geometry - hand2mind · 2014. 3. 17. · 126. geometry. is the study of points, lines, angles, and...

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Geometry is the study of points, lines, angles, and shapes. Students in fourth grade draw points, lines, and angles, and they deepen their understanding of two-dimensional shapes. In particular, they focus on the classification of shapes and on symmetry.

In their study of lines and angles, students become familiar with the concepts of parallel and perpendicular lines and learn to classify angles by their size, using the terms right, acute, and obtuse. They classify two-dimensional figures based on the presence or absence of parallel and perpendicular lines and on the types and sizes of angles. Students learn that parallel lines are those that never intersect and are always equidistant. They learn that perpendicular lines intersect to form right angles. They also learn that right triangles have one right (90-degree) angle, recognize right triangles as a category for classification, and identify different types of right triangles.

Mathematically proficient fourth-graders model with mathematics. They experiment with representing problem situations in multiple ways by drawing pictures, using objects, and communicating about numbers and other mathematical concepts. They make and explain connections—for instance, between rays and angles and between angles, lines, and shapes. They use representations to further their mathematical understanding as needed, which in turn helps them apply what they know to everyday situations. The following hands-on activities will enable teachers to provide meaningful activities that further students’ development of geometry concepts.

The Grade 4 Common Core State Standards for Geometry specify that students should—

Geometry

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GeometryContents

Lesson 1 Parallel and Perpendicular Lines . . . . . . . . . . . . . . . . . . . . 128

Objective: Identify parallel and perpendicular lines.

Manipulative: Geoboard

Lesson 2 Plane Shapes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132

Objective: Identify and compare plane shapes.

Manipulative: Attribute Blocks

Lesson 3 Identify and Classify Triangles . . . . . . . . . . . . . . . . . . . . . 136

Objective: Identify and classify triangles.

Manipulative: AngLegs®

Lesson 4 Line Symmetry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 140

Objective: Identify and draw lines of symmetry in polygons.

Manipulative: Pattern Blocks, GeoReflector™ Mirror

Lesson 5 Symmetrical Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 144

Objective: Explore the meaning of symmetry and make symmetrical figures.

Manipulative: Pattern Blocks

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Objective

Common Core State Standards

■ 4.G.1

Geometry

Parallel and Perpendicular LinesWhen students understand relationships between lines, they are better equipped to describe shapes and other spatial relationships. Parallel lines are lines in the same plane that do not intersect—that is, do not meet or cross. Perpendicular lines are lines that intersect to form a right angle at the point where they cross.

Talk About ItDiscuss the Try It! activity.

■ Ask: How can you tell if lines are perpendicular to each other?

■ Ask: If a line is parallel to the x-axis, what is its relationship to the y-axis?

■ Have students identify parallel lines and perpendicular lines in the classroom.

■ Say: Name some other polygons that have parallel and perpendicular lines.

Solve ItReread the problem with students. Have students draw a map of the park and show the paths. Then have them describe the layout of the paths using the terms parallel and perpendicular.

More IdeasFor other ways to teach about parallel and perpendicular lines—

■ Have students use AngLegs® to create polygons. Have them identify parallel and perpendicular sides.

■ Create other polygons on a Geoboard and have students describe the relationships of the lines to one another.

Formative AssessmentHave students try the following problem.

Which segments are perpendicular?

A . AB and AC

B . AB and BC

C . BC and AC

D . None of these

1

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Materials• Geoboards (1 per group)• rubber bands (6 per group)• masking tape (1 roll per group)• paper (1 sheet per group)• pencils (1 per group)

Try It! 20 minutes | Groups of 4

Here is a problem about identifying parallel and perpendicular lines.

Cassidy is landscaping a new city park on a square plot of land. She makes a map

of the park to decide where the main paths should lie. She decides to put paths

around the perimeter and from the four corners through the center of the park.

How can she describe the relationships of the paths to one another?

Introduce the problem. Then have students do the activity to solve the problem. Distribute Geoboards, rubber bands, tape, paper, and pencils to students. Say: The Geoboard represents the park and rubber bands will represent the paths.

3. Have students connect the four points to form a square. Have them connect the diagonals of the square. Ask: What do you notice about the segments formed by the diagonals?

2. Have students show segments between points A and B and points B and C. Say: Compare these two segments. Define perpendicular lines. Have students model other perpendicular segments.

Have students who confuse the terms parallel and perpendicular make vocabulary cards and use them to find examples in letters and mathematical symbols. For example, the plus sign has perpendicular lines and the equal sign has parallel lines.

