78

U N I T Geometry

• sort and name polygons by sides and angles

• measure, name, and construct angles

• construct triangles, given sideand angle measures

• identify and construct nets of solids

Pratt Truss

Double Warren Truss

Howe Truss

Howe Truss with counter braces

Learning Goals

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polygon

equilateral triangle

isosceles triangle

scalene triangle

acute angle

right angle

obtuse angle

perpendicular

acute triangle

right triangle

obtuse triangle

regular polygon

irregular polygon

net

Key Words

• Text• What is the most common geometric figure

you see in the bridges?How many triangles can you count in each bridge?How are the triangles the same? How are they different?

• What other geometric figures do you see?How are they the same? How are they different?

• Which bridge do you think would support the greatest mass? Why?

These are different types of truss bridges.

They were built during the great age of trains, about a hundred years ago.

A truss is a framework.

It is made of wooden beams or metal bars.

The bridges are light, strong,and rigid.

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Naming and Sorting Polygonsby Sides

L E S S O N

80 LESSON FOCUS Name and sort polygons by number of sides and by side length.

A polygon is a closed figure with 3 or more sides.

A quadrilateral is a polygon with 4 sides.What are the attributes of these quadrilaterals?

Rectangle Parallelogram

Trapezoid Kite

You will investigate polygons with 3 sides in Explore.

You will need a millimetre ruler and scissors.Your teacher will give you a large copy of these triangles.

Share the work.➤ How are the triangles alike?

How are they different?➤ Measure the lengths of

the sides of each triangle.What do you notice?

➤ Cut out the triangles.Sort the triangles by the number of equal sides.

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Show and Share

Compare your sorting with that of another pair of students.Did you sort the triangles the same way? Explain.

Here are some ways to name polygons.

➤ Name polygons by the number of sides.

➤ Name polygons by their vertices.Label each vertex with a different capital letter.

This is triangle ABC. This is quadrilateral MNPQ.

Use the letters to name the sides of the polygon.Triangle ABC has 3 sides: AB, AC, and BC

➤ Name triangles by the number of equal sides.

A trianglehas 3 sides.

A pentagonhas 5 sides.

A hexagonhas 6 sides.

An octagonhas 8 sides.

An equilateral trianglehas all sides equal.

An isosceles trianglehas 2 sides equal.

A scalene trianglehas no sides equal.

AB = BC = AC DE = DF

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1. Use a geoboard, geobands, and dot paper.a) Make 3 different scalene triangles.

Record each triangle on dot paper.How do you know each triangle is scalene?

b) Make 3 different isosceles triangles.Record each triangle on dot paper.How do you know each triangle is isosceles?

c) Try to make an equilateral triangle.What do you notice?

2. Measure the sides of each triangle.Name each triangle as equilateral, isosceles, or scalene.a) b) c)

3. a) Name each polygon.

b) Sort the polygons by the number of sides.c) Sort the polygons by the number of vertices.d) Compare the two sortings. What do you notice?

Do you think this is always true? Explain.

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B

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How can you use side lengths to name a triangle?Use words and pictures to explain.

Calculator Skills

Find 3 odd numbersthat have a product of 693and a sum of 27.

Numbers Every Day

83ASSESSMENT FOCUS Question 5

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D

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0 1 2 3 4 5 6 7 8 9

cm

4. a) Which triangles are isosceles? How do you know?

b) For each triangle, name the sides that are the same length.c) Find the perimeter of each triangle.

5. You need drinking straws,scissors, and pipe cleaners.Cut the straws into 8 pieces as shown.Use pieces of pipe cleaner as joiners.a) Make each triangle.

Trace and label your results.• an equilateral triangle• an isosceles triangle with the least perimeter• a scalene triangle with the greatest perimeter

b) Which straws could not be used together to make a triangle? Explain.

6. Use a geoboard, geobands, and dot paper.a) Make an isosceles triangle. Record the triangle on dot paper.b) Use the triangle from part a.

Change the triangle so it is scalene.Describe the changes you made.

