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7/28/2019 Introduction to Drawing Shapes
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Geometry and Measurement, Grades 6-8 1
Geometry and
Measurement
TABLE OF CONTENTS1 Measure and Map the Room 2
2 Scale Drawings .6
3 Perimeter ..104 Area 145 Exploring Angles .206 Investigating Turns 26
7 Coordinate Graphs ..29
8 Similarity and Ratio 32
9 Circles .37Student Pages....43
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Geometry and Measurement, Grades 6-8 2
Measure and Map
the Room
Mathematical Focus
8 Estimate measurements
8 Select appropriate units of measurement
8 Use techniques and tools accurately to determine measurements
In this activity, students begin by creating their own tape measure.
The process of creating the tape measure helps them form a mental
picture of the size of a unit. Students use their mental picture of a
unit as well as other strategies to visually estimate the height or
length of different objects and distances in the room. They then check
their estimates with the tape measure. To conclude, students create a
top-view sketch of the room and record the measurements of different
objects in the room on their sketches.
Preparation and Materials
Before the session, gather the following materials:
8 Ruler
8 Tape
8 two-inch wide strips of paper (more than enough to create a six-
foot long tape)
Before beginning this activity, ask students whether they usually use
the English or the metric system of measurement in school (inchesand feet or centimeters and meters). This activity is written using the
English system, but can be changed to use the metric system if
appropriate.
Activity 2 builds on the work that students do in this activity. Save
the top-view sketch of the room that students create for use in Activity
2. Also save students tape measures for use in future activities.
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Geometry and Measurement, Grades 6-8 3
Construct a Measuring Tape
1. Use strips of paper to make a tape measure.
Give students a ruler, some tape, and the two-inch strips of paper.
Challenge them to create a six-foot measuring tape that shows one-
quarter-inch increments. As students begin the challenge, ask
questions such as:
Can you show me with your hands about how long a foot is?
How many strips of paper will you need to make a six-foot
tape measure? (Students should see that they need to begin
by determining the length of each individual strip.)
Willyou tape the strips end-to-end, or will there be a bit of
overlap where you tape them together?
2. Review common measurement benchmarks.
Review with students common measurement benchmarks, such as 12
inches = 1footand 36 inches = 3 feet = 1 yard. Have students mark
the larger units of measurement on their tape measure.
3. Estimate the length of objects in the room and thenfind the actual measurement using tape measure.
Create a three-column chart similar to the one pictured below. Have
students pick five or six objects from around the room to measure.
List those objects on the chart. Ask students to estimate the length of
each object and record their estimate on the chart. Have them check
their estimate by measuring the object and then record the actual
measurement on the chart.
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Geometry and Measurement, Grades 6-8 4
Object Estimate Actual
Measurement
Length of table
Width of doorway
Length of
chalkboard
Height of mentor
4. Suggest additional estimating challenges.Present students with opportunities to practice estimating.
Challenges might include the following:
Pick an object on your chart. Can you find something that
you havent measured yet that is about the same length as
that object? Write the name of the object and your estimate
on the chart. Then measure it.
Can you find something that is about three feet long?
Can you find something that is approximately two yards
long?
Estimate the height of the ceiling. What strategy did you
use for making this estimate?
Map the Room
1. What is a floor plan?
Talk with students about floor plans and top-view representations ofthree-dimensional objects. Invite them to share what they know.
Who uses floor plans?
What are they used for?
What kinds of things does a floor plan show? What doesnt
it show?
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Geometry and Measurement, Grades 6-8 5
2. Create a top-view sketch of the classroom.
Explain that a floor plan is like a top-view map of a room or building.
Challenge students to create a top-view sketch of the room in which
you are working. To help students begin, ask questions such as:
Can you draw an outline of the walls of this room? Where
are the doorways and windows?
Which objects in the room should be included in the sketch?
Should the desks and tables be included?
Which objects should not be included? Why do you think
so?
3. Measure and record room and furniture dimensionson the top-view sketch.
Have students measure the dimensions of the room and record the
measurements on their sketches. Ask them to measure the pieces of
furniture they included on their sketch and record those
measurements on the sketch as well. Remind students to only
measure the parts of the furniture that are seen from a top view. Ask:
Do you need to measure the height of a desk? [no]
Do you need to measure the width of a desk? [yes]
Have students save their sketch of the room to use in Activity 2: Scale
Drawings.
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Geometry and Measurement, Grades 6-8 6
Scale Drawings
Mathematical Focus
8 Scale drawings
8 Use of scale rulers
In this activity, students explore the concept of scale, and use their
top-view classroom sketches from Activity 1 as a guide for creating a
scale drawing of the room. They learn to use a scale ruler as a tool
for creating scale drawings and investigate the use of quarter-inchgraph paper as another technique for making scaled representations.
Students then employ their new knowledge to determine whether
pictured objects drawn to different scales represent real objects or
toys.
Preparation and Materials
Before the session, gather the following materials:
8 Student Page 1: Scale Ruler
8 Student Page 2: Half-Inch Graph Paper, several copies
8 Student Page 3: Quarter-Inch Graph Paper, several copies
8 Student Page 4: Real, Toy, or Giant Size?
8 Classroom map from Activity 1
Cut out the scale ruler from Student Page 1. Fold and tape the ruler
together ahead of time.
Save students scale drawings of the room for use in Activities 3 and 4.
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Geometry and Measurement, Grades 6-8 7
Scale Rulers and Grid Paper
1. Share ideas about scale drawings.
Ask students questions such as:
What is a scale drawing? [A representation of a real object
or space that maintains the proportions of the original]
Can you think of some examples?[floor plans, maps]
Who uses them?
What are they used for?
How are they created?
Is the top-view sketch of the classroom you made in Activity1 a scaled drawing? Why or why not?
