G7 3 Measuring an angle

Transcription

1 1 Geometry 1 Introduction In this unit, students will use protractors, rulers, set squares, and various strategies to construct related lines using angle properties investigate geometric properties related to symmetry, angles, and sides sort and classify triangles and quadrilaterals by their geometric properties Protractors If you need additional protractors for individual students, you can photocopy a protractor (or LM Protractors, p. 9) onto a transparency and cut it out. Such protractors are also convenient to use on an overhead projector. Paper Folding Many ctivities in these lessons involve paper folding. Unless otherwise noted, the starting shape is a regular 8½ 11 sheet of paper. Sometimes the starting shape is an oval or a cloud, to make sure there are no angles for students to start with or refer to. Questions for Extra Practice Many lessons include questions for extra practice. You can write these questions on the board or photocopy them onto transparencies and use an overhead projector to display them. Summary LMs Step-by-step instructions for constructions used in the unit are summarized on LMs, for easy reference. This chart lists the summary LMs available and the lesson(s) they relate to. Summary LM Lesson(s) Constructions Measuring and Drawing ngels and Triangles (p 1) Drawing Perpendicular Lines and isectors (p 2) G7 3 Measuring an angle G7 4 G7 5 Drawing an angle Drawing lines that intersect at an angle Drawing a triangle Drawing a line segment perpendicular to through point P (using a set square, using a protractor) Drawing the perpendicular bisector of line segment Drawing Parallel Lines (p 3) G7 6 Drawing a line parallel to through point P (using a set square, using a protractor)

2 2 G7 1: Points and Lines Workbook pages: Goals: Students will review the basic geometric concepts of (and notation for) points, lines, line segments, and rays. They will also review intersecting lines and lines segments. Prior Knowledge Required: Can use a ruler to draw lines Vocabulary: point line line segment ray intersect intersection point endpoint Curriculum Expectations: Ontario: conceptual review; 7m2 WNCP: conceptual review, [R] Review the concepts of a point, line, line segment, and ray. This dot represents a point. point is an exact location. It has no size no length, width, or height. The dot has size, or you couldn t see it, but real points do not. line extends in a straight path forever in two directions. It has no ends. Lines that are drawn have a thickness, but real lines do not. line segment is the part of a line between two points, called endpoints. It has a length that can be measured. ray is part of a line that has one endpoint and extends forever in one direction. Review naming a point, line, ray, and line segment. F G line or ray (not ) line segment FG or GF point is named with a capital letter. To name a line, give the names of any two points on the line. To name a ray, give the name of the endpoint and any point on the line. To name a line segment, give the names of the endpoints.

3 3 Point out that you don t need to draw arrows and/or dots at the ends of lines, rays, or line segments unless you especially want to show that something is a line, ray, or line segment. Introduce the intersection point a point that lines, line segments, or rays have in common. Draw several examples of intersecting line segments, including the following: Invite volunteers to identify the intersection point in each picture. Review intersecting lines and line segments. Draw the picture at right on the board and SK: Does intersect CD? The answer depends on whether and CD are lines, line segments, or rays. Check all possible combinations of lines and line segments, extending the lines to show intersection. NSWERS: If either one of these is a line segment, they do not intersect. Lines and CD intersect. If students are engaged, ask them to check all four possible rays, as well as all combinations of rays and lines or rays and line segments. PE Revisiting conjectures that were true in one context NSWERS: Out of four possible rays, only and DC intersect. lso, line intersects with ray DC and ray intersects with line CD. Examples: C D C D D C rays and DC do not intersect line and ray DC intersect line and ray CD do not intersect C D

4 4 G7 2: ngles and Shapes Workbook pages: Goals: Students will review the naming of angles and shapes. Prior Knowledge Required: Knows what an angle is Can identify polygons Knows the terms clockwise, counter-clockwise Vocabulary: angle vertex arms line segment Curriculum Expectations: Ontario: review WNCP: 6SS1 Review the concept of an angle and how to name it by following the progression on the worksheet. n angle is formed by two rays with the same endpoint. The endpoint is the vertex of the angle. The two rays are the arms of the angle. Point out that the letter for the point at the vertex has to be in the middle of the name for an angle. vertex Y X Z arms XYZ or ZYX (not ZXY or YXZ) Extra practice: 1. Name the angle in all possible ways. K E Q 2. a) Which of the following are possible names for this angle? CT, CTP, PT, PTU, UT, UTC, TCP b) Write three more different names for the angle. C P U T

5 5 Point out that sometimes, when there is no chance of confusion, only the vertex letter is used to name an angle. For example, there is only one possible D, but there are three possibilities for : D or C or DC. C D Review the concept of a polygon. polygon is a closed 2-D (flat) shape has sides that are straight line segments has each side touching exactly two other line segments, one at each of its endpoints The point where two sides of a polygon meet is called a vertex. (The plural of vertex is vertices.) Polygons vertex Not polygons Have students explain (using the definition above) why each of the shapes below is not a polygon. Review naming polygons. C D C S F D M Q Start at any vertex. Choose a letter to name the vertex. Go around the polygon clockwise or counter-clockwise, labelling the other vertices. To name the polygon, write the letters at the vertices in order. For example, you could name this rhombus CD or DC or DC, but not DC. You can choose any sequence of letters. Extra practice: a) Circle the correct names for this polygon. KLXY LYXK KLYX KXYL XYLK X K b) Write another correct name for this polygon. Y L

