Download Polygons or Not Polygons

Polygons
Polygons or Not Polygons
Come to a consensus on the definition of a polygon (include all necessary properties):
Sketch a polygon that is different from the ones you were given. Then sketch a shape that is not a polygon.
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Polygons
Note: You can use your Geometry to Go book to look up definitions and concepts, too.
Polygons can be classified according to the number of sides they have.
Polygons with more than 10 sides are not usually given special names. A polygon with 11 sides is
described as an 11-gon, a polygon with 12 sides as a 12-gon (officially dodecagon), and so on. When
people talk about a general polygon—one where you don’t know the exact number of sides, they refer to it
as an n-gon.
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Polygons
Classifying Triangles
The following shows how triangles can be classifies according to some of their features:
Angles:
Right Triangles
One right angle (90º)
Acute Triangles
All angles less than 90º
Obtuse Triangles
One angle more than 90º
Sides:
Equilateral Triangles
All three sides the same
length.
Isosceles Triangles
At least two sides the same
length.
Size:
Similar Triangles
Same shape, possibly different size.
Congruent Triangles
Same size and shape.
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Scalene Triangles
All three sides different
lengths.
Polygons
Classifying Triangles
Use this recording sheet to combine two features (angle and side length) into one triangle if possible.
If the combination is possible, draw a sketch in the box. If not, explain why not. You may want to
use spaghetti to make a model.
Acute
Right
Obtuse
Equilateral
Isosceles
Scalene
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Polygons
Building Triangles
Materials: Spaghetti
Question: Do you think you can make a triangle using any three lengths for the sides? With three side
lengths, can you make more than one triangle? (Short 2-3 minute discussion with your group.)
Do as a class: Using spaghetti, build a 6-8-12 cm triangle. Use the following chart to record this answer in
the row marked Triangle #1. Suggestion: Measure and mark the length with a pen or pencil, use a scissors
to cut the spaghetti, and label the length.
Triangle
Length
of
Side 1
Length
of Side
2
Length
of Side
3
#1
6
8
12
#5
4
5
10
#6
7
7
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Makes a
Triangle?
(Yes or No)
If yes, is the
triangle acute,
right, or obtuse?
If yes, can you make
a different
Triangle?
#2
#3
#4
Choose a set of three numbers—each less than 25 cm—to be the lengths of the sides of a triangle. Use
straws or spaghetti to test the three lengths to see whether or not they will make a triangle. If so, will they
make a different triangle? Record your results in the row marked Triangle #2.
Chose sets of three numbers and repeat your experiment for two or three more triangles.
Complete rows 5 and 6.
Based on the information in column 5 “Makes a Triangle?” write a summary of your conclusions (a rule
about the lengths of the sides of triangles):
Whole group summary.
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Polygons
Building Quadrilaterals
Question: Do you think you can make a quadrilateral using any four lengths for the sides? Again record
your answers in a chart. Repeat your experiment for several more quadrilaterals.
Materials: Use poly-strips for this activity. Attach the strips using the end holes, and measure the
length from the center of one end hole to the center of the other.
Length Length Length Length
Makes a
If yes, can you make
Quadrilateral
of
of
of
of
quadrilateral?
a different
Side 1
Side 2
Side 3
Side 4
(Yes or No)
quadrilateral? How
many?
#1
#2
#3
What polystrips could you choose to make a parallelogram? A rectangle? A square? A rhombus? A
trapezoid? Test your conjectures and sketch your results on dot paper.
Definition: A conjecture is an educated guess about what you think is true based on observations.
What are the attributes of a parallelogram?
A rectangle?
A square?
A rhombus?
A trapezoid?
A kite?
Is a rectangle a parallelogram? Explain.
Is a square a rectangle? Explain.
Is a rectangle a square? Explain.
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Polygons
Write definitions for the quadrilaterals listed below. Make your definitions as concise as possible. For
example, if you have already defined parallelogram, then you can use the word parallelogram when
defining other quadrilaterals, if appropriate.
You may check in Geometry to Go after you’ve written your ideas.
1. Parallelogram
1. Rectangle
1. Square
1. Rhombus
1. Trapezoid
1. Kite
Use the graphic organizer below to show how the above quadrilaterals are related to each other. Write the
correct term on each line and draw one example in each section.
Quadrilaterals
Note: Some books define trapezoid as a quadrilateral with at least one pair of opposite sides parallel, which changes
everything! We won’t use that definition. (See blue highlighted paragraph in section 165 of “Geometry to Go”.)
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Polygons
ALWAYS TRUE, SOMETIMES TRUE, OR NEVER TRUE?
Suggestion: Use poly-strips for this activity.
• If the statement is true, give a convincing argument to show your reasoning.
• If the statement is never true, provide a counterexample and explain your reasoning.
• If the statement is sometimes true, explain that also.
1. If a quadrilateral is a square, then it is a rhombus.
1. If a polygon is a right triangle, then the other two angles are acute.
1. Parallelograms have congruent diagonals.
1. Quadrilaterals with congruent diagonals are parallelograms.
1. If the opposite angles of a quadrilateral are congruent, then it is a parallelogram.
1. The diagonals of a parallelogram bisect each other (bisect means to cut into two equal parts).
1. The diagonals of a parallelogram are perpendicular to each other.
1. If a quadrilateral has at least one right angle, then it’s a rectangle.
1. An equiangular quadrilateral is a square.
1. The diagonals of a trapezoid are congruent.
1. A right triangle can have three congruent sides.
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Angles
What is an Angle?
Small group discussion: Make a list of things you know about angles. Include such ideas as the definition
of an angle, types of angles, angle notation, measuring angles, benchmark angles, and so on.
Note: See pages 62 and 63 in Geometry to Go for correct notation when naming and measuring angles.
Pattern Block Angles
Angles as Wedges
Materials: Pattern Blocks. Hinged mirrors.
Directions: Find the measure of each angle in your set of pattern blocks. Note: Do not use a protractor or
angle ruler to do this activity. Record your results below and be ready to explain how you determined your
answers.
Angles
Sum of all the angles
The size of a right angle is the only prior knowledge you
Orange Square
may use.
Green Triangle
Blue Rhombus
Tan Rhombus
Red Trapezoid
Yellow Hexagon
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Angles
Making a Protractor
Angles as rotations
Materials: Compass, large index card, ruler, pattern blocks, patty paper.
Make a “protractor” using angles found in the pattern blocks and/or with other methods.
•
Mark a point somewhere near the center of your index card.
•
Using that point as the center, use a compass to draw a circle 4 inches in diameter and cut it out.
•
Draw in a radius.