2. Introduce the scale ruler.
Show students the scale ruler you made from Student Page 1. Explain
that this three-sided ruler is specially made for creating scale
drawings. The side marked Scale: inch = 1 foot is used to make
drawings at a quarter-inch-to-a- foot scale. The side marked Scale:
inch = 1 foot is used to make drawings at a half-inch-to-a-foot scaleThe side marked Standard Ruler is simply a one-inch ruler.
3. Demonstrate how the scale ruler works.
Ask students to look at the classroom map they created in Activity 1.
Tell them that they can use the actual dimensions of the room and the
furniture written on the map to create scale drawings. Explain that
one half- inch on the scale ruler represents one foot. Pick a desk from
the students map and demonstrate how to use the half-inch scale rulerby drawing a line that represents the length of the desk. Have
students then draw the rest of it.
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Geometry and Measurement, Grades 6-8 8
4 ft (2 on the half-inch scale ruler)2 ft
4. Use graph paper to create a scaled drawing.
Draw the desk again on a copy of Student Page 2: Half- Inch Graph
Paper. Explain that the graph paper can be used in the same way as
the ruler to create a scale drawing: the length of each square on the
graph paper (in this case, one half-inch) represents one foot.
Scale Drawing of the Room
1. Use a scale ruler or graph paper to create a scaledrawing of the classroom.
Challenge students to use their map from Activity 1 to create a
quarter-inch scaled floor plan of the classroom. Tell them that one
strategy for doing this is to create scaled top-view representations of
all the major pieces of furniture in the room on a copy of Student Page
3: Quarter-Inch Graph Paper, cut out the furniture, and then place it
on a scaled drawing of the room. Encourage students to develop their
own strategies for creating the scale drawing, exploring the use of
both the scale ruler and quarter-inch graph paper.
Scale: inch = 1 foot
Scaled top-view drawing.
2 inches
1 inch
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Geometry and Measurement, Grades 6-8 9
2. Share experiences of creating scale drawings.
When students have finished, ask questions such as:
Whatstrategy did you use for creating your scale drawing
of the room? What did you do first? What did you do next?
What was difficult about this activity? What did you like
about it?
If someone asked you to create a scaled floor plan of another
room, how would you go about it? Would you use the same
strategy or change your strategy?
Suppose someone wanted to create a scale drawing of a
room but did not know how. How would you explain the
process?
Suppose you wanted to create a scaled floor plan of this
building (or one floor of this building); how would you goabout it? What would you do first?
Real, Toy, or Giant-Size?
1. Use mathematical reasoning to determine whetherscale drawings represent life-size objects, toyobjects, or giant-size objects.
Give students Student Page 4: Real, Toy, or Giant Size? Have them
look at the scale drawings of various objects. Call their attention to
the scale below each object; for example, below the hammer, the scale
says inch = 1 inch. Ask: Is the hammer toy size, real size, or giant
size?
Have students use a ruler to measure each of the pictures and then
use the given scale to determine the actual dimensions of the object.
Ask students to record the actual dimensions on the student page and
then determine if the object is toy size, real size, or giant size. Have
them explain their thinking.
Look at several examples of real maps (or, if available, floor
plans, scale drawings of furniture, etc.). Discuss the scale for
each representation of space. Calculate distances on the maps
or actual dimensions from the floor plans.
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Geometry and Measurement, Grades 6-8 10
Perimeter
Mathematical Focus
8 Understand the attribute of perimeter
8 Use different tools and techniques to determine perimeter
In this activity, students explore the attribute of perimeter. They
estimate, measure, and compare the perimeter of various objectswithin the room, as well as the perimeter of the room itself.
Preparation and Materials
Before the session, gather the following materials:
8 Tape measure from Activity 1
8 Scale drawings from Activity 2
8 Ruler, piece of string, or other things students might use tomeasure perimeter
8 Map of your state (optional)
8 Map of the United States (optional)
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Geometry and Measurement, Grades 6-8 11
The Perimeter of the Room
1. Discuss the concept of perimeter.
Invite students to share what they know about the attribute of
perimeter. Ask questions such as:
Can you describe the perimeter of an object or space?
What is the perimeter of this room?
How could you estimate the perimeter of this room?
The perimeteris the measurement around the edge of something. The
perimeter of the room is the measurement around the edge of the
room. One way of estimating the perimeter of the room is to take a
walk around it, using footsteps to approximate standard linear feet.
2. Estimate the perimeter of the room.
Have students visually estimate the perimeter of the room. Then have
them walk around it and make a second estimate. Record both
estimates on a piece of paper. Ask:
What strategies did you use? (Some students may estimatethe length of each wall and then add the lengths together to
get their estimates, others may keep a running total as they
walk around the room.)
What problems did you encounter in estimating the
perimeter of this room? (Maneuvering around furniture
may be the greatest difficulty in making the estimate.)
How accurate do you think your estimate is?
How could you get a more accurate measurement of the
perimeter of this room?
What unit of measurement would you use?
3. Use the scale drawing to determine the perimeter ofthe room.
Use students scale drawing of the room from Activity 2. Have them
look at the drawing to determine the perimeter of the room. Ask:
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Geometry and Measurement, Grades 6-8 12
How can you use your scale drawing to determine the
perimeter of this room?
What other information do you need?
Have students compare their perimeter calculation from the scale
drawing with the measurement they got by walking around the room.Ask: Are the measurements close?
4. Use a tape measure to determine the perimeter ofthe room.
Ask students to use their tape measure from Activity 1 to measure the
perimeter of the room to the nearest inch (giving measurements in
feet and inches). Have students record their calculation and compare
it with the measurement they got from the scale drawing and from
walking around the room.
Estimating and Comparing the Perimeter ofObjects and Spaces
1. List objects in the classroom to be measured.
Have students create a three-column chart in which they list the
objects such as books, rugs, or tables for which they will estimate and
then measure the perimeter. Label the columns as follows:
Object PerimeterEstimate
Perimeter
Notebook
Table top
Circular rug
2. Estimate the perimeter of each object in the chartand then measure it.
Discuss different strategies for finding the perimeter of the objects on
the chart. Students may suggest using their tape measure, a piece of
string, or a ruler. Encourage them to explore different strategies.
