Download Making Squares

Shaping Up – Activity 1
Based in part on Marilyn Burns About Teaching Mathematics page 322
Current Standard(s): 3MG2.1; 3MG2.2; 3MG2.3; 4MG3.3; 4MG3.7; 4MG3.5; 4MG3.8; 4MG1.4;
5MG2.1; 5MG2.2; 5MG1.1; 6MG2.3;
Common Core Standard(s): 3.G.CA-1; 4.G.1; 4.G.2; 4.G.3; 4.MD.7; 5.G.CA-1; 5.G.CA-2; 5.G.CA3; 6.G.1;
Focus: Helping students recognize relationships between the area of the individual polygons.
Guiding students in a vocabulary rich discussion in which they share their observations about how
to determine the area of each of the tangram pieces.
Students need:
 Tangrams
Task 1: Draw a representation of each of the seven tangram pieces in your math journal. Correctly label
each of the seven polygons (i.e. scalene triangle, square, etc.).
square
Task 2: If the small square tangram piece has an area of 1 square unit, determine the area of the six other
tangram pieces. Record the area on each of the polygons you drew in your journal.
square
1
Discussion: What are some of the relationships you noticed about the area of these polygons?

½
½



½
½


The square is made up of two smaller congruent right isosceles
triangles.
½+½=1
The two triangles = 1 square so we can say that when we put the two
triangles together we make a square that is congruent with the square
in our tangram set.
The medium triangle is made up of two congruent smaller triangles.
Each of those triangles equals ½ and I know that ½ + ½ = 1. This
explains why the medium triangle has an area of 1.
I can also say that two small triangles stuck together would make one
triangle congruent to the medium triangle.
I can also say that the smaller right isosceles triangle is similar to the
medium isosceles triangles.
Shaping Up – Activity 1 Continued
Based in part on Marilyn Burns About Teaching Mathematics page 322

½

½

½
½
The larger triangle is made up of four congruent smaller triangles. Each
of those triangles equals ½ and I know that ½ + ½ + ½ + ½ = 2. This
explains why the larger triangle has an area of 2.
I can also say that four small triangles stuck together would make one
triangle congruent to the large triangle.
I can also say that the smaller right isosceles triangle is similar to the
larger isosceles triangles.