6 6 G7 3: Measuring and Drawing ngles and Triangles Workbook pages: Goals: Students will measure and draw angles. Prior Knowledge Required: Knows what an angle is Can name an angle and identify a named angle Vocabulary: angle vertex arms acute obtuse Materials: dice geoboards and elastics grid paper Curriculum Expectations: Ontario: 5m51, 5m52, 5m54, 6m48, 6m49, 7m46 WNCP: 6SS1 Review the concept of an angle s size. Draw two angles: SK: Which angle is smaller? Which corner is sharper? The diagram on the left is larger, but its corner is sharper, and mathematicians say that this angle is smaller. The distance between the ends of the arms in both diagrams is the same, but this does not matter; angles are made of rays and these can be extended. What matters is the sharpness. The sharper the corner on the outside of the angle, the narrower the space between the angle s arms. Explain that the size of an angle is the amount of rotation between the angle s arms. The smallest angle is closed; both arms are together. Draw the following picture to illustrate what you mean by smaller and larger angles.

7 7 You can show how much an angle s arm rotates with a piece of chalk. Draw a line on the board then rest the chalk along the line s length. Fix the chalk to one of the line s endpoints and rotate the free end around the endpoint to any desired position. You might also illustrate what the size of an angle means by opening a book to different angles. Draw some angles and ask your students to order them from smallest to largest. Define acute and obtuse angles in relation to right angles. Obtuse angles are larger than a right angle; acute angles are smaller than a right angle. Extra practice: 1. Copy the shapes onto grid paper and mark any right, acute, and obtuse angles. Which shape has one internal right angle? What did you use to check? 2. Which figures at left have a) all acute angles? b) all obtuse angles? c) some acute and some obtuse angles? Introduce protractors. On the board, draw two angles that are close to each other say, 50 and 55 without writing the measurements and in a way that makes it impossible to compare the angles visually. SK: How can you tell which angle is larger? Invite volunteers to try different strategies they suggest (such as copying one of the angles onto tracing paper and comparing the tracing to the other angle, or creating a copy of the angle by folding paper). Lead students to the idea of using a measurement tool. Explain that to measure an angle, people use a protractor. protractor has 180 subdivisions around its curved edge. These subdivisions are called degrees ( ). Degrees are a unit of measurement, so just as we write cm or m when writing a measurement for length, it is important to write the symbol for angles.

8 8 Using protractors. Show your students how to use a protractor on the board or on the overhead projector. Identify the origin (the point at which all the degree lines meet) and the base line (the line that goes through the origin and is parallel to the straight edge). When using a protractor, students must place the vertex of the angle at the origin; position the base line along one of the arms of the angle. You could draw pictures (see below) to illustrate incorrect protractor use, or demonstrate it using an overhead projector and a transparent protractor. origin base line Introduce the degree measures for right angles (90 ), acute angles (less than 90 ), and obtuse angles (between 90 and 180 ). Point out that there are two scales on a protractor because the amount of rotation can be measured clockwise or counter-clockwise. Students should practise choosing the correct scale by deciding whether the angle is acute or obtuse, then saying whether the measurement should be more or less than 90. (You may want to do some examples as a class first.) Then have students practise measuring angles with protractors. Include some cases where the arms of the angles have to be extended first. Introduce angles in polygons, then have students measure the angles in several polygons. Students can draw polygons (with both obtuse and acute angles) and have partners measure the angles in the polygons. Drawing angles. Model drawing angles step by step (see p 104 in the Workbook or LM Measuring and Drawing ngles and Triangles, p 1). Emphasize the correct position of the protractor. Have students practise drawing angles. You could also use ctivity 2 for that purpose. Have students practise drawing lines that intersect at a given angle. nother way to practise drawing angles is to construct triangles with given angle measures. ctivities 1. Students can use geoboards and elastics to make right, acute, and obtuse angles. When students are comfortable doing that, they can create figures with different numbers of given angles. Examples: a) a triangle with 3 acute angles b) a quadrilateral with 0, 2, or 4 right angles c) a quadrilateral with 1 right angle d) a shape with 3 right angles e) a quadrilateral with 3 acute angles Sample NSWERS: d) e)

9 9 2. Students will need a die, a protractor, and a sheet of paper. Draw a starting line on the paper. Roll the die and draw an angle of the measure given by the die; use the starting line as your base line and draw the angle counter-clockwise. Label your angle with its degree measure. For each next roll, draw an angle in the counter-clockwise direction so that the base line of your angle is the arm drawn at the previous roll. The measure of the new angle is the sum of the result of the die and the measure of the angle in the previous roll. Stop when there is no room to draw an angle of the size given by the roll. For example, if the first three rolls are 4, 5, and 3, the picture will be as shown. Extensions 1. ngles on an analogue clock. What is the angle between the hands at 12:24? 13:36? 15:48? (Draw the hands first!) To guide students to the answers, draw an analogue clock that shows 3:00 on the board. sk your students what angle the hands create. What is the measure of that angle? If the time is 1:00, what is the measure of the angle between the hands? Do you need a protractor to tell? sk volunteers to write the angle measures for each hour from 1:00 to 6:00. Which number do they skip count by? n hour is 60 minutes and a whole circle is 360. What angle does the minute hand cover every minute? (6 ) How long does it take the hour hand to cover that many degrees? How do you know? (12 minutes, because the hour hand covers only one twelfth of the full circle in an hour, moving 12 times slower than the minute hand) If the time is 12:12, where do the hour hand and the minute hand point? What angle does each hand make with a vertical line? What is the angle between the hands? (NSWER: The hour hand points at one minute and the angle that it makes with the vertical line is 6. The minute hand points at 12 minutes and the angle that it makes with the vertical line is 12 6 = 72. The angle between the hands is 72 6 = 64.) 2. Some scientists think that moths travel at a 30 angle to the sun when they leave home at sunrise. Note that the sun is far away, so all the rays it sends to us seem parallel. a) What angle do the moths need to travel at to find their way back at sunset? Hint: Where is the sun in the evening?