Discuss appropriate units of measurement (i.e. inches, feet, yards).
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Geometry and Measurement, Grades 6-8 13
Encourage students to extend their thinking about perimeter by
considering the following challenges:
Draw a large rectangle on a piece of paper. Can you drawanother shape within the rectangle that has a perimeter
longer than this rectangle? Explain your thinking. Then
draw the shape you have in mind, and measure and
compare its perimeter to the perimeter of the rectangle.
Example of a shape within a rectangle with a larger
perimeter than that rectangle:
What do you think the perimeter of our state is in miles?
Use a map of the state to estimate its perimeter.
Look at a map of the United States. Which state do you
think has the greater perimeter, Maine or Tennessee?
Explain your thinking. (Repeat this challenge with other
pairs of states.)
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Geometry and Measurement, Grades 6-8 14
Area
Mathematical Focus
8 Understand the concept of area
8 Use different tools and techniques to determine area
In this activity, students explore the attribute of area. They
investigate various methods for measuring the area of rectangles and
develop a formula for finding area. They apply what they know about
the area of rectangles to develop formulas for finding the area ofparallelograms, triangles, and trapezoids.
Preparation and Materials
Before the session, gather the following materials:
8 Student Page 5: Rectangles, 2 copies
8 Student Page 6: Triangles and Trapezoids, 2 copies
8 Ruler
8 Scissors
8 Tape
8 Scale drawings from Activity 2
8 Graph paper
Cut out the rectangles from one copy of Student Page 5 ahead of time.
Leave the other copy intact.
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Geometry and Measurement, Grades 6-8 15
Investigating the area of simple shapes
1. Discuss the concept of area.
Talk with students about area. Ask questions such as:
What are some units of measurement for area? [Square
feet, square inches, square miles, etc.]
What unit of measurement would you use to measure the
area of this room?
How does the area of this room compare to the area of the
gym (or some other large room that students are familiar
with)?
What unit of measurement would you use to measure the
top of a desk?
2. Find the area of different rectangles.
Give students the shapes you cut out from Student Page 5:
Rectangles. Ask them to use a ruler to find the area of each rectangle
in square inches and write it on the rectangle. Ask: How did you find
the area of each rectangle? (Some students may already be familiar
with the formula for finding the area of a rectangle: length x width.
Other students may use a counting strategy for determining the area.)
Ask: Can you write a rule for finding the area of a rectangle?
3. Transform a rectangle into a parallelogram anddetermine the area.
Ask students to select one of the rectangles they have been working
with. On it, have them draw a line from one corner to a point on the
opposite side, making a right triangle within the rectangle. Ask: If
you cut off the triangle, slide it over to the other side, and reattach it,
what new shape is formed? [a parallelogram]. Have students cut off
the triangle and reattach (tape) it on the other side. Ask: What is the
area of the new shape? [The same area]
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Geometry and Measurement, Grades 6-8 16
Transform the other rectangles into parallelograms by cutting off a
triangle and taping it on the other side.
4. Write a rule for finding the area of a parallelogram.
Have students use the words base(the width) and heightto describe
the parallelograms. Have them write base and height on the
appropriate spot on each parallelogram. Ask: Can you write a rule for
finding the area of a parallelogram? [The area of a parallelogram is
equal to the area of a rectangle with the same base and height: A = b
x h]
5. Divide a parallelogram into congruent triangles andfind the area of each triangle.
Have students choose one of their parallelograms. Ask:
What is the length of its base? What is the height? What is
the area? Can you divide the parallelogram into two congruent, or
equal, triangles? How many ways could you divide the
parallelogram into congruent triangles?
Students may need to draw lines on their parallelograms to see how
many ways there are of dividing them. Have students choose one way
and then cut the parallelogram into two triangles. Ask:
Are the triangles congruent? How can you test for
congruence?
What is the area of each triangle? How do you know?(the
area of each triangle is simply equal to half the area of the
parallelogram).
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Geometry and Measurement, Grades 6-8 17
6. Discuss strategies for finding the area of a triangle.
Give students two copies of Student Page 6: Triangles and
Trapezoids. Ask:
What strategy can you use for finding the area of a
triangle?[Double the shape]
Can you write a rule for finding the area of a triangle?
A=b x h or l x w
2 2
7. Explore strategies for finding the area of trianglesand trapezoids.
Ask students to cut out the shapes from one copy of Student Page 6:Triangles and Trapezoids. Challenge them to apply what they know
about finding the area of rectangles and parallelograms to finding the
area of triangles and trapezoids.
One strategy for finding the area of a triangle is to add an identical or
congruent triangle to the existing one to form a rectangle or
parallelogram. Students can then easily find the area of the rectangle
or parallelogram and divide it by 2.
A strategy for finding the area of a trapezoid is to cut off a right
triangle from one side of the trapezoid, flip it vertically and slide it
over to the other side. Again the shape has been transformed into a
rectangle, a shape for which the area can be easily calculated.
Encourage students to explain their strategies as they work.
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Geometry and Measurement, Grades 6-8 18
Investigating the area of a Space
1. Estimate the area of the classroom.
Return to the students scale drawings of the room from Activity 2.
Using their scale drawings, have students estimate the area of theroom in square feet.
Exploring the Relationship Between Areaand Perimeter
1. Use graph paper to explore the relationship betweenarea and perimeter.
Give students several sheets of graph paper and present the following
challenges:
How many different figures can you draw with an area of
five squares? Ten squares? What is the perimeter of each
figure? (Have students number each figure and use a table
to keep track of the area and perimeter.) Example:
Figure # Area Perimeter
1 5 10
2 5 12
3
4
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Geometry and Measurement, Grades 6-8 19
How many different figures can you draw with a perimeter
of 12? A perimeter of 20? What is the area of each figure?
