Survey
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
Polygons Polygons or Not Polygons Come to a consensus on the definition of a polygon (include all necessary properties): Sketch a polygon that is different from the ones you were given. Then sketch a shape that is not a polygon. 1 Polygons Note: You can use your Geometry to Go book to look up definitions and concepts, too. Polygons can be classified according to the number of sides they have. Polygons with more than 10 sides are not usually given special names. A polygon with 11 sides is described as an 11-gon, a polygon with 12 sides as a 12-gon (officially dodecagon), and so on. When people talk about a general polygon—one where you don’t know the exact number of sides, they refer to it as an n-gon. 2 Polygons Classifying Triangles The following shows how triangles can be classifies according to some of their features: Angles: Right Triangles One right angle (90º) Acute Triangles All angles less than 90º Obtuse Triangles One angle more than 90º Sides: Equilateral Triangles All three sides the same length. Isosceles Triangles At least two sides the same length. Size: Similar Triangles Same shape, possibly different size. Congruent Triangles Same size and shape. 3 Scalene Triangles All three sides different lengths. Polygons Classifying Triangles Use this recording sheet to combine two features (angle and side length) into one triangle if possible. If the combination is possible, draw a sketch in the box. If not, explain why not. You may want to use spaghetti to make a model. Acute Right Obtuse Equilateral Isosceles Scalene 4 Polygons Building Triangles Materials: Spaghetti Question: Do you think you can make a triangle using any three lengths for the sides? With three side lengths, can you make more than one triangle? (Short 2-3 minute discussion with your group.) Do as a class: Using spaghetti, build a 6-8-12 cm triangle. Use the following chart to record this answer in the row marked Triangle #1. Suggestion: Measure and mark the length with a pen or pencil, use a scissors to cut the spaghetti, and label the length. Triangle Length of Side 1 Length of Side 2 Length of Side 3 #1 6 8 12 #5 4 5 10 #6 7 7 14 Makes a Triangle? (Yes or No) If yes, is the triangle acute, right, or obtuse? If yes, can you make a different Triangle? #2 #3 #4 Choose a set of three numbers—each less than 25 cm—to be the lengths of the sides of a triangle. Use straws or spaghetti to test the three lengths to see whether or not they will make a triangle. If so, will they make a different triangle? Record your results in the row marked Triangle #2. Chose sets of three numbers and repeat your experiment for two or three more triangles. Complete rows 5 and 6. Based on the information in column 5 “Makes a Triangle?” write a summary of your conclusions (a rule about the lengths of the sides of triangles): Whole group summary. 5 Polygons Building Quadrilaterals Question: Do you think you can make a quadrilateral using any four lengths for the sides? Again record your answers in a chart. Repeat your experiment for several more quadrilaterals. Materials: Use poly-strips for this activity. Attach the strips using the end holes, and measure the length from the center of one end hole to the center of the other. Length Length Length Length Makes a If yes, can you make Quadrilateral of of of of quadrilateral? a different Side 1 Side 2 Side 3 Side 4 (Yes or No) quadrilateral? How many? #1 #2 #3 What polystrips could you choose to make a parallelogram? A rectangle? A square? A rhombus? A trapezoid? Test your conjectures and sketch your results on dot paper. Definition: A conjecture is an educated guess about what you think is true based on observations. What are the attributes of a parallelogram? A rectangle? A square? A rhombus? A trapezoid? A kite? Is a rectangle a parallelogram? Explain. Is a square a rectangle? Explain. Is a rectangle a square? Explain. 6 Polygons Write definitions for the quadrilaterals listed below. Make your definitions as concise as possible. For example, if you have already defined parallelogram, then you can use the word parallelogram when defining other quadrilaterals, if appropriate. You may check in Geometry to Go after you’ve written your ideas. 1. Parallelogram 1. Rectangle 1. Square 1. Rhombus 1. Trapezoid 1. Kite Use the graphic organizer below to show how the above quadrilaterals are related to each other. Write the correct term on each line and draw one example in each section. Quadrilaterals Note: Some books define trapezoid as a quadrilateral with at least one pair of opposite sides parallel, which changes everything! We won’t use that definition. (See blue highlighted paragraph in section 165 of “Geometry to Go”.) 7 Polygons ALWAYS TRUE, SOMETIMES TRUE, OR NEVER TRUE? Suggestion: Use poly-strips for this activity. • If the statement is true, give a convincing argument to show your reasoning. • If the statement is never true, provide a counterexample and explain your reasoning. • If the statement is sometimes true, explain that also. 1. If a quadrilateral is a square, then it is a rhombus. 1. If a polygon is a right triangle, then the other two angles are acute. 1. Parallelograms have congruent diagonals. 1. Quadrilaterals with congruent diagonals are parallelograms. 1. If the opposite angles of a quadrilateral are congruent, then it is a parallelogram. 1. The diagonals of a parallelogram bisect each other (bisect means to cut into two equal parts). 1. The diagonals of a parallelogram are perpendicular to each other. 1. If a quadrilateral has at least one right angle, then it’s a rectangle. 1. An equiangular quadrilateral is a square. 1. The diagonals of a trapezoid are congruent. 1. A right triangle can have three congruent sides. 8 Angles What is an Angle? Small group discussion: Make a list of things you know about angles. Include such ideas as the definition of an angle, types of angles, angle notation, measuring angles, benchmark angles, and so on. Note: See pages 62 and 63 in Geometry to Go for correct notation when naming and measuring angles. Pattern Block Angles Angles as Wedges Materials: Pattern Blocks. Hinged mirrors. Directions: Find the measure of each angle in your set of pattern blocks. Note: Do not use a protractor or angle ruler to do this activity. Record your results below and be ready to explain how you determined your answers. Angles Sum of all the angles The size of a right angle is the only prior knowledge you Orange Square may use. Green Triangle Blue Rhombus Tan Rhombus Red Trapezoid Yellow Hexagon 9 Angles Making a Protractor Angles as rotations Materials: Compass, large index card, ruler, pattern blocks, patty paper. Make a “protractor” using angles found in the pattern blocks and/or with other methods. • Mark a point somewhere near the center of your index card. • Using that point as the center, use a compass to draw a circle 4 inches in diameter and cut it out. • Draw in a radius. • Mark benchmark angles and any other angles that can be found using the pattern blocks or other methods. Which pattern blocks can be used to make a full 360º rotation? Mark angles until the circle is completed. Possible strategies: o Combine two angles to get a third o Draw an angle on patty paper and fold it in half. • What does it mean to say that there are 360º in a circle? Where are these degrees? • Use an angle ruler or protractor to check. How accurate were you? Which angle is bigger? Angles as two rays meeting at a common vertex Discuss the question, “Which angle is bigger?” with your group. A B Use your angle ruler to check. How might this result contribute to your understanding of the meaning of "angle?" (Why do many students say that ∠B is bigger?) 