1. Have students locate points at (0, 0), (0, 4), (4, 4), and (4, 0) on the Geoboard and label them A, B, C, and D, respectively. Say: Use rubber bands to connect points A and B and points C and D. Have students compare the segments by noticing the distance between them. Define parallel lines. Have students show other segments that are parallel to AB.

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Hands-On Standards, Common Core Edition

1

Use a Geoboard to model each set of lines. Identify the relationship of the lines.

1.

0 1 2 3 4

4

3

2

1

0

ba

c

Lines a and b are ____________ .

Lines c and b are ____________ .

Lines c and a are ____________ .

Using a Geoboard, model each pair of lines named. Sketch the model.

Identify each pair of lines as parallel lines or perpendicular lines.

2.

0 1 2 3 4

4

3

2

1

0

w

s

t

z

Lines t and w are ____________ .

Lines s and z are _____________ .

3. perpendicular lines

0 1 2 3 4

4

3

2

1

0

4

3

2

1

0

4. parallel lines

0 1 2 3 4

4

3

2

1

0

4

3

2

1

0

5.

_________________________

6.

_________________________

Answer Key

Download student pages at hand2mind.com/hosstudent.

(Check students’ work.)

parallel

parallelperpendicular

perpendicular

perpendicular

Check students’ sketches.

parallel perpendicular

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Challenge! At what times are the hands of a clock perpendicular? Explain your answer.

Answer Key


Challenge: Check students’ list. Samples include 3:00 and 9:00.

132

Objective


■ 4.G.2

2Geometry

Plane ShapesThe reasoning skills that students develop at this age allow them to explore more complex geometric problems and properties. They should develop more precise ways to describe and classify shapes, with an emphasis on using the specialized vocabulary associated with the geometric attributes and properties. The words parallel, perpendicular, and vertex should be introduced and used repeatedly in context when discussing two-dimensional shapes.


■ Draw a triangle, circle, square, and rectangle on the board. Point to each shape. Ask: How many vertices does this shape have? How many sides? How many parallel sides? What is the name of this shape?

■ Display one Attribute Block. Ask: What shape is this? How can you tell? Encourage students to describe the shape using the correct geometric vocabulary. Then challenge students to name correct shapes by giving them clues using attributes.

Solve ItWith students, reread the problem. Have students draw the shape Nick and his friends were supposed to make. Instruct students to label each side and vertex and write a sentence telling how they knew which shape was being described. (square or rhombus)

More IdeasFor other ways to teach about identifying and comparing plane shapes—

■ Name a two-dimensional shape, then have students build it on a Geoboard. Instruct students to compare results with a partner and explain the attributes of the shape. Point out that while the Geoboard shapes may vary in size, side length, and position, they should all have the same number of sides, vertices, and so on.

■ Play “What Shape Am I?” by telling riddles about two-dimensional shapes. For example, say: I have 3 sides and 3 vertices. None of my sides are parallel. Ask: What shape am I? Have students hold up a Pattern Block of that shape to show their answer.


Which shape has parallel lines?

A . B . C . D .

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Materials• Attribute Blocks (1 circle, 1 square,

1 rectangle, and 1 triangle per student)• Plane Shapes Recording Sheet (BLM 10;

1 per student)• pencils (1 per student)

Try It! 20 minutes | Pairs

Here is a problem about identifying and comparing plane shapes.

Mr. Roberts asks Nick and his friends to help him get the playground ready

for a game the class will play during recess. He asks the students to use chalk

to make a large shape on the pavement. The shape should have two sets of

parallel sides that are all the same length. It should have four corners. How

can the boys figure out what shape Mr. Roberts is asking them to make?

Introduce the problem. Then have students do the activity to solve the problem. Distribute materials to students. Introduce and define the terms vertex (the point where two sides join), perpendicular (lines that intersect and form only right angles), and parallel (lines that never cross and stay the same distance apart at all points) for students.

1. Instruct students to select a circle, square, rectangle, and triangle block, and discuss similarities and differences between the shapes. Have students complete the first and second rows of the Plane Shapes Recording Sheet.

3. Have students look for parallel sides in the shapes. Instruct students to write the number of parallel sides in the fifth row for each shape.

2. Explain that the plural of vertex is vertices. Instruct students to write the number of sides in the third row and the number of vertices in the fourth row for each shape.