Have students number each figure and record its perimeter
and area on their charts.
Challenge students to investigate questions such as the following:
How could you find the area of your footprint?
How could find the area of a puddle?
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Geometry and Measurement, Grades 6-8 20
Exploring Angles
Mathematical Focus
8Angles in shapes
8Angle relationships
Students explore angle measurements in a collection of shapes.
Beginning with their knowledge of the angle measurements of a
square (90), students use angle and shape relationships to find the
angle measurements of the shapes in the collection. After checking
the measurements with a circle protractor, students classify the
angles as right, acute, obtuse, straight or reflex and then use the
shapes to construct new angles. Students investigate the sum of the
interior angles of different types of shapes and look for patterns in
their findings.
Preparation and Materials
Before the session, gather the following materials:
8 Student Page 7: Shapes
8 Student Page 8: Angles In and Around Shapes
8 Student Page 9: Circle Protractor
8 Student Page 10 and 11 : Dot Paper, several copies
Cut out shapes from Student Page 7 ahead of time. Copy the circle
protractor (Student Page 9) onto a transparency.
If you have access to pattern blocks, these can be used in place of the
cut-out shapes from Student Page 7. Similarly, an actual circleprotractor can be used in place of the protractor on Student Page 9.
The protractor will be used again in Activities 6 and 8.
In Activities 5 and 6, students use a circle protractor to measure
angles. If your students are unfamiliar with this task, Activity 2 in
the Grades 35 Geometry and Measurement unit provides a good
introduction.
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Geometry and Measurement, Grades 6-8 21
Shapes and Angles
1. Identify angles less than, greater than, and equal to90.Give students a set of shapes cut out from Student Page 7, or a set of
pattern blocks, and a copy of Student Page 8: Angles In and Around
Shapes. Ask:
Can you show me a shape in your collection that has a
right, or 90, angle? [the square]
Can you show me a shape with an angle that is smaller
than 90? An angle that is larger than 90?
2. Given only one angle measurement, usemathematical reasoning to determine all he otherangle measurements of shapes in a collection.
Give students a copy of Student Page 8: Angles in and Around Shapes.
Demonstrate how three of the thin rhombi fit into the 90 corner of the
square. Ask: How could this help you determine the angle
measurements of the two small angles of each rhombus? Explain that
an indirect way of measuring angles is to compare one or more shapeswhose angles are known with a shape whose angles are not known. In
this case, the 90 angle of the square is covered exactly with the three
thin rhombi. 90 divided by 3 is 30; therefore, the angle measurement
of each small angle is 30.
30
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Geometry and Measurement, Grades 6-8 22
Challenge students to use the square and the thin rhombi to
determine the angle measurements of all of the other shapes. Have
them record the angle measurements on Student Page 8 and use
sketches or words to explain their thinking. When students have
found all of the angle measures, ask:
What strategies did you use for determining the angle
measure of the different shapes?
Which shapes have angles that are the same size?
3. Use a protractor to check angle measurements.
Show students how to check the measurements of each shape with a
circle protractor. Position one shape so that its vertex is at the center
of the protractor and one of its edges is along the 0 line. Explain that
the number at the edge of the protractor is the angle measurement.
*** THIS GRAPHIC NEEDS TO BE MADE CLEARER!! ***
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Geometry and Measurement, Grades 6-8 23
4. Classify angles as right, acute, or obtuse.
Remind students that right angles are exactly 90, acute angles are
less than 90, and obtuse angles are greater than 90 but less than
180. Ask students to identify the angles in each shape as right,
acute, or obtuse.
5. Make new angles of varying sizes.
Challenge students to use the shapes or pattern blocks to make new,
different size angles. Ask: How many different size angles can you
make using two pieces? Three pieces? Encourage students to show
how each angle was made by tracing the shapes they put together and
recording the resulting angle measure. Explain that there are two
additional types of angles: straightand reflex. Straight angles are
180; reflex angles are greater than 180. Have students classify the
angles they made as acute, right, obtuse, straight, or reflex.
Sum of the Angles in Shapes
1. Investigate the sum of angles in triangles.
The angle measurements students found for the triangle on Student
Page 8 are 60, 60, and 60. Ask: What is the sum of the angles in
this triangle? [180] Give students copies of Student Pages 10 and
11: Dot paper, and ask them to draw 5 different triangles on the dotpaper. Have students use a circle protractor to find the angle
measurements for each triangle. Next have them calculate and record
the sum of the angles for each triangle. Ask: What do you notice?
[The sum of the angles is always 180] Talk with students about
accuracy in measuring and the fact that measurements may be off a
little, but that the sum of the angles in any triangle is always 180.
Can you draw a triangle whose angle measurements do not
add up to 180?
30
120
30
60
60
60 30
60
90
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Geometry and Measurement, Grades 6-8 24
2. Explore the sum of angles in squares.
Have students consider the square on Student Page 8. Ask:
What is the sum of the angles in a square?[360]
What is the sum of the angles in a rectangle?[360].
Explain that both squares and rectangles are quadrilaterals, shapes
with four sides and four angles. Have students look back at the
shapes from Student Page 7 and identify the quadrilaterals. Ask
them to find the sum of the angles for each quadrilateral. Ask: Do
you think the sum of the angles in anyquadrilateral, or four-sided
shape, is 360? Have students use dot paper to construct a variety of
different quadrilaterals. For each quadrilateral they make, ask
students to determine the angle measure of each angle and then find
the sum of the angles.
3. Look for patterns in the sum of angles for differentpolygons.
Have students make a chart similar to the one below. Invite them to
explore the sum of the interior angles of the other polygons in the way
they investigated the triangles and quadrilaterals. Have them use dot
paper to construct at least three polygons of each type they
investigate.
