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Classroom Strategies Blackline Master Page 35II - 1

Construct a Gift Box

The plan below shows how to construct a gift box. Change the measurements on the plan to

make the box fit your item. Then construct your box out of tag board.

3 cm8 cm

5 cmFront

Top

Top

BottomLeft Right

Front

Back Flaps for taping.

Lid flap

Classroom Strategies Blackline Master II - 2

Ladders and Saws

Reprinted with permission from Woodrow Wilson Foundation -- 1991 Mathematics Institute

Cut out any type of triangle from stiff paper, and color each angle a different color. (If each student has a different triangle, the conclusions reached seem even more powerful to them.) Draw a straight base line fairly near one edge of a blank sheet of unlined paper and line one edge of the triangle up along the base line. Trace one edge of the triangle away from the base line, and mark the point where the third vertex meets the base line. (See Figure 1.)

Then slide the triangle down the base line to the marked point and repeat -- trace the top edge and mark the lower vertex as shown in figure 2. Color in each angle as you make it.

Y

R G

Mark this point.

Base lineFigure 1

Trace this side.

Continue this process all the way down the base line, creating a series of parallel lines, as shown in Figure 3. The resulting figure is called a "ladder."

R R R

Y

R G

Mark this point.

Base lineFigure 2

Trace this side.

R

Figure 3

Base line

Base line

Classroom Strategies Blackline Master 7 II - 3

Ladders and Saws continuedTo create a "saw," line up the triangle along the base line and trace its other edges. Color in the angles as you go. (See figure 4).

Y

R G

Figure 4

R

Y Y

GG

Now, rotate the triangle so that the vertex marked Y fits snugly against the base line, and the side marked YG fits against its tracing as shown in Figure 5. Then trace the top side.

Y

R G

Figure 5

R

Y Y

GG

G

Y

R

The "ladders" in Figure 3 illustrate corresponding angles formed by parallel lines cut by a transversal. The "saw" in Figure 4 shows alternate interior angles formed by parallel lines and parallel transversals. What conclusions can you make about each type of angle?

On a blank piece of paper, start with a base line, and make a row of ladders, coloring the angles as you go. Then turn your triangle around to complete the saws, again coloring in angles to match the original triangle. Repeat until the entire page is covered. Parts of two rows are shown in Figure 6.

Y

R GR

Y Y

GG

G

Y

RR R

Y Y

G GG R

RY

G R

Y Figure 6

R

G R

YR


Ladders and Saws continued

Once the page is covered with ladders and saws, look carefully at the figure and make as many conclusions as you can. You may want to outline or highlight certain parts of the figure, or cover up certain parts to make your conclusions stand out. How many conclusions can you make based on your diagram? Do your conclusions hold when you look at someone else's diagram? What would happen if you started with a special kind of triangle? Can you make any special conclusions about these figures?

Some seventh grade students have come up with as many as 15 different conclusions. Can you top them?

Some Observations

1. The sum of the angles of any triangle equals 180°.2. The sum of the angles of any quadrilateral is 360°.3. The sum of the angles of any polygon is (n - 2)180°.4. The opposite angles of a parallelogram are always equal.5. Angle size is preserved in similar figures.6. Compare areas of similar figures.7. Circles are 360°.8. Vertical angles are equal.9. Alternate interior angles are equal.10. Alternate exterior angles are equal.11. Corresponding angles are equal.12. Supplements of the same angle are equal.13. An exterior angle of a triangle is equal to the sum of the two non-adjacent interior angles. 14. Triangles will tessellate.15. Diagonals in a parallelogram bisect each other.16. The diagonal of a parallelogram divides it into two congruent triangles.17. Explore transformations, reflections, and rotations.


Guess Which Figure

1 432

5 876

9 121110

13 161514

17 201918

ab

c abc

abc a

bc

ab

c ab

cab

c

abc

abc

ab

ca

bc a

bc

abc

a

bc a

b

c ab

c

a

bca

b

ca

b

ca

bc


Draw it Again, Sam

IIIIII IV

A

C

D

EF

G

BSlide Sam into quadrant IV. Then flipSam upside down into quadrant III.List the coordinates.

I IV IIIA (-3,6) (9, -6) (-3, -6)B _____ _____ _____

C _____ _____ _____

D _____ _____ _____

E _____ _____ _____

F _____ _____ _____

G _____ _____ _____

III

III IV

Now draw your own design in quadrantII. Slide it into quadrant IV. Then flipit upside down into quadrant III.List the coordinates.

I IV IIIA _____ _____ _____

B _____ _____ _____

C _____ _____ _____

D _____ _____ _____

E _____ _____ _____

F _____ _____ _____

G _____ _____ _____

Name_______________________________________Date___________


“Mira” ActivityName_______________________________________Date___________


Name_______________________________________Date___________

Quilter Challenge

Here is one way to create a design on a grid using the two shapes shown below.

A

B

C

D

E F

G

Use the shapes to create your own design. List the coordinates of each triangle and each parallelogram used.

Parallelogram coordinatesA B C D

Triangle CoordinatesE F G


x

y

Geometric Transformations - 7a Name________________________ Date_______________

1. Draw the pentagon below after it has been reflected over the y- axis. Draw this figure in red.

2. Draw the pentagon below after it has been reflected over the x-axis. Draw this figurein blue.


x

y


1. Connect these points in order to make a quadrilateral. (-1,2), (7, -3), (15,2), (7, 10)

2. Draw the quadrilateral after it has been reflected over the y- axis. Draw this quadrilateral in red. List the new coordinates.