10 Angles Degrees in a Triangle Activity • Using a straightedge, draw a triangle on an index card and cut it out. One person in your group may draw a right triangle, but everyone else should draw a different type of triangle. • Label the three angles 1, 2, and 3. • Tear off the three angles (don’t cut them off—you want a jagged edge to avoid confusion about which angles are the actual angles of the triangle). • Move the three angles together so that their vertices and sides are touching. Compare your results with others in your group. • Make a conjecture about the number of degrees in any triangle. (You are using “inductive reasoning”—using lots of examples as an informal justification. Although it is not a proof, use your conjecture in the next activity as if it were always true. We will prove your conjecture later.) Interior Angles of Polygons You made a conjecture about the sum of the interior angles in any triangle. In this activity, you will make conjectures about the sum of the interior angles of other polygons. • Use a straightedge to draw various polygons: one with 4 sides (quadrilateral), one with 5 sides (pentagon), and one with 6 sides (hexagon). They do not need to be regular polygons (congruent sides and angles). • Determine the number of degrees in each interior angle. (Use your angle ruler to measure.) Add your results to find the sum of the degrees for each polygon. Record your results in a table. Leave the last column blank for now. Experimental results Type of polygon Number of Sides & Angles Sum of Interior Angles • Triangle Quadrilateral Pentagon 3 Hexagon n-gon Any number 180º Compare your results with other in your group. If your angle sums are slightly different, to what can you attribute this difference? Come to a consensus about which sum you will use. 11 Angles Interior Angle Sums A. Amy divides polygons into triangles by drawing all the diagonals of the polygon from one vertex. Pick any vertex on each polygon below and draw in all the diagonals from just that one vertex. How can you use Amy’s method to find the sum of the interior angles of each polygon? You may want to make a chart relating the # of sides, the # of triangles, and the sum of the interior angles. • Does Amy’s method work for any polygon? Explain. • Using Amy’s method, write a rule, based on the number of sides, n, for finding the sum of the interior angles of any size polygon (n-gon). • How could you use her rule to find each angle any regular polygon? B. Cole also discovered a method for finding the angle sum of any polygon. He marks an arbitrary point in the interior of the polygon, and draws line segments from that point to each vertex. Follow Cole’s method to mark the polygons below. How can you use Cole’s method to find the sum of the interior angles of each polygon? • Does Cole’s method work for any polygon? Explain. • Using Cole’s method, write a rule, based on the number of sides, n, for finding the sum of the interior angles of any size polygon (n-gon). 12 Angles C. Keisha says it doesn’t matter how you draw in the diagonals as long as they don’t intersect. For example, she says you can divide up the polygon below as shown and Amy’s method will still work. Do you agree? Explain. Extension A: Would the rules that you wrote in Parts A and B (second bullet) work for concave polygons? Illustrate why or why not with examples. Extension B: Find the sum of the exterior angles of any convex polygon. Exterior angles are angles outside a polygon that lie between one side and the extension of its adjacent side. Mark three more exterior angles on the pentagon below. 13 Angles Vertical Angles The pairs of opposite angles formed by two intersecting lines are called vertical angles. Which angles in the diagram below are vertical angles? By looking at the diagram, make a conjecture about the measures of vertical angles. You may want to copy the diagram onto patty paper. The pairs of adjacent angles formed by two intersecting lines, such as angles 1 and 2 above, form a straight angle (a straight line) whose measure is 180º. Name the pairs in the diagram above that form straight angles? Write an explanation describing how you know your conjecture about vertical angles is always true (prove your conjecture). Other Interesting Angles When two lines (m and n) are cut by a transversal (t), eight angles are formed. Name the pairs of vertical angles in the following diagram. Angles can be paired in different ways. Other angle pairs are called: • alternate interior angles • alternate exterior angles • corresponding angles Just by thinking about the names, try to determine which pairs of angles go with each name. Hint: Angles 2 and 6 are one pair of corresponding angles. You can check yourself on page 74 of Geometry to Go. 14 Angles Parallel Lines Cut by a Transversal Materials: Patty paper. A. Draw two parallel lines on one piece of patty paper. Make sure they are really parallel! (You may discuss with your group possible ways to ensure that you end up with parallel lines.) Draw a transversal (a third line that intersects both parallel lines). Do not make the transversal perpendicular to the parallel lines. After you complete the activity, think about why you were asked not to draw a perpendicular transversal. A. Number your angles and list the pairs of: • vertical angles • alternate interior angles • alternate exterior angles • corresponding angles A. Make an exact copy of your diagram, including the numbers, on a second piece of patty pap r. A. Use your patty paper diagrams to make some conjectures about the measures of the alternate interior angles, the alternate exterior angles, and the corresponding angles. 15 Angles E. A postulate is a statement that is assumed to be true, but is never proved. They are accepted without proof. One such postulate is the Corresponding Angles Postulate, which states: If two parallel lines are cut by a transversal, then the pairs of corresponding angles are congruent. All your other conjectures can be proved using what you already proved about vertical angles being congruent and this postulate. Prove your conjectures about alternate interior and alternate exterior angles. It is perfectly acceptable to write your “proof” as an explanation. If you’d prefer to use a 2-column proof that you learned about in high school, that’s O.K., too! Extension (If your group finishes before the other groups are done): Make some conjectures about same-side interior angles and same-side exterior angles. Prove your conjectures. 16 Angles A Proof that there are 180º in any Triangle • Draw a triangle in the space below. • Label the angles with the numbers 1, 2, and 3. • Draw a line parallel to the base of your triangle through the vertex above the base. • We know the “Corresponding Angles Postulate” and we already proved our conjectures about vertical angles, alternate interior angles, and alternate exterior angles. Use any of the information that you already know to prove that every triangle has 180º. 17 Congruence, Symmetry and Transformations Reflection/Line Symmetry with the Alphabet If you can reflect (or flip) a figure over a line and the figure appears unchanged, then the figure has reflection symmetry or line symmetry. The line that you reflect over is called the line of symmetry. A line of symmetry divides a figure into two mirror-image halves. The dashed lines below are lines of symmetry: Draw ALL the lines of symmetry on each of the letters below. Circle the ones with no lines of symmetry. 