Polygon Number of
Sides
Total Interior
Angle
Triangle 3 180
Quadrilateral 4 360
Pentagon 5
Hexagon 6
Octagon 8
n-gon
When they have finished, ask: Can you write a rule for finding the
sum of the interior angles of any polygon? [180 x (number of sides
2]
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Geometry and Measurement, Grades 6-8 25
The sum of the interior angles in any simple closed polygon (closed
shape in which none of the line segments cross one another) is
related to the number of sides. The sum of the interior angles in any
triangle is 180; in any quadrilateral, 360; in any pentagon, 540;
and so forth. The relationship between the sum of the interior angles
of a polygon and its number of sides can be described by the formula
180 x (N 2), where N is the number of sides in the polygon. While
it is not necessary for students to derive this formula, encourage
them to look for patterns in their findings.
Some students may be able to see the pattern in the sum of interior
angles, but unable to generalize it to a rule; however, by eighth grade
most students should be ready for this level of abstraction.
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Investigating Turns
Mathematical Focus
8 Estimate and measure turns and angles
8 Explore the relationship between turns and resulting angles
In this activity, students investigate the idea of turns as a change in
orientation or direction. They begin by traveling around shapes and
measuring the degree of turn needed at each vertex point in order to
stay on the shapes path. Students then use their knowledge of turnsto navigate through the waters of a crowded harbor.
Preparation and Materials
Before the session, gather the following materials:
8 Student Page 1: Scale Ruler
8 Student Page 8: Angles In and Around Shapes
8 Student Page 9: Circle Protractor
8 Student Page 12: Harbor Map8 Student Page 13: Directions8 Straight edge or ruler
Re-use the scale ruler made for Activity 2 or make another scale ruler
from Student Page 1 ahead of time.
Re-use the circle protractor from Activity 5 or copy Student Page 9
onto a transparency. A commercially available protractor could also
be used.
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Geometry and Measurement, Grades 6-8 27
Traveling Around Shapes
1. Investigate the relationship between turns andresulting angles.
Distribute Student Page 8: Angles In and Around Shapes and have
students look at the trapezoid. Draw a point on the lower left vertex
of the trapezoid, and present students with the following scenario:
Suppose I had a miniature car that I could drive along the lines of this
shape. If I started at this point and drove up to the next vertex point
(move pencil along edge of shape as indicated below), how many
degrees would I need to turn my car to continue traveling along the
path of the trapezoid? Explain your thinking.
In order to determine the number of degrees in the turn, students may
use a straightedge or ruler to extend the line. Give students a circle
protractor, and have them place it on the shape to measure the
number of degrees the car must turn.
?
Have students continue traveling around the shape. At each vertex
point, ask them to determine how much they must turn to continue
traveling along the shape and record the number of degrees beside
each vertex point. Once they reach the starting vertex point, have
students determine how much they need to turn to return to their
starting orientation. Ask: How many degrees did you turn altogether
to travel around the trapezoid?
2. Look for patterns in the total number of degreesturned when traveling around different shapes.
Challenge students to travel around the rest of the shapes on Student
Page 8, recording the number of degrees turned at each vertex point.
For each shape, ask: How many degrees did you turn altogether when
traveling around the shape?
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Geometry and Measurement, Grades 6-8 28
After students have investigated a few shapes, ask:
What do you notice about the total number of degrees
turned when traveling around the different shapes? [360
degrees] How can you explain this?
Do you think this will be true for any shape? Can youconvince me with examples?
Navigating the Waters
1. Use knowledge of turns and angles to get from onepoint to another on a map.
Give students a copy of Student Page 12: Harbor Map, Student Page
12: Directions, and a scale ruler. Discuss the scale of the map ( inch
= 1 mile), the positions of Boats 1 and 2, and the landmarks on themap. Challenge students to read all the directions on Student Page 13
and decide which directions will lead Boat 1 to the harbor and which
will lead Boat 1 to island A. Then ask:
How did you think about the problem?
Which directions can be used for getting from Boat 2s
current location to Island A?
Could any of the other directions be used? Why or why not?
2. Give directions for getting from one point to anotheron a map.
Challenge students to give directions to the skipper of Boat # 2 for
getting from Boat 2s current location to the dock and from Boat 2s
current location to the harbor. Remind them to avoid getting
shipwrecked on the rocks by navigating around the buoys. Have one
student pretend to be the skipper and, with a pencil, follow the
directions given by the other students.
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Geometry and Measurement, Grades 6-8 29
Coordinate Graphing
Mathematical Focus
8 Specify locations and determine spatial relationships on a
coordinate system.
In this activity, students locate landmarks on a coordinate map and
gain practice in naming points on a coordinate grid as they play the
strategy game Capture!
Preparation and Materials
Before the session, gather the following materials:
8 Student Page 14: Harbor Map with Grid
8 Student Page 15: Capture!Game Board, 1 for each student
8 Student Page 16: Coordinate Grid, 2 copies for each student
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Geometry and Measurement, Grades 6-8 30
Using Coordinates to Describe Locations
1. Use coordinate points to describe the location oflandmarks on a map.
Give students a copy of Student Page 14: Harbor Map with Grid.
Begin by having students find Boat 1. Ask: Can you describe this
boats location, using coordinate points? The boat is at (7, 2). Explain
that the first coordinate tells how far to move horizontally and the
second coordinate tells how far to move vertically. Have students
write Boat 1s coordinates on the map beside the boat.
Ask students to find the landmarks at the coordinates below and then
record the coordinates on the map beside the landmark:
(4, 7) -- [The lighthouse]
(8, 4) -- [Buoy]
Ask students to find and record the coordinates of each of the
following landmarks:
Buoy 2
Buoy 3 Rocks
Boat 2
Dock
Capture!
1. Play the game ofCapture!
Introduce the game ofCapture!, similar to the game Battleship.
