Unit 7 Geometry: Shapes

Transcription

1 Unit 7 Geometry: Shapes Learning Geometric Terms Most students find elementary geometry easy; what they find hard is all the new terminology. Students won t be able to describe or classify shapes if they can t spell the relevant terms. Therefore, JUMP recommends that you teach the spelling of geometric terms during spelling lessons. The most important terms are: side edge vertex, vertices line of symmetry right angle triangle square rectangle rhombus parallelogram quadrilateral pentagon hexagon trapezoid Attributes and Properties Students will be introduced to attributes and properties through examples. Color, gender, food, age, and length are attributes; red, green, boy, girl, fruit, vegetable, age 9, and 4 cm long are properties. Properties are specific instances of attributes. Parallelograms and Trapezoids There are two possible definitions of a trapezoid: - a quadrilateral with exactly one pair of parallel sides - a quadrilateral with at least one pair of parallel sides When using the first definition, parallelograms are different from trapezoids, whereas when using the second definition, all parallelograms are trapezoids. Both definitions are legitimate. JUMP materials use the first definition. Geometry H-1

2 G4-1 Introduction to Classifying Data Page 161 STANDARDS preparation for 4.G.A.2 Goals Students will group data into categories. Vocabulary attribute classify data PRIOR KNOWLEDGE REQUIRED Recognizes patterns and attributes Sorting into two groups without common elements. Have eight volunteers stand up. Ask students to suggest ways in which to sort or classify the volunteers into two groups (for example, long or short hair, boy or girl, nine years old or ten years old, wearing jeans or not wearing jeans, wearing yellow or not wearing yellow). Then have one student classify the eight volunteers without telling the class how he or she chose to classify them. The student tells each of the eight volunteers which side of the room to stand on. Each remaining student in the class then guesses which group he or she belongs to. The student who classified the volunteers then puts the student in the right group. Stop when students have guessed correctly five consecutive times. The last student to guess correctly appoints each remaining student to either of the two groups and is told if they re right or not. Repeat this entire exercise several times. Note that the student classifying the students never reveals the classification. To make guessing the classification harder, students may sort the volunteers using two or more attributes. For example, they may group boys not wearing yellow and boys wearing yellow and girls. Or they may group boys or anyone wearing yellow and girls or anyone not wearing yellow. Have students identify attributes shared by all members of the following groups: a) gray, green, grow, group (Sample answer: starts with gr, one syllable, word) b) pie, pizza, peas, pancakes (Sample answer: food, starts with p) c) 39, 279, 9, 69, 889, 909 (Sample answer: odd, ones digit 9, less than 1,000) d) 5,412, 9,807, 7,631, 9,000, 9,081 (Sample answer: number, four digits, greater than 5,000) e) hat, cat, mat, fat, sat, rat (Sample answer: three-letter word, rhymes with at) Have students write two attributes for each group: a) 42, 52, 32, 62, 72 b) lion, leopard, lynx Write the numbers below and ask how they can be categorized. Group A: 321, 725, 129, 421 Group B: 38, 54, 16, 70, 82 H-2 Teacher s Guide for AP Book 4.1

3 Have a volunteer add one more number to each group to help the class identify categories. Then other students can guess the categories. The student who guesses correctly chooses another way to categorize the groups (but doesn t reveal the categories!), gives a new number that fits into each group, and asks the rest of the class to guess the new categories. Play the game long enough to allow a variety of categorizations. Possible categories include: Two-digit numbers, three-digit numbers Tens digit 2, tens digit other than 2 Even numbers, odd numbers More than 100, less than 100 Play the same game with these words: trick, train, sick, stain, drain, drink, sink, think, thick. Start by asking a volunteer to sort the words into two or three groups. Possible categories include: First letter is d, s, or t Rhymes with rain, rink, or brick One vowel, two vowels Four letters, five letters Last letter is n, last letter is k Extension a) Give pairs of students a copy of BLM Shapes (p. H-33), which is composed of 16 shapes that are either striped or plain, big or small, triangles or squares. The BLM has two of each shape, so students can work individually. (Pairs can also use a Set card game.) Ask students to separate the striped triangles from the rest of the shapes, then challenge them to describe the remaining group of shapes (plain or square). Allow them to use or or and in their answers, but do not encourage use of the word not at this point since, in this case, not striped simply means plain. If students need help, offer them choices: plain or triangle, striped and square, plain or square. b) Challenge students to describe the remaining group of shapes when they separate out the following: i) small squares ii) plain squares iii) big triangles iv) shapes that are small or triangular v) shapes that are striped or triangular vi) shapes that are striped or big vii) shapes that are both striped and big viii) shapes that are both plain and big ix) shapes that are striped or small Geometry 4-1 H-3