•
Mark benchmark angles and any other angles that can be found using the pattern blocks or other
methods. Which pattern blocks can be used to make a full 360º rotation? Mark angles until the
circle is completed.
Possible strategies:
o Combine two angles to get a third
o Draw an angle on patty paper and fold it in half.
•
What does it mean to say that there are 360º in a circle? Where are these degrees?
•
Use an angle ruler or protractor to check. How accurate were you?
Which angle is bigger?
Angles as two rays meeting at a common vertex
Discuss the question, “Which angle is bigger?” with your group.
A
B
Use your angle ruler to check. How might this result contribute to your understanding
of the meaning of "angle?" (Why do many students say that ∠B is bigger?)
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Angles
Degrees in a Triangle Activity
•
Using a straightedge, draw a triangle on an index card and cut it out. One person in your group may
draw a right triangle, but everyone else should draw a different type of triangle.
•
Label the three angles 1, 2, and 3.
•
Tear off the three angles (don’t cut them off—you want a jagged edge to avoid confusion about
which angles are the actual angles of the triangle).
•
Move the three angles together so that their vertices and sides are touching. Compare your results
with others in your group.
•
Make a conjecture about the number of degrees in any triangle. (You are using “inductive
reasoning”—using lots of examples as an informal justification. Although it is not a proof, use your
conjecture in the next activity as if it were always true. We will prove your conjecture later.)
Interior Angles of Polygons
You made a conjecture about the sum of the interior angles in any triangle. In this activity, you will make
conjectures about the sum of the interior angles of other polygons.
•
Use a straightedge to draw various polygons: one with 4 sides (quadrilateral), one with 5 sides
(pentagon), and one with 6 sides (hexagon). They do not need to be regular polygons (congruent
sides and angles).
•
Determine the number of degrees in each interior angle. (Use your angle ruler to measure.) Add
your results to find the sum of the degrees for each polygon. Record your results in a table. Leave
the last column blank for now.
Experimental results
Type of polygon
Number of
Sides & Angles
Sum of Interior
Angles
•
Triangle
Quadrilateral
Pentagon
3
Hexagon
n-gon
Any number
180º
Compare your results with other in your group. If your angle sums are slightly different, to what
can you attribute this difference? Come to a consensus about which sum you will use.
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Angles
Interior Angle Sums
A. Amy divides polygons into triangles by drawing all the diagonals of the polygon from one vertex.
Pick any vertex on each polygon below and draw in all the diagonals from just that one vertex.
How can you use Amy’s method to find the sum of the interior angles of each polygon? You may
want to make a chart relating the # of sides, the # of triangles, and the sum of the interior angles.
• Does Amy’s method work for any polygon? Explain.
• Using Amy’s method, write a rule, based on the number of sides, n, for finding the sum of the interior
angles of any size polygon (n-gon).
• How could you use her rule to find each angle any regular polygon?
B. Cole also discovered a method for finding the angle sum of any polygon. He marks an arbitrary
point in the interior of the polygon, and draws line segments from that point to each vertex. Follow
Cole’s method to mark the polygons below. How can you use Cole’s method to find the sum of the
interior angles of each polygon?
• Does Cole’s method work for any polygon? Explain.
• Using Cole’s method, write a rule, based on the number of sides, n, for finding the sum of the interior
angles of any size polygon (n-gon).
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Angles
C. Keisha says it doesn’t matter how you draw in the diagonals as long as they don’t intersect. For
example, she says you can divide up the polygon below as shown and Amy’s method will still work.
Do you agree? Explain.
Extension A: Would the rules that you wrote in Parts A and B (second bullet) work for concave polygons?
Illustrate why or why not with examples.
Extension B: Find the sum of the exterior angles of any convex polygon. Exterior angles are angles
outside a polygon that lie between one side and the extension of its adjacent side. Mark three more exterior
angles on the pentagon below.
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Angles
Vertical Angles
The pairs of opposite angles formed by two intersecting lines are called vertical angles. Which angles in
the diagram below are vertical angles?
By looking at the diagram, make a conjecture about the measures of vertical angles. You may want to copy
the diagram onto patty paper.
The pairs of adjacent angles formed by two intersecting lines, such as angles 1 and 2 above, form a
straight angle (a straight line) whose measure is 180º.
Name the pairs in the diagram above that form straight angles?
Write an explanation describing how you know your conjecture about vertical angles is always true
(prove your conjecture).
Other Interesting Angles
When two lines (m and n) are cut by a transversal (t), eight angles are formed. Name the pairs of vertical
angles in the following diagram.
Angles can be paired in different ways. Other angle pairs are called:
•
alternate interior angles
•
alternate exterior angles
•
corresponding angles
Just by thinking about the names, try to determine which pairs of angles go with each name. Hint: Angles
2 and 6 are one pair of corresponding angles. You can check yourself on page 74 of Geometry to Go.
14
Angles
Parallel Lines Cut by a Transversal
Materials: Patty paper.
A. Draw two parallel lines on one piece of patty paper. Make sure they are really parallel! (You may
discuss with your group possible ways to ensure that you end up with parallel lines.) Draw a
transversal (a third line that intersects both parallel lines). Do not make the transversal
perpendicular to the parallel lines. After you complete the activity, think about why you were asked
not to draw a perpendicular transversal.
A. Number your angles and list the pairs of:
•
vertical angles
•
alternate interior angles
•
alternate exterior angles
•
corresponding angles
A. Make an exact copy of your diagram, including the numbers, on a second piece of patty pap r.
A. Use your patty paper diagrams to make some conjectures about the measures of the alternate
interior angles, the alternate exterior angles, and the corresponding angles.
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Angles
E. A postulate is a statement that is assumed to be true, but is never proved. They are accepted
without proof. One such postulate is the Corresponding Angles Postulate, which states: If two
parallel lines are cut by a transversal, then the pairs of corresponding angles are congruent.
All your other conjectures can be proved using what you already proved about vertical angles being
congruent and this postulate.
Prove your conjectures about alternate interior and alternate exterior angles. It is perfectly
acceptable to write your “proof” as an explanation. If you’d prefer to use a 2-column proof that you
learned about in high school, that’s O.K., too!
Extension (If your group finishes before the other groups are done): Make some conjectures about
same-side interior angles and same-side exterior angles. Prove your conjectures.
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Angles
A Proof that there are 180º in any Triangle
•
Draw a triangle in the space below.
•
Label the angles with the numbers 1, 2, and 3.
•
Draw a line parallel to the base of your triangle through the vertex above the base.
•
We know the “Corresponding Angles Postulate” and we already proved our conjectures about vertical
angles, alternate interior angles, and alternate exterior angles. Use any of the information that you
already know to prove that every triangle has 180º.