Give each player a copy of the Capture! game board. Have playersplace a barrier, such as a book or notebook, between them so they
cannot see each others boards. Explain that the grids are bodies of
water. Have students place three boats on their game board. A boat
is made by circling three adjacent points on the grid that form a
horizontal, vertical, or diagonal line. The goal of the game is to
capture an opponents boats by guessing the boats locations. Players
take turns calling out coordinate points.
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Geometry and Measurement, Grades 6-8 31
If Player 1 calls out a coordinate point of one of Player 2s boats, then
player 2 must say, Boat. If the point is not on a boat, Player 2 says,
Water. Have students use the bottom grid on the game board to
record boat or water guesses. The first player to capture all three
of the opponents boats wins the game.
Relationships on a Coordinate Grid
1. Plot points on a coordinate grid.
Ask students to plot the set of points below on a copy of Student Page
16: Coordinate Grid. Explain that the x-axis represents how manybuckets of popcorn were sold by a vendor, and the y-axis represents
how many dollars the vendor earned by selling buckets of popcorn.
5 buckets of popcorn, $10
2 buckets of popcorn, $4
8 buckets of popcorn, $16
10 buckets of popcorn, $20
4 buckets of popcorn, $8
2. Use a linear pattern on a coordinate grid to solveproblems.
Ask students to determine how much money the vendor would earn if
he sold 9 buckets of popcorn, 12 buckets, or 1 bucket by looking at the
graph. Have students show where on the graph they would look to see
how many buckets of popcorn were sold for the vendor to earn: $6,
$18, or $30. Ask: Is it possible for the vendor to sell enough buckets
of popcorn to earn exactly $8.50? Why or why not? Have students try
to create a rule for the number of dollars earned based on the number
of buckets of popcorn sold.
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Geometry and Measurement, Grades 6-8 32
Similarity and Ratio
Mathematical Focus
8 Ratios
8 Similar triangles
8 Properties of similar shapes
In this activity, students explore the concept of ratio. They measure
lengths, heights, and perimeters of various shapes and compare ratios
of measurements. Students then apply their knowledge of ratios to aninvestigation of similar triangles (triangles with corresponding
angles that are equal, and corresponding sides that are in the same
ratio). They conclude the activity by using their knowledge of
similarity and ratio to determine the height of a streetlight.
Preparation and Materials
Before the session, gather the following materials:
8 Student Page 17: Staircases
8 Student Page 18: Shape Pairs
8 Student Page 19: Streetlight Height
8 Tape measure
8 Circle protractor or Student Page 9
8 Ruler (optional)
Re-use the circle protractor from Activity 5 or copy Student Page 9
onto a transparency. A commercially available protractor could also
be used.
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Geometry and Measurement, Grades 6-8 33
Ratio
1. Discuss the concept of ratio.
Ask students if they have heard the term ratio used and, if so, in what
context. Explain that a ratiois the comparison of two numbers using
division. Give students an example such as the ratio of the length of a
side of a square to its perimeter. Ask students to draw a square and
find this ratio.
Example:
2. Explore the ratio ofside length: perimeterfordifferent size squares.
Ask if this ratio will be the same for other squares? Have students
draw some more squares and measure their side lengths and
perimeters. Ask if the ratio is the same for these squares. If the
numbers are different, then how can you tell if the ratio is the same?
Help students reduce the ratio like they would reduce a fraction to its
lowest terms to see if the ratios are the same.
3. Investigate the ratio ofheight: thumb length.
Ask students to investigate another ratiothe ratio of a persons
height to the length of his or her thumb. Have them use a tape
measure to measure your height, their heights, your thumb length,
and their thumb lengths. Ask them to create a table that shows
heights and thumb lengths. (If possible, gather some more data for
the table by measuring more peoples heights and thumb lengths.)
Have students compare the ratios to determine whether they are the
same. Ask questions such as:
2 units
Length of side = 2Perimeter = 8
Ratio of side to perimeter = 2:8 or 2/8Ratio reduced to lowest terms = 1:4 or 1/4.
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Why do you think different people have slightly different
ratios of thumb length to height?
What do you think the range of ratios of thumb length to
height would be for the general population? Why?
Try this activity with other ratios, such as foot length to height.
4. Interpret and explore a building code that states theratio ofrise height: tread lengthfor stairs.
Tell students that a local building code states that the ratio of the rise
of a stair to its tread cannot exceed 3:4. Ask: What does this building
code mean? Explain that the riseof a stair is the height of each step,
and the treadof a stair is the length of the top of a step.
5. Determine which staircases meet the building code.
Distribute Student Page 17: Staircases and ask students to determine
which staircases meet the building code and which do not. Have
students explain their thinking as they work. Ask questions such as:
Why is a building code such as this important? What might
happen with steps that do not meet the building code?
Can you think of staircases where the ratio is very different
from the steps on this worksheet? (Students may have seen
very shallow steps on an outdoor staircase or steep steps in
an old building)
Ask students to measure the stairs in your building, at home,
or wherever you can find them to see if they pass the code.
Rise
Tread
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Similarity of Shapes
1. Discuss the concept of similarity.
Give students Student Page 18: Shape Pairs. Have them start with
the first pair of triangles, and measure the lengths of their sides. Askif the corresponding sides of the triangle are in the same ratio.
Explain that two triangles are called similarif their sides are in the
same ratio. Ask students to measure each of the angles in the two
triangles and write the measurements inside the angles. When both
triangles are labeled, ask: What do you notice about the angles of the
two triangles? Do you think that this relationship would hold true for
other similar triangles?
2. Determine whether different pairs of triangles aresimilar.
Ask students to look at the next pair of triangles and determine
whether they are similar. When students have finished, ask them if
the angles of the triangles are the same. Have students check this
with the circle protractor.
Explain that similar triangles have corresponding sides with equal
ratios and corresponding angles that are the same size. Help students
understand that any two triangles that both have one of the traits
(corresponding sides that are in equal ratios, or corresponding angles
that are equal) are similar triangles and therefore will both have the
other trait. Ask students to draw two different triangles that both
have the same size angles. Ask: Are the sides of these two triangles
in the same ratio? How do you know?