4 G4-2 Venn Diagrams Pages STANDARDS preparation for 4.G.A.2 Goals Students will sort data using Venn diagrams. Vocabulary and not or property Venn diagram PRIOR KNOWLEDGE REQUIRED Identifying attributes Classifying according to attributes Words like and, or, and not MATERIALS cardboard box a variety of small objects, toys and not toys pens, pencils, and other objects, some of them blue two Hula-hoops Sorting using one circle. Show students a variety of small objects, some toys and some not toys. Label a cardboard box toy box, then ask students if various items belong inside or outside the box. Then tell students that you want to classify items as toys without having to put them in a box. Draw a circle, write toys in the circle, and tell students that you want all of the toy names written inside the circle, and all of the other items names written outside the circle. Ask students to tell you what to write inside the circle. To save space, have students problem-solve a way to avoid writing the entire word inside the circle. If they suggest you write just the first letter, ask them what will happen if two toys start with the same letter. Explain to them that you will instead assign a different letter to each word. Example: A. Lego piece B. bowl C. toy car D. pen Which of these letters go inside the circle, and which of the letters go outside the circle? (Answer: A, C inside, B, D outside) Draw several shapes and label them with letters: Shapes Triangles A. B. C. D. E. F. Draw the diagram in the margin. Ask students if all the letters belong inside the box (yes) and invite them to explain why. Which letters belong inside the circle? Which letters belong outside the circle but still inside the box? Then ask students why the circle is inside the box. Are all of the triangles shapes? Explain that everything inside the box is a shape, but in order to be inside the circle, the shape has to be a triangle. Change the word inside the circle and repeat the exercise. (Suggested words to use include dark, light, quadrilaterals, polygons, circles, dark triangles, dark circles.) ASK: Why is the circle empty when it s labeled dark circles? H-4 Teacher s Guide for AP Book 4.1

5 What other property could you use to label the circle so that it remains empty? Remind students that the word inside the circle should reflect the fact that the entire box consists of shapes, so rockets isn t a good choice, even though the circle would still be empty. ACTIVITY 1 Create a large circle on the floor (using masking tape, for example) and have students who are nine years old stand inside the circle. Label the circle 9 years old. Repeat with several examples: wearing yellow, girl, wearing blue, takes the school bus. Have students stand inside the circle when the property applies to them and outside the circle when it does not. Play a game: Invite students to suggest properties and challenge them to sort themselves correctly before you finish writing the label. Students should learn to strategically pick properties that take a long time to write. Introduce Venn diagrams. Clearly label one Hula-hoop pens and another blue, then ask students to assign several colored pens (black and red) and pencils (blue, red, and yellow) to the proper position inside one of the hoops or outside both of them. Do not overlap the hoops at this point. Then present students with a blue pen and explain that it belongs in both hoops. Allow students to problem-solve a way to move the hoops so that the blue pen is circled by both hoops at the same time. Have a volunteer show the others how it s done. If students move the hoops closer together without overlapping them, so that part of the blue pen is circled by the blue hoop and part of the blue pen is circled by the pens hoop, be sure to ask them if it makes sense for part of the pen to be outside of the pens hoop. Shouldn t the entire pen be in the pens hoop? Draw and label the same shapes used previously: A. B. C. D. E. F. Shapes Dark Triangles Then draw the diagram in the margin. Explain to students that this picture is called a Venn diagram. ASK: Why do the two circles overlap? Have a volunteer shade the overlapping area of the circles and ask the class which letters go in that area. Have a second volunteer shade the area outside the circles and ask the class which letters go in that area. Finally, ask students which shape belongs in the dark circle and which shape belongs in the triangle circle. Change the labels for both circles and repeat the exercise. (Suggested words to use include light and quadrilateral, light and dark, polygon and light, circles and light.) Geometry 4-2 H-5

6 Connection Social Studies, Health, Language Arts This is a good opportunity to tie in ideas learned in other subjects. For example, have students categorize words by their first or last letters, by their sound, by the number of syllables, etc. ( rhymes with tin and 2 syllables : A. begin, B. chin, C. mat, D. silly). Start with four words, then add four more words to be sorted into the same categories. Encourage students to suggest words and their place in the diagram. (Cities, states, or food groups are also good categories.) ACTIVITY 2 Repeat Activity 1 using two overlapping circles. Example: circle 1: wearing blue; circle 2: has dark hair. Extensions 1. Have students sort the eight shapes from BLM Shapes (p. H-33) into Venn diagrams. (The BLM has two of each shape, so you ll need one copy for every two students.) Possible categories include: a) Shapes Striped or Triangle Big squares b) plain or triangle; plain squares c) plain triangles; big or plain d) small or triangle; striped and big e) small or striped; big or square f) big squares; small squares (MP.6) In addition, you can distribute copies of BLM Circles (p. H-34) one copy for every four students and have students sort the shapes from both BLMs into Venn diagrams. Possible new categories include: g) plain or triangle; not a circle h) striped, but not a square; circle or big 2. Combine the shapes BLM Circles and BLM Shapes and have students repeat Extension b) from G4-1. Allow them to use the word not when describing the separated shapes. For example, when you separate the plain triangles, the remaining shapes can be described as striped or not a triangle or striped or square or circle. Although not a plain triangle is also correct, students should be encouraged to use a more precise description. Students can complete BLM Venn Diagram (Advanced) (p. H-35). H-6 Teacher s Guide for AP Book 4.1