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L E S S O N

84 LESSON FOCUS Use a protractor to measure and construct angles.

Is each angle greater than 90°, less than 90°, or equal to 90°?What is the measure of each angle?

Measuring and Constructing Angles

You will need a ruler and a protractor.

➤ Use a ruler to draw an angle.

➤ Have your partner:• estimate the size of the angle,

in degrees• measure the angle with a protractor• record the estimate and the

angle measure

➤ Trade roles. Continue until you have 6 different angles.Try to make angles that are less than 90°, greater than 90°,and equal to 90°.

➤ Order the angles from least to greatest.

Show and Share

Show your work to another pair of students.How did you use the measure of one angle to estimate the measure of another angle?

The angle is just greaterthan 90°. I estimate its

size to be 110°.

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➤ We name angles according to their size.

The measure of The measure of The measure of anan acute angle a right angle obtuse angle isis less than 90°. is 90°. between 90° and 180°.

➤ Use a ruler and a protractor to construct an angle with a given measure.Follow these steps to construct an angle that measures 145°.

Step 1

Use a ruler.Draw one arm of the angle.

Step 2

Place the protractor on the arm.One end of the arm is at the centre of the protractor.The arm lines up with the base line of the protractor.Start at 0° on the arm along the base line.Count around the protractor until you reach 145°.Make a mark at 145°.

Step 3

Remove the protractor.Draw a line to join the end of the arm at the centre of the protractor with the mark at 145°.Label the angle with its measure.

centre baseline

145°

You can measure from 0° to 180° clockwise or

counterclockwise. Remember to start at 0° when

you draw an angle.

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1. For each angle:• estimate the size of the angle, in degrees• use a protractor to find the angle measure• tell whether the angle is acute, obtuse, or righta) b)

c) d)

e)

f)

2. Measure each angle.Do the angles in each pair have the same measure?a)

b)

Do the lengths of the arms affect the measure of the angle? Explain.

Number Strategies

Estimate each sum.Which strategies did you use?

$4.89 + $15.09

$97.76 + $12.12

$4.50 + $78.49

$34.78 + $67.76

Numbers Every Day

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3. a) Use a geoboard and geobands or square dot paper.Construct each angle:• an angle greater than 90° • an angle less than 90°

b) Measure each angle with a protractor.

4. Use a ruler and a protractor. Construct an angle with each measure.a) 80° b) 30° c) 100° d) 10° e) 180°How might you name the angle in part e? Explain.

5. The lines in each pair are perpendicular.

The lines in each pair are not perpendicular.

Explain what you think perpendicular means.

Draw an angle.Explain how to use a protractor to measure the angle.Use words and pictures to explain.

87ASSESSMENT FOCUS Question 4

45˚

BA C

ED F

Math LinkYour World

The angle of a kick helpsdetermine how far the ball will travel.A 45° angle allows the ball to travel the greatest distance.

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What do you know?• There are 6 different Pattern Blocks.• You can use one or more blocks to

construct an angle that measures 150°.

Think of a strategy to help you solve theproblem.• You can make an organized list.• Use different blocks to make angles

that measure 150°.

L E S S O N

88 LESSON FOCUS Interpret a problem and select an appropriate strategy.

You will need a tangram and a protractor.

How many different angles can you construct using one or more tans?Is it possible to construct an angle that measures 150°? How do you know?Record your work.

Show and Share

How do you know you have found all the possible angles?

Strategies

• Make a table.

• Use a model.

• Draw a diagram.

• Solve a simpler problem.

• Work backward.

• Guess and check.

• Make an organized list.

• Use a pattern.

• Draw a graph.

You will need Pattern Blocks and a protractor.How many different ways can you construct an angle thatmeasures 150°, using one or more Pattern Blocks?Explain.

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Trace or sketch each block.Use a protractor to measure the angles in each Pattern Block.Record the angle measures on your sketch.Choose 1 or more blocks you think you can arrange to form an angle that measures 150°.Record your arrangement onyour list.Continue to build, sketch,measure, and record until youhave found all the possible arrangements.