The next activity can be done outside with a real pole or another tall
object (such as a streetlight, a flagpole, the pole holding up a slide,
the pole from a swing set, or the trunk of a tree). Measure the actual
shadow created by this pole and a smaller stick.
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Geometry and Measurement, Grades 6-8 36
How Tall Is It?
1. Use knowledge of similar triangles to determine theheight of a streetlight.
Use Student Page 19: Streetlight Height. Ask students how tall theythink the streetlight is, and have them explain how they made their
guess. Help students notice that the suns light makes a shadow of
the pole on the ground. Ask students to trace the triangle formed by
the ray of light from the sun, the streetlight, and the streetlights
shadow. Ask: Can you think of a way to use a short stick to
determine the height of the streetlight? How could you use what you
know about similar triangles? If students have trouble with this
question, ask:
What can you say about the angles of two triangles that are
similar?
What can you say about the lengths of the sides of two
triangles that are similar?
Is there a way to make a triangle that would be similar to
the triangle formed by the sun, the streetlight, and the
shadow?
How would this triangle help you find the length of the
streetlight?
Which side of the triangle would you make the short stick
be? What would form the other two sides of the triangle?
How would you determine that the two triangles are
similar?
Have students try constructing the two triangles, either on paper or
with a real pole and shadow. Let them use a ruler or the tape
measure to measure the stick, the shadows, and the distance from the
tip of the sticks shadow to the tip of the stick. Ask: What is the
length of the streetlight? How do you know?
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Geometry and Measurement, Grades 6-8 37
Circles
Mathematical Focus
8 Identify components and properties of circles
8 Find the circumference and area of circles.
In this activity, students discuss the properties of circles. They learn
about pi, explore the ratio between the circumference and diameter of
a circle, and develop formulas for finding the circumference and area
of a circle using pi.
Preparation and Materials
Before the session, gather the following materials:
8 Student Page 20: Circles
8 Student Page 21: Circle Cards
8 Student Page 22: Circle Chart
8 Student Page 23: Circle Puzzles
8 Several circular items, if available
8 Tape measure and/or string
8 Calculator
8 Ruler
Cut out circle cards from Student Page 21 ahead of time.
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Examining Circles
1. Discuss the properties of circles.
Draw a circle on a piece of paper and then ask students to describe
what makes this shape different from other shapes. Ask students to
look around the room and find some examples of circles. Talk about
the properties of circles, asking some of the following questions:
What do we call the distance around the outside of the
circle? [Circumference, as well as perimeter]
If you rotated a circle, what would it look like?
If you slid a circle over, what would it look like?
If you flipped a circle, what would it look like? Do circles have a line of symmetry? Do they have more
than one?
Tell students that a line drawn from one point on the circle to any
other point on the circle is called a chordof the circle. Have students
draw three chords inside the circle. Ask them where in the circle they
would find the longest chord. Ask: What are the longest chords in a
circle called? [Diameters]
Diameters and Circumferences1. Measure the diameter and circumference of different
circles.
Give students several circular objects and/or a copy of Student Page
20: Circles. Ask students to measure the diameter of and the distance
around each circle and to make a chart with columns for these two
numbers.
diameter
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Have students record the diameters and circumferences on the chart.
Students can use their tape measure or a piece of string (which they
wrap around the circle and then measure) to measure the circles.
2. Discuss the relationship between the diameter andcircumference of a circle.
Ask students questions such as:
Whats the relationship between the circumference and the
diameter of the first circle?
About how many times bigger is the circumference of the
second circle than its diameter?
Does this relationship hold for all of the diameters and
circumferences you measured?
3. Introduce pi.
Explain that this relationship is a constant calledpi, a number that
represents the ratio between a circles circumference and its diameter.
Have students use a calculator to divide some of the circumferences by
the diameters to see more of the digits of pi calculated out. Show
students that there is a button (on most calculators) that represents
pi.
4. Solve circumference and diameter challenges.
Have students find some more circles in the room (e.g., tabletops,
bicycle wheels, pictures in magazines or books). Ask them to measure
the diameters of these circles and then figure out what the
circumferences are without measuring. Once they have made their
predictions, they can measure around the circles to confirm that the
ratio works.
Give students some of the circumference/diameter challenges that
follow. Ask them to explain their thinking and draw pictures as they
solve each problem:
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If you are eating a slice of pizza that is 1/8 of a 12inch
pizza, then what is the length of the crust on the piece you
have?
If the circumference of the front wheel of a bicycle is 100
inches, how long would one of the spokes be?
If there are 25 people standing shoulder-to-shoulder around
a circus ring watching a lion tamer, about how far is one of
these spectators from the lion, if the lion is on a pedestal in
the center of the ring? Assume that each person is about 18
inches from shoulder to shoulder.
How far does a wheel travel in one full turn, if its diameter
is three feet?
Area of a Circle1. Estimate the area of a circle.
Return to Student Page 20 and ask students to think about how they
might estimate the area of these circles. If theyre having trouble, ask
them to think about how they find areas of other shapes, such as
squares. Encourage them to discuss their ideas. Students may
suggest that they could trace the circles onto graph paper and count
how many little squares are inside the circle. They could also draw a
large square around each circle traced on graph paper, so that the
circle just fits inside the square, and then use the area of the bigsquare to estimate the area of the circle.
2. Find a formula for the area of circles.
Ask students to make a three-column chart that shows the radius of
each of the circles from Student Page 20, the radius squared, and the
estimated area. Ask:
How are pi and the radius of the first circle squared related
to your estimated area for the first circle?
Does this relationship hold up for the other circles?
Have students use a calculator to find the area of the circles by using
pi. Tell them that the formula they should use is area = pi x r2. Ask:
Are the areas that you calculated close to your estimated area?