18 Congruence, Symmetry and Transformations Transformations Translations (slides), Reflections (flips), Rotations (turns) Read pages 273-4 in “Geometry to Go”. Pretend that the two-dimensional shapes below are on transparencies for an overhead projector (or use patty paper if needed to check). Which shapes are the same (congruent)? What transformation(s) were used to get from a shape to its congruent “mates”? Draw the reflection of the figure on the other side of the vertical line. Use a mirror or patty paper to check. 19 Congruence, Symmetry and Transformations Draw the reflection of the figure on the other side of the diagonal line (the diagonal line is the line of reflection). Label the triangle ABC. Translate (slide) the figure to a different place on the coordinate plane in the direction indicated by the arrows (vector). Label the corresponding points on the translated image A'B'C' (read as A prime, B prime, C prime). Check yourself by copying the figure and the line on a piece of patty paper, placing it over the original, and sliding the copy along the line. Do your images match up? 20 Congruence, Symmetry and Transformations Label the figure ABCDE as described in the last problem. You have several tasks on the next image: 1) Reflect (flip) the original figure across the y-axis. 2) Reflect the original figure across the x-axis. 3) Translate the original figure in the direction indicated by the arrow. Label the reflected and translated images using “prime” notation as described above. 21 Congruence, Symmetry and Transformations Rotational Symmetry Rotational Symmetry: A figure has rotational symmetry if it can be rotated less than a full turn (360º) about a fixed interior point and the rotated image matches, or coincides with, the original figure. Center of Rotation: A fixed point about which a figure is rotated. 1. Copy each figure inside the circle onto a piece of patty paper. Lightly draw in all the lines of symmetry. 2. Make a mark at the top of each figure on your patty paper so you can remember the original orientation of the traced figure. • Place the patty paper figure over the original, hold it in place with the tip of a pen on the center dot, and rotate it until the rotated figure matches the original. • If the figure has rotational symmetry, write down all the angles of rotation. Extension (If your group finishes before the other groups are done): Find all the lines of symmetry and all the angles of rotational symmetry of a regular hexagon. You can use the yellow hexagon to trace a copy onto your paper. Try a rectangle? 22 Congruence, Symmetry and Transformations Complete the following steps for a clockwise rotation (turn) of the figure below: • Label the original figure. • Place a piece of patty paper on top of the original. Trace the figure, the vertex (point A) of the angle, and the top ray of the angle. • Place your pencil tip on the vertex of the angle (the center of rotation). Turn the patty paper sheet until the ray on your patty paper is aligned with the other ray of the original angle. • Mark the points of the rotated image on the coordinate grid and connect those points. Label the rotated image. Where would you place a mirror to justify that the diagonals of a square have the same length? Extension (next page) 23 Congruence, Symmetry and Transformations Extension (If your group finishes before the other groups are done): Draw your own figure in the space below or on another piece of paper and translate it in a direction of your choice, reflect it across a line of your choice, and rotate it about a point of your choice. 24 Perimeter and Area Launch questions: What characteristics must a “unit” have in order to measure area? What does it mean to find the area of something? Pattern Block Areas: Nonstandard Units Use the green triangle, the blue rhombus, the red trapezoid, and the yellow hexagon for this activity. 1. If the area of the green triangle is 1, what is the area of each of the other pieces? 2. If the area of the blue rhombus is 1, what is the area of each of the other pieces? 3. If the area of the red trapezoid is 1, what is the area of each of the other pieces? 4. If the area of the yellow hexagon is 1, what is the area of each of the other pieces? Use Geoboards as needed for any of the activities in this section. Grid Areas: Standard Units Be ready to explain your thinking. 25 Perimeter and Area Developing Formulas 1. Draw a rectangle on a piece of grid paper and find the area. If you counted the number of squares to find the area, can you find a quicker method? Write an algorithm (a description in words) for your method. Compare your ideas with others in your group. Will your algorithm always work for any rectangle? Formula (your algorithm in symbols): ___________________________________ 2. Draw several parallelograms on grid paper. Figure out a formula for finding the area of your parallelograms, using what you know about rectangles. Hint: you may want a scissors for this activity. Be ready to explain your thinking. Formula: ___________________________________ • Use your formula to find the area of these parallelograms: Will your formula always work? 3. Draw a right triangle on grid paper. Figure out a formula for the area of your triangle using what you know about rectangles. Will your formula always work for any right triangle? Formula: ___________________________________ 4. Draw several non-right triangles on grid paper. Make sure one side is horizontal or vertical on the paper. Develop a formula that will work for any triangle. How do you know it will always work? Be ready to explain your thinking. Formula: ___________________________________ • Use the formula to find the area of these triangles: 26 Perimeter and Area 5. Sometimes a shape is irregular and none of the formulas will work so a different strategy is needed. Find the area of the following figure using the following two methods: • Decomposing the figure’s interior: Cutting the figure into pieces for which you know the formulas, or for which you can count squares. Subdivide the figure into pieces, find the area of each piece, and combine them for the total area. • The rectangle surround method: Surround the figure with a rectangle and find its total area. Find the areas of the pieces inside the rectangle, but not inside the original figure, and subtract them from the area of the rectangle. 27 Perimeter and Area 6. Sometimes one or the other of the two methods used in the last activity is easier to use. Using both methods, try to find the areas of the triangles below. Which method seems easier to use and which one doesn’t? Why? • What is it about these two triangles that makes it difficult to use the formula (without measuring with a ruler) to find the area? 7. The triangles below have been constructed between two parallel lines. Using any method you want, find the area of each triangle. • If you didn’t use the formula to find the area, sketch in the height of each triangle and check yourself using the formula. • Make a conjecture about the areas of triangles with the same base drawn between two parallel lines, as shown. Explain why your conjecture makes sense. 8. Construct the following shapes on grid paper: • A triangle with an area of 6 square units. A different triangle with an area of 6 square units. • A triangle, a square, and a parallelogram with equal areas • An irregular figure with an area of 12 square units. 