The geometric terms introduced in this lesson may be difficult to remember for some students. If so, draw a square or rectangle on the board. Write the terms side, vertex, perpendicular lines, and parallel lines on the board next to the shape, and draw arrows to parts of the shape to indicate examples of each term. Emphasize the parallel and perpendicular lines on the shape using thicker lines. Have students use masking tape to label the sides, vertices, perpendicular lines, and parallel lines on the blocks.

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2

Choose an Attribute Block that matches each shape. Name the shape. Fill in the blanks with the correct numbers.

1.

Name __________

Sides __________

Vertices __________

Parallel Sides _____

2.

Name __________

Sides __________

Vertices __________


3.

Name __________

Sides __________

Vertices __________


Using Attribute Blocks, select each shape named. Then complete the table.

Name of Shape

Sketch of Shape

Number of Sides

Number of Vertices

Number of Parallel Sides

4. triangle

5. hexagon

Sketch each shape described. Name the shape.

6. 4 Sides

4 Vertices

2 Parallel Sides

________________

7. 0 Sides

0 Vertices

0 Parallel Sides

________________

8. 3 Sides

3 Vertices

0 Parallel Sides

________________

Answer Key



square

3

6

3

6

0

3

circle rectangle

4 0 4

4 0 4

2 0 2

rectangle or ; square ; circle ; triangle

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Challenge! Name a three-sided figure and explain why it cannot have any parallel sides. Draw a picture to help.

Answer Key


Challenge: (Sample) A triangle has three sides. At each vertex, two of the three sides meet, so each side touches the other two sides.

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Objective


■ 4.G.2

3Geometry

Identify and Classify TrianglesClassifying triangles helps students develop reasoning skills that they will use when they study similar and congruent triangles. By building triangles, students visualize possible and impossible combinations of angles and side lengths. They also can learn to make generalizations about the properties of triangles.


■ Ask: Is it possible to build a triangle with two right angles? Two obtuse angles? A right angle and an obtuse angle? What do you notice when you build a triangle that has a right or an obtuse angle?

■ Ask: Looking at the equilateral, isosceles, and scalene triangles, can you suggest any generalizations about these triangles?

Solve ItReread the problem with students. Have students describe the triangles they built and discuss the different ways students could sort the triangles.

More IdeasFor other ways to teach about identifying and classifying triangles—

■ Have students work in pairs. One student makes a triangle on a Geoboard and the other classifies the triangle by its angles and its sides.

■ Draw a shape on the board, such as a rectangle bisected by two diagonals. Ask students to duplicate the shape on the Geoboard and find as many triangles as they can. Have students classify the triangles.


Which best describes the triangle?

A . acute, equilateral C . acute, scalene

B . obtuse, isosceles D . obtuse, scalene

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Materials• AngLegs® (1 set per group)• 4-Column Recording Chart

(BLM 6; 3 per group)• pencils (1 per group)


Here is a problem about identifying and classifying triangles.

Andrew wants to make a quilt using triangles. How many types of triangles

could Andrew use?

Introduce the problem. Then have students do the activity to solve the problem. Distribute AngLegs, recording charts, and pencils. Explain that all triangles have three sides and three angles. Say: Triangles can be classified by their angles, their sides, or both. Have students start two charts with the headings Name of Triangle, Angles, and Sketch on one chart and Name of Triangle, Sides, and Sketch on the other to record their results.

1. Say: Triangles can be classified by their angles as acute, obtuse, or right. Review the types of angles. Have students build a right triangle they can use to judge the angles in other triangles. Say: Build examples of each type of triangle and record their properties.

3. Say: Build all the different types of triangles classified by their angles and sides, for example acute scalene. Have students start a third chart to record their findings. Ask: How many types of triangles can you build?

2. Say: Triangles can be classified by their sides as equilateral, isosceles, or scalene. Describe the triangles according to the number of congruent sides. Have students build and record each type of triangle. Ask: Is an equilateral triangle also isosceles?

Some students may have difficulty building and sorting triangles classified by both angles and sides. Suggest that they start with the first type of angle on their recording sheet and then build as many triangles as possible using that type of angle and the three different types of sides. Then have them repeat this for each of the other types of angles.

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3

Use AngLegs to model each triangle. Identify the triangle as acute, obtuse, or right. Also identify the triangle as equilateral, isosceles, or scalene.

1.

orange

green

pur

ple

__________

__________

Using AngLegs, model each of the triangles named. Sketch the model.

4. acute, scalene

7. obtuse, scalene

The side lengths of triangles are given. Idenfity each triangle with as many names as you can.

