangular prism by applying various strategies.
For example: Use a net or decompose the surface into rectangles.
5.1.3.4
Solve real-world and mathematical problems requiring addition and subtraction of decimals, fractions and mixed
numbers, including those involving measurement, geometry and data.
7.3.2.1
Describe the properties of similarity, compare geometric figures for similarity, and determine scale factors.
For example: Corresponding shapes in similar Tangram sets are the same size.
Launch: What can you make out of your 7 pieces? Who thinks we could make 10
different shapes and designs? What is a rectangle, a trapezoid, and pentagon?
How do we know what shape to call something when we make it? If I designed a
bird, what would that be and could I measure that using Tangram pieces?
Explore: Materials needed are each students’ package of Tangram pieces, a
notebook, a pencil, a ruler, an overhead projector. Students will also need to
remember what was discovered the lesson before, during their creations of the
pieces.
Activities:
Make a square with the medium-size triangle and the two small congruent
triangles. What is the area of this square? How do you know? Sketch the
square in your notebook and record its area.
Make a rectangle with the parallelogram and the two small congruent
triangles. What is the area of this rectangle? How do you know? Sketch the
rectangle in your notebook and record its area.
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Construct a triangle congruent to the large triangle shown below without
using the small square. Sketch the large triangle in your notebook and record
its area
Make a square congruent to the square shown below without using a large
triangle. Sketch the square in your notebook and record its area.
Construct a square using all seven Tangram pieces. What is its area? How do
you know? Sketch this large square in your notebook and record its area.
Find a trapezoid congruent to the trapezoid shown below. What is the area
of this trapezoid? How do you know? Sketch the trapezoid in your notebook
and record its area.
Find a trapezoid that is similar (but not congruent) to the trapezoid shown
above. What is the area of this trapezoid? How do you know? Sketch the
trapezoid in your notebook and record its area.
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Create a pentagon congruent to the pentagon shown below. What is the area
of this pentagon? How do you know? Sketch the pentagon in your notebook
and record its area.
Create a pentagon congruent to the pentagon shown below without using the
small square. What is the area of this pentagon? How do you know? Sketch
the pentagon in your notebook and record its area.
Share: Have students share their discoveries and polygons that were created.
Students can put them on the overhead and show their peers. What kind of
formulas did we come up with to measure the area and perimeter without using a
measurement tool? What did some of you choose to use as your measurement tool?
Which piece was easiest to use and why? What was the area of the polygons, using
just the other shapes for information?
Summarize: Today we learned how calculate the area for each piece individually
and the area for a whole shape. (An extension of the activity would be to use
actual measuring tools or saying the large square has an area of 12 square units.)
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Lesson 9
Objective: Students will use the information they learned in the previous lesson on
area to measure and record the area of the shapes and figures that they filled in
and created on lessons 6 and 7. Students will also compare the areas to see if
there are any similarities or differences.
MN Standards:
Geometry & Measurement
5.3.2.1
5.3.2.2
Develop and use formulas to determine the area of triangles, parallelograms and figures that can be decomposed
into triangles.
Determine the surface area of a rectangular prism by applying various strategies.
For example: Use a net or decompose the surface into rectangles.
5.1.3.4
Solve real-world and mathematical problems requiring addition and subtraction of decimals, fractions and mixed
numbers, including those involving measurement, geometry and data.
7.3.2.1
Describe the properties of similarity, compare geometric figures for similarity, and determine scale factors.
For example: Corresponding shapes in similar Tangram sets are the same size.
Launch: As we ended our lesson the other day on filling in the shapes with our
Tangram pieces, I posed the questions to you about the area of those shapes and if
they would all be different or all the same and if we could find it. Today you will
see if it is possible to find area of such strange looking shapes.
Explore: Students will need their notebooks with their shapes recorded from
lessons 6 and 7. Students will also need their area information from lesson 8.
Students may need copies of the shapes again. Have students find all of the areas
that they can during the class period.
Share: Discuss amongst each other if different shapes had the same areas. Why
or why not could that be possible? Show some examples on an overhead or elmo.
Any other discoveries??
Summarize: So as we can see, all different shapes can have the same area or even
be created in different ways. I just wonder if you could create a community of
some sort using these shapes and areas.
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Lesson 10
Objective: Students will manipulate Tangrams by using flips and rotations to help
create a specific design. Students will also create a collaborative classroom
bulletin board project to display what they have learned about area and perimeter
by using colorful Tangrams.
MN Standards:
Geometry & Measurement
4.3.3.1
Apply translations (slides) to figures.
4.3.3.2
Apply reflections (flips) to figures by reflecting over vertical or horizontal lines and relate reflections to lines
of symmetry.
4.3.3.3
Apply rotations (turns) of 90˚ clockwise or counterclockwise.
5.3.2.1
Develop and use formulas to determine the area of triangles, parallelograms and figures that can be decomposed
into triangles.
5.1.3.4
Solve real-world and mathematical problems requiring addition and subtraction of decimals, fractions and mixed
numbers, including those involving measurement, geometry and data.
7.3.2.1
Describe the properties of similarity, compare geometric figures for similarity, and determine scale factors.
For example: Corresponding shapes in similar Tangram sets are the same size.
Launch: Yesterday we briefly talked about a community. Do any of you have any
ideas as to what type of community we could create? Brainstorm ideas and create
a list on the board. Record the list for possible future use. (transportation, birds,
letters, animals, etc.) Where do you usually find different animals in the same
location? Where do you pay to see animals? Show the page with the animals on the
overhead. Also make a few copies of this page to pass around the classroom and
possibly a poster to hang up on a different wall in the classroom.
Explore: Students will each choose an animal to create for the Zoometry
classroom bulletin board. They may create more than one animal if they would like
to. They may also create and name their own animal.
Each student will get one 17 x 22 white piece of paper. They will also choose the
color that they need, or would like for their animal. (They may use more than one
color or swap pieces with friends)
Each student will once again cut out a set of Tangrams for themselves by using
colored construction paper this time (refer back to Lesson 2 for directions). Once
the shapes are cut out students proceed by manipulating the shapes on their white
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piece of paper to create the shape of their chosen animal. Once they have figured
it out they will glue their animal to their paper. They will label their exhibit by
gluing on a small sign in front of their animal naming the animal, the area of the
animal, and the perimeter of the animal.
Share: Discuss amongst each other if different animals have different areas and
perimeters. Which animals are different but have the same area. Do you think a
zookeeper would know how big an exhibit would be for a giraffe? What about for
the monkeys? Do they need the same size area to live in? How about their body
perimeter? Are they all the same size?
Summarize: How many flips and rotations do you think you had to use to make your
animal for the zoo? Do you feel the last two weeks of working with the Tangrams
familiarized you in creating different designs with an assortment of polygons?
What other things might you create with these shapes?
Once all of the animals are complete the class will compare and contrast each
exhibit. They will find out the animal with the greatest area and the animal with
the greatest perimeter. They could also find the exhibit with the best background
and most realistic background (variation examples).
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Zoometry
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Additional Tangram Designs
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