28 Perimeter and Area More Polygon Areas Find the area of each polygon. Use either Decomposing the Figure’s Interior or the Rectangle Surround Method. Show your work. 29 Perimeter and Area Extension1 (If your group finishes before the other groups are done): Go back and find the areas of the figures formed by the reflections on pages 19-20. Extension 2: The Land Problem: Two farmers own land that is divided by a fence (as shown below). They need to build a new fence and have decided to make it in one straight line across their property. Of course, neither one wants to lose any land, so where can they build the fence so they each still own exactly the same amount of land? (There are hints at the bottom of the page, but try to figure it out on your own first.) (Hint #1: Use what you learned about parallel lines cut by a transversal to help you figure it out.) (Hint #2: Use what you learned in Problem #7 of this set to help you figure it out.) 30 Perimeter and Area Discovering More Formulas • Choose one polygon from the diagrams below. (If you have time, do more than one.) • Invent a formula to find the area of the polygon you have chosen using only the dimensions labeled. Feel free to use patty paper, scissors, or any other tool that may help you. • Can you invent more than one method for the area of the polygon you have chosen? • Will your formula work for all sizes and shapes of the polygon you have chosen? Trapezoid Square Kite 31 Perimeter and Area Measurement Concepts with Tangrams: More Nonstandard Units 1. Arrange the 7 tangram pieces in order by their perimeters. 2. There are three sizes of triangles in your Tangram set. Use the two small triangles and one medium triangle for this activity. Using all three triangles (you must use all three each time), build the polygons listed below. Sketch your solution to each one. • Square • Rectangle that is not a square • Parallelogram that is not a rectangle • Triangle • Trapezoid 3. Without using measuring tools, can you determine which of the polygons in #2 has the greatest area? Why do you think you were asked this question? 4. Assemble all seven Tangram pieces into a square. (Don’t spend too much time doing this on your own. The solution is on the bottom of the page.) If the area of this large square is 1 square unit, what is the area of each of the other pieces? How do you know you are right? Extension 1: (Extension of #4). Assign a different piece to have an area of 1 and determine the area of the other pieces based on your new “unit.” Extension 2: Predict: True or False? Since the polygons from #2 have the same area, they will also have the same perimeter. Without using a ruler to measure, place the polygons from #2 in order, from least to greatest, by the length of their perimeters. How did you determine which shape has the smallest perimeter? (See hint below if needed.) Hint for Extension 2: Use a comparing method by assigning “short,” “medium,” and “long” to the three different lengths. 32 Perimeter and Area Scale Factors and Area Use your Pattern Blocks for the following activities. Sketch each solution and record your results in the table. 1. Directions for the orange square: • We will define the area of the orange square as having as area of 1 square unit, as recorded in the table. • Build another square in which each side of the square is twice as long as the original square (a scale factor of 2:1). Make a sketch; then record the area of your new square in the table. • What is the relationship between the scale factor and the area of the enlarged figure? • • Build a third square in which each side is three times as long as the original square (a scale factor of 3:1). Make a sketch and record the area of your new square in the table. What is the relationship between the scale factor and the area of this enlarged figure? Shape Area Orange Square Green Triangle Blue Rhombus Red Trapezoid Tan Rhombus * Yellow Hexagon (Extension: see #11) Scale Factor of 2: Area of the Enlargement in Terms of the Original Shape General rule for 2:1 (Area) Scale Factor of 3: Area of the Enlargement in Terms of the Original Shape General rule for 3:1 (Area) 1 unit of area 1 unit of area 1 unit of area 1 unit of area 1 unit of area 1 unit of area 2. Follow the same procedures for the Green Triangle, then the Blue Rhombus, the Red Trapezoid, and the Tan Rhombus, defining the area of each pattern block as 1 each time. You may want to split up the work with others in your group. Continue to sketch your solutions, and record the areas in the table. 3. What is the general rule for the relationship of the areas between two similar figures when the scale factor is 2? When the scale factor is 3? Record your rules in the table above. 4. What is the relationship of the perimeters to the scale factor? 33 Perimeter and Area 2 5. If the area of a polygon were 8 cm , what would be the area of an enlargement with a scale factor of 2? An enlargement with a scale factor of 3? Explain your thinking. 6. Do you think your rule will hold true for a different type of triangle? To test your conjecture, draw a triangle of your choice and copy it 3 more times. Can you assemble these 4 triangles into a triangle similar to the original that has a scale factor of 2? Does your rule hold true? 7. What would you predict the area rule would be for a scale factor of 4? Explain. 8. What happens to the area of a shape if you decrease its size by a certain scale factor, for example by half? Does the same rule apply? 9. In general, if the scale factor of an enlargement is k, what is area of the enlargement in terms of k? 10. In this activity, the word similar was used. Similar has a very specific meaning in mathematics. What properties, as far as sides and angles, did each set of similar shapes have in common? After we do the next sets of activities, you may want to come back and refine your thoughts about similarity. 11. Extension (If your group finishes before the other groups are done): The Yellow Hexagon. Create a new hexagon with a scale factor of 2:1. You will need to use other pattern blocks pieces to complete this problem. Does your rule work for this one? 34 Similarity and Scaling Similar Rectangles Are all rectangles similar? Make a prediction. You will explore this question in the Similar Rectangles activity: • Use the rectangles provided. Measure to the nearest half centimeter. • Use the table below to record your information • Group discussion Look-Alike Rectangles Three Groups and an Odd Ball Rectangles Group 1 (Letter of rect.) Short side Measures in cm. Long side Ratio of sides Short/Long Rectangles Group 2 (Letter of rect.) Short side Measures in cm. Long side Ratio of sides Short/Long Rectangles Group 3 (Letter of rect.) Short side Measures in cm. Long side Ratio of sides Short/Long Short side Measures in cm. Long side Ratio of sides Short/Long Odd Ball (Letter of rect.) 35 Similarity and Scaling Similarity—continued: Using a Coordinate Grid For this activity, use the picture of the cat’s head (it’s really a polygon!) on the coordinate grid below. 1. Copy the picture of the cat’s head onto a coordinate grid, and find the coordinates of each point. 2. Multiply the coordinates of each point by 2, creating points A' (read as A prime) through H'. Use the rule (x,y) (2x,2y). Plot these new points on the same grid, and connect them in the same order. Is your new figure similar to the original picture? How do you know? 3. Find the area of the original and the area of the enlargement. How do they compare? What is the relationship between the area of the enlargement and the scale factor? Is your result consistent with your findings in the Scale Factors and Area activity with pattern blocks? Explain. 