10 10 b) moth sees the light from the candle flame and thinks it s the sun. The candle is very near to us, and the rays it sends to us go out in all directions. Where does the moth end up? Draw the moth s path.

11 11 G7 4: Perpendicular Lines Workbook pages: Goals: Students will identify and draw perpendicular lines. Prior Knowledge Required: Knows what a right angle is Can name an angle and identify a named angle Vocabulary: angle arms right angle perpendicular slant line Curriculum Expectations: Ontario: 7m46 WNCP: 6SS1, 7SS3, [CN, V] Introduce perpendicular lines (lines that meet at 90 ) and show how to mark perpendicular lines with a square corner. Draw several pairs of intersecting lines on the board and have students identify the perpendicular lines. Include pairs of lines that are not horizontal and line segments that intersect in different places and at different angles (see examples in the margin). Invite volunteers to check whether the lines are perpendicular using a corner of a page, a protractor, and/or a set square. sk students where they see perpendicular lines or line segments also called perpendiculars in the environment (sides of windows and desks, intersections of streets, etc.). Extra practice: Draw square corners to show any lines that look like they are perpendicular. a) b) c) d) ONUS perpendicular through a point. Explain that sometimes we are interested in a line that is perpendicular to a given line, but we need an additional condition the perpendicular should pass

12 12 through a given point. In each diagram below, have students identify first the lines that are perpendicular to the segment, then the lines that pass through point P, and finally the one line that satisfies both conditions. P P Constructing a perpendicular through a point. Model using a set square and a protractor to construct a perpendicular through a point that is not on the line. Emphasize the correct position of the set square (one side coinciding with the given line, the other touching the given point) and the protractor (the given line should pass through the origin and through the 90 mark). Have students practise the construction. Circulate among the students to ensure that they are using the tools correctly. Then invite volunteers to model the construction of a perpendicular through a point that is on the line. (Emphasize the difference in the position of the set square: the square corner is now at the point, though one arm still coincides with the given line.) Then have students practise this construction as well. Extra practice: Draw a pair of perpendicular slant lines (i.e., lines that are neither vertical nor horizontal) and a point not on the lines. Draw perpendiculars to the slant lines through the point. What quadrilateral have you constructed? (rectangle) ONUS Draw a slant line and a point not on the line. Using a protractor and a ruler or set square, draw a square that has one side on the slant line and one of the vertices at the point you drew. (NSWER: Draw a perpendicular through the point to the given line. Measure the distance from the point to the line along the perpendicular. Then mark a point on the given line that is the same distance from the intersection as the given point. Now draw a perpendicular to the given line through this point as well. Finally, draw a perpendicular to the last line through the given point.) Why perpendiculars are important. (Connection Science) Discuss with students why perpendiculars are important and where are they used in real life. For example, you can explain that it is easy to determine a vertical line (using gravity just hang a stone on a rope and trace the rope), but you need a right angle to make sure that the floor of a room is horizontal. Extension Discuss with your students whether there can be more than one line perpendicular to a given line through a given point, and whether such a perpendicular always exists. You can use the diagrams below to help students visualize the answers. PE Visualization, communicating P P

13 13 G7 5: Perpendicular isectors Workbook pages: Goals: Students will identify and draw perpendicular bisectors. Prior Knowledge Required: Can identify and construct a perpendicular using a set square or a protractor Can identify and mark right angles Can name line segments and identify named line segments Can draw and measure with a ruler Vocabulary: line segment midpoint right angle perpendicular Materials: paper circles or LM Circles (p 10) Curriculum Expectations: Ontario: 7m48; 7m1, 7m3 WNCP: 6SS1, 7SS3, [C, R, V] Introduce the notation for equal line segments. Explain that when we want to show that line segments are equal, we add the same number of marks across each line segment. This is particularly useful for sketches, when you are not drawing everything exactly to scale. Introduce the midpoint. Model finding the midpoint of a segment using a ruler, and mark the halves of the line segment as equal, then have students practise this skill. Introduce bisectors. Explain that a bisector of a line segment is a line (or ray or line segment) that divides the line segment into two equal parts. There can be many bisectors of a line segment. sk students to draw a line segment with several bisectors. SK: Can a line have a bisector? What about a ray? (no, because lines and rays have no midpoints) Extra practice: Example: Draw a scalene triangle. Choose a side and draw a bisector to that side that passes through the vertex of the triangle that is opposite that side. Repeat with the other sides. What do you notice? (NSWER: ll three bisectors, called medians, pass through the same point.)