7 G4-3 Sides and Vertices of 2-D Shapes Pages STANDARDS preparation for 4.G.A.2 Goals Students will identify polygons, sides, and vertices and will distinguish polygons according to the number of sides. Vocabulary hexagon pentagon polygon quadrilateral side three-dimensional, 3-D triangle two-dimensional, 2-D vertex, vertices PRIOR KNOWLEDGE REQUIRED Can count to ten Can distinguish a straight line MATERIALS large paper shapes (see vocabulary and lesson) various polygons from BLM Polygons (pp. H-36 H-37) geoboards pattern blocks or tangrams or BLM Pattern Blocks (p. I-1) or BLM Tangrams (p. H-38) (MP.8) Sides and vertices. Draw and label a polygon with the words sides and vertex/vertices. Remind students of what a side and vertex are. Show students how to count sides marking the sides as you count then have them count the sides and the vertices of several polygons. Ask students if they can see a pattern between the number of vertices and the number of sides. Be sure that all students are marking sides properly and circling the vertices, so they don t miss any sides or vertices. Construct a large triangle, quadrilateral, pentagon, and hexagon using construction paper or bristol board. Label each shape with its name and display the shapes on the board. Explain that gon means corner (vertex) and lateral means side. You might want to leave these shapes posted on a wall throughout the geometry unit. Later (G4-8), you can add special quadrilaterals. not polygons Polygons. Explain that flat shapes such as squares, triangles, and circles are called two-dimensional, or 2-D for short. That is because they have two dimensions: length and width. Shapes such as cubes (show a cube) have length, width, and height, and so are called three-dimensional or 3-D. Divide the board in two. On one side, draw a variety of polygons, including regular polygons (shapes with equal sides and equal angles) and irregular polygons. Label the shapes Polygons, reading the label aloud. Label the other side of the board Not Polygons. Draw a pentagon with a little part of one side missing on the Not Polygons side and ASK: How is this different from the polygons? (It is a line with endpoints, not a shape.) Repeat with a shape that has one curved side, a shape that has a tail (or loose endpoint), and a shape with self-intersecting sides (it looks like a combination of two shapes; it has a vertex where more than two edges meet). Add a variety of other non-polygons and ASK: What is the same about all the polygons? Do they all have straight sides? (yes) Do they have endpoints? (no) Are they all Geometry 4-3 H-7

8 flat? (yes) Conclude by telling students that all 2-D shapes that have straight sides and no loose endpoints are called polygons. Explain that poly means many, then ask students what the word polygon might mean. (many corners, many vertices) shapes with indentations Draw several more shapes and ask students to decide whether each is a polygon or not. Include shapes that have indentations. Students can point to the correct side of the board to indicate whether a shape is a polygon or not. Sorting polygons by the number of sides. Give students an assortment of polygons (use BLM Polygons) and have them count the sides of the polygons. Ask students to sort the shapes by the number of sides. (Help struggling students by providing group names, such as 3 sides, 4 sides, 5 sides, 6 sides, more than 6 sides.) After giving students time to work, make a chart on the board with columns labeled according to the number of sides and display larger copies of the same shapes. Have students show in which column each shape belongs (they can raise the number of fingers equal to the number of sides). Introduce the names of polygons. Explain that mathematicians give special names to polygons according to the number of sides and vertices they have. Present the names (see vocabulary) on cards, explain each one (e.g., hexagons have 6 straight sides and 6 vertices), and invite volunteers to assign each card as the label for a column. Students may notice that the column for shapes with 4 sides, labeled quadrilateral, includes shapes that they would call squares and rectangles. Explain that, just as there is more than one word that describes a student (e.g., boy/girl, child, person), many shapes have more than one name. Tell students that squares and rectangles are special types of quadrilaterals. (MP.6) Draw a pentagon on the board. ASK: How many sides does this polygon have? What is the name for this polygon? How do you know that this is a pentagon? If students do not say that it has 5 sides and 5 vertices and all sides are straight, draw another pentagon (different size, color, or pattern) and ask what the two pentagons have in common. Repeat with a triangle and a quadrilateral that is neither a rectangle nor a square (e.g., a parallelogram, rhombus, or trapezoid). Finally, draw a square and a rectangle. For each, ASK: How many sides does this polygon have? What is a polygon with 4 sides called? (quadrilateral) What is another name for this shape? (square/rectangle) ACTIVITY 1 Use a geoboard to create a shape (do each shape introduced in the lesson). Ask students to re-create the exact same shape on their geoboards. Draw a group of shapes on the board and ask students to count the sides and sort the shapes according to the number of sides. Also ask them to draw a triangle, a pentagon, a figure with six sides, a figure with four angles, and a figure that is not a polygon but has vertices. H-8 Teacher s Guide for AP Book 4.1

9 Bonus Draw a figure that has: a) two curved sides and three straight sides b) two straight sides and three curved sides Students who need additional practice can do these Exercises: 1. Draw a polygon with seven sides. 2. Draw a quadrilateral. How many vertices does it have? 3. Altogether, how many vertices are there in two pentagons and a triangle? 4. Draw a shape that is not a polygon and explain why it is not a polygon (for instance, the rectangle with rounded edges at left does not have proper vertices). (MP.1) ACTIVITY 2 Give each student a set of pattern blocks or several tangram pieces with the following instructions (answers are shown): a) Group your pieces according to how many sides they have. How many of each type do you have? b) Can you make a shape with four sides using two triangles? Three triangles? c) Can you make a large triangle using four triangles? d) Can you make a triangle from two small tangram triangles and a square? e) Can you make a pentagon with pattern blocks? (MP.8) (MP.3) Extensions 1. Count the sides of a paper polygon. Count the vertices. Cut off one of the vertices. Count the sides and vertices again. Cut off another vertex. Repeat the count. Do you notice a pattern? (The number of sides will increase by one and the number of vertices will increase by one.) 2. BLM Word Search (Shapes), p. H-39. This can also be assigned as homework. 3. After students have sorted polygons by the number of sides, ask if anybody has a polygon with two sides only. Then suggest that students try to draw a polygon with two sides. Let them discuss in pairs why this is impossible. Debrief as a class. Geometry 4-3 H-9