3. Reflect the original quadrilateral over the x-axis. Draw this quadrilateralin blue. List the new coordinates.


x

y


1. Connect these points in order to make a quadrilateral: (-12,-2), (-12,10), (-6,10), (-6, 5), (-12,5)2. Translate the figure according to the rule: (x, y) (x + 8, y - 10)


Solution:

1 2

3 4

5

c

2 1

5

4 3

c

c

ab

CUT AND TAPE PROOF OF THE PYTHAGOREAN THEOREM2.3 Activity B


Name_______________________________________Date___________

Pythagorean Theorem with Tiles

Materials needed: square tiles, centimeter paper, scissorsRecord your results in the table below.

Procedure: Build squares along the sides of the triangle shown below. One square will

have side length a, one b, and one c.

a

b

c

Legs Hypotenuse Area of Squares a b c a2 b2 c2

3 4 _______ ____ ____ ____

Make other triangles with grid paper and complete the chart below.

6 8 _______ ____ ____ ____

5 12 _______ ____ ____ ____

8 15 _______ ____ ____ ____

What pattern do you see?


Centimeter Grid PaperName_______________________________________Date___________

Classroom Strategies Blackline Master Page 49 II - 15

First make a square from a rectangle. Give each student a rectangular piece of paper. Fold andcut as follows:

a. b. c. d.Unfoldand cut

Fold

Fold

Now use the square to make a set of tangram pieces. Cut and fold as instructed below:

a. Fold square into two large congruent triangles. Cut apart on the fold.

b. Fold both large triangles into two right triangles. Cut one of them on the fold and setthese two pieces aside.

c. Fold the second large triangle so that the right angle corner touches the midpoint of thebase. Unfold and cut the triangle from the trapezoid. Set the triangle aside.

d. Cut the trapezoid in half (on the fold).

e. Fold one of the quadrilaterals into a square and a right triangle. Cut apart and set thesepieces aside.

f. Fold the remaining quadrilateral into a parallelogram and a right triangle. Cut along thefold to complete the set.

c. e. f.d.b.a.

TANGRAM FOLDName_______________________________________Date___________


Name_______________________________________Date___________THE AMAZING PYTHAGORAS

Study the triangles on the nextpage. Triangles A, B, E, F, G, and I areparticular kinds of triangles. Drawanother triangle that also fits thiscategory.

Place a check in front of the 6letters in the table that represent righttriangles.

1. Measure the sides of eachtriangle to the nearest half-centimeter. Place yourmeasurements in the left partof the table.

2. Make the computations necessary to complete the right side of the table.

3. Study the results in the right side of the table. Look closely at the numbers that correspond with the right triangle. Write about any observations you've made.

4. Do you think a 9x10x11 cm triangle is a right triangle? First make aprediction. Then check your prediction by making a drawing.

5. Is a 9x12x15 cm triangle a right triangle? First make a prediction. Then check yourprediction by making a drawing.

B

C

D

E

F

G

H

I

Triangle

A

a b c c2 2a2 b

Needed: Centimeter ruler

Pythagoras was a Greek mathematician who lived about 2500 years ago. He made manymathematical discoveries. One of his most famous discoveries was about certain kinds of triangles.

Classroom Strategies Blackline Master Page 51

The Amazing Pythagoras continued

A

B

C D

E

F

G

H

I

a

bc

a

bc

a

b c

a

bc

a

b

c

a

bc

a

b

c

a

bc

a

b

c

II - 17

Name_______________________________________Date___________


Name_______________________________________Date___________

Pythagorean Construction - A

Materials needed: Centimeter grid paper and scissors

Procedure:

1. Cut two congruent rectangles of any size from your grid paper. Let a represent the

length of the shorter side of the rectangle, and b represent the length of the longerside.

2. Now cut out a square that has each side equal to the length a+b. Arrange the

rectangles on the square as shown in this diagram.

What is the area of each of the two smaller squares?

Denote these with a2 and b2.

3. Now cut each rectangle diagonally into two right

triangles. Let c represent the length of the

hypotenuse of each of these triangles.

Arrange the triangles as shown here.

Why must it be true that a2 + b2 = c2 ?

a

b

a2

b2a

b

a

b

c


Pythagorean Theorem Flip Book

Have students cut out the following diagrams and staple them into a flip book. The

resulting animation is a proof of the Pythagorean Theorem.St

aple

Her

eSt

aple

Her

e

1

2

Stap

le H

ere

Stap

le H

ere

3

4

II - 19

Classroom Strategies Blackline Master



resulting animation is a proof of the Pythagorean Theorem.

Stap

le H

ere

Stap

le H

ere

5

6

Stap

le H

ere

Stap

le H

ere

7

8

II - 20




resulting animation is a proof of the Pythagorean Theorem.St

aple

Her

eSt

aple

Her

e

9

10

Stap

le H

ere

Stap

le H

ere

11

12

II - 21

Classroom Strategies Blackline Master



resulting animation is a proof of the Pythagorean Theorem.

Stap

le H

ere

13

II - 22


Pythagorean Construction -B

Begin with any right triangle. Construct squares on the sides as shown.

Find the center of the square built on the longer leg. Call this point O.

Now construct a line through O perpendicular to the hypotenuse of the right triangle.

Then construct another line through O perpendicular to the first line you drew. Draw each

of these lines long enough to cut through the square.

Cut this square into the four parts created by the two perpendicular

lines. These four parts and the smaller square should fit inside the

square on the hypotenuse.

O

O

II - 23

Name_______________________________________Date___________

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