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Congruence, Symmetry and Transformations
Reflection/Line Symmetry with the Alphabet
If you can reflect (or flip) a figure over a line and the figure appears unchanged, then the figure has
reflection symmetry or line symmetry. The line that you reflect over is called the line of symmetry. A line
of symmetry divides a figure into two mirror-image halves. The dashed lines below are lines of symmetry:
Draw ALL the lines of symmetry on each of the letters below. Circle the ones with no lines of symmetry.
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Congruence, Symmetry and Transformations
Transformations
Translations (slides), Reflections (flips), Rotations (turns)
Read pages 273-4 in “Geometry to Go”.
Pretend that the two-dimensional shapes below are on transparencies for an overhead projector (or use patty
paper if needed to check). Which shapes are the same (congruent)? What transformation(s) were used to
get from a shape to its congruent “mates”?
Draw the reflection of the figure on the other side of the vertical line. Use a mirror or patty paper to check.
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Congruence, Symmetry and Transformations
Draw the reflection of the figure on the other side of the diagonal line (the diagonal line is the line of
reflection).
Label the triangle ABC. Translate (slide) the figure to a different place on the coordinate plane in the
direction indicated by the arrows (vector). Label the corresponding points on the translated image A'B'C'
(read as A prime, B prime, C prime).
Check yourself by copying the figure and the line on a piece of patty paper, placing it over the original, and
sliding the copy along the line. Do your images match up?
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Congruence, Symmetry and Transformations
Label the figure ABCDE as described in the last problem. You have several tasks on the next image:
1) Reflect (flip) the original figure across the y-axis.
2) Reflect the original figure across the x-axis.
3) Translate the original figure in the direction indicated by the arrow.
Label the reflected and translated images using “prime” notation as described above.
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Congruence, Symmetry and Transformations
Rotational Symmetry
Rotational Symmetry: A figure has rotational symmetry if it can be rotated less than a full turn (360º)
about a fixed interior point and the rotated image matches, or coincides with, the original figure.
Center of Rotation: A fixed point about which a figure is rotated.
1. Copy each figure inside the circle onto a piece of patty paper. Lightly draw in all the lines of
symmetry.
2. Make a mark at the top of each figure on your patty paper so you can remember the original
orientation of the traced figure.
• Place the patty paper figure over the original, hold it in place with the tip of a pen on the center
dot, and rotate it until the rotated figure matches the original.
• If the figure has rotational symmetry, write down all the angles of rotation.
Extension (If your group finishes before the other groups are done): Find all the lines of symmetry and
all the angles of rotational symmetry of a regular hexagon. You can use the yellow hexagon to trace a copy
onto your paper. Try a rectangle?
22
Congruence, Symmetry and Transformations
Complete the following steps for a clockwise rotation (turn) of the figure below:
• Label the original figure.
• Place a piece of patty paper on top of the original. Trace the figure, the vertex (point A) of the
angle, and the top ray of the angle.
• Place your pencil tip on the vertex of the angle (the center of rotation). Turn the patty paper sheet
until the ray on your patty paper is aligned with the other ray of the original angle.
• Mark the points of the rotated image on the coordinate grid and connect those points. Label the
rotated image.
Where would you place a mirror to justify that the diagonals of a square have the same length?
Extension (next page)
23
Congruence, Symmetry and Transformations
Extension (If your group finishes before the other groups are done): Draw your own figure in the
space below or on another piece of paper and translate it in a direction of your choice, reflect it across a line
of your choice, and rotate it about a point of your choice.
24
Perimeter and Area
Launch questions: What characteristics must a “unit” have in order to measure area? What does it mean
to find the area of something?
Pattern Block Areas: Nonstandard Units
Use the green triangle, the blue rhombus, the red trapezoid, and the yellow hexagon for this activity.
1. If the area of the green triangle is 1, what is the area of each of the other pieces?
2. If the area of the blue rhombus is 1, what is the area of each of the other pieces?
3. If the area of the red trapezoid is 1, what is the area of each of the other pieces?
4. If the area of the yellow hexagon is 1, what is the area of each of the other pieces?
Use Geoboards as needed for any of the activities in this section.
Grid Areas: Standard Units
Be ready to explain your thinking.
25
Perimeter and Area
Developing Formulas
1. Draw a rectangle on a piece of grid paper and find the area. If you counted the number of squares to
find the area, can you find a quicker method? Write an algorithm (a description in words) for your
method. Compare your ideas with others in your group. Will your algorithm always work for any
rectangle?
Formula (your algorithm in symbols): ___________________________________
2. Draw several parallelograms on grid paper. Figure out a formula for finding the area of your
parallelograms, using what you know about rectangles. Hint: you may want a scissors for this activity.
Be ready to explain your thinking.
Formula: ___________________________________
• Use your formula to find the area of these parallelograms: Will your formula always work?
3. Draw a right triangle on grid paper. Figure out a formula for the area of your triangle using what you
know about rectangles. Will your formula always work for any right triangle?
Formula: ___________________________________
4. Draw several non-right triangles on grid paper. Make sure one side is horizontal or vertical on the
paper. Develop a formula that will work for any triangle. How do you know it will always work? Be
ready to explain your thinking.
Formula: ___________________________________
• Use the formula to find the area of these triangles:
26
Perimeter and Area
5. Sometimes a shape is irregular and none of the formulas will work so a different strategy is needed.
Find the area of the following figure using the following two methods:
• Decomposing the figure’s interior: Cutting the figure into pieces for which you know the formulas,
or for which you can count squares. Subdivide the figure into pieces, find the area of each piece,
and combine them for the total area.
•
The rectangle surround method: Surround the figure with a rectangle and find its total area. Find
the areas of the pieces inside the rectangle, but not inside the original figure, and subtract them from
the area of the rectangle.
27
Perimeter and Area
6. Sometimes one or the other of the two methods used in the last activity is easier to use. Using both
methods, try to find the areas of the triangles below. Which method seems easier to use and which one
doesn’t? Why?
• What is it about these two triangles that makes it difficult to use the formula (without measuring with a
ruler) to find the area?
7. The triangles below have been constructed between two parallel lines. Using any method you want,
find the area of each triangle.
• If you didn’t use the formula to find the area, sketch in the height of each triangle and check yourself
using the formula.
• Make a conjecture about the areas of triangles with the same base drawn between two parallel lines, as
shown. Explain why your conjecture makes sense.
8. Construct the following shapes on grid paper:
• A triangle with an area of 6 square units. A different triangle with an area of 6 square units.
• A triangle, a square, and a parallelogram with equal areas
• An irregular figure with an area of 12 square units.