14 14 Introduce perpendicular bisectors. Of all the bisectors of a line segment only one is perpendicular to the line segment, and it is called the perpendicular bisector. The perpendicular bisector of a line segment divides the line segment into two equal parts ND intersects the line segment at right angles (90 ). Point out that there are two parts in the definition, and both must be true. SK: How can we draw a perpendicular bisector? How is that problem similar to constructing a bisector? Constructing a perpendicular? Lead students to the idea that they should first find the midpoint of the line segment, then construct a perpendicular through that point. Have students practise drawing perpendicular bisectors using set squares and protractors. PE Changing into a know problem Present several diagrams that combine intersecting line segments, perpendicular lines, and bisectors, such as the one at right, and have students identify equal segments, then find perpendicular lines and bisectors. (Example: Find a bisector of EG. Is it a perpendicular bisector?) sk other questions about the diagram, such as: J is the midpoint of what segment? (CG) Why not D? How do you know? (We do not have any information about the lengths of DJ and J. They look equal, but might have different lengths.) Name three line segments GJ is perpendicular to. Finally, have students identify all the perpendicular bisectors and the line segments they bisect. (NSWERS: D, J, and JD bisect line segment CG; CJ, JG, and CG bisect line segment FH) sk students to find examples of equal segments, midpoints, and perpendicular bisectors in the classroom or elsewhere, such as in letters of the alphabet, in pictures or photographs, and so on. E F G H D J C ctivities 1. Paper folding and line segments Draw a line segment dark enough that you can see through the paper. Fold the paper so that meets. What line has your crease made? (NSWER: a perpendicular bisector) Use a ruler and protractor to check your answer. 2. Paper folding and circles Give each student a circle (you can use LM Circles) and ask students to draw and label a triangle on their circle. a) Fold the circle in half so that meets. Look at the line that the crease in your fold makes. Is it a bisector of angle C? Is it a perpendicular bisector of line segment? C

15 15 b) Fold the circle in half again, this time making meet C. What two properties will the crease fold have? c) Repeat, making meet C. t what point in the circle will all three perpendicular bisectors meet? P Workbook Question 8f) [C,], 7m3 Extensions 1. Given a triangle C, how can you use perpendicular bisectors to help you draw the circle going through the points,, and C? Draw an acute scalene triangle. Draw the perpendicular bisectors by hand do not cut and fold the triangle. 2. Start with a paper circle. Choose a point C on the circle and draw a right angle so that its arms intersect the circle. Label the points where the arms intersect the circle and and draw the line segment. Repeat with several circles to produce different right triangles. (You can use LM Circles for this Extension.) Fold the circle in two across the side C so that falls on C (creating a perpendicular bisector of C). Mark the point where falls on the circle. Repeat with the side C, marking the point where falls on the circle. What do you notice? (The image of is the same as the image of.) What type of special quadrilateral have you created? (a rectangle) Repeat the exercise starting with an obtuse or an acute angle C. Do the images of and coincide? C

16 16 G7 6: Parallel Lines Workbook pages: Goals: Students will identify and draw parallel lines. Prior Knowledge Required: Can identify and construct a perpendicular using a set square or a protractor Can identify and mark right angles Can name a line segment and identify a named line segment Can draw and measure with a ruler Vocabulary: line segment parallel right angle perpendicular Materials: LM Distance etween Parallel Lines (p 6) Curriculum Expectations: Ontario: 7m46; 7m1, 7m2 WNCP: 5SS5, 7SS3, [C, R, PS] Introduce parallel lines straight lines that never intersect, no matter how D much they are extended. Show how to mark parallel lines with the same number of arrows. C Have students identify parallel lines in several diagrams. Then have students identify and mark parallel sides of polygons. Include polygons that have pairs of parallel sides that are neither horizontal nor vertical. Introduce the symbol for parallel lines, label the vertices of the polygons used above, and have students state which sides are parallel using the new notation (Example: CD). sk students to think about where they see parallel lines. Some examples of parallel lines in the real world are a double centerline on a highway and the edges of construction beams. Determining if two lines are parallel. Have students draw a triangle on grid paper. Then ask them to draw line segments that are parallel to the sides of the triangle. For each pair of parallel lines (say, m and n) ask students to draw a perpendicular (say, p) to one of the lines (m) so that it intersects both lines. sk students to predict what the angle between n and p is. sk students to explain their prediction. Then have them measure the angle between the lines. Was the prediction correct? Have students check the prediction using the other two pairs of parallel lines they drew. Explain that this property If one line in a pair of parallel lines interests a third line at a right angle, the other? m p n

17 17 parallel line also makes a right angle with the same line allows us to check whether two lines are parallel and to construct parallel lines. P Workbook Question 8 [R, C], 7m2 Drawing a line segment parallel to a given line segment. Have students problem-solve how to construct parallel lines using what they ve just learned about a perpendicular to parallel lines. s a prompt, you could use the rectangle and square problems from G7-4 (Extra practice and onus, p 12 students are required to draw a rectangle or a square using a point and a pair of parallel lines). SK: What do you know about the sides of a rectangle? How does constructing a rectangle mean that you constructed a pair of parallel lines? (P [PS], 7m1) Model drawing the line parallel to a given line through a point using a protractor (see p 111 in the Workbook or LM Drawing Parallel Lines, p 3), and then model doing the same thing using a set square. Have students practise drawing parallel lines using both tools. ctivities 1. Paper folding Draw a line dark enough so you can see it through the page. Fold the paper so that you can find a line perpendicular to that does not bisect. How can you use this crease to find a line parallel to? How can you use a ruler or any right angle to find a line parallel to? NSWER: Fold the paper so that the perpendicular to falls onto itself. The crease is perpendicular to the perpendicular, so it is parallel to. 2. Students can investigate distances between parallel lines with LM Distance etween Parallel Lines. Extensions 1. plane is a flat surface. It has length and width, but no thickness. It extends forever along its length and width. Parallel lines in a plane will never meet, no matter how far they are extended in either direction. Can you find a pair of lines not in a plane that never meet and do not intersect? 2. Draw: P 7m1, [PS] a) a hexagon with three parallel sides b) an octagon with four parallel sides c) a heptagon with three pairs of parallel sides d) a heptagon with two sets of three parallel sides Sample NSWERS: e) a polygon with three sets of four parallel sides f) a polygon with four sets of three parallel sides a) b) c) d) e) f)