10 G4-4 Right Angles Pages STANDARDS 4.G.A.2 Goals Students will identify right angles in drawings and objects. Vocabulary perpendicular lines perpendicular sides right angle PRIOR KNOWLEDGE REQUIRED Can identify polygons MATERIALS rectangular sheets of paper paper polygons (see lesson) cards from BLM Polygons (pp. H-36 H-37) geoboards Introduce right angles. Explain that a right angle is the corner of a square (there is no need to define it properly at this stage). Ask students where they can see right angles in real life (corners of a sheet of paper, doors, windows, etc.). Draw a right angle and show students how to mark right angles with a small square. Explain to students that not all angles are right angles; some are sharper than a right angle, some are less sharp. Tell them to think of corners the sharper the corner is, the smaller the angle is. Demonstrate how to compare angles to a corner of a piece of paper by superimposing. Emphasize that you need to match one side of the angle to a side of the piece of paper and that the corner of the piece of paper should match the vertex of the shape (or the angle). If the paper or the angle are sticking out, the angle is not a right angle. Have students compare various corners with a corner of a piece of paper to decide whether the corners are right angles. B Explain to students that when they want to check whether the angles of a polygon are right angles, they need to look at the parts that are inside the shape. Draw the shape at left and explain that angle B of this polygon is not a right angle. Show the same shape made from bristol board, and compare the angle with a corner of a piece of paper. The angle inside the shape is visibly larger than a right angle. The angle outside the shape might be a right angle, but that s not the angle we re interested in! Give students cards from BLM Polygons and have them compare the angles of the polygons on the cards with a corner of a sheet of paper. Show students how to mark right angles with a small square and have them mark the right angles on the cards. Students who have trouble identifying the correct angles in shapes with indentations will benefit from cutting out the shapes. Draw several angles and ask volunteers to identify and mark the right angles. For a short assessment, you can also draw several shapes (see below) and ask students to signal how many right angles there are. H-10 Teacher s Guide for AP Book 4.1

11 Do not mark the right angles in the diagram. Introduce perpendicular lines. Explain to students that when two lines meet and make a right angle, they are called perpendicular lines. Write the term on the board and draw several pairs of lines intersecting at a right angle as in Question 6 on AP Book 4.1 p Show how to compare the angles with a square corner again. Draw several pairs of intersecting lines on the board (perpendicular and not) and have students predict whether the lines are perpendicular. Students can show thumbs up for perpendicular lines and thumbs down for lines that are not perpendicular. Include pairs of lines that are not horizontal and line segments that intersect in different places and at different angles (see examples below). Invite volunteers to check whether the lines are perpendicular using the corner of a page. Perpendicular sides. Explain that sides of polygons that make a right angle are called perpendicular sides. So when you are looking for perpendicular sides in a polygon, you are looking for right angles. Draw several polygons (without indentations) and number the sides. Have students identify pairs of sides that are perpendicular (e.g., side 1 and 2 are perpendicular). Return to the shape from the beginning of the lesson, where the sides that make a reflex angle (greater than 90 ) inside the shape are perpendicular. Trace the two sides that make a reflex angle on the board, remove the shape and ASK: Are these lines perpendicular? (yes) Explain that although the angle of the polygon is not a right angle, the sides are still perpendicular, because the angle on the outside of the shape is a right angle. So when we are looking for perpendicular sides, it is a good strategy to look for right angles, but we also might have some other perpendicular sides. ACTIVITY Have students use geoboards to make shapes with a given number of right angles: a) 4 right angles b) 2 right angles c) 1 right angle Geometry 4-4 H-11

12 Bonus d) 3 right angles e) 6 pairs of perpendicular sides and 5 right angles Sample answers: a) rectangle or square, b) right trapezoid c) d) e) Extension Sometimes you need to extend the sides of a shape to check whether they are perpendicular. Have students individually extend the sides of an octagon to check which sides are perpendicular. Do one pair of sides with them: H-12 Teacher s Guide for AP Book 4.1

13 G4-5 Parallel Lines Pages STANDARDS 4.G.A.1 Goals Students will identify and mark parallel lines. Vocabulary parallel lines PRIOR KNOWLEDGE REQUIRED Can identify straight lines MATERIALS grid paper rulers cards from BLM Polygons (pp. H-36 H-37) pictures from magazines or newspapers that show examples of parallel lines Lines that are the same distance away at all points. Tell students that an engineer wants to build a bridge over a river. Draw a picture of the river, as shown, on the board. The engineer wants the bridge to be as short as possible. Where should he build the bridge? Have students come over and show where they would build a bridge. Have students measure the width of the river at different points. Where should the engineer build a bridge? Repeat with another picture of a river whose width varies. Finally, present a river with straight, parallel banks. Have students measure its width in different places. ASK: What do you notice about the distance between the banks? (it is always the same) Is there one place that is best to build a bridge? (no) Parallel lines. Explain that lines that look like the banks of the last river are called parallel lines. Parallel lines are straight lines that are always the same distance apart. Like railway tracks, they never meet. Ask students where they can see parallel lines in the classroom (shelves, table sides or legs, the lines where the walls meet, etc.). Draw several parallel lines on the board. Show how to mark parallel lines with arrows. Draw a pair of lines that are not parallel but will intersect if extended on the board. ASK: Could a train go along a pair of tracks like this? Why not? (they are getting closer together, and the wheels of a train do not get closer together) Explain that lines can be extended in either direction, and parallel lines will never meet, even if extended. Invite volunteers to extend the lines so that they clearly intersect. ASK: Are these lines parallel? (no) Geometry 4-5 H-13