½
½
The parallelogram is made up of two smaller congruent right isosceles
triangles.
 ½+½=1
 The two triangles = 1 parallelogram so we can say that when we put the
two triangles together we make a parallelogram that is congruent with
the parallelogram in our tangram set.
What relationship exists between the square, medium triangle, and the parallelogram?
 Even though they are different shapes, they all have an area of one.
 They are all equal to two of the smallest triangles.
Extension:
Tell students that the smallest triangle has an area of 1. What would the areas of the other tangram pieces
be?
What if the largest triangle had an area of 8? What would the areas of the other tangram pieces be?
Shaping Up – Activity 2
Based in part on Marilyn Burns About Teaching Mathematics page 322
Current Standard(s): 3MG2.1; 3MG2.2; 3MG2.3; 4MG3.3; 4MG3.7; 4MG3.5; 4MG3.8; 4MG1.4;
5MG2.1; 5MG2.2; 5MG1.1; 6MG2.3;
Common Core Standard(s): 3.G.CA-1; 4.G.1; 4.G.2; 4.G.3; 4.MD.7; 5.G.CA-1; 5.G.CA-2; 5.G.CA3; 6.G.1;
Focus: Developing students’ visual-spatial sense by having them construct squares using specific shapes.
Helping students recognize relationships between the area of the individual polygons and the area
of the squares they made.
Guiding students in a vocabulary rich discussion about the properties of the polygons used to
construct the squares (symmetry, congruence, flip, rotation, angles, area, and etc.).
Students need:
 Tangrams
 Representations from Shaping Up Activity 1
Task 3: Below is a representation of how you could use 2 of the trangrams to create a square (draw this
on the board for the students). Have students choose two other tangrams to create a square. Have
students record a representation of their square in their journal.
½
½
Task 4: Have students make a square using any 3 tangram pieces. Then they should make a square using
any 4 tangram pieces, any 5 tangram pieces, and then all 7 tangram pieces (some solutions are below).
1
½
½
½
1
1
1
2
½
2
½
½
1
½
1
2
1
½
1
Challenge Task: Creating a square using six of the tangram pieces is impossible. Look at the area of each
of your squares, and each of your tangram pieces. Think about why it is impossible to make a square
using six pieces.
Shaping Up – Activity 3
Based in part on Marilyn Burns About Teaching Mathematics page 111
Current Standard(s): 3MG2.1; 3MG2.2; 3MG2.3; 4MG3.3; 4MG3.4; 4MG3.5; 5MG1.1; 5MG2.1;
5MG2.2; 6MG2.2; 6MG2.3
Common Core Standard(s): 3.G.CA-1;4G.3; 5.G.CA-1; 5.G.CA-2; 5.G.CA-3; 7.G.2
Focus: Developing students’ visual-spatial sense by having them construct specific polygons.
Guiding students in a vocabulary rich discussion about the properties of the polygons constructed
(symmetry, congruence, flip, rotation, angles, area, and etc.).
Students need:
 One set of tangrams
 Representations from Shaping Up Activities 1 and 2
Task 1: Using the three smallest triangles make the shapes listed in the table. Have students record
representations of their work in their math journal by copying the chart below (it works well if you have
them turn their journal sideways so they can make a longer chart). Be sure to have them include the
area on each of the pieces they used in their representation.
Square
3 small
triangles
Rectangle
Isosceles right
triangle
Trapezoid
Parallelogram
½
1
½
½
½
1
1
½
½
1
1
½
½
1
½
½
Discussion: What are some of the patterns and relationships that you noticed about the square?
 When I made the square using the three smaller triangles I noticed that the two smallest triangles
are equal to one of the bigger triangle (they are congruent). That makes sense because the smaller
triangles each have an area of ½ and ½ + ½ = 1 which is the area of the bigger triangle
 The two smaller triangles are congruent.
 The smaller triangles are similar to the bigger triangle (they are both isosceles right triangles).
 If I were to glue the two smaller triangles together I would create a triangle that is congruent to the
larger triangle.
 The two smaller triangles are congruent but the top one has been rotated ¼ of a turn from the
bottom one to complete the square.
 All three of these triangles are isosceles right triangles.
 If I glued all three pieces together I can see that the outside angles would all equal 90°.
 Since the two smaller triangles are congruent and the bottom right angle is equal to 90°, then each
of the angles in that corner must equal 45° because 90° ÷ 2 = 45°
Shaping Up – Activity 3 Continued
Based in part on Marilyn Burns About Teaching Mathematics page 111
Discussion: What are some of the patterns and relationships that you noticed about the rectangle?



When I added ½ to each side of the bigger triangle I created a rectangle.
The two smaller triangles are congruent but I flipped the one on the left to complete the rectangle.
If I drew a line down the middle of the larger triangle I would have four congruent smaller
triangles.
You can hold similar discussions for each of the shapes made. Focus the discussion so that it meets your
grade level standards.
Task 2: Using the five smallest tangram pieces (all but the two large triangles) make the shapes listed in
the table. Record representations of your work in your math journal, by adding onto the table you have
already created.
Square
Rectangle
Isosceles right
triangle
Trapezoid
Parallelogram
5 small pieces
Hold discussions similar to the ones you had when they used only the three triangles to make the shapes.
Task 3: Using all seven tangram pieces make the shapes listed in the table. Record representations of
your work in your math journal, by adding onto the table you have already created. Hold discussions
similar to the ones you had when they used only the three triangles to make the shapes.
Square
Rectangle
Isosceles right triangle
Trapezoid
Parallelogram
All 7 pieces
Hold discussions similar to the ones you had when they used only the three triangles to make the shapes.
Solutions Shaping Up Activity 3
5 Pieces
½
1
1
½
1
½
1
1
1
½
½
1
1
½
½
1
½
1
1
1
½
1
1
1
½
7 Pieces
2
1
1
2
2
2
1
½
1
1
½
1
½
½
1
½
1
½
1
1
½
1
1
½
2
2
2
2
2
1
1
½
½
1
2