36 Similarity and Scaling 4. Repeat the activity, but instead of multiplying by 2, add 2 to each of the coordinates. Use the rule (x,y) (x+2,y+2). Plot the new points on a coordinate grid. Is this figure similar to the original picture? Explain. 5. Explain why, if you want to create a similar figure with sides twice as long as the original, you don’t double all the angles as well. 6. Repeat the activity, multiply the x coordinate by 2 as in question #2 and leave the y coordinate as is. Use the rule (x,y) (2x,y) Plot the new points on a coordinate grid. Is this figure similar to the original picture? Explain your results. 7. If you added 2 to the lengths of each side of the “cat” instead of adding 2 to the coordinates, as in #4, would the new figure still be similar to the original? Explain. 8. What does it mean for two figures to be similar? You may want to go back and refine your definition of similarity that you wrote in #9 of the Scale Factors and Area activity. 37 Similarity and Scaling Similarity—continued: Make a Smaller Puzzle • Copy the square puzzle below (not drawn to scale) onto centimeter grid or dot paper. • Draw new puzzle pieces such that all the pieces are similar to the original figure, but with a scale factor such that the segment that is marked 4 cm long is 3 cm. • Divide up the work with other members of your group. Each person should complete one or two pieces until all the pieces have been made. When you are done, you should be able to assemble all your pieces as in the original puzzle. • In the Similar Rectangles activity, you found that all rectangles are not similar. What about squares? Are all squares similar? Explain. Extension: You have a picture that you want to mat before you frame it. You plan to use a mat that is 3 inches wide (see sketch). Will the matted picture be similar to the original picture? Explain. 38 Measurement Relationships Measurement Relationships: Area and Perimeter If you have 24 meters of fencing and you want to use it to make a rectangular pen for your puppy, you must consider both the perimeter of the pen and its area. What relationships exist between these two measures? Do shapes with the same perimeter have the same area? Let’s investigate. 1. You want to use all 24 meters of fencing to make the rectangular pen, that the dimensions of the pen in meters will be whole-number values, and that you want the maximum area for your puppy. a. What are the dimensions of each of the possible rectangular pens? Sketch each pen on grid paper. (You may want to use toothpicks to model the pens.) b. What is the area of each pen? 2. All of the pens have a perimeter of 24 m., yet the areas of the pens differ. What do you notice about the shapes of the pens with small areas as opposed to those with large areas? What are the characteristics of a rectangular shape with the greatest area? 3. Another situation involving perimeter and area: Joe, a student in a sixth grade class, was asked to find the approximate area of his hand traced onto a sheet of paper. He took a string and placed it around the perimeter of his hand. He then took the length of string that represented the perimeter of his hand, reshaped it into a rectangle, and found the area of the rectangle. Joe concluded that the area of his hand and the area of the rectangle were the same. Will his strategy work? Why or why not? 39 Measurement Relationships 4. Looking at perimeter from another direction: Do figures with the same area always have the same perimeter? Use 12 one-inch square tiles to explore this question. Arrange, and rearrange, all 12 square tiles to make figures in which each tile must share at least one side with another tile. (The shapes do not have to be rectangles.) Record your shapes on grid paper. Then determine the perimeter of each shape you create. a. What is the smallest perimeter possible using all 12 square tiles? b. What is the largest possible perimeter? 5. Using any number of one-inch color tiles, is it possible to make rectangles with perimeters of 14 through 26 inches? If not, choose one perimeter that you could not make and explain why it is impossible. 6. In real life, under what circumstances might you want the smallest perimeter? 7. Consider this rectangle drawn on grid paper. What is the perimeter? A common misconception made by some students is to think that the perimeter is 16. What do you think those students are doing wrong? What do you think they are doing wrong if they say the perimeter is 24? How about 21? Extension: Draw a right triangle that also has a 30º angle. a. Can you draw a different triangle with these same constraints? b. Do they have the same area? c. Do they have the same perimeter? d. Are the triangles similar? Explain. 40 Circles Circles—Discovering π How does the Circumference of a Circle Compare with its Diameter? 1. Use the circular object that was provided by your instructor for this activity. 2. Measure the circumference of your object. Use string, a tape measure, a ruler, or whatever method you want. Using the same unit of measure, measure the diameter of your circle. 3. Compare the circumference to the diameter. Record your results in the table below. If time, do one or two more circular objects. What patterns do you notice, if any? Measure accurately! Name of circular object Circumference Diameter C+D C-D C*D C÷D Decimal form 4. Record your group’s results on the class chart and plot your points using one coordinate graph for the whole class. Use the horizontal axis (x) for diameter and the vertical axis (y) for circumference. 5. Discuss your findings with the whole group. 6. The formula for finding the circumference of any circle if you know the diameter is C = πd. Explain how this formula makes sense in light of the activity measuring circles. 7. How would you write this formula if you only knew the radius? _________ (This is typically what we see in textbooks.) 8. Pi (π) is an irrational number, which means it cannot be written as the quotient of any two whole numbers. Its decimal approximation is known to more than a million decimal places. Yet we sometimes see π written as 3.14. Why is that? 9. When mathematicians are asked to determine the circumference of a circle, say with a diameter of 5, they say the circumference is 5π. Why do you think they record the answer this way? Why not use the π key on the calculator to find a numerical value for the circumference? 41 Circles Finding the Area of a Circle Listed below are a number of activities that will help you figure out the area of a circle. Choose 3 or 4 of the investigations to do, and be ready to report your findings to the whole group. Everyone should do #6, which relates specifically to “discovering” the formula for area of a circle. Method 1: Counting Squares You need: a sheet of centimeter grid paper and a compass. 1. Draw a circle with a diameter of 12 centimeters on centimeter grid paper. 2. Count all the whole square centimeters that lie completely inside the circle. This underestimates the area of the circle. 3. Count the number of squares that lie partly inside and partly outside the circle. Add this number the number you counted in Step 2. This total overestimates the area of the circle. 4. Average the two estimates. This is the approximate area of the circle. Method 2: Inscribing and Circumscribing Squares You need: a compass, a compass or angle ruler, and a straight ruler. 