POSTER PROLEMS Triangles To Order Seventh Grade Poster Problem Geometry In this poster problem, students try to build triangles to particular specifications (specs). The specs can include side lengths, angles, or a combination of both. Students go on to generalize, and develop an understanding about when they can determine a triangle from partial information and when they cannot. hallenges Using protractors. Many students have little experience using protractors or with angle measure at all, for that matter. Ultimately, we want students to be able to use a protractor to construct or measure angles up to 180 to the nearest degree. See Triangle onventions and Mechanics. The issue of whether two triangles are the same (i.e., congruent). We do not make a big deal about that here, but if it comes up, there are two main ways to look at it, described in Triangles and onstraints. The main point is that two triangles are the same if they have the same size and shape. nother way to say it is that they re the same if you can cut one triangle out (or make a transparency of it) and move it, turn it, and flip it to match the other exactly. Materials: Scissors, Rulers, Protractors, Scratch paper. Lots of scratch paper. Learning Objectives Students construct triangles according to specifications (angle and/or side measures). Students recognize that with three conditions, you can often but not always determine a triangle. Students begin to understand the rules for when three conditions determine a triangle. The way this works: one lesson in six phases LUNH Teachers set the stage by leading an introductory discussion that orients students to the context of the problem. POSE PROLEM Teachers introduce a mathematical way of thinking about the context and engage students in a preliminary approach that opens the door to the workshop phase. WORKSHOP The workshop starts with a more challenging and more openended extension of the problem. In teams, students plan and produce mathematical posters to communicate their work. POST, SHRE, OMMENT Teams display their posters in the classroom, get to know other teams posters, and attach questions/comments by way of small adhesive notes (or similar). Day 1 FLEXILE ommon ore State Standards for Mathematics: 7.G..2 STRTEGI TEHER-LED DISUSSION Day 2 Teacher Tune Up: See Triangles and onstraints See Triangle onventions and Mechanics Teachers then compare, contrast, and connect several posters. In the process they highlight a progression from a more basic approach to a more generalizable one. y doing this, teachers emphasize standards-aligned mathematics using studentgenerated examples. FOUS PROLEM: SME ONEPT IN NEW ONTEXT Serving as a check for understanding, this more focused problem gives teachers evidence of student understanding. SERP 2014!!! Triangles To Order - Seventh Grade Poster Problem!!!! 1
1. LUNH This lesson will be all about triangles. Explain why triangles are important, for example, triangles are at the center of geometry. If you understand triangles really understand them you ll be in great shape. One reason is that you can make things out of triangles. In fact, most realistic computer graphics, from games to movies, are made up of triangles. 3D Graphics use Triangles Show Slide #1: These are wireframe drawings. n animator would color the triangles and use more of them to make smoother, more realistic heads. sk, What do you need to know to determine a triangle? fter all, rectangles are easy: if you know the height and the width, you have the rectangle. ut what do you need to know for a triangle? Show Slide #2: sk: what are all the numbers on this diagram? What do they mean? 5 cm is the length of and. 7.1 cm is the approximate length of. 45 is the measure of or. 90 is the measure of. POSTER PROLEMS - TRINGLES TO ORDER SLIDE #1 Slide #1 Suppose you re on the phone with somebody and they re trying to describe this triangle. What do they have to tell you before you can draw the triangle? Here s the point, which you can elicit in discussion: they don t need to tell you all of the information in the picture. Suppose they left out the 7.1 cm for the long side. You could still draw the triangle using the rest of the information. How little information do you need? Slide #2 Possible mini-activity: If students need additional reinforcement Have students work in pairs. The teller faces the screen or board, and the drawer faces away. Teacher draws or displays a different triangle (for example, the one on Slide #3) The teller tries to get the drawer to draw the triangle using as few clues as possible. ( clue is one of the numbers on the slide.) Slide #3 SERP 2014!!! Triangles To Order - Seventh Grade Poster Problem!!!! 2
2. POSE PROLEM Make sure students have rulers, protractors, scissors, scratch paper. Each group gets Handout #1 and several copies of the recording sheet, Handout #2. (If a group runs out of Handout #2, they can use scratch paper if they record carefully.) Have students cut out the cards at the bottom of Handout #1 (leaving the form in the top half of the handout intact). Mix them up and put them face down. Triangles to Order 1 Triangle onditions Handout The Form? º? cm? cm? º? º? cm Explain the setting: Imagine your friend is trying to tell you how to draw a particular triangle. He/she give you three pieces of information as a specification. Tell students to draw three cards and place them on the form. Explain that this represents your friend telling you the three parts of the specification. Lengths go in the three rectangles labeled,, or. Those are side lengths. ngles (with ) go where you see,, or. The group must agree about whether the specifications tell you what you need to make the triangle. The group must agree whether the specifications will let you make just one triangle, two triangles, no triangles at all, or many triangles an infinite number. If you can make just one triangle, you should draw it carefully using the ruler and protractor. If you can make two triangles, you should draw them both. If you can make