28
Perimeter and Area
More Polygon Areas
Find the area of each polygon. Use either Decomposing the Figure’s Interior or the Rectangle Surround
Method. Show your work.
29
Perimeter and Area
Extension1 (If your group finishes before the other groups are done): Go back and find the areas of the
figures formed by the reflections on pages 19-20.
Extension 2:
The Land Problem:
Two farmers own land that is divided by a fence (as shown below). They need to build a new fence and
have decided to make it in one straight line across their property. Of course, neither one wants to lose
any land, so where can they build the fence so they each still own exactly the same amount of land?
(There are hints at the bottom of the page, but try to figure it out on your own first.)
(Hint #1: Use what you learned about parallel lines cut by a transversal to help you figure it out.)
(Hint #2: Use what you learned in Problem #7 of this set to help you figure it out.)
30
Perimeter and Area
Discovering More Formulas
•
Choose one polygon from the diagrams below. (If you have time, do more than one.)
•
Invent a formula to find the area of the polygon you have chosen using only the dimensions labeled.
Feel free to use patty paper, scissors, or any other tool that may help you.
•
Can you invent more than one method for the area of the polygon you have chosen?
•
Will your formula work for all sizes and shapes of the polygon you have chosen?
Trapezoid
Square
Kite
31
Perimeter and Area
Measurement Concepts with Tangrams: More Nonstandard Units
1. Arrange the 7 tangram pieces in order by their perimeters.
2. There are three sizes of triangles in your Tangram set. Use the two small triangles and one medium
triangle for this activity. Using all three triangles (you must use all three each time), build the polygons
listed below. Sketch your solution to each one.
• Square
• Rectangle that is not a square
• Parallelogram that is not a rectangle
• Triangle
• Trapezoid
3. Without using measuring tools, can you determine which of the polygons in #2 has the greatest area?
Why do you think you were asked this question?
4. Assemble all seven Tangram pieces into a square. (Don’t spend too much time doing this on your own.
The solution is on the bottom of the page.) If the area of this large square is 1 square unit, what is the
area of each of the other pieces? How do you know you are right?
Extension 1: (Extension of #4). Assign a different piece to have an area of 1 and determine the area of the
other pieces based on your new “unit.”
Extension 2: Predict: True or False? Since the polygons from #2 have the same area, they will also have
the same perimeter.
Without using a ruler to measure, place the polygons from #2 in order, from least to greatest, by the length
of their perimeters. How did you determine which shape has the smallest perimeter? (See hint below if
needed.)
Hint for Extension 2: Use a comparing method by assigning “short,” “medium,” and “long” to the three different lengths.
32
Perimeter and Area
Scale Factors and Area
Use your Pattern Blocks for the following activities. Sketch each solution and record your results in the
table.
1. Directions for the orange square:
• We will define the area of the orange square as having as area of 1 square unit, as recorded in the
table.
• Build another square in which each side of the square is twice as long as the original square (a scale
factor of 2:1). Make a sketch; then record the area of your new square in the table.
• What is the relationship between the scale factor and the area of the enlarged figure?
•
•
Build a third square in which each side is three times as long as the original square (a scale factor of
3:1). Make a sketch and record the area of your new square in the table.
What is the relationship between the scale factor and the area of this enlarged figure?
Shape
Area
Orange Square
Green Triangle
Blue Rhombus
Red Trapezoid
Tan Rhombus
*
Yellow Hexagon
(Extension: see #11)
Scale Factor of 2:
Area of the
Enlargement in
Terms of the
Original Shape
General
rule for
2:1
(Area)
Scale Factor of 3:
Area of the
Enlargement in
Terms of the
Original Shape
General
rule for
3:1
(Area)
1 unit of
area
1 unit of
area
1 unit of
area
1 unit of
area
1 unit of
area
1 unit of
area
2. Follow the same procedures for the Green Triangle, then the Blue Rhombus, the Red Trapezoid, and the
Tan Rhombus, defining the area of each pattern block as 1 each time. You may want to split up the
work with others in your group. Continue to sketch your solutions, and record the areas in the table.
3. What is the general rule for the relationship of the areas between two similar figures when the scale
factor is 2?
When the scale factor is 3?
Record your rules in the table above.
4. What is the relationship of the perimeters to the scale factor?
33
Perimeter and Area
2
5. If the area of a polygon were 8 cm , what would be the area of an enlargement with a scale factor of 2?
An enlargement with a scale factor of 3? Explain your thinking.
6. Do you think your rule will hold true for a different type of triangle? To test your conjecture, draw a
triangle of your choice and copy it 3 more times. Can you assemble these 4 triangles into a triangle
similar to the original that has a scale factor of 2? Does your rule hold true?
7. What would you predict the area rule would be for a scale factor of 4? Explain.
8. What happens to the area of a shape if you decrease its size by a certain scale factor, for example by
half? Does the same rule apply?
9. In general, if the scale factor of an enlargement is k, what is area of the enlargement in terms of k?
10. In this activity, the word similar was used. Similar has a very specific meaning in mathematics. What
properties, as far as sides and angles, did each set of similar shapes have in common? After we do the
next sets of activities, you may want to come back and refine your thoughts about similarity.
11. Extension (If your group finishes before the other groups are done): The Yellow Hexagon. Create
a new hexagon with a scale factor of 2:1. You will need to use other pattern blocks pieces to complete
this problem. Does your rule work for this one?
34
Similarity and Scaling
Similar Rectangles
Are all rectangles similar? Make a prediction.
You will explore this question in the Similar Rectangles activity:
• Use the rectangles provided. Measure to the nearest half centimeter.
• Use the table below to record your information
• Group discussion
Look-Alike Rectangles
Three Groups and an Odd Ball
Rectangles
Group 1
(Letter of rect.)
Short side
Measures in cm.
Long side
Ratio of sides
Short/Long
Rectangles
Group 2
(Letter of rect.)
Short side
Measures in cm.
Long side
Ratio of sides
Short/Long
Rectangles
Group 3
(Letter of rect.)
Short side
Measures in cm.
Long side
Ratio of sides
Short/Long
Short side
Measures in cm.
Long side
Ratio of sides
Short/Long
Odd Ball
(Letter of rect.)
35
Similarity and Scaling
Similarity—continued: Using a Coordinate Grid
For this activity, use the picture of the cat’s head (it’s really a polygon!) on the coordinate grid below.
1. Copy the picture of the cat’s head onto a coordinate grid, and find the coordinates of each point.
2. Multiply the coordinates of each point by 2, creating points A' (read as A prime) through H'. Use the
rule (x,y)  (2x,2y).