Check your work.How do you know that you have found all the angles? Explain.

Blocks Total AngleMeasure

1. Use 2 or more of each type of Pattern Block.How many different angles can you construct? Show your work.

2. Use only red Pattern Blocks.How many different angles can you construct? How do you know that you found all of them?

3. Use 6 green Pattern Blocks.Find all the different figures you can make using all 6 blocks.Record each figure.

Choose one of the

Strategies

How did you use an organized list to solve a problem?Use an example to explain.

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L E S S O N

90 LESSON FOCUS Name and sort polygons by number of angles and by angle measure.

Naming and Sorting Polygonsby Angles

You will need a protractor.Your teacher will give you a large copy of the triangles.

➤ Measure each angle in each triangle.Record the angle measures.

➤ Sort the triangles according to the measures of their angles.How are the triangles in each group the same?How are they different?

Show and Share

Share your work with another pair of students.Did you sort the triangles the same way? Explain.

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➤ We can sort and name triangles by angle measure.

➤ We can sort and name quadrilaterals by angles.

➤ We can sort polygons by the numbers of equal sides and equal angles.

A regular polygon has all sides equal and all angles equal.

An equilateral triangle is a A square is a regular rectangle.regular triangle. It has 3 equal sides. It has 4 equal sides.Each angle measures 60°. Each angle measures 90°.

An irregular polygon does not have all sides equal and all angles equal.

An acute triangle has all angles less than 90°.

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A right triangle has one 90° angle.

An obtuse triangle has one angle greater than 90°.

A rectangle has 4 right angles.

A parallelogram has 2 pairs of equal angles.

A kite has 1 pair ofequal angles.

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�G = 90°The symbol �means angle.

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1. You will need a geoboard, geobands, and dot paper.a) Make 3 different acute triangles.

Record each triangle on dot paper.How do you know each triangle is acute?

b) Make 3 different obtuse triangles.Record each triangle on dot paper.How do you know each triangle is obtuse?

c) Make 3 different right triangles.Record each triangle on dot paper.How do you know each triangle is right?

2. Use a protractor.Measure the angles in each triangle.Name each triangle as acute, obtuse, or right.a) b)

c) d)

3. Is each polygon regular or irregular? How do you know?a) b)

c) d)

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4. Make a large copy of this Venn diagram.Sort the figures.

Explain how you know where to place each figure.

5. Use a geoboard or square dot paper.Make polygons that have:a) exactly 2 obtuse anglesb) no more than 3 acute anglesc) more than 1 right angled) 1 acute, 1 obtuse, and 1 right angleHow many different polygons can you make in each case?Name each polygon.

6. Is it possible for a triangle to have:a) more than 1 obtuse angle?b) 2 right angles?c) 3 acute angles?Explain your thinking.Use pictures and words.

How can you use angles to sort polygons?Use pictures, numbers, and words to explain.

Number Strategies

Order the numbers in each set fromgreatest to least.

• 1284, 4182, 1428, 1248, 2148

• 9090, 9009, 9990, 9099, 999

• 6789, 7689, 6897, 6987, 7869

Numbers Every Day

93ASSESSMENT FOCUS Question 5

Has a right angle Has an obtuse angle

Has an acute angle

B

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H I

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L E S S O N

94 LESSON FOCUS Construct triangles given side and angle measures.

Constructing Triangles

You will need:• a millimetre ruler• triangles from a geometry set

➤ Construct triangle DEF.The measure of �D is 30°.The measure of �E is 60°.The measure of �F is 90°.Which side will be the longest?Try to make more than one triangle DEF.

➤ Construct triangle ABC.The length of AB is 66 mm.The measure of �A is 120°.The length of AC is 66 mm.How long is side BC?What are the measures of �B and �C?

➤ Name each triangle 2 ways.Record your work.

90˚

90˚60˚

30˚ 45˚ 45˚

These triangles come with a geometry set.The measure of each angle is shown.