14 (MP.7) Discuss with students how they can draw parallel lines. One strategy could be to draw a line using a ruler, then slide the ruler without turning it and draw another line. Point out that just as railway tracks go in the same direction, parallel lines also go in the same direction. So if one parallel line on grid paper goes 3 squares up and 4 squares left, the other parallel line will also go 3 squares up and 4 squares left. Ask students to draw a line (first horizontal, then vertical, then diagonal) on grid paper and to draw another one parallel to it. Mark the parallel lines with arrows. On a grid on the board draw a right-angled but not isosceles triangle (sides of 4 and 3 squares). Ask students to copy the triangle and to draw lines parallel to each of the sides. The purpose of this exercise is to draw a line parallel to a diagonal line that does not pass through every point of intersection on the grid. Draw several shapes that contain parallel sides on the board (it is important to have parallel lines with different slopes and of different lengths use various trapezoids, parallelograms, and hexagons, and not only rectangles and parallelograms) and ask volunteers to mark pairs of parallel sides. You may also do the opposite task draw a pair of parallel line segments and ask students to join the ends to make a quadrilateral. Students who need additional practice can identify parallel sides on polygons from BLM Polygons. ACTIVITY In a picture from a magazine or a newspaper, find as many parallel lines as you can, marking each set with a different color. Extension (MP.1) Draw a polygon that has: a) more than one pair of parallel sides. b) more than two pairs of parallel sides. c) three parallel sides. Sample answers a) b) c) H-14 Teacher s Guide for AP Book 4.1

15 G4-6 Quadrilaterals Page 170 STANDARDS preparation for 4.G.A.2 Goals Students will distinguish quadrilaterals from other polygons. Vocabulary quadrilateral PRIOR KNOWLEDGE REQUIRED Can identify polygons Can measure sides with a ruler Can count the sides of a polygon MATERIALS cards from BLM 2-D Shape Sorting Game (pp. H-40 H-41) cards from BLM Polygons (pp. H-36 H-37) larger cards from the same BLMs for demonstration (see below) pattern blocks or BLM Pattern Blocks (p. I-1) toothpicks modeling clay (MP.6) Introduce quadrilaterals. Draw a quadrilateral on the board. Ask how many sides it has. Write the word quadrilateral on the board and explain what it means. Explain that quad means four and lateral means side in Latin. Ask if students have ever encountered any other word having either of these parts in it. (square, quadruple) You might also mention that tri means three and ask if students know what the French words are for 3 and 4. Emphasize the similarities: tri and trois, quad and quatre. Draw several polygons on the board and ask whether they are quadrilaterals. Write the number of sides for each and mark the answer on the board. (MP.6) Give students shape cards from BLM 2-D Shape Sorting Game (you can add cards from BLM Polygons as well). Have students sort the shapes individually into quadrilaterals and non-quadrilaterals using a chart with two columns, then check the answers on the board using a larger version of the same cards with tape on the back side. (To make larger cards, photocopy the BLM onto A3 paper (11 1/2 by 17) using enlargement factor 150%.) For each card, ask students to explain their choice. Repeat with polygons and non-polygons. Finally, have students sort the shapes by the number of sides. For additional practice, you may ask students to sort pattern block pieces into quadrilaterals and non-quadrilaterals. Include the word quadrilateral on your next spelling test. Geometry 4-6 H-15

16 ACTIVITIES 1 2 (MP.3) (MP.1, MP.3) 1. Give each student 4 toothpicks and some modeling clay to hold them together at vertices. Ask students to create a shape that is not a quadrilateral. (It might be either a three-dimensional shape or a self-intersecting one.) Also ask students to make several different quadrilaterals. 2. Give each student 10 toothpicks and ask them to check how many different triangles they can make with these toothpicks. Each triangle should use all the picks. Then ASK: How many different quadrilaterals can you make using 10 toothpicks? The answer to the first problem is two: 2, 4, 4, and 3, 3, 4. The answer to the second problem is infinity a slight change in the angles will make a different shape. H-16 Teacher s Guide for AP Book 4.1

17 G4-7 Properties of Shapes Pages STANDARDS 4.G.A.2 Goals Students will sort shapes according to the number of sides, parallel sides, or right angles. Vocabulary equilateral hexagon parallel lines pentagon quadrilateral triangle PRIOR KNOWLEDGE REQUIRED Polygons Quadrilaterals Parallel lines Right angles MATERIALS cards from BLM Polygons (pp. H-36 H-36) cards from BLM 2-D Shape Sorting Game (p. H-40 H-41) toothpicks geoboards Sort shapes by properties. Review with students how to check that sides of a polygon are parallel (e.g., by sliding a ruler from one side to the other if you can slide the ruler without turning, and both sides line up with the ruler, the sides are parallel). Review how to check that angles in a polygon are right angles (by using a corner of a page). Give each student several shapes from BLM Polygons and have students identify parallel sides and right angles on each shape. Then ask them to sort the shapes: a) by name (triangle, quadrilateral, pentagon, hexagon, other) b) by the number of right angles (0 right angles, 1 right angle, 2 right angles, more than 2 right angles) c) by the number of pairs of parallel sides (0 pairs, 1 pair, more than 1 pair) Draw the table below on the board. Leave enough space in each cell to draw a shape and to write a few names. Do not fill in the table with the numbers shown they are included for your convenience. Property No parallel lines 1 pair of parallel lines 2 or more pairs of parallel lines No right angles 1, 7, 10, 12, 13 3, 23 8, 11, 14, 15 1 right angle 4, 9 17, 18 16, 21 2 or more right angles 19, 24 6, 22 2, 5, 20 Geometry 4-7 H-17

