Introduction to Drawing Shapes

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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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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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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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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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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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.



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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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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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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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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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.



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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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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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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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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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.



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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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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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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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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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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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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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.



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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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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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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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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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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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.



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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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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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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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!



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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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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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. 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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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.



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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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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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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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.



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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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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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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Half-Inch Graph Paper


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Quarter-Inch Graph Paper


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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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Rectangles


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Triangles and Trapezoids


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Shapes


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Angles In and Around Shapes


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Geometry and Measurement, Grades 6-8Student Page 51

Final Protractor


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Dot Paper I

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Dot Paper II


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Harbor Map

Scale: 1/4 inch = 1 mile


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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 8 miles northwest


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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1


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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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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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Staircases


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Shape Pairs

Shape Pair#1

Shape Pair#2

Shape Pair#3

Shape Pair#4


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Streetlight Height


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Circles


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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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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

Introduction to Drawing Shapes

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Transcript of Introduction to Drawing Shapes