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Show and Share

Compare your triangles with those of another pair of students.Is it possible to make different triangles ABC? DEF? Explain.

You can use a ruler and a protractor to construct a triangle.

Construct triangle MNP.The length of MN is 4.5 cm.The measure of �M is 40°.The length of MP is 3.7 cm.

Step 1

Sketch the triangle first.Label each side and angle.This sketch is not accurate.It shows each given measure.

Step 2

Use a ruler to draw side MN 4.5 cm long.

Step 3

Place the protractor on MN,with its centre at M.From 0° on the inner circle,measure an angle of 40° at M.

0 1 2 3 4 5 6 7 8

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Step 4Remove the protractor.Join M to the mark at 40°.Measure 3.7 cm from M.Mark the point P.

Step 5

Use a ruler to join P to N to form side NP.Label the triangle with its measures.

1. Use a ruler and a protractor.Construct each triangle.Sketch the triangle first.a) Triangle RST

The length of side TS is 5.2 cm.The measure of �T is 26°.The length of side RT is 3.4 cm.

b) Triangle VWXThe length of side VW is 7 cm.The measure of �V is 60°.The measure of �W is 50°.

Label each triangle with the measures of all the sides and angles.

2. Use a geoboard or dot paper.Construct a triangle with two 45° angles.Do this 3 times to construct 3 different triangles.How are the triangles the same? Different?

Number Strategies

Use addition, subtraction,multiplication, or division.Find 10 different ways tomake 42.

Numbers Every Day

0

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3. Use a ruler and a protractor.Construct a triangle with angles 40°, 60°, and 80°.Compare your triangle with that of a classmate.Are your triangles congruent?How could you find out?

4. Construct triangle GHK.The measure of �H is 45°.The length of side HK is 64 mm.The length of side HG is 46 mm.a) What is the measure of �K?

What is the length of side GK?b) Suppose the length of side HG is 7 cm.

What happens to the measure of �K?What happens to the length of GK?

Show your work.

5. Construct a right triangle with two angles of 55° and 35°.Can you make more than one triangle? Explain.

6. Try to construct triangle ABC.Draw AB 42 mm long.The measure of �A is 90°.The measure of �B is 95°.Can you construct triangle ABC?How do you know?

7. Can you construct a triangle with three 45° angles? Explain your thinking.

Which measures do you need to knowto be able to draw a triangle? For each example,draw the triangle.

97ASSESSMENT FOCUS Question 4

Look for triangles in your home.They could be pictures orobjects with triangular faces.Name each triangle 2 ways.Choose 1 triangle. Draw it.

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L E S S O N

98 LESSON FOCUS Identify and construct nets of solids.

Making Nets

Label each face of the solid with a letter.➤ Trace each face of the solid.

Label each tracing with its letter.➤ Cut out your tracings.

Tape them together at the edges to make a figure.Arrange the faces so the figure can be folded to make a model of your solid.How many different ways can you do this?

➤ Fold the figure to build the model.

Show and Share

Share your work with another pair of students.How did you decide how to arrange the faces?

Number Strategies

Write each number as a decimal.• one and one-hundredth• twelve and twelve-hundredths• three hundred three and

three-tenths• four hundred forty and

four-hundredths

Numbers Every Day

You will need:• a pyramid• tape• a millimetre

ruler• scissors

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A net shows all the faces of a solid, joined in one piece.It can be folded to form the solid.

Here are 2 ways to construct a net for a solid.

➤ Construct a net for this cube. The cube is a cardboard carton.Carefully cut the cube apart along its edges so it is in one flat piece.

➤ Construct a net for this triangular pyramid.Label each face of the solid.

Trace one face.

Trace each other face.Arrange the faces so when they are folded,they make the pyramid.

A

A

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B

Faces A and B form one edge.

I will draw faces A and B so they share one side.

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You will need a variety of solids.