18 18 3. Lines are parallel if they point in the same direction that s why we use arrows to show parallel lines! We can regard direction as an angle with a horizontal line. For example, if two lines are both vertical, they both make a right angle with a horizontal line, and they are parallel. The choice of a horizontal line as a benchmark is arbitrary it is just a convention; any line could be used for that purpose. Indeed, any two lines perpendicular to a third line are parallel. Students can complete the Investigation on LM Properties of Parallel Lines (p 4) to learn what happens when parallel lines meet a third line at different angles. 4. sk students if they can draw a parallelogram that's not a rectangle using only a ruler and a set square. SOLUTION:

19 19 G7 7: ngle Relationships Workbook pages: Goals: Students will find angles in triangles, discover the sum of the angles in a triangle, and use this sum to solve problems. Prior Knowledge Required: Can use a protractor to measure angles Vocabulary: straight angle acute, obtuse, right angle adjacent angles intersecting lines linear pair Materials: LM Sum of the ngles in a Triangle (p 6 7) Curriculum Expectations: Ontario: 7m43; 7m3, 7m7 WNCP: 6SS2, 7SS1, [C, R, V] Introduce straight angles, adjacent angles, and linear pairs. straight angle is formed when the arms of the angle point in exactly opposite directions and form a straight line through the vertex of the angle. djacent angles share an arm and a vertex. Two adjacent angles are a linear pair if, together, they form a straight angle. a b a b c a d b f Have students name all the pairs of adjacent angles in the picture at left. SK: Which angles make a linear pair? Which two combinations of angles make a straight angle? ngle measures in adjacent angles add to the measure of the large angle. Draw a line segment divided into two smaller segments. SK: What is the length of the whole line segment? How do you know? What do you do with the lengths of the smaller line segments to obtain the length of the whole line segment? 1.5 cm 2 cm Point out that the same procedure applies to capacities, volumes, and areas: + = 1 3 cup 1 2 cup 5 6 cup 4 cm cm 3 = 6 cm 3

20 20 What happens with angles? Their measures are also added: the measure of the large angle at right is = 75. Sum of the angles around a point. sk students to draw a pair of intersecting lines, measure the angles, and write the measures on the picture. Then ask them to add up the measures. What is the sum of the angles around the point? Did they all get the same answer? Show the picture at right (mention that these angles were made by two intersecting lines) and ask students how they can tell that whoever measured the angles made a mistake. (ny two adjacent angles in this picture make a linear pair, so their sums should all be 180 ). Sum of the angles in a triangle. Have students complete the Investigation on LM Sum of the ngles in a Triangle. What is the sum of the angles in a triangle? Then have students draw several triangles, measure their angles, and add the measures. Did they get the same sum every time? Explain that the sums might not add to 180 due to mistakes in measurement. The protractor, though a convenient tool, is imprecise. Finding the measure of the angles using the sum of the angles in a triangle. Draw a triangle on the board and write the measure of two of the angles in the triangle. SK: How can I find the measure of the third angle? (180 minus the sum of the other two angles) Have students find the measures of the angles in several problems of this sort, then proceed to more complicated questions, such as the following: ll the angles in a triangle are equal. What is the size of each angle? One angle of a triangle is 30. The other two angles are equal. What is the size of these angles? triangle has two equal angles. One of the angles in this triangle is 90. What are the sizes of the other two angles? With the last question, ask whether the equal angles can be 90 each. Have students explain why this is not possible. (The two angles would already add to 180, leaving no room for the third angle.) PE Reflecting on the reasonableness of the answer Invite a volunteer to draw the correct triangle on the board and mark the measures of the angles. Then present a similar problem: triangle has two equal angles. One of the angles in this triangle is 50. What are the sizes of the other two angles? SK: How is this problem different from the previous problem? (The given angle is an acute angle, not a right angle.) Can a triangle have two angles of 50? What is the third angle then? (80 ) Draw an acute isosceles triangle on the board and ask volunteers to mark the angles on the picture. Then draw another acute isosceles triangle, mark the base angles as equal, and mark the unequal angle as 50. Can this situation happen? What are the measures of the other two angles? PE Reflecting on other ways to solve the problem P Workbook, Question 11 [C, V], 7m7 Extra practice: a) If half an angle is 20, the whole angle is. b) If one-third of an angle is 30, the whole angle is. c) If one-quarter of an angle is 25, the whole angle is. d) What is half of 90? ONUS What is one-third of 120?