1. Draw a circle with a diameter of 12 centimeters on grid paper. 2. Circumscribe a square around the circle as shown. Find its area. 3. Inscribe a square inside the circle. Find its area. 4. Average the areas of the two squares. This is the approximate area of the circle. Method 3: Using the Octagonal (Egyptian) Method You need: a compass, a compass or angle ruler, and a straight ruler. 1. Draw a circle with a diameter of 12 centimeters on grid paper. 2. Circumscribe a square around the circle as shown. 3. Divide the square into nine congruent squares, as shown 4. Construct an octagon by drawing a diagonal in each corner square. 5. Find the area of the octagon. This is an approximation of the area of the circle. 42 Circles Method 4: Weighing the Circle You need: a linoleum circle with a diameter of 12 centimeters, some linoleum rectangles measured in square centimeters, some 1 x 1 centimeter linoleum squares, and a pan balance. 1. Place the linoleum circle on one side of the pan balance. 2. Balance the circle with linoleum rectangular and square pieces placed on the opposite side of the pan balance. 3. Count the number of rectangular and square pieces you used. The total area of all the pieces approximates the area of the circular region. Method 5: Using a Curvy Parallelogram You need: a compass, a scissors, and tape. 1. Draw a circle with a diameter of 12 centimeters on plain paper. 2. Fold the circle in half horizontally and vertically. 3. Cut the circle into four wedges on the fold lines. Then fold each wedge into quarters. Cut each wedge on the fold lines. You will have 16 wedges. 4. Tape the wedges to a piece of paper to form the following figure: Notice that we have a crude parallelogram with a height approximately equal to the radius of the original circle. 5. Find the approximate area of the curvy parallelogram. This approximates the area of the circle. Note: you will need to figure out the length of the base, b, of the parallelogram. How does the base relate to the circumference of the circle? 43 Circles Method 6: Examining the Formula Using Radius Squares You need: a sheet of white centimeter grid paper, a sheet of colored cm grid paper, and a compass. 1. Draw a circle with a diameter of 12 centimeters on white centimeter grid paper. 2. On a piece of colored centimeter grid paper, draw a square such that one side of the square is equal to the length of the radius, as shown in Circle 1 (not to scale). 3. Cut out several copies of the “radius squares” from the colored paper. Determine the number of radius squares it takes to cover your circle from #1. You may cut the radius squares into parts if you need to. 2 4. If you were to estimate the area of a circle in radius squares (r ), what would you report as the best estimate? Would your estimate work for Circle 2? For any circle? 2 5. The formula for the area of a circle is A = πr . Explain how this formula makes sense in light of the activity with radius squares. Extension: When you enlarge a circle so that the radius is twice as long (a scale factor of 2), what do you think happens to the circumference and the area? Do they double? What happens if you enlarge your circle by a scale factor of 3? Are your results consistent with your work on scale factors in polygons? Explain. 44 Circles Applying What You Know about Circles Reminder: use correct labels. Leave your answers in terms of π. 1. Area = _____________ Perimeter = ______________ Approximate your answer using the π key on a calculator and then using 3.14 as an estimate for π. Why are these two answers slightly different? 2. Un-shaded Area = _____________ Circumference of each circle: Large circle = _____________ Small circle = _______________ 3. Area = ________________ Arc length = _________________ Extension: A circle with radius r is inscribed in a square. The shaded part (the circle) is what percentage of the square? Do not use an approximation for π (3.14) until you are ready to write the percent. 45 The Pythagorean Theorem The Pythagorean Theorem 1. Looking for Squares: On 5 dot-by-5 dot grids, draw squares of various sizes by connecting dots. Draw squares with as many different areas as possible. Label each square with its area. Include any possible squares whose sides are not horizontal and vertical. 46 The Pythagorean Theorem 2. Square Roots: The area of a square is the length of a side multiplied by itself. This can be expressed 2 by the formula A = s * s, or A = s . How is this related to the formula that you developed for a rectangle: A = lw? If you know the area of a square, you can work backward to find the length of a side. For example, suppose a square has an area of 9 square units. To find the length of a side, you need to figure out what positive number multiplied by itself equals 9. Because 3 * 3 = 9, the side length is 3 units. We call 3 a square root of 9: written as = 3. Find the square in problem #1 that has an area of 9 square units, and confirm that the side length is 3. Note: Numbers that have whole-number square roots, are called perfect squares or square numbers. You may recall the pattern problem we did in “Algebra Camp” using square numbers: What is the area (small squares) of each of these figures? Why do you think those areas are called square numbers? What are the next three square numbers is the pattern? What is the rule for finding the number of squares in the nth figure? Revisit the rest of your squares in problem #1. Find the square root of each area to find the side length of that square. Are the areas of your “tippy squares” square numbers? How would you find the square roots of these numbers? Example: √5 ≈ 2.24 47 The Pythagorean Theorem 3. More Areas of “Tippy Squares”: Subdivide the squares below into right triangles and squares using horizontal and vertical lines. Combine the areas of all the pieces to find the total area of each square. After you have found the area, find the side length, s, of each square using square roots. Label the side of the square with the length. You can check yourself with a ruler. The first one has been done as an example. 48 The Pythagorean Theorem 4. The Pythagorean Theorem: Recall that a right triangle is a triangle with a right, or 90º, angle. The longest side of a right triangle is the side opposite the right angle and is called the hypotenuse of the triangle. The other two sides are called the legs. The right angle of a right triangle is often marked with a square. Each leg of the right triangle on the left below has a length of 1 unit. Suppose you draw squares on the hypotenuse and legs of the triangle, as shown on the right. How are the areas of the three squares related? In this activity, you will look for a relationship among the areas of squares drawn on the sides of right triangles. Do all three steps for the first pair of leg lengths a and b. Then repeat for each of the other pairs: • Draw a right triangle with the given leg lengths on dot paper. • Draw a square on each side of the triangle. Hint: Use an index card to help draw right angles. • Find the area of the squares, write the area inside the square, and record the results in the table. Length of Length of Leg 1: a Leg 2: b (units) (units) 1 1 2 1 2 3 3 a • • • Area of Square on Leg 1 2 (square units: a ) Area of Square on Leg 2 2 (square units: b ) Area of Square on Hypotenuse: c 2 (square units: c ) 1 2 2 3 3 3 4 b Make a conjecture about the relationship among the areas of the three squares drawn on the sides of any right triangle. Draw a right triangle with side lengths that are different than those given in the table. Use your triangle to test your conjecture. Write your conjecture using the symbols in the table: _______________________ You have “discovered” the Pythagorean Theorem! 