Plot these new points on the same grid, and connect them in the same order.
Is your new figure similar to the original picture? How do you know?
3. Find the area of the original and the area of the enlargement. How do they compare? What is the
relationship between the area of the enlargement and the scale factor?
Is your result consistent with your findings in the Scale Factors and Area activity with pattern blocks?
Explain.
36
Similarity and Scaling
4. Repeat the activity, but instead of multiplying by 2, add 2 to each of the coordinates. Use the rule
(x,y)  (x+2,y+2).
Plot the new points on a coordinate grid. Is this figure similar to the original picture? Explain.
5. Explain why, if you want to create a similar figure with sides twice as long as the original, you don’t
double all the angles as well.
6. Repeat the activity, multiply the x coordinate by 2 as in question #2 and leave the y coordinate as is.
Use the rule (x,y)  (2x,y)
Plot the new points on a coordinate grid. Is this figure similar to the original picture? Explain your
results.
7. If you added 2 to the lengths of each side of the “cat” instead of adding 2 to the coordinates, as in #4,
would the new figure still be similar to the original? Explain.
8. What does it mean for two figures to be similar? You may want to go back and refine your definition
of similarity that you wrote in #9 of the Scale Factors and Area activity.
37
Similarity and Scaling
Similarity—continued: Make a Smaller Puzzle
•
Copy the square puzzle below (not drawn to scale) onto centimeter grid or dot paper.
•
Draw new puzzle pieces such that all the pieces are similar to the original figure, but with a scale factor
such that the segment that is marked 4 cm long is 3 cm.
•
Divide up the work with other members of your group. Each person should complete one or two pieces
until all the pieces have been made. When you are done, you should be able to assemble all your pieces
as in the original puzzle.
•
In the Similar Rectangles activity, you found that all rectangles are not similar. What about squares?
Are all squares similar? Explain.
Extension: You have a picture that you want to mat before you frame it. You plan to use a mat that is 3
inches wide (see sketch). Will the matted picture be similar to the original picture? Explain.
38
Measurement Relationships
Measurement Relationships: Area and Perimeter
If you have 24 meters of fencing and you want to use it to make a rectangular pen for your puppy, you must
consider both the perimeter of the pen and its area. What relationships exist between these two measures?
Do shapes with the same perimeter have the same area? Let’s investigate.
1. You want to use all 24 meters of fencing to make the rectangular pen, that the dimensions of the pen in
meters will be whole-number values, and that you want the maximum area for your puppy.
a. What are the dimensions of each of the possible rectangular pens? Sketch each pen on grid
paper. (You may want to use toothpicks to model the pens.)
b. What is the area of each pen?
2. All of the pens have a perimeter of 24 m., yet the areas of the pens differ. What do you notice about the
shapes of the pens with small areas as opposed to those with large areas? What are the characteristics
of a rectangular shape with the greatest area?
3. Another situation involving perimeter and area: Joe, a student in a sixth grade class, was asked to find
the approximate area of his hand traced onto a sheet of paper. He took a string and placed it around the
perimeter of his hand. He then took the length of string that represented the perimeter of his hand,
reshaped it into a rectangle, and found the area of the rectangle.
Joe concluded that the area of his hand and the area of the rectangle were the same. Will his strategy
work? Why or why not?
39
Measurement Relationships
4. Looking at perimeter from another direction: Do figures with the same area always have the same
perimeter? Use 12 one-inch square tiles to explore this question. Arrange, and rearrange, all 12 square
tiles to make figures in which each tile must share at least one side with another tile. (The shapes do
not have to be rectangles.) Record your shapes on grid paper. Then determine the perimeter of each
shape you create.
a. What is the smallest perimeter possible using all 12 square tiles?
b. What is the largest possible perimeter?
5. Using any number of one-inch color tiles, is it possible to make rectangles with perimeters of 14
through 26 inches? If not, choose one perimeter that you could not make and explain why it is
impossible.
6. In real life, under what circumstances might you want the smallest perimeter?
7. Consider this rectangle drawn on grid paper. What is the perimeter?
A common misconception made by some students is to think that the perimeter is 16. What do you
think those students are doing wrong? What do you think they are doing wrong if they say the
perimeter is 24? How about 21?
Extension: Draw a right triangle that also has a 30º angle.
a. Can you draw a different triangle with these same constraints?
b. Do they have the same area?
c. Do they have the same perimeter?
d. Are the triangles similar? Explain.
40
Circles
Circles—Discovering π
How does the Circumference of a Circle Compare with its Diameter?
1. Use the circular object that was provided by your instructor for this activity.
2. Measure the circumference of your object. Use string, a tape measure, a ruler, or whatever method you
want. Using the same unit of measure, measure the diameter of your circle.
3. Compare the circumference to the diameter. Record your results in the table below. If time, do one or
two more circular objects. What patterns do you notice, if any?
Measure accurately!
Name of
circular object
Circumference
Diameter
C+D
C-D
C*D
C÷D
Decimal
form
4. Record your group’s results on the class chart and plot your points using one coordinate graph for the
whole class. Use the horizontal axis (x) for diameter and the vertical axis (y) for circumference.
5. Discuss your findings with the whole group.
6. The formula for finding the circumference of any circle if you know the diameter is C = πd. Explain
how this formula makes sense in light of the activity measuring circles.
7. How would you write this formula if you only knew the radius? _________ (This is typically what we
see in textbooks.)
8. Pi (π) is an irrational number, which means it cannot be written as the quotient of any two whole
numbers. Its decimal approximation is known to more than a million decimal places. Yet we
sometimes see π written as 3.14. Why is that?
9. When mathematicians are asked to determine the circumference of a circle, say with a diameter of 5,
they say the circumference is 5π. Why do you think they record the answer this way? Why not use the
π key on the calculator to find a numerical value for the circumference?
41
Circles
Finding the Area of a Circle
Listed below are a number of activities that will help you figure out the area of a circle. Choose 3 or 4 of
the investigations to do, and be ready to report your findings to the whole group. Everyone should do #6,
which relates specifically to “discovering” the formula for area of a circle.
Method 1: Counting Squares
You need: a sheet of centimeter grid paper and a compass.
1. Draw a circle with a diameter of 12 centimeters on centimeter grid paper.
2. Count all the whole square centimeters that lie completely inside the circle. This underestimates the
area of the circle.
3. Count the number of squares that lie partly inside and partly outside the circle. Add this number the
number you counted in Step 2. This total overestimates the area of the circle.
4. Average the two estimates. This is the approximate area of the circle.
Method 2: Inscribing and Circumscribing Squares
You need: a compass, a compass or angle ruler, and a straight ruler.