21 ? ctivities 1. Create right angles by paper folding Give your students a piece of paper that has no corners (e.g., an oval or a cloud shape). sk students to create a right angle on the paper by folding. (POSSILE NSWER: Fold once, unfold to see your straight-line crease, then fold again across the crease so that the sides of the crease coincide.) SK: When you folded the paper the first time, what is the size of the angle you created? (180, a straight angle) What fraction of the straight angle is the right angle? (one half) Use your answers to these two questions to explain why your angle is a right angle. (NSWER: right angle is 90, so two right angles are 180, or a straight angle. right angle is exactly half of a straight angle. In other words, when you fold the paper the second time and the halves of the first crease coincide, you know that the two angles you have created are equal. Together, the angles make a straight angle, so they have to be right angles.) P 7m7, [C] 2. Start with a cloud- or oval-shaped piece of paper. How can you fold the paper to create the following? a) a 45 angle b) a square c) a rectangle Students can work independently or in pairs to try and create the angle and shapes listed. Invite students to share strategies and solutions with the class, then, if necessary, give students the correct sequence of steps to follow. Do not unfold the paper unless it is mentioned as part of instructions. The instructions might be easier to understand if you actually perform the folding as you go along. NSWERS: a) Make a 45 angle: Create a right angle by the method of ctivity 1. Fold the right angle through the vertex so that its arms fall onto each other. This divides the angle into two equal parts, so the angle is a 45 angle. b) Make a square: i) Fold the paper to make a line segment. Fold the line segment onto itself to create a right angle; the vertex of the right angle will be one of the vertices of the square. Sides of square ii) Fold the right angle in two to make a 45 angle.

22 22 iii) iv) Fold the corner of the 45 angle onto one of the arms, so that parts of that arm meet each other. Trace the fold you created with a pencil. Unfold once and trace the crease that is revealed with a pencil. v) Unfold once again and flip the paper over. The lines you drew form two more sides of the square. c) Make a rectangle: i) Repeat step i) from part b). Sides of rectangle ii) Choose a point on one of the arms of your right angle to be the second vertex of the rectangle. Fold the paper through the chosen point so that the arm of the right angle you created folds onto itself. This will be the third side of the rectangle. iii) Flip the paper over. Fold the bottom part of the strip backwards so that both creases forming the sides of the strip fold onto themselves. Second corner of the rectangle. Third side of rectangle

23 23 G7 8: Triangle Properties Workbook pages: Goals: Students will classify triangles by side and angle measures. Prior Knowledge Required: Knows the sum of the angles in a triangle Can draw triangle using given sides and angles Vocabulary: straight angle acute, obtuse, right angle/triangle scalene, isosceles, equilateral triangle Materials: dice, spinners, rulers, and protractors (see ctivity 1) rulers, scissors, straws (see ctivity 3) Curriculum Expectations: Ontario: 7m47, 7m48; 7m3, 7m7 WNCP: 6SS4, [C, CN, R, V] Do Investigations 1 and 2 as a class. fter the second investigation, have students work in groups of three. Each student has to draw three triangles using a protractor: Student 1 Student 2 Student 3 Triangle 1 two angles of 60 two angles of 60 two angles of 60 Triangle 2 two angles of 30 two angles of 40 two angles of 25 Triangle 3 two angles of 55 two angles of 75 two angles of 70 sk students to find the measure of the third angle in each triangle and to classify the triangles they created by their angle measures. Then ask students to cut the triangles out and fold them, so as to check whether they have any equal sides. Compare the findings in each group. Students should understand that when two angles are equal, there will be two equal sides, regardless of the measure of the third angle. If all three angles are equal, there will be three equal sides. ONUS What should the measure of the two equal angles should be in order for the triangle to be an isosceles right triangle? P Worksheet, Question 7 [C, R], 7m3, 7m7 ctivities

24 24 1. Each student will need a die, a ruler, a protractor, and a spinner divided into three parts and labelled with the types of triangles they have been studying (acute triangle, obtuse triangle, right triangle). The object of the game is to build a set of triangles around any vertex that fills all 360. Students start with a horizontal line of 5 cm. Students roll the die and spin the spinner each turn. The spinner gives the type of triangle to be constructed. The result of the die multiplied by 10 gives the size of one angle of the triangle. (The other angles should also be multiples of 10.) Students have to draw the first triangle using the 5 cm line as the base. Each new triangle should use one of the sides of an existing triangle as a side. The position of the angles is up to the student. The triangles cannot overlap. Students may find it convenient to write the sizes of the angles on the triangles they draw. Let students know that there is one combination of the results of the spinner and the die that makes it impossible to draw a triangle with angles that are all multiples of 10. sk students to figure out what combination that is. (1 and acute) When this combination is rolled, students have to roll again. SMPLE GME: This game started with 5 and acute triangle, which gave Triangle 1. The next roll was 2 and the next spin was acute triangle, so Triangle 2 had to be isosceles with angles of 80 at the base. The student decided to put the 80 angle next to the 50 angle. The next two turns were 6, right triangle (Triangle 3) and 1, right triangle (Triangle 4), and the game ended with 6, acute triangle (Triangle 5). In the last turn, the 60 angle did not fit in the remaining 50 gap, but a triangle with 50 and 60 angles is still acute, so the student used the 50 angle to fill the gap and end the game. If the final spin had given obtuse triangle instead of acute triangle, the student would have a choice of drawing the next triangle either around a different vertex or place the smallest angle of the triangle (which could be only 10 or 20 ) in the remaining gap. In any case the game would have continued Paper folding First, create a square from a rectangular sheet of paper: a) Fold the short side of the paper onto the long side (to create a right trapezoid with a 45 angle and a triangle with a 45 angle). b) Fold the extra part of the page a rectangle over the triangle. c) Unfold the paper and cut off the rectangle. unfold Now, create a triangle: a) Fold the square in half (vertically, not diagonally) so that a crease divides the square into two rectangles. b) Fold the top right corner of the square down so that the top right vertex meets the crease. The vertex should be slightly above the bottom edge.