2.

purple

purp

le

purple

__________

__________

3.

__________

__________

5. right, isosceles

8. obtuse, isosceles

6. acute, equilateral

9. right, scalene

10. 4 ft, 4 ft, 4 ft

________________

________________

11. 6 in., 10 in., 12 in.

________________

________________

12. 12 cm, 12 cm, 5 cm

________________

________________

yello

w

redpurple

Answer Key



right acute obtuse

scalene equilateral scalene

Check students’ sketches. Check students’ sketches. Check students’ sketches.

Check students’ sketches.Check students’ sketches.Check students’ sketches.

acute obtuse acute

equilateral scalene isosceles

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Challenge! Can you draw or build an equilateral obtuse triangle? Explain why or why not. Draw a picture to help.

Answer Key


Challenge: (Sample) No; An obtuse triangle must have an angle that has a measure greater than 90˚. An equilateral triangle has three angles that are congruent. The sum of the angles of the triangle is 180˚. Three obtuse angles cannot equal 180˚.

140

Objective


■ 4.G.3

4Geometry

Line SymmetryIdentifying lines of symmetry in polygons helps students recognize symmetry in the world around them. Line symmetry is the basis of reflection, one of the transformations used in geometric thinking. The concept of line symmetry complements the concepts of congruence and similarity.


■ Ask: Are all lines that divide a shape in half called lines of symmetry?

■ Ask: Why is a line of symmetry sometimes called a mirror line or a line of reflection?

■ Ask: How many lines of symmetry does the hexagon have?

■ Say: Three of the lines of symmetry divide the hexagon into trapezoids. Ask: What shapes is the hexagon divided into by the other lines of symmetry? Students should see that the shapes are irregular pentagons.

Solve ItReread the problem with students. Have them write a paragraph to explain what they know about line symmetry and how they used the mirror to find lines of symmetry.

More IdeasFor other ways to teach line symmetry—

■ Have students use Pattern Blocks and a GeoReflector™ Mirror to construct more shapes using the same method they used to construct the hexagon in Step 2 of the Try It! activity. Have them draw the shapes that they create.

■ Have students create figures with line symmetry using AngLegs®. Students can make two halves separately and snap the halves together to make a symmetrical shape.


Which figure has three lines of symmetry?

A . B . C . D .

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Materials• Pattern Blocks (1 hexagon and

7 trapezoids per group)• GeoReflector™ Mirror (1 per group)• paper (1 sheet per group)• pencils (1 per group)


Here is a problem about line symmetry.

Kim’s company makes Pattern Blocks. They make trapezoid blocks by cutting

hexagon blocks in half. How many ways can a hexagon block be cut in half to

make a pair of trapezoid blocks? Consider other lines that divide the hexagon

into halves. In all, how many different lines produce halves that are mirror

images of each other?

Introduce the problem. Then have students do the activity to solve the problem. Distribute Pattern Blocks, GeoReflector Mirrors, paper, and pencils to students.

1. Have students use a hexagon block to trace three hexagons onto a sheet of paper, all oriented the same way. Say: Use trapezoid blocks to fill in the hexagons. Show the different ways a hexagon can be cut in half. Guide students to orient the “cut” lines three different ways.

3. Say: Trace another hexagon onto your sheet of paper. Use the mirror to find and draw all the lines of symmetry of the shape.

2. Have students select another trapezoid block and use it with the GeoReflector Mirror to “construct” a regular hexagon. Say: Half of the hexagon is formed by the trapezoid block and the other half is formed by the image in the mirror. Introduce the concepts of symmetry, line of symmetry, and mirror image.

Students might think that a line of symmetry is any line that divides a shape into two equal halves. Reiterate that the halves must be mirror images of each other in order for the dividing line to be classified as a line of symmetry. Have students investigate this idea with a simple shape, such as a rectangle, and a GeoReflector Mirror.

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4

Use Pattern Blocks to model an equilateral triangle. Trace the triangle on paper. Use the GeoReflector Mirror to find all lines of symmetry. How many lines of symmetry are there?

1.

__________________________

Using Pattern Blocks and the GeoReflector Mirror, model all lines of symmetry of a square. One line of symmetry is shown in the GeoReflector. Sketch the model and draw the lines of symmetry. How many lines of symmetry are there?

2.

__________________________

How many lines of symmetry does each shape have?

3.

____________________________

4.