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Geometry and Measurement, Grades 6-8 41
3. Solve are challenges for circles.
Challenge students with some of the following questions, asking them
to explain their thinking and draw pictures as they solve the
problems:
If a tablespoon of jelly is enough to cover three square
inches of an English muffin, then how many tablespoons of
jelly will you need for an English muffin with a two-inch
radius approximately?
If two people are sitting opposite each other at a circular
table, how far apart are they if the table has an area of
approximately 201 square feet?
If you bake a cake in a nine-inch diameter pan, how could
you figure out how many square inches of cake you will
need to cover with frosting (assuming that you are frosting
the top of the cake, but not the sides)?
Circle Games
1. Play circle games.
Make a pile of the cut-out cards from Student Page 21: Circle Cards.
Give each player a copy of Student Page 22: Circle Chart. Explain
that each player starts with a circle of radius 1. All players should fill
in the first column of their charts to show the radius, diameter,
circumference, and area of their starting circles. Player 1 draws acard from the pile of Circle Cards, follows the directions on the card,
and fills in the information about the resulting circle in the next
column of his or her chart.
Then it is Player 2s turn. After all players have taken three turns
each, the player who has the circle with the largest radius wins.
2. Solve circle puzzles.
Give students a copy of Student Page 23: Circle Puzzles. Have
students use a ruler to find the area of the shaded part of the first two
pictures (a donut and a face). Ask students to estimate the area by
tracing the pictures onto graph paper, and then to find the actual area
by measuring the circles and using the formula, Area = pi x radius x
radius. Ask students to think about and explain how they find the
area of just the shaded part, without the white parts.
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Geometry and Measurement, Grades 6-8 42
Next, ask students to calculate the circumference of the clock in the
third picture, given the measurement that is shown. Then have
students try to match the radii, circumferences, and areas that go
with one another in the fourth puzzle. As students work, ask them to
explain their thinking and to draw pictures.
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Geometry and Measurement, Grades 6-8Student Page 1 43
Scale Ruler
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Geometry and Measurement, Grades 6-8Student Page 2 44
Half-Inch Graph Paper
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Geometry and Measurement, Grades 6-8Student Page 3 45
Quarter-Inch Graph Paper
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Geometry and Measurement, Grades 6-8Student Page 4 46
Real, Toy, or Giant-Size?
Scale: inch = 1 inch Scale: inch = foot
inch = 1 inch inch = 2 feet
inch = inch inch = 5 feet
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Geometry and Measurement, Grades 6-8Student Page 5 47
Rectangles
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Geometry and Measurement, Grades 6-8Student Page 5 48
Triangles and Trapezoids
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Geometry and Measurement, Grades 6-8Student Page 7 49
Shapes
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Geometry and Measurement, Grades 6-8Student Page 8 50
Angles In and Around Shapes
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Geometry and Measurement, Grades 6-8Student Page 51
Final Protractor
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Geometry and Measurement, Grades 6-8Student Page 52
Dot Paper I
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Dot Paper II
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Geometry and Measurement, Grades 6-8Student Page 53
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Geometry and Measurement, Grades 6-8Student Page 54
Harbor Map
Scale: 1/4 inch = 1 mile
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Geometry and Measurement, Grades 6-8Student Page 55
Sets of Directions
Harbor Map with Grid
8 miles east
8 miles north
7 miles northeast
9 miles northeast
4 miles north
10 miles east
3 miles south
20 miles northeast
3 miles north
Travel 11 miles north
Travel 4 miles northwest
Travel 6 miles southwest
Travel 6 miles west
Travel 11 miles north
Travel 8 miles northwest
Travel 4 miles north
Travel 10 miles east
Travel 3 miles south
0 2 3 4 5 6 7 8 9018765432910
Scale: 1/4 inch = 1 mile
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Geometry and Measurement, Grades 6-8Student Page 56
1
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Geometry and Measurement, Grades 6-8Student Page 57
Capture! Game Board
0
1
2
3
4
5
6
7
8
321 4 5 6 7 80
0
1
2
3
4
5
6
7
8
321 4 5 6 7 80
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Geometry and Measurement, Grades 6-8Student Page 58
Coordinate Grid
10 201918171615141312119876543210
$10
$8
$6
$4
$2
0
$12
$14
$16
$18
$20
$22
$24
$26
$28
$30
$32
$34
$36
$38
$40
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Geometry and Measurement, Grades 6-8Student Page 59
Staircases
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Geometry and Measurement, Grades 6-8Student Page 60
Shape Pairs
Shape Pair#1
Shape Pair#2
Shape Pair#3
Shape Pair#4
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Geometry and Measurement, Grades 6-8Student Page 61
Streetlight Height
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Geometry and Measurement, Grades 6-8Student Page 62
Circles
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Geometry and Measurement, Grades 6-8Student Page 63
Circle Cards
Add two
to the
Diameter
Double
the
Area
Increase the
Circumference
by one
Take 1/2
away from
the radius
Take 1 square
unit away
from the area
Cut the
diameter
in half
Add 2 units
to the
circumference
Add three to
the radius
Cut the
radius
in half
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Geometry and Measurement, Grades 6-8Student Page 64
Circle Chart
Radius Diameter Circumference Area Picture(Draw on largerpaper if necessary
Start 1 Inch
AfterFirstTurn
After
Second
Turn
AfterThird
Turn
(Finish)
Circle Puzzles
Puzzle 1: Find the shaded areaPuzzle 2: Find the shaded area
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Puzzle 3: Find the circumference of
the clockPuzzle 4: Draw a line from each radius to
the matching circumference and area for that
circle
Circumference Radius Area
10 inches 2 inches square inches
2 inches 5 inches a little morethan 314
square inches
almost 63 10 inches 4 squareinches inches
between 81 1 inch about 78
and 82 inches square inches
12
1
2
3
4
5
6
7
8
9
10
11
1.5 inches