18 Go through the table cell by cell and ask volunteers to draw an example of a shape that has these properties. Remind students to mark parallel lines and right angles on their pictures. Then assign each student a single card from BLM Polygons, and have them decide how many pairs of parallel sides and how many angles the polygon has. Ask students to decide which cell their shape belongs to and write their name in the cell. This will separate students into groups according to the numbers on the cards as written in the table. If you want to obtain 9 pairs, include only two cards from each group in the activity. (MP.8) Find sorting rules for a pre-sorted group of shapes. After the sorting activity above, let students play a game. Students will need a set of shapes from BLM 2-D Shape Sorting Game in addition to those they were given for the sorting activity. Player 1 (or Team 1) lays down 3 shapes that have a common feature (for instance, they might all have the same number of sides). Player 2 or Team 2 adds a fourth shape with the same feature. Player 1 tells Player 2 if the addition is correct. If it is, Player 2 guesses what the shapes have in common. If the addition is incorrect, Player 1 adds another shape with the same feature, to give Player 2 more information, and Player 2 guesses again. Players continue adding shapes and guessing until Player 2 correctly identifies what the shapes have in common. Advanced variation: Player 1 puts down 3 shapes that have a common feature and 1 extra shape that doesn t have that feature, and Player 2 tries to find the shape that doesn t belong and explains the choice. (MP.3) Shapes with all equal sides. Draw a rectangle and a square on the board. ASK: How are these shapes different? (one has two pairs of longer sides and two pairs of shorter sides, and the other has all sides equal) Draw an equilateral triangle and ask students if it is more like a rectangle or a square. (square) Why? (it has all sides equal) Point out that a square and a rectangle look the same from many directions. For example, if you turn a square on its side, it will look exactly the same. ASK: Do all shapes with all sides equal have this property? To help students realize that shapes with all sides equal do not have to look the same from many directions, provide students with at least 5 toothpicks of the same length. Ask them to check that the toothpicks are all the same length. Ask students to create several polygons using all of their toothpicks. Can they make polygons that do not look the same from all sides? Can they make ugly polygons? Have students trace the polygons they created and then share their results with the class. Ask students to sort the shapes from BLM Polygons or BLM 2-D Shape Sorting Game into shapes that have all sides of the same length (cards 1, 2, 11, 13, 15) and shapes that have some sides that are not the same lengths as others. Point out that the shapes in the second group still might have some equal sides, but not all sides are the same length. H-18 Teacher s Guide for AP Book 4.1

19 ACTIVITY Ask students to make each shape shown below with an elastic on a geoboard. Then ask them to connect two pins with another elastic to change the given shape into: a) two triangles b) a quadrilateral and a triangle Extension Find the names of polygons for all polygons with up to 20 sides. If you are looking for names of polygons on the Web, check at least three sources to see that they give the same name for the same polygon. Are there coins that have polygons on them or that are in the shape of a polygon? Which (not-so-common) US coin has a polygon on it? What coins from other countries are polygons, and what polygons are they? Answers: The US $1 coin with the portrait of Susan B. Anthony has a hendecagon (11 sides) drawn on it. Many other countries have polygonal coins. For example, a Canadian $1 coin is a hendecagon. A UK 20-pence coin is a heptagon (7 sides). Geometry 4-7 H-19

20 G4-8 Special Quadrilaterals Pages STANDARDS 4.G.A.2 Goals Students will distinguish between special quadrilaterals. Vocabulary parallelogram rectangle rhombus square trapezoid PRIOR KNOWLEDGE REQUIRED Quadrilaterals Parallel sides Right angles Can measure sides with a ruler MATERIALS large paper quadrilaterals (see vocabulary and lesson) cards from BLM Polygons (pp. H-36 H-37) cards from BLM 2-D Shape Sorting Game (pp. H-40 H-41) geoboards pictures from magazines or newspapers with shapes containing quadrilaterals Introduce special quadrilaterals (parallelograms, rectangle, rhombus, square, trapezoid). Use large special quadrilaterals made from construction paper or bristol board to introduce these shapes to students. Label each shape with its name. Ask students where they have seen these shapes in real life. (e.g., a rhombus may be found in the diamond shape on playing cards) Add the special quadrilaterals to any other shapes already posted on the wall, and leave them up for as long as you are studying geometry. Include the names of the special quadrilaterals in your next spelling text. (MP.8, MP.5) Give students cards with quadrilaterals from BLM Polygons and BLM 2-D Shape Sorting Game and have them sort the shapes into a chart with headings Parallelogram, Trapezoid, Other. Check answers on the board with large copies of the cards. Then SAY: I want to sort the shapes into rectangles, rhombuses, and squares. Will I find these shapes in all columns? (no) Which column are they all in? (parallelograms) Why? (because they are all parallelograms they all have two pairs of parallel sides) Have students sort the parallelograms into squares, rectangles, rhombuses, and other parallelograms. Have students fill in the following table with their parallelograms: 4 equal sides 4 right angles ASK: What name can we attach to the shapes in the first row? (rhombuses) In the second row? (rectangles) What problem did students encounter? Which shape should be placed in both rows at the same time? (square) Which sorting tool would be more convenient than this table? (a table with H-20 Teacher s Guide for AP Book 4.1