1. Identify each solid.How many faces does it have?Sketch each face.a) b)

c) d)

2. Choose 2 solids.Make sure the solids are different from the solid in Explore.Construct a net for each solid.

3. Which solid would each net make? How do you know?a) b)

c) d)

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4. Which diagrams show nets?Identify the solid.If a diagram does not show a net, how could you change it to make a net?a) b)

c) d)

5. The net for a cube has 6 congruent squares.a) How many different ways can you arrange

6 squares to form a net for a cube?Record each way on grid paper.

b) How do you know each arrangement forms a net?Show your work.

6. Identify each solid.a) It has 6 congruent square faces.b) The faces are 2 congruent triangles

and 3 congruent rectangles.c) The faces are 3 pairs of congruent rectangles.d) Two faces are congruent hexagons

and 6 faces are congruent rectangles.

ASSESSMENT FOCUS Question 5

How can you tell if an arrangement of figures is a net?Use words and pictures to explain.

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Using a Computer to Explore Nets

Work with a partner.

Use AppleWorks.Follow these steps to create a net for a cube.

1. Open a new drawing document in AppleWorks. Click:

2. If a grid appears on the screen, go to Step 3.

If not, click: , then click:

3. Check that Autogrid is on. Click:If Turn Autogrid Off appears in the menu, Autogrid is on.

If not, click:

4. Check the ruler settings. Click:

Click: , then click:

Choose these settings:

Click:

5. To draw a square, click the Rectangle Tool:

The cursor will look like this:

Hold down the Shift key while you click and hold down the mouse button.

Drag the cursor. Release the mouse button.

102 LESSON FOCUS Use a computer to create nets.

TE

CHN OL OGY

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6. To change the size of the square, click the square to select it.

Click:

Then click:

Enter 6 cm for the width and 6 cm for the height.

Click: . This closes the Object Size box.

7. To move a square, click the square.

Click and hold down the mouse button.

Drag the square to where you want it.Release the mouse button.

8. To copy a square, click the square.

Click: , then click:

Click: , then click:

The copy shows on top of the square.Click and drag the square to where you want it.

9. Follow Steps 5 to 8 to create squares.Then arrange them to form a net for a cube.

10. Save your net.

Click: , then click:

Name your file, then click:

11. Print your net.

Click: , then click:

Click:

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Follow these steps to create a net for a square pyramid.

12. Repeat Steps 5 to 7 to draw a square.

13. To draw a triangle, click the Regular Polygon tool:

Click: , then click:

Enter 3 for Number of sides. Click:

The cursor will look like this:

Click one of the vertices of the square.Drag along the edge to another vertex to make a triangle.Release the mouse button.Do this three more times.

14. To change the size or shape of the triangle, click the triangle to select it.Click a black square, hold down the mouse button, and drag until the triangle is the size and shape you want.

How did you decide how to arrange the squares to makea net for a cube?Use words and pictures to explain.

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What’s My Rule? Ga mes

You will need a set of What’s My Rule? game cards,scissors, 2 labels, and two 1-m lengths of string.

Cut out the game cards.Spread them out, face up.Use the string to make 2 loops.Label one loop “Matches”and one loop “Discards.”

➤ Player A thinks of a secret rule that describes some of the polygons on the cards.The rule could be:• all triangles with a

right angle; or• all regular polygons; or• all quadrilaterals with 1 pair of parallel sides

➤ Player A chooses 2 game cards.One card must fit the rule.He places it face up inside the “Matches” loop.The other card must not fit the rule.He places it face up in the “Discards” loop.

➤ Player B chooses a game card.If she thinks the card fits the rule, she places it inside the “Matches” loop.Otherwise, she places it in the “Discards” loop.

➤ Player A tells Player B whether her placement is correct.If the placement is correct, she can guess the rule.If the placement is not correct, she cannot make a guess.

➤ Players C and D continue until someone guesses the secret rule.

➤ Switch roles. Another player thinks of a secret rule.The other players take turns trying to guess the new rule.

The winner takes the fewest turns to guess the rule.

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1. Name each triangle as scalene, isosceles, or equilateral.Tell how you know.