49 The Pythagorean Theorem 5. A Proof of the Pythagorean Theorem: The pattern you discovered in the last activity is a famous theorem named after the Greek mathematician Pythagoras (although the Chinese used the idea as far back as 1000 B.C. A theorem is a general mathematical statement that has been proven true. The Pythagorean Theorem is one of the most famous theorems in mathematics. Over 300 different proofs have been given for the Pythagorean Theorem. You will explore one of these proofs in the next activity. • Use the puzzle pieces provided. Cut out the pieces and look at how the triangle and the three squares are related: how do the side lengths of the squares compare to the side lengths of the triangle? • Label the hypotenuse of one of triangles as c. Label the two legs a and b. • Label the square with a side length equal to the length of the hypotenuse as c . Label the two 2 2 2 2 smaller squares a and b . (Make sure the square labeled a goes with the leg labeled a.) Why do these labels make sense? • Arrange the 11 puzzle pieces to fit exactly into the two puzzle frames. Use four triangles in each frame. • What conclusions can you draw about the relationship among the areas of the three squares? • What does the conclusion you reached mean in terms of the side lengths of the triangles. (How 2 2 2 does your conclusion prove a + b = c ?) Extension: James Garfield (1831-1881) was the 20th president of the United States. Five years before becoming president, he discovered this proof of the Pythagorean Theorem. What is the reasoning behind his proof? Hint: Compute the area of the figure two ways—first by finding and adding the areas of the triangles, and second by finding the area of the whole thing, which is a trapezoid. (Refer back to the formula for area of a trapezoid.) What must be true of the two areas? (You’ll need some algebra.) 50 The Pythagorean Theorem 6. Applying the Pythagorean Theorem: Use correct labels. a. Find the length of the missing side: b. What is the length of the diagonal of a square that has a side length of 8? (Hint: draw a sketch.) c. Shaded area = ______________________ d. What are the lengths of the line segments below? 51 The Pythagorean Theorem e. Find the area and the perimeter of the figure below (without using a ruler): Perimeter = _______________________ Area = _____________________ f. A new and larger triangle is similar to the triangle below, with the size transformation having a scale factor of 4. What is the area enclosed by the new triangle? Explain how you can use what you know about the impact of scale factor on area to find the area of the new triangle without actually using the dimensions of the new triangle: ___________________ ________________________________________________________________________________ 7. The converse of the Pythagorean Theorem: The converse of the Pythagorean Theorem is also true. 2 2 2 In other words, if the three side lengths of a triangle satisfy the formula a + b = c then the triangle is a right triangle. Determine if the triangles listed below are right triangles. (Actually, one of the sets of side lengths below won’t make a triangle at all. Which one is it? How do you know?) Side Lengths Do the side lengths satisfy Is the triangle a right triangle? 2 2 2 (units) a +b =c ? 6, 8, 10 5, 6, 10 5, 5, 5 1, 2, 3 52 The Pythagorean Theorem 8. If time: Tri-Square Rug Games: Al and Betty have designed a game that uses “tri-square rugs.” Their rugs are made by sewing three separate square rugs together at their corners, with an empty triangular space between them. (See the examples below.) In Al and Betty’s game a dart falls randomly on the tri-square rug. • • • If the dart hits the largest of the three squares, Al wins. If the dart hits either of the other two squares, Betty wins. If the dart misses the rug, they just let another dart fall. Your goal is to determine which of the rugs you would prefer if you were Al and which you would prefer if you were Betty, and if there are any rugs that lead to a fair game. On chart paper show some rugs that are wins for Al in the long run, rugs that are wins for Betty, and rugs that are fair games. When you have several examples for each category, look for a pattern in your results. Your goal is to find a way to tell, just by looking at a tri-square rug, who will win in the long run and why that’s true. Summarize your results below: Adapted from Interactive Mathematics Program, Year 2, Key Curriculum Press 53 Polyhedra Polyhedra 1. Look at the nets (outlines) for shapes A-K, which are provided on card stock. Try to imagine what the final, folded-up shape will look like before you cut out and assemble them. 2. Cut out and assemble the polyhedra so that the letter of the shape is visible. Hint: Use a sharp pen and straightedge to score each fold line; the folds will be easy and accurate! 3. Each shape is an example of a polyhedron (plural: polyhedra or polyhedrons): a closed surface made up of planar regions (flat pieces). The planar regions in a polyhedron are faces of the polyhedron, the line segments where faces meet are edges, and the points at which edges meet are vertices (singular: vertex) of the polyhedron. Choose one of the polyhedra and count the number of faces, edges, and vertices. 4. Sort your completed polyhedra into different sets. What criteria did you use? Compare your sets with another group’s sets. Are they the same? Discuss the attributes you used. 5. Sort your polyhedra into the following three groups: • Group 1: Shapes A, D, G, and I • Group 2: Shapes B, C, E, and F • Group 3: The others—set aside for now a. Group 1 shapes are called pyramids. Come to a consensus on a definition for pyramid, without looking it up. One face is called the base of the pyramid; which one do you think it is for each shape? Since the shape of its base names each pyramid, give the name for each pyramid in Group 1. For example, shape I is a square pyramid. b. Group 2 shapes are called prisms. Give a definition for prism. Two faces of each prism are called the bases of the prism; which ones are the bases for each shape? How are the bases related to each other? Which prisms can be reoriented so that the two faces called the bases are different? Write the name for each prism in Group 2. For each prism, how do the lateral edges (the edges not on a base) appear to be related? What shape are all the lateral faces (the ones that aren’t the bases)? 54 Polyhedra 6. Some of your pyramids and prisms are right pyramids and right prisms. a. Which ones? Why do you think that? (You may check the definitions in Sections 326 and 327 in Geometry to Go.) b. What shape are all the lateral faces of a right prism? c. What shape are all the lateral faces of a right pyramid? 7. Why are the shapes in Group 3 not included as pyramids or prisms? 8. Which of the shapes, A-K, meet the criteria in each part? a. All the faces are parallelograms (including special parallelograms, like rectangles and squares). b. A base is a pentagonal region. c. All the triangular faces are equilateral. d. None of the triangular faces is equilateral. e. Two faces are parallel and congruent, but the shape is not a prism. f. Some of the faces are isosceles trapezoids. g. All the edges are congruent. h. No pair of edges is parallel. 