21 three rows or a Venn diagram) Point out that just as a person can be given several names (boy, David, McDonald) because he has the properties for each of these names (he is male, his name is David, and he belongs to the McDonald family), the same quadrilateral can be given different names because it has the properties that define each name. AP Book 4.1 Questions 6 and 7 on p. 174 may be done with a geoboard. Students who need additional practice can answer these questions: What shape am I? a) I am a quadrilateral and all my sides are equal. None of my angles are right angles. b) I have four sides and all my angles are the same. One of them is a right angle. c) I have four sides and two of them are parallel. The other two are not. d) I am a quadrilateral and my opposite sides are parallel. e) I am a rhombus that has equal angles. f) I am a quadrilateral with two parallel sides. One of my parallel sides is twice as long as the other one. Draw me first! Answers: a) rhombus, b) rectangle or square, c) trapezoid, d) parallelogram, e) square, f) trapezoid ACTIVITIES Ask students to mark as many special quadrilaterals as possible on a picture from a magazine or a newspaper. Make sure that there are examples of regular parallelograms as well (some train or car windows, for instance) Ask students to play with quadrilaterals on this Web site and to sketch various shapes they obtain for each special quadrilateral. Extensions 1. There are three vertices of a square on the grid paper. Can you finish the square? 2. Can you draw a square without vertical sides on blank (non-grid) paper? Use a ruler. Geometry 4-8 H-21

22 3. Miss Maple is a Fourth Grade teacher. She wants to teach her students about Venn diagrams. To help her students sort shapes, she decides to label all the regions of the diagram, and not only the circles. The first diagram she makes looks like this: Names Boys names Girls names Names that both boys and girls can have Miss Maple starts another Venn diagram but doesn t finish it. Place the words parallelograms, rectangles, rhombuses, and squares in the right place in an empty Venn diagram. Answer: Parallelograms Rectangles Rhombuses Squares H-22 Teacher s Guide for AP Book 4.1

23 G4-9 Symmetry Pages STANDARDS preparation for 4.G.A.3 Goals Students will identify shapes with a line of symmetry. Vocabulary horizontal line of symmetry symmetrical vertical PRIOR KNOWLEDGE REQUIRED Can identify figures of the same shape and size MATERIALS large paper right triangle paper squares shapes from BLM Shapes to Fold (pp. H-42 H-43) Miras cards from BLM Polygons (pp. H-36 H-37) Folding shapes so that parts match exactly. Show students a right triangle created from a rectangular sheet of paper by cutting the rectangle along the diagonal. Fold the triangle once as shown below. Show students how the two parts do not match one part sticks out. From one side, the wider part of the triangle covers the narrower part, but when you flip the folded shape over, the narrower part doesn t cover the wider part. Fold Nothing sticks out Flip This part sticks out (MP.3) Give each student a paper square. Ask students to fold their square so that one of the vertices matches up with any other vertex. (All students don t have to and likely won t fold the square the same way.) ASK: Does the top part of your folded square cover the bottom part? Flip the folded square over does the top part cover the bottom part now, too? Unfold your square what parts do you see? Are the shapes of the same kind? Are the shapes of the same size? How do you know? (nothing sticks out ) What can you say about the parts of the square? (They are the same, or equal.) Explain that in this case we say that the parts of the square match exactly. Ask students to check whether they folded the square so that the parts match exactly. If not, ask them to fold the square again so that the parts match exactly. ACTIVITY 1 Give students several shapes from BLM Shapes to Fold and ask them to fold the shapes along the dotted lines. Do the parts match exactly? For the shapes in which the parts do not match, discuss how students can trace and cut to make the shapes match, then have them test their predictions. Geometry 4-9 H-23

24 Bonus Find another way to fold the square so that the parts match exactly. How many different ways can you find to fold a square so that the parts match exactly? E Miras. Ask students what object they know that can show an exact match of anything. (a mirror) Give students Miras. If students have not used Miras before, ASK: What does this kind of mirror do? Let students experiment with personal objects. When students understand that this mirror is transparent, show them the way to check whether parts of objects match exactly. For example, you could write a large E (as shown) on the board and ask whether the top part is exactly the same as the bottom part. ASK: if you could fold the board, would the top of this E fall precisely on the bottom? How could we check? Take guesses, then show students how to check this using the Mira. Students should clearly see that the parts do not match. ACTIVITY 2 Have students use Miras to check whether some objects, such as letters on the cover of their JUMP Math AP Book 4.1, have matching parts. Introduce lines of symmetry. Have students unfold their squares and explain that the line along the fold is called a line of symmetry. This line indicates that the parts of the shape match exactly when folded over. Explore lines of symmetry. Show students a rectangular sheet of paper. Invite a volunteer to fold it in half so that the vertices match. Point out that when the sheet is folded in this way, the sides match also. The parts of the rectangle match exactly one part is exactly the same as the other. The line along the fold is a line of symmetry. Now fold the rectangle along the diagonal. Do the parts match exactly? (no) Is this line a line of symmetry? (no) Are both parts the same size and shape? (yes) Explain that even though the shapes would match if cut and turned, cutting is not allowed in symmetry. Only folding is allowed. Introduce symmetrical shapes. Explain that a shape that has a line of symmetry is called symmetrical. Check symmetry with a Mira. Draw a symmetrical heart or a stick person on the board and ask students where and how they would place a Mira to check that this picture has two matching parts. Invite a volunteer to show how to place a Mira and to draw its location on the board. ASK: What do we call the line you drew? (a line of symmetry) Explain that a line of symmetry is also called a mirror line. H-24 Teacher s Guide for AP Book 4.1