25 25 c) Mark the point where the corner meets the crease and trace a line along the folded side of the square with a pencil. d) Unfold. e) Repeat steps b) d) with the top left corner of the square. The corner will meet the crease at the same point, which is the vertex of your triangle. f) Cut the triangle out along the traced lines. Which triangle have your created? Explain. NSWER: You started with a square and ensured in b) that the sides of the triangle are equal. It is an equilateral triangle. 3. The triangle inequality. Give students a ruler, scissors, and some straws. Have students measure and cut a set of 10 straws: one straw of each of the following lengths: 10 cm, 9 cm, 8 cm, 6 cm, 3 cm two straws of 4 cm three straws of 5 cm Questions a) How many (distinct) right triangles can you make using the straws? NSWER: 2 distinct triangles with sides of length 3, 4, 5 and 6, 8, 10. b) How many isosceles triangles can you make? NSWER: 9 triangles with the following side lengths: 3, 4, 4 3, 5, 5 4, 5, 5 5, 5, 5 5, 4, 4 6, 4, 4 6, 5, 5 8, 5, 5 9, 5, 5 Note that equilateral triangles are also isosceles this is a special case. c) Why can you not make triangles with side lengths 8, 4, 4; 9, 4, 4; 10, 4, 4; and 10, 5, 5? If you had two straws of length 3 cm, could you make a triangle with sides 3 cm, 3 cm, and 6 cm? If you had two straws of length 6 cm, could you make a triangle with sides 3 cm, 6 cm, and 6 cm? (This question provides a hint for Question 7i) on the worksheet. If students have trouble producing the triangle in 7i), they should create the triangle with straws and trace it.) Finish the activity by discussing this question: What could be a rule for determining the sets of straws that will make a triangle and those that won t? The rule is known as the triangle inequality. It says that for three lengths to make a triangle, the sum of the lengths of any two sides must be greater than the length of the third side. a c b a + b > c

26 26 G7 9: ngle isectors Workbook page: 117 Goals: Students will identify and draw angle bisectors. Prior Knowledge Required: Can divide 2-digit and 3-digit numbers by 2 Can double 2-digit numbers Can use a protractor for measurement and construction Vocabulary: straight angle acute, obtuse, right angle/triangle scalene, isosceles, equilateral triangle Materials: dice Curriculum Expectations: Ontario: 7m47, 7m48; 7m2, 7m5 WNCP: 6SS4, 7SS3 [C, CN, PS, R, V] Introduce angle bisectors. n angle bisector is a ray that cuts an angle exactly in half, making two equal angles. Point out that an angle only has one bisector. sk students to think about where they see angle bisectors in the real world. For example, angle bisectors are often seen in corners on furniture and picture frames. Explain that we can usually use an equal number of small arcs or lines to show that the angles in a diagram are equal. Show the diagrams below and SK: Which diagrams look like they show an angle bisector? Invite volunteers to mark the angles that look like equal angles. Draw several bisected angles on the board and give the measure of one angle (as in Question 2 on the worksheets). Have students determine the measure of the second angle and of the whole (unbisected) angle. Then do the reverse provide the measure for the whole angle and have students determine the measures of the parts. ONUS Find the size of each of the equal angles. a) b) 80 60

27 27 Constructing angle bisectors. Draw an angle on the board, then have a volunteer measure it and write the measurement. SK: If you were to draw an angle bisector, what would be the degree measure of each half? Model drawing the bisector. Emphasize that the bisector lies between the arms of the angle. Have students practise bisecting different angles, including obtuse, right, and acute angles. (You might use ctivity 1 at this point.) sk students to identify both the angles they drew and the halves as acute or obtuse angles. Can they get two obtuse angles after bisecting an angle? (no) Why not? (ecause double the obtuse angle is more than 180. You might point out that such angles are called reflexive angles.) If students need a prompt, have them bisect a straight angle or draw two equal obtuse angles with a common arm. P Worksheet, Question 6 7m2, [R,C] P Worksheet, Question 7 [V] Extra practice: Draw a scalene triangle and bisect each of the angles. If the bisection is performed correctly, the bisectors should meet at the same point. ctivities 1. Students will each need two dice of different colours. If two dice are unavailable, students can roll one die twice, taking the first roll as the result on the red die (r) and the second roll as the result on the blue die (b). Students roll the dice and draw an angle with the degree measure equal to 25r + b. (Example: If you roll 2 on the red die and 6 on the blue die, you draw an angle of 56.) Then they bisect the angle. PE Connecting (algebra) ONUS What is the largest angle you can create with this rule? (156 ) What is the smallest angle? (26 ) How many ways can you create an angle that is a multiple of 5? (6 ways: 25r is a multiple of 5 for any r, so to get the whole sum to divide by 5 you need b to be 5.) 2. Paper folding Make an angle using folds. To bisect your angle, fold the paper through the point of intersection so that the creases that form the two arms of the angle fall one on top of the other. Extensions 1. You are given an angle and a transparent mirror (a Mira). How can you find the angle bisector? P [CN, PS], 7m5 Hint: n angle bisector is a line of symmetry. 2. The incentre of a triangle a) Draw a triangle. b) Draw the angle bisectors for all the angles of the triangle. What do you notice? (The bisectors all pass through the same point, called the incentre of the triangle.) c) Label the point where the bisectors intersect O. Draw perpendiculars through O to each of the sides of the triangle. Measure the distances along the perpendicular lines from O to the sides of the triangle. What do you notice? (the distances are all the same)