1. Draw a circle with a diameter of 12 centimeters on grid paper.
2. Circumscribe a square around the circle as shown. Find its area.
3. Inscribe a square inside the circle. Find its area.
4. Average the areas of the two squares. This is the approximate area of the circle.
Method 3: Using the Octagonal (Egyptian) Method
You need: a compass, a compass or angle ruler, and a straight ruler.
1. Draw a circle with a diameter of 12 centimeters on grid paper.
2. Circumscribe a square around the circle as shown.
3. Divide the square into nine congruent squares, as shown
4. Construct an octagon by drawing a diagonal in each corner square.
5. Find the area of the octagon. This is an approximation of the area of the circle.
42
Circles
Method 4: Weighing the Circle
You need: a linoleum circle with a diameter of 12 centimeters, some linoleum rectangles measured in
square centimeters, some 1 x 1 centimeter linoleum squares, and a pan balance.
1. Place the linoleum circle on one side of the pan balance.
2. Balance the circle with linoleum rectangular and square pieces placed on the opposite side of the
pan balance.
3. Count the number of rectangular and square pieces you used. The total area of all the pieces
approximates the area of the circular region.
Method 5: Using a Curvy Parallelogram
You need: a compass, a scissors, and tape.
1. Draw a circle with a diameter of 12 centimeters on plain paper.
2. Fold the circle in half horizontally and vertically.
3. Cut the circle into four wedges on the fold lines. Then fold each wedge into quarters. Cut each
wedge on the fold lines. You will have 16 wedges.
4. Tape the wedges to a piece of paper to form the following figure:
Notice that we have a crude parallelogram with a height
approximately equal to the radius of the original circle.
5. Find the approximate area of the curvy parallelogram. This approximates the area of the circle.
Note: you will need to figure out the length of the base, b, of the parallelogram. How does the base
relate to the circumference of the circle?
43
Circles
Method 6: Examining the Formula Using Radius Squares
You need: a sheet of white centimeter grid paper, a sheet of colored cm grid paper, and a compass.
1. Draw a circle with a diameter of 12 centimeters on white centimeter grid paper.
2. On a piece of colored centimeter grid paper, draw a square such that one side of the square is equal
to the length of the radius, as shown in Circle 1 (not to scale).
3. Cut out several copies of the “radius squares” from the colored paper. Determine the number of
radius squares it takes to cover your circle from #1. You may cut the radius squares into parts if
you need to.
2
4. If you were to estimate the area of a circle in radius squares (r ), what would you report as the best
estimate? Would your estimate work for Circle 2? For any circle?
2
5. The formula for the area of a circle is A = πr . Explain how this formula makes sense in light of the
activity with radius squares.
Extension: When you enlarge a circle so that the radius is twice as long (a scale factor of 2), what do
you think happens to the circumference and the area? Do they double? What happens if you
enlarge your circle by a scale factor of 3? Are your results consistent with your work on scale factors
in polygons? Explain.
44
Circles
Applying What You Know about Circles
Reminder: use correct labels. Leave your answers in terms of π.
1. Area = _____________
Perimeter = ______________
Approximate your answer using the π key on a calculator and then using 3.14 as an estimate for π. Why
are these two answers slightly different?
2. Un-shaded Area = _____________
Circumference of each circle: Large circle = _____________ Small circle = _______________
3. Area = ________________
Arc length = _________________
Extension: A circle with radius r is inscribed in a square. The shaded part (the circle) is what percentage
of the square?
Do not use an approximation for π (3.14) until you are ready to write the percent.
45
The Pythagorean Theorem
The Pythagorean Theorem
1. Looking for Squares: On 5 dot-by-5 dot grids, draw squares of various sizes by connecting dots.
Draw squares with as many different areas as possible. Label each square with its area. Include any
possible squares whose sides are not horizontal and vertical.
46
The Pythagorean Theorem
2. Square Roots: The area of a square is the length of a side multiplied by itself. This can be expressed
2
by the formula A = s * s, or A = s . How is this related to the formula that you developed for a
rectangle: A = lw?
If you know the area of a square, you can work backward to find the length of a side. For example,
suppose a square has an area of 9 square units. To find the length of a side, you need to figure out what
positive number multiplied by itself equals 9. Because 3 * 3 = 9, the side length is 3 units. We call 3 a
square root of 9: written as
= 3. Find the square in problem #1 that has an area of 9 square units,
and confirm that the side length is 3.
Note: Numbers that have whole-number square roots, are called perfect squares or square numbers.
You may recall the pattern problem we did in “Algebra Camp” using square numbers:
What is the area (small squares) of each of these figures? Why do you think those areas are called
square numbers?
What are the next three square numbers is the pattern?
What is the rule for finding the number of squares in the nth figure?
Revisit the rest of your squares in problem #1. Find the square root of each area to find the side length
of that square. Are the areas of your “tippy squares” square numbers? How would you find the square
roots of these numbers?
Example: √5 ≈ 2.24
47
The Pythagorean Theorem
3. More Areas of “Tippy Squares”: Subdivide the squares below into right triangles and squares using
horizontal and vertical lines. Combine the areas of all the pieces to find the total area of each square.
After you have found the area, find the side length, s, of each square using square roots. Label the side
of the square with the length. You can check yourself with a ruler. The first one has been done as an
example.
48
The Pythagorean Theorem
4. The Pythagorean Theorem: Recall that a right triangle is a triangle with a right, or 90º, angle. The
longest side of a right triangle is the side opposite the right angle and is called the hypotenuse of the
triangle. The other two sides are called the legs. The right angle of a right triangle is often marked with
a square.
Each leg of the right triangle on the left below has a length of 1 unit. Suppose you draw squares on the
hypotenuse and legs of the triangle, as shown on the right.
How are the areas of the three squares related?
In this activity, you will look for a relationship among the areas of squares drawn on the sides of right
triangles. Do all three steps for the first pair of leg lengths a and b. Then repeat for each of the other
pairs:
• Draw a right triangle with the given leg lengths on dot paper.
• Draw a square on each side of the triangle. Hint: Use an index card to help draw right angles.
• Find the area of the squares, write the area inside the square, and record the results in the table.
Length of Length of
Leg 1: a
Leg 2: b
(units)
(units)
1
1
2
1
2
3
3
a
•
•
•
Area of Square
on Leg 1
2
(square units: a )
Area of Square
on Leg 2
2
(square units: b )
Area of Square
on Hypotenuse: c
2
(square units: c )
1
2
2
3
3
3
4
b
Make a conjecture about the relationship among the areas of the three squares drawn on the
sides of any right triangle.