9. How many of the angles on all the faces of a prism could be right angles, if… a. The prism is a right triangular prism? (Hint: consider bases that are both right and non-right triangles.) b. The prism is a right quadrilateral prism? (Hint: some quadrilaterals have right angles and some don’t.) 10. Pick one of the shapes. What attributes of this shape can be measured and/or counted? Pick another one of the shapes. Are the same quantities relevant to it also? Explain. 55 Polyhedra Representing and Visualizing Polyhedra What do you see when you look at ? Did you see a regular hexagon with some segments drawn to its center? Or did you see a cube? (Do you see both now?) Three-dimensional shapes are often represented by drawings in two dimensions. Without a context suggesting three dimensions, these drawings can be ambiguous. How to “read” such drawings is important for understanding the ideas they represent, and how to make such drawings is a useful skill that is often needed to represent and solve a problem. Different representations of a polyhedron Physical model “Polyhedron” (word) Net Drawing 1. Nets a. When you assembled shapes A-K, you were working from nets. A net is a two-dimensional pattern that can be folded to form a three-dimensional solid. Which one of your polyhedra is a cube? Draw a net for it. Can you draw a different one? (Hint: there are eleven possible nets for a cube.) b. If these nets were folded up to give cubes, which pairs of faces would be opposite each other? c. Using a centimeter ruler to measure, find the surface area (the total area of all the faces) of shape E. Explain how you figured it out. d. Find the surface area of the right rectangular prism in your set. Measure to the nearest half centimeter. How are the dimensions of the prism related to the dimensions of its faces? e. Explain how you would find the surface area of the right triangular prism in your set. Don’t actually find the surface area; just explain the process for finding it. 56 Polyhedra 2. Drawings or sketches a. Using isometric dot paper, make a sketch of a cube. Represent the “hidden” edges with dotted lines. Can you draw this same cube on plain paper? b. Draw a cube that has an edge length that is twice as long as the cube you drew in part a. c. Draw a right rectangular prism, such as shape F, on isometric dot paper. Extension: Draw the same prism from a different viewpoint. You may want to hold up your physical model in order to view it from different angles. 3. Surface area (continued) A company wants to arrange 24 alphabet blocks in the shape of a rectangular prism and then package them in a box that exactly fits the prism. Each block is a cube with 1-inch edges, so each block has a volume of 1 cubic inch. a. Find all the ways 24 cubes can be arranged into a rectangular prism. Make a sketch of each arrangement. Record the dimensions and surface area and volume in the table below. Length Width Height Volume Surface Sketch Area b. Which arrangement would you recommend to the company? Why? c. Why do you think the company makes 24 alphabet blocks rather than 26? 57 Polyhedra Volume You may use cubes for the following activities. Let the dimensions of your cube be 1 x 1 x 1. 1. What is the surface area of one cube? What is the volume? What label should you use for each measurement? 2. On the next page are the nets of some “open” boxes—boxes without tops or lids. A. If you were to cut out each net, fold it into a box, and fill the box with cubes, how many cubes would it take to fill the box? Make a quick prediction and then use two different approaches to find the number of cubes. You may want to cut out the actual nets, fold them up, and tape them into boxes. B. What strategies did you use to determine the number of cubes that filled each box? 3. Build some rectangular prisms with your cubes. Find the volume of each. Generalize an approach for finding the number of cubes (the volume) in any right rectangular prism. Write your rule in symbols using the dimensions: length, width, and height. 4. • If the box below was completely filled with cubes, how many cubes would be in the bottom layer? • How could you figure this out without counting? (This number is the same as the _____ of the base.) • How many layers tall is the box? • What is the total number of cubes, including the cubes already shown, needed to fill the box below? Check yourself using the rule you wrote in #3. Do you think this “layering method” would work for other prisms, such as triangular prisms, hexagonal prisms, etc? Explain. 5. Find the surface area and volume of the box below: 58 Polyhedra 59 Polyhedra 6. What is the relationship between scale factor and volume? Use the sketches below to help you answer this question. (First, determine the volume of each cube.) What would be the volume of an image with a scale factor of n? (What is the rule for any scale factor?) 7. What if you were working with rectangular solids that were not cubes? Would your rule still work? For example, if you have a right rectangular prism with dimensions of 2 x 3 x 5 and you doubled all the dimensions (a scale factor of 2), does your general rule for a scale factor of n still work? 8. How is the relationship between scale factor and volume the same as the relationship between scale factor and area that we worked on earlier in this course? How is it different? 9. Extension A: Would your rule work for “odd-shaped” figures, such as the one below? 10. Extension B: Build one or two odd-shaped figures with cubes and sketch them on isometric dot paper. Make at least one figure that is more than one cube tall. 11. Extension C. Draw a net for a cylinder, such as the one below. (Hint: you might want to “build” a cylinder out of paper first.) How would you find the surface area? The volume? (Hint: Think about the layering method you used in Problem 4.) 60 Polyhedra Representing and Visualizing Polyhedra (revisited) Orthogonal Views 1. Build the shape below with your cubes. Move your body so that you are at “eye level” with the front of your shape, in order to see the “front view”. Does your eye level view match the drawing of the front view? Try the other views as pictured; do you see these views as shown? 2. Using your cubes, construct a model from this building plan (which is also the top view). Each number tells you how many cubes go in that column as you are building it. Note: This indicates the front of the building, not the front view. Sketch each view in the spaces provided: Front View Back View Right View Left View 61 Polyhedra 3. Below are base plans for four different buildings. Match each view with the correct set of drawings below. (Can you do it without constructing a model?) (Check your answers on the last page when you are done ) 62 Polyhedra 4. Sketch the front, right-side, and top views for each of these cube arrangements. You may want to build the arrangements first. 5. Sketch the view from the left of the shapes in parts a and b. How are they related to the views from the right? (See also, #2.) 6. Use cubes to make a model of a building for EACH set of plans below. On the base outline record your base plan for each building by marking in the number of cubes you used. Compare your models with those of your group. There may be more than one correct answer. (The answers are on the next page, but don’t peek until you are sure you have one that fits!) Extension: Sketch some of your building on isometric dot paper. 63 Polyhedra Answers to Problem 3 Building A matches Set 3 Building C matches Set 4 Building B matches Set 1 Building D matches Set 2 Answers to Problem 6 64