zero or a large number of triangles, you should be ready to explain how you know. When that s done, set the little cards aside and start a new triangle using the face-down cards. Try to solve as many different configurations as you can. ut these out 2 cm 4 cm 2 cm 5 cm 3 cm 6 cm 3 cm 4 cm 6 cm 6 cm 7 cm 7 cm 90º 90º 30º 45º 45º 15º 60º 60º 60º 90º 120º 105º Triangles to Order Triangle Record Sheet Handout Here are our specifications (from the cards): If we can make one or two triangles, we draw them below. If making a triangle is impossible, we write and draw our explanation of why it is impossible below. With these specifications, we could make no triangles two triangles one triangle many triangles Handout #1 2 When to end the student activity This is a judgment call: ideally every group has encountered a situation in which they made a single triangle, and another situation in which the triangle was impossible. Here are our specifications (from the cards): If we can make one or two triangles, we draw them below. If making a triangle is impossible, we write and draw our explanation of why it is impossible below. With these specifications, we could make no triangles two triangles one triangle many triangles Handout #2 SERP 2014!!! Triangles To Order - Seventh Grade Poster Problem!!!! 3
3. WORKSHOP rrange students in groups. They should still have their form and the cards from Phase 2 of the lesson (Pose a Problem). They should also have records of specifications they used. First hold a brief discussion about what happened during Phase 2. You might ask: Did you find specifications that gave you just one triangle? Did it matter which points they referred to? (In general, yes) What order did you draw this triangle in? That is, which clue did you use first? Which was second? s students report, be alert for misunderstandings: e sure that students always fill three cells out of the six available on the form. e sure sides and angles go in the right slots. e sure students are paying attention to which vertices are which. Distribute Handout #3. Explain that each group will divide their poster into two columns and follow the instructions on the handout. Help for two and many Some groups may have trouble finding examples of specs that yield two or more triangles. Here are ideas about how to help. Suggest that they try specifications with three angles. nd if the triangle is impossible, they should ask themselves what would have to be different in the spec to make the triangle work. Look to see if they have an SS situation one with two sides and the angle that is not between the two sides, e.g.,,, and. Have them play with that, experimenting with different values. If they still have trouble, suggest they use a very small angle, such as 15. Triangles to Order Workshop & Poster Handout Suppose you have three cards on the form. Those three numbers might be lengths of sides, or measures of angles, or some combination. These are your specifications for a triangle. Sometimes, with three cards, you can find a unique triangle. That is, you can make a triangle, and there is no other triangle that fits all the specifications on your form. Sometimes you can t. Sometimes it s because there is more than one triangle that works with the specs. nd sometimes it s impossible: no triangles work. The Problem Your job, as a group, is to figure out how to tell whether you can make a unique triangle when you have placed three cards on the form. What You Do Explore! Use the form and turn cards from Handout #1 face-up. You may use whatever cards you want, but every specification you make must have three cards. Divide your poster into two columns. Label them no triangles and triangles. - On the no triangles side, list specifications where you cannot make any triangles at all. - On that same side, list the reasons why you can t make triangles with those specifications. If you generalize (see below), you can probably put the specifications into groups and explain them all at once. - On the triangles side, list specifications under which you can make triangles. - On that same triangles side, explain how you made the triangles, especially if it was difficult. It will help if you can group triangles that work the same way. Try to generalize. For example, there may be several specs where you can t make triangles. Even though the specific numbers are different, the actual reason you can t make a triangle may be the same. Similarly, sometimes the reason you can make a triangle is the same even though the details are different. If you come across situations where the triangles are not unique, that is, you can make two or more triangles from a spec, give them a special section. Explain them as well as you can. Why can you make more than one triangle in these cases and not in the others? 3 Handout #3 SERP 2014!!! Triangles To Order - Seventh Grade Poster Problem!!!! 4
4. POST, SHRE, OMMENT Have students post their work around the classroom. Encourage students to view the posters other groups created. Encourage students to write questions for other groups by attaching small adhesive notes. During this time, the teacher reviews the posters and considers which ones to highlight during discussion in the next phase. Sample Posters: In Poster, the students recognize that they can t make a triangle with 7-4-2 as the side lengths, and correctly reason that the 4 and 2 were too short. ut they don t generalize. On the right side, they correctly construct a triangle with two sides and an angle, but again do not generalize. Poster identifies a nice impossible situation, and correctly recognizes that the problem is that the angles are too big. The generalization isn t quite correct, however; you can still make triangles with obtuse angles. The students have a similar problem on the yes side: the generalization is pretty good but not completely correct. They do recognize that their two specifications are basically the same: two angles and the included side. Poster generalizes the problem with three angles: they have to add to 180 or the edges can t line up. The students then note that if the sum of angles is 180, you can make many triangles. They also take on SS, the situation with two sides and the included angle. SERP 2014!!! Triangles To Order - Seventh Grade Poster Problem!!!! 5