____________________________

Answer Key




3

4

2 8

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Challenge! Can a shape have both a horizontal line of symmetry and a vertical line of symmetry? Draw an example. Can a shape have a horizontal line of symmetry and not have a vertical line of symmetry? Draw an example.

Answer Key


Challenge: (Sample) Yes; a square has both lines of symmetry; yes; a trapezoid can have only one line of symmetry, either vertical or horizontal.

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Objective


■ 4.G.3

5Geometry

Symmetrical FiguresIn this lesson, students solve problems involving symmetry. Because relationships across a line of symmetry correspond exactly in terms of size, form, and arrangement, students can begin to develop a sense of formal balance, which will serve as a foundation for geometric equations.


■ Say: An image that has symmetry can be divided into two halves that are mirror images of each other. Ask: What is a mirror image?

■ Draw two shapes on the board, one that is symmetrical and one that is not. Point to the symmetrical shape. Ask: Is this shape symmetrical? How can you tell? Point to the asymmetrical shape. Ask: Is this shape symmetrical? How can you tell?

Solve ItWith students, reread the problem. Have students create their own symmetrical masks by folding construction paper in the center and placing Pattern Blocks along the fold to make the first half of the mask. Students then trace the blocks and unfold the paper. Using blocks, they should complete the other half of the mask so that it is symmetrical and trace the other half.

More IdeasFor other ways to teach about symmetry—

■ Have students use a GeoReflector™ Mirror on the shapes they made in the lesson. With one half of the shape in place, students can place the mirror along the line of symmetry and observe the reflection.

■ Have students work in pairs. Students should divide a Geoboard in half with a rubber band. One student then makes a shape or pattern on one half of the Geoboard. The other student must then make a mirror image of the first shape or pattern on the second half of the Geoboard so that the design is symmetrical. Students then switch roles and repeat the activity.

■ Have students make symmetrical designs out of Centimeter Cubes. Students can then use crayons to draw their designs on a Centimeter Grid (BLM 12).


Which shows a line of symmetry?

A . B . C . D .

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Materials• Pattern Blocks (50 assorted per pair)• crayons (10 per pair)• paper (5 sheets per pair)• pencils (1 per student)

Try It! 30 minutes | Pairs

Here is a problem about symmetry.

Andrew wants to make a face mask of his favorite action hero to wear on

the night of his class costume party. The mask he wore last year was torn in

half, but Andrew is hoping to use it as a pattern. How can symmetry help

Andrew create a whole face mask from the half he has?

Introduce the problem. Then have students do the activity to solve the problem. Give Pattern Blocks, crayons, and blank sheets of paper to each pair. Introduce and model the concept of symmetry.

1. Take the blue rhombus. Use the green triangles to show a line of symmetry. Invite students to trace the blue rhombus and draw other lines of symmetry. Do the same with the yellow hexagon and red trapezoid.

3. Students should trace their figures onto the paper and color the shapes. Have students check to make sure that they can draw a line of symmetry that will divide the design into two halves that are mirror images of each other. Partners then switch roles and repeat the activity.

2. Say: We can make a group of shapes that has symmetry, too. Have one student in each pair create one half of a shape; then the other student should complete a symmetrical side. Have students draw a vertical line on their paper to help.

Some students may try to complete the pattern by repeating the shapes (i.e., triangle, trapezoid, triangle, trapezoid) instead of creating a mirror image (triangle, trapezoid, trapezoid, triangle). You may wish to use a hand mirror or GeoReflector™ Mirror to help students check their design for symmetry. Hold the mirror so that students can see half their design reflected along the line of symmetry. Then remove the mirror and have students check that the other half of the design matches what they just saw in the mirror.

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5

Use Pattern Blocks to model the symmetrical figure. Name the figure and the two parts.

1.

____________________________

3.

____________________________

5.

7.

2.

____________________________

4.

____________________________

6.

8.

Using Pattern Blocks, model a symmetrical figure. Sketch the shape. Name the shape that you sketched. Name the shape formed by both sides together.

Sketch the other half of each shape to make a symmetrical figure.

Answer Key




hexagon; trapezoid rectangle; square

square; rectangle triangle; rhombus

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Challenge! If a shape is drawn on a piece of paper, explain how you can tell if the shape is symmetrical. Draw a picture to help.

Answer Key


Challenge: (Sample) Fold the paper so that the fold line is along the center line of the shape. If the two halves of the shape match, the shape is symmetrical.

geometry - hand2mind · 2014. 3. 17. · 126. geometry. is the study of points, lines, angles, and...

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