Draw a right triangle with side lengths that are different than those given in the table. Use your
triangle to test your conjecture.
Write your conjecture using the symbols in the table: _______________________
You have “discovered” the Pythagorean Theorem!
49
The Pythagorean Theorem
5. A Proof of the Pythagorean Theorem: The pattern you discovered in the last activity is a famous
theorem named after the Greek mathematician Pythagoras (although the Chinese used the idea as far
back as 1000 B.C. A theorem is a general mathematical statement that has been proven true. The
Pythagorean Theorem is one of the most famous theorems in mathematics.
Over 300 different proofs have been given for the Pythagorean Theorem. You will explore one of these
proofs in the next activity.
•
Use the puzzle pieces provided. Cut out the pieces and look at how the triangle and the three
squares are related: how do the side lengths of the squares compare to the side lengths of the
triangle?
•
Label the hypotenuse of one of triangles as c. Label the two legs a and b.
•
Label the square with a side length equal to the length of the hypotenuse as c . Label the two
2
2
2
2
smaller squares a and b . (Make sure the square labeled a goes with the leg labeled a.) Why do
these labels make sense?
•
Arrange the 11 puzzle pieces to fit exactly into the two puzzle frames. Use four triangles in each
frame.
•
What conclusions can you draw about the relationship among the areas of the three squares?
•
What does the conclusion you reached mean in terms of the side lengths of the triangles. (How
2
2
2
does your conclusion prove a + b = c ?)
Extension: James Garfield (1831-1881) was the 20th president of the United States. Five years before
becoming president, he discovered this proof of the Pythagorean Theorem. What is the reasoning behind
his proof?
Hint: Compute the area of the figure two ways—first by finding and adding the areas of the triangles, and
second by finding the area of the whole thing, which is a trapezoid. (Refer back to the formula for area of a
trapezoid.) What must be true of the two areas? (You’ll need some algebra.)
50
The Pythagorean Theorem
6. Applying the Pythagorean Theorem: Use correct labels.
a. Find the length of the missing side:
b. What is the length of the diagonal of a square that has a side length of 8? (Hint: draw a sketch.)
c. Shaded area = ______________________
d. What are the lengths of the line segments below?
51
The Pythagorean Theorem
e. Find the area and the perimeter of the figure below (without using a ruler):
Perimeter = _______________________ Area = _____________________
f. A new and larger triangle is similar to the triangle below, with the size transformation having a
scale factor of 4. What is the area enclosed by the new triangle?
Explain how you can use what you know about the impact of scale factor on area to find the area of
the new triangle without actually using the dimensions of the new triangle: ___________________
________________________________________________________________________________
7. The converse of the Pythagorean Theorem: The converse of the Pythagorean Theorem is also true.
2
2
2
In other words, if the three side lengths of a triangle satisfy the formula a + b = c then the triangle is
a right triangle. Determine if the triangles listed below are right triangles. (Actually, one of the sets of
side lengths below won’t make a triangle at all. Which one is it? How do you know?)
Side Lengths
Do the side lengths satisfy
Is the triangle a right triangle?
2
2
2
(units)
a +b =c ?
6, 8, 10
5, 6, 10
5, 5, 5
1, 2, 3
52
The Pythagorean Theorem
8. If time: Tri-Square Rug Games: Al and Betty have designed a game that uses “tri-square rugs.”
Their rugs are made by sewing three separate square rugs together at their corners, with an empty
triangular space between them. (See the examples below.) In Al and Betty’s game a dart falls
randomly on the tri-square rug.
•
•
•
If the dart hits the largest of the three squares, Al wins.
If the dart hits either of the other two squares, Betty wins.
If the dart misses the rug, they just let another dart fall.
Your goal is to determine which of the rugs you would prefer if you were Al and which you would
prefer if you were Betty, and if there are any rugs that lead to a fair game.
On chart paper show some rugs that are wins for Al in the long run, rugs that are wins for Betty, and
rugs that are fair games. When you have several examples for each category, look for a pattern in
your results. Your goal is to find a way to tell, just by looking at a tri-square rug, who will win in
the long run and why that’s true.
Summarize your results below:
Adapted from Interactive Mathematics Program, Year 2, Key Curriculum Press
53
Polyhedra
Polyhedra
1. Look at the nets (outlines) for shapes A-K, which are provided on card stock. Try to imagine what the
final, folded-up shape will look like before you cut out and assemble them.
2. Cut out and assemble the polyhedra so that the letter of the shape is visible. Hint: Use a sharp pen and
straightedge to score each fold line; the folds will be easy and accurate!
3. Each shape is an example of a polyhedron (plural: polyhedra or polyhedrons): a closed surface made
up of planar regions (flat pieces). The planar regions in a polyhedron are faces of the polyhedron, the
line segments where faces meet are edges, and the points at which edges meet are vertices (singular:
vertex) of the polyhedron. Choose one of the polyhedra and count the number of faces, edges, and
vertices.
4. Sort your completed polyhedra into different sets. What criteria did you use? Compare your sets with
another group’s sets. Are they the same? Discuss the attributes you used.
5. Sort your polyhedra into the following three groups:
• Group 1: Shapes A, D, G, and I
• Group 2: Shapes B, C, E, and F
• Group 3: The others—set aside for now
a. Group 1 shapes are called pyramids. Come to a consensus on a definition for pyramid, without
looking it up. One face is called the base of the pyramid; which one do you think it is for each
shape?
Since the shape of its base names each pyramid, give the name for each pyramid in Group 1. For
example, shape I is a square pyramid.
b. Group 2 shapes are called prisms. Give a definition for prism. Two faces of each prism are called
the bases of the prism; which ones are the bases for each shape?
How are the bases related to each other?
Which prisms can be reoriented so that the two faces called the bases are different?
Write the name for each prism in Group 2.
For each prism, how do the lateral edges (the edges not on a base) appear to be related?
What shape are all the lateral faces (the ones that aren’t the bases)?
54
Polyhedra
6. Some of your pyramids and prisms are right pyramids and right prisms.
a. Which ones? Why do you think that? (You may check the definitions in Sections 326 and 327
in Geometry to Go.)
b. What shape are all the lateral faces of a right prism?
c. What shape are all the lateral faces of a right pyramid?
7. Why are the shapes in Group 3 not included as pyramids or prisms?
8. Which of the shapes, A-K, meet the criteria in each part?
a. All the faces are parallelograms (including special parallelograms, like rectangles and squares).
b. A base is a pentagonal region.
c. All the triangular faces are equilateral.
d. None of the triangular faces is equilateral.
e. Two faces are parallel and congruent, but the shape is not a prism.
f. Some of the faces are isosceles trapezoids.
g. All the edges are congruent.
h. No pair of edges is parallel.