5. STRTEGI TEHER-LED DISUSSION Select a sequence of posters to use during the teacher-led discussion that will help move all students from their current thinking (often Levels 1 3 below) up to 4 or 5. Level 1: Students found examples of specs that make no triangles and one triangle. ut the explanations (the why and how) are specific to the numbers and do not address the reasons. ( Our sides were 5, 3, and 1. With those lengths, we couldn t make a triangle. ) Level 2: Like Level 1, but the group uses the numbers to create correct reasons. ( Our sides were 5, 3, and 1. Since 3 + 1 = only 4, the two short sides are not long enough to make a triangle with 5 on the other side. ) Level 3: Students move beyond the specific numbers in their explanations ( Our sides were 5, 3, and 1, but to make a triangle the long side has to be less than the sum of the other two ) but do not group triangles together with others in the same category (e.g., one with sides 7-4-2). Level 4: Students create coherent reasons and descriptions that go beyond the specific numbers and recognize that other triangles are in the same categories. ( ny time you have side with angle and angle [and + is less than 180 ], just draw side first, then make the two angles. Where the anglelines meet is point. onnect them up to make your triangle. ) Students may explain situations with two or more triangles. Level 5: Students recognize that it s not the specific points and labels that matter, but rather the relationship between them. ( ny time you have a side and the two angles on either end of it, draw the side first that give you two points and then the angles. The angle-lines meet at the third point ) Students can explain the cases where you get two and many triangles coherently. Supporting Generalization Help early presenters generalize. For impossible triangles, ask,!s there a way I could change one of these numbers and have the specs make a triangle? From there, help the class figure out the criteria for impossibility. For a unique triangle, ask the class, re these specifications the same sort as the others we ve seen, or are they different? ould they use the same procedure to draw the triangle as these other triangles? That way, the class can see that making a triangle by specifying 3 sides is different from 2 sides and the included angle, etc. You will need to update them as students generalize better; avoid imposing your own wisdom. How can three clues fail to make a triangle? Triangle inequality: with three sides, the longest is longer that the sum of the other two. ngles too large: the sum of two angles exceeds 180. Swing-and-a-miss: suppose you have = 3 cm, = 4 cm, and angle is 90. If you make first, then angle, it s clear that can t reach to the other side. nd how can three clues make a triangle (provided that they don t meet one of the impossibility criteria above)? Three sides (SSS). This can be difficult without a compass; use guess and check to get close. Two sides and the angle between (SS). lways works. Order doesn t matter. Two sides and a different angle. Depends on the details; you could get zero, one, or two. Start with the side connected to the angle. Two angles and the included side (S). Works if the angles aren t too big. Draw the side first. Two angles and a different side (S). Hard unless you know about 180 ; guess and check. Three angles: the miracle of 180 Students may not know the 180 rule; this is a good time to expose them to it especially if they discover it themselves. One consequence: If your clues are three angles that add to 180, you get an infinite number of triangles. ll the triangles will be the same shape, but of any size. They re similar instead of congruent. Some students may object to calling the similar triangles different. There are at least two responses: one is to explain that it s a convention. nother, perhaps more to the point, is that if you were building something (a table top, say) and you made it a different size, it wouldn t matter that it was the right shape it would not fit. How you get two possibilities One way is on Handout #4, problem #5. See the answers on the next page. onnecting across groups s students present, keep a list of the types of impossible specifications and the ways you can make a unique triangle. SERP 2014!!! Triangles To Order - Seventh Grade Poster Problem!!!! 6
6. FOUS PROLEM: SME ONEPT IN NEW ONTEXT Pass out Handout #4 for this problem ( Zero, One, Two, Many ) nswers 1. One triangle (S). 2. One triangle (SS). 3. One triangle (Hypotenuse-leg; angle is 90 ). 4. One triangle (Two angles give a third 60 and you have S. This result is similar to #3). 5. Two triangles, at last! This is SS. See the illustration below. If you make side (6 cm) and angle (15 ), you now have to figure out side (3 cm). Imagine it swinging from point. It can rest its end on the line coming out of in two places, marked 1 and 2. Either of these could be the third vertex of the triangle. Triangles to Order Zero, One, Two, Many Handout Instructions Tell whether you can make 0, 1, 2, or many triangles to meet each set of specifications. If 1, make the triangle. If 2, make both of them. If many, make 3 different triangles that fit all the conditions. nd if 0, explain why it s impossible to make such a triangle (a diagram can help!). 1. 2. 4 cm 45 º 90 º 60 º 2 cm 3 cm 3. 4. 30 3 cm 30 º 90 30º 6 cm 5 cm 5. 6. 3 cm 15º 120º 90º 6 cm 6 cm 4 6. No triangles. Sum of angles is too great. 7. 8. 75 7. Many triangles, all similar (Sum of angles is 180 ). 8. No triangles, because of the triangle inequality: the two 2- cm sides are not long enough to meet opposite the 50- cm side. 2 cm 2 cm 60 45 5 cm Handout #4 More about nswer #5: SERP 2014!!! Triangles To Order - Seventh Grade Poster Problem!!!! 7