28 28 G7 10 Quadrilateral Properties Workbook pages: Goals: Students will investigate properties of quadrilaterals related to angles and sides and sort quadrilaterals according to these properties. Prior Knowledge Required: Is familiar with special quadrilaterals Can identify equal sides and angles Can identify and mark parallel lines Can identify and mark right angles Vocabulary: acute, obtuse, right angle equilateral special quadrilaterals: trapezoid, parallelogram, rectangle, rhombus, square, kite, right trapezoid, isosceles trapezoid opposite sides Materials: LM Quadrilaterals (pp 11-12) LM Straw Quadrilaterals (p 8) straws Curriculum Expectations: Ontario: 7m47, 7m48; 7m2 WNCP: 5SS6, 6SS5, [R, V] Introduce quadrilaterals (polygons with four sides). Explain that some quadrilaterals have special properties. This means that they have certain attributes that apply to all of the quadrilaterals of that type. sk students which special quadrilaterals they know and what special properties these quadrilaterals have. Introduce any special quadrilaterals that are not mentioned (see Vocabulary). Make sure students understand the meaning of opposite sides and adjacent sides. Sort quadrilaterals by properties of sides and angles. Give your students some paper parallelograms, rhombuses, rectangles, squares, and trapezoids (see LM Quadrilaterals for a sample of shapes). Discuss with students how they can check for equal sides or angles using folding, and check whether angles add to 180 using rolling or folding (the example at right shows how to check that for a parallelogram). Review with students how they can check whether sides are parallel by drawing (or folding) a perpendicular to one of the sides. If the perpendicular makes a right angle with the second side as well, the sides are parallel. sk students to sort the shapes into this table:

29 29 Properties of sides No equal sides One pair of equal sides Two pairs of equal sides Equal sides are adjacent Equal sides are opposite Four equal sides Shapes with the property Properties of angles No equal angles One pair of equal angles Two pairs of equal angles Equal angles are adjacent Equal angles are opposite Four equal angles One pair of parallel sides No pairs of angles add to 180 Two pairs of parallel sides One pair of angles add to 180 Two pairs of angles add to 180 Four pairs of angles add to 180 Shapes with the property sk students to look at the table closely. re there any properties that go together? For example, all quadrilaterals that have two pairs of parallel sides also have four pairs of angles that add to 180. sk students to pick at least one such pair of properties and to try to draw (using protractors and rulers) a quadrilateral that shares one of these properties but not the other. For some properties, such as the pair in the example above, this is impossible, but in some cases students might be able to come up with a shape. PS Making and investigating conjectures Debrief as a class. Example: One pair of parallel sides might seem to go with two pairs of angles add to 180 ; however, a kite with two right angles as shown has two pairs of angles adding to 180 and no parallel sides. ctivity Students can create quadrilaterals from straws and investigate them using LM Straw Quadrilaterals. Extensions 1. You can build a diagram that shows relationships between different special quadrilaterals. If a shape has all the properties of some other shape, it is drawn inside a more general shape. Start with a rectangle. ll squares are rectangles, so we can draw a square as part of a rectangle: ll squares are also rhombuses. Let s draw a square as part of a rhombus:

30 30 Can you find a shape that is a rectangle and a rhombus at the same time but is NOT a square? No. Let s show that a square is the intersection of the rhombus and the rectangle: ll rectangles and rhombuses are also parallelograms, so we can draw the shape from the last step inside a parallelogram: rhombus is a parallelogram that is also a kite. Can you find a parallelogram that is also a kite but NOT a rhombus? No, so the rhombus is the intersection of a kite and a parallelogram. 2. For 3 lengths to make a triangle, the sum of any 2 lengths must be longer than the third length.. Predict the rule that describes when 4 lengths will make a quadrilateral and when they will not.. Test your prediction. Try to make a quadrilateral using 3 small paperclips and a long pencil as the sides. Did it work? Why or why not? C. Cut various lengths from straws and do some more tests. Sketch the results of your tests. D. Check off the correct ending for the statement. For 4 lengths to make a quadrilateral, 2 sides must be equal in length. any 3 sides must together be longer than the fourth side. each of 2 sides must be longer than the other 2 sides. the sum of 2 sides must be longer than the sum of the other 2 sides. PE Making and investigating conjectures 3. Draw a parallelogram. isect each angle of the parallelogram and extend each bisector so that it intersects two other bisectors. What geometric shape can you see in the middle of the parallelogram? (a rectangle) Use the sum of the angles in the shaded triangle in the picture to explain why this is so. (NSWER: The acute angles of the triangle are both half of the angles of a parallelogram. The adjacent angles of a parallelogram add to 180, so their halves add to 90. y the sum of the angles in a triangle, the third angle is a right angle. Since there are three other triangles like the shaded triangle in the parallelogram, the shape in the middle has four right angles and must be a rectangle.) PE Connecting