9. How many of the angles on all the faces of a prism could be right angles, if…
a. The prism is a right triangular prism? (Hint: consider bases that are both right and non-right
triangles.)
b. The prism is a right quadrilateral prism? (Hint: some quadrilaterals have right angles and some
don’t.)
10. Pick one of the shapes. What attributes of this shape can be measured and/or counted?
Pick another one of the shapes. Are the same quantities relevant to it also? Explain.
55
Polyhedra
Representing and Visualizing Polyhedra
What do you see when you look at
?
Did you see a regular hexagon with some segments drawn to its center? Or did you see a cube? (Do
you see both now?) Three-dimensional shapes are often represented by drawings in two dimensions.
Without a context suggesting three dimensions, these drawings can be ambiguous. How to “read” such
drawings is important for understanding the ideas they represent, and how to make such drawings is a
useful skill that is often needed to represent and solve a problem.
Different representations of a polyhedron
Physical model
“Polyhedron”
(word)
Net
Drawing
1. Nets
a. When you assembled shapes A-K, you were working from nets. A net is a two-dimensional pattern
that can be folded to form a three-dimensional solid. Which one of your polyhedra is a cube? Draw
a net for it. Can you draw a different one? (Hint: there are eleven possible nets for a cube.)
b. If these nets were folded up to give cubes, which pairs of faces would be opposite each other?
c. Using a centimeter ruler to measure, find the surface area (the total area of all the faces) of shape E.
Explain how you figured it out.
d. Find the surface area of the right rectangular prism in your set. Measure to the nearest half
centimeter. How are the dimensions of the prism related to the dimensions of its faces?
e. Explain how you would find the surface area of the right triangular prism in your set. Don’t
actually find the surface area; just explain the process for finding it.
56
Polyhedra
2. Drawings or sketches
a. Using isometric dot paper, make a sketch of a cube. Represent the “hidden” edges with dotted
lines. Can you draw this same cube on plain paper?
b. Draw a cube that has an edge length that is twice as long as the cube you drew in part a.
c. Draw a right rectangular prism, such as shape F, on isometric dot paper. Extension: Draw the
same prism from a different viewpoint. You may want to hold up your physical model in order to
view it from different angles.
3. Surface area (continued)
A company wants to arrange 24 alphabet blocks in the shape of a rectangular prism and then
package them in a box that exactly fits the prism. Each block is a cube with 1-inch edges, so each block
has a volume of 1 cubic inch.
a. Find all the ways 24 cubes can be arranged into a rectangular prism. Make a sketch of each
arrangement. Record the dimensions and surface area and volume in the table below.
Length Width Height Volume Surface
Sketch
Area
b. Which arrangement would you recommend to the company? Why?
c. Why do you think the company makes 24 alphabet blocks rather than 26?
57
Polyhedra
Volume
You may use cubes for the following activities. Let the dimensions of your cube be 1 x 1 x 1.
1. What is the surface area of one cube? What is the volume? What label should you use for each
measurement?
2. On the next page are the nets of some “open” boxes—boxes without tops or lids.
A. If you were to cut out each net, fold it into a box, and fill the box with cubes, how many cubes
would it take to fill the box? Make a quick prediction and then use two different approaches to find
the number of cubes. You may want to cut out the actual nets, fold them up, and tape them into
boxes.
B. What strategies did you use to determine the number of cubes that filled each box?
3. Build some rectangular prisms with your cubes. Find the volume of each. Generalize an approach for
finding the number of cubes (the volume) in any right rectangular prism. Write your rule in symbols
using the dimensions: length, width, and height.
4. • If the box below was completely filled with cubes, how many cubes would be in the bottom layer?
• How could you figure this out without counting? (This number is the same as the _____ of the base.)
• How many layers tall is the box?
• What is the total number of cubes, including the cubes already shown, needed to fill the box below?
Check yourself using the rule you wrote in #3.
Do you think this “layering method” would work for other prisms, such as triangular prisms,
hexagonal prisms, etc? Explain.
5. Find the surface area and volume of the box below:
58
Polyhedra
59
Polyhedra
6. What is the relationship between scale factor and volume? Use the sketches below to help you answer
this question. (First, determine the volume of each cube.)
What would be the volume of an image with a scale factor of n? (What is the rule for any scale factor?)
7. What if you were working with rectangular solids that were not cubes? Would your rule still work?
For example, if you have a right rectangular prism with dimensions of 2 x 3 x 5 and you doubled all the
dimensions (a scale factor of 2), does your general rule for a scale factor of n still work?
8. How is the relationship between scale factor and volume the same as the relationship between scale
factor and area that we worked on earlier in this course? How is it different?
9. Extension A: Would your rule work for “odd-shaped” figures, such as the one below?
10. Extension B: Build one or two odd-shaped figures with cubes and sketch them on isometric dot paper.
Make at least one figure that is more than one cube tall.
11. Extension C. Draw a net for a cylinder, such as the one below. (Hint: you might want to “build” a
cylinder out of paper first.) How would you find the surface area? The volume? (Hint: Think about
the layering method you used in Problem 4.)
60
Polyhedra
Representing and Visualizing Polyhedra (revisited)
Orthogonal Views
1. Build the shape below with your cubes. Move your body so that you are at “eye level” with the front of
your shape, in order to see the “front view”. Does your eye level view match the drawing of the front
view? Try the other views as pictured; do you see these views as shown?
2. Using your cubes, construct a model from this building plan (which is also the top view). Each number
tells you how many cubes go in that column as you are building it.
Note: This indicates the front of the building, not the front view.
Sketch each view in the spaces provided:
Front View
Back View
Right View
Left View
61
Polyhedra
3. Below are base plans for four different buildings.
Match each view with the correct set of drawings below. (Can you do it without constructing a model?)
(Check your answers on the last page when you are done )
62
Polyhedra
4. Sketch the front, right-side, and top views for each of these cube arrangements. You may want to build
the arrangements first.
5. Sketch the view from the left of the shapes in parts a and b. How are they related to the views from the
right? (See also, #2.)
6. Use cubes to make a model of a building for EACH set of plans below. On the base outline record your
base plan for each building by marking in the number of cubes you used.
Compare your models with those of your group. There may be more than one correct answer.
(The answers are on the next page, but don’t peek until you are sure you have one that fits!)
Extension: Sketch some of your building on isometric dot paper.
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Polyhedra
Answers to Problem 3
Building A matches Set 3
Building C matches Set 4
Building B matches Set 1
Building D matches Set 2
Answers to Problem 6
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