2. Measure the angles in the triangles in question 1.Order the angles from least to greatest.

3. Use a protractor.Draw an angle with each measure.a) 65°b) 135°c) 95°Name each angle as acute, obtuse, or right.

4. Name each triangle 2 ways.How did you choose each name?

5. Is it possible to draw a quadrilateral with:a) 2 obtuse angles?b) 3 obtuse angles?Use pictures and words to explain.

6. Is a rhombus a regular polygon?How do you know?

Show What You Know

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LESSON

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Use dot paper to draw the quadrilaterals.

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5

6

LESSON

Learning GoalsU N I T

✓

7. Use a ruler and a protractor.a) Construct triangle ABC.

The length of AB is 56 mm.The measure of �A is 35°.The measure of �B is 90°.

b) What are the lengths of AC and BC?What is the measure of �C?

8. Try to construct triangle QRS.Draw QR 5 cm long.The measure of �Q is 110°.The measure of �R is 75°.Can you construct triangle QRS? How do you know?

9. Which arrangements of figures show a net for a rectangular prism?How do you know?a)

b)

c)sort and name polygons by sides and anglesmeasure, name, and construct anglesconstruct triangles, given sideand angle measuresidentify and construct nets of solids

✓

✓

✓

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Bridges

108

Pratt Truss

Double Warren Truss

You will need:• Bristol board• a hole punch

or a compass• paper fasteners• a centimetre ruler• centimetre cubes

or standard masses

Part 1

Choose one type of bridge truss to build.Your bridge must:• span a 35-cm gap• support a load• stand up by itself

Your teacher will give you a copy of the truss pieces.Use the truss pieces to cut strips of Bristol board.How many of each size of strip do you need?Cut a strip of Bristol board 14 cm wide for the roadway.How long does the road need to be?Draw a line 2 cm in from each long edge.Fold along the lines.

Build the bridge.How will you brace the top?

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Check List

How can you use what you know about triangles and other polygons to• draw nets of solids?• identify solids from their nets?

109

Part 2

Look at your bridge.Identify as many of these attributes as you can:• congruent figures• scalene, equilateral, and isosceles triangles• obtuse, acute, and right angles• equal anglesName different polygons you see.Are any of them regular? Explain.

Part 3

Use two desks or some textbooks to make a 35-cm gap.Place your bridge across the gap.Find the load your bridge can support.

Compare your bridge with those of other groups.Which type of bridge can support the greatest mass?

Write about the bridges and the attributes that make them strong.

Howe Truss Howe Truss with counter braces

Your work should showa clear explanation of whatyou did and whyas many attributes as possiblehow you used what youknow about geometryhow you found the greatestmass your bridge couldsupport

✓

✓

✓

✓

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You will need a ruler, several sheets of grid paper, and scissors.

Triangle, Triangle, Triangle

110 FOCUS Cross Strand Performance Assessment

Part 1

➤ On grid paper, draw a large right triangle. Make sure its base is along a grid line and the thirdvertex is at a grid point.Estimate the area of the triangle.

➤ On another sheet of grid paper,draw a congruent triangle.

➤ Cut out both triangles.Place the triangles edge to edge to make a rectangle.

➤ Write a multiplication statement to find the area of the rectangle.Calculate the area of the rectangle.Compare the area of the rectangle to the area of the triangle.

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Display Your WorkCreate a summary of your work.Show all your calculations. Explain your thinking.

Take It Further➤ Draw an obtuse triangle on grid paper.

Predict its area.➤ How did you use what you learned about acute and

right triangles to make your prediction?➤ Find a way to check your prediction.

Part 2

➤ Draw a large acute triangle with its base along a grid line and the third vertex at a grid point.Estimate the area of the triangle.

➤ Draw a congruent triangle.➤ Cut out both triangles.

Then, cut along a grid line on each triangle to make 2 triangles.

➤ Arrange the 4 triangles edge to edge to make a rectangle with no gaps or overlaps.

➤ Write a multiplication statement to find the area of the rectangle.Calculate the area of the original acute triangle.

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