25 ACTIVITY 3 (MP.3) Give students an assortment of shapes from BLM Polygons and ask them to determine which shapes from their collection have lines of symmetry. Let them check the shapes with Miras, trace the shapes in their notebooks, and use a ruler or straight edge to draw the lines of symmetry. Before assigning pages in AP Book 4.1, review with students the meaning of the terms horizontal and vertical. Bonus Add something to the figures in Questions 6 and 7 on AP Book 4.1 p. 176 that don t already have both horizontal and vertical lines of symmetry so that they do. Extension Which states have symmetrical flags? Search the Web for United States sub national flags and check. Geometry 4-9 H-25

26 G4-10 More Symmetry Page 177 STANDARDS 4.G.A.3 Goals Students will find lines of symmetry and identify shapes that have lines of symmetry. Vocabulary hexagon horizontal line of symmetry parallelogram pentagon rectangle rhombus square vertical PRIOR KNOWLEDGE REQUIRED Special quadrilaterals Can identify and draw a line of symmetry MATERIALS large paper rectangle paper and scissors sheets of paper with a straight line down the middle (see Activity 1) and pattern blocks (or BLM Pattern Blocks p. I-1) BLM Lines of Symmetry (p. H-44) BLM Paper Quadrilaterals (p. H-45) Miras pictures from magazines or newspaper (see Activities 4 and 5) Creating symmetrical shapes and patterns. Hold up a large paper rectangle. Fold it in four along the lines of symmetry. Ask students to identify the folded shape. Unfold once and ask students to identify the new shape. Unfold completely and ask students to identify the shape again. Refold the paper and cut the folded vertex as shown (each end of the cut-line should be the same distance from the vertex). cut (MP.1) ASK: What shape will the hole be when I unfold the paper? Unfold once and let students adjust their predictions. Then unfold completely to check their predictions. Ask students to predict how they could produce holes of different shapes (e.g., rhombus, circle, rectangle, square). Let them experiment with paper and scissors. Bonus How would you cut the folded paper to create a star-shaped hole? Bonus Making snowflakes. Fold a sheet of paper three times (into eight parts). Cut out some small pieces from the sides. For each piece you cut out, predict the shape of the hole it produces. Unfold in stages and check your predictions. How does the way you fold the sheet of paper affect the number of lines of symmetry of your snowflake? H-26 Teacher s Guide for AP Book 4.1

27 ACTIVITY 1 Give pairs a sheet of paper with a straight line drawn down the middle (either vertically or horizontally) and a set of pattern blocks. Player 1 places a pattern block so that it touches the line. Player 2 places a block on the other side of the line to make the design symmetrical. Then Player 2 places a second block on his or her side of the page such that the two blocks have a common side. Now Player 1 must add a block to his or her side of the page to maintain the symmetry and then add another block. Players continue placing pairs of pattern blocks in turn. At the end, ask students how they could check whether the design is symmetrical (sample answer: using a Mira). Example (three turns were taken): Have pairs display their designs and describe them to classmates, naming the shapes used and explaining how they know the design is symmetrical. Ask students to consider whether any of the designs have more than one line of symmetry. Students who need extra help can complete BLM Lines of Symmetry. Remind students that a line of symmetry can also be found by folding. Take a large paper shape with several lines of symmetry (such as a regular pentagon or hexagon) and fold it to find any lines of symmetry. Students can use paper shapes from BLM Paper Quadrilaterals to complete Question 1 on AP Book 4.1 p. 177). ACTIVITIES Using each pattern block shape at least once, create a figure that has at least one line of symmetry. Choose one line of symmetry and explain why it is a line of symmetry. Draw your shape in your notebook. 3. Using exactly four pattern blocks, build as many shapes as you can that have at least one line of symmetry. Draw your shapes in your notebook. 4. Find a picture in a magazine that has a line of symmetry and mark the line with a pencil. Is it a horizontal or a vertical line? Try to find a picture with a slanted line of symmetry. 5. Cut out half an animal or human face from a magazine and glue it on a piece of paper. Draw the missing half to make a complete face. Geometry 4-10 H-27

28 Extensions 1. The figure shown has four lines of symmetry. a) Show how you could move one square so the resulting figure has no lines of symmetry. b) Show two different ways to move a square so the resulting figure has one line of symmetry. You are allowed to move the center square. (MP.1, MP.3) 2. Make a shape on a geoboard (or on grid paper) with the given number of lines of symmetry: a) a quadrilateral with 1 line of symmetry b) a quadrilateral with 4 lines of symmetry c) a triangle with 1 line of symmetry d) a triangle with 3 lines of symmetry Bonus: Is it possible to make e) a triangle with exactly 2 lines of symmetry? f) a quadrilateral with exactly 3 lines of symmetry? 3. Draw a shape that has a horizontal and vertical line of symmetry, but no diagonal line of symmetry. 4. How many lines of symmetry does an oval have? reflection 5. Sudha drew a mirror line on a square. Then she drew and shaded the reflection of the corner of the square in the mirror line. a) Try Sudha s method. Draw a large square on grid paper. Draw a mirror line on the square and reflect part of the square in the line. (MP.1) b) Repeat part a) but move the mirror line. Can you place the mirror line so that the two parts of the square on either side of the mirror line make: i) a rectangle ii) a hexagon iii) an octagon H-28 Teacher s Guide for AP Book 4.1