Lesson Transcript

G7M6 Lesson 11: Conditions on Measurements that Determine a Triangle

1. / Mrs. S: / As we know, a triangle is a polygon with exactly three sides. In our first activity we are going to answer the question “Given three line segments, can we always construct a triangle?”
Before we begin, give the question some independent think time and then write down your conjecture. Try to include some supporting detail to your yes or no conjecture.
2. / Mrs. S / The plastic bag on your table contains wooden dowels which have been cut to different sizes. Select three dowels at a time and try to construct a triangle. Repeat this process until you feel confident that you have an answer to our question.
[The students work in groups of three. As they work, Mrs. S circulates around the room watching the students work with the dowels and listening to their conversations. After three minutes she sees that all of the groups have found at least one case where they couldn’t construct a triangle and are starting to make conjectures about the determining factor]
3. / Mrs. S / Who can answer the question? Were you always able to construct a triangle?
[All of the students raise their hands. Mrs. S uses her equity sticks to choose a student to share his/her observation] (equity sticks are popsicle sticks, each with a student’s name written on it. As students enter the room, they pick up their equity stick and place it in a container. Mrs. S takes roll by marking students whose equity stick still remain on the table at the start of class absent)
Lizzy, what did you observe?
4. / Lizzy / Most of the time we could make a triangle but a couple of times we couldn’t.
5. / Mrs. S / [to the class]
Did you all experience the same thing Lizzy’s group experienced? [the students indicated that they did]
Look back at the conjecture you wrote before we did the activity. Was this the result you were expecting? Turn to your shoulder partner: 1’s tell 2’s what their conjecture was and then explain why the results of the activity supported or refuted this conjecture. Then 2’s do the same for 1’s.
6. / Mrs. S / [Mrs. S walks around the room listening to the students for about 1 minute]
If we can’t always construct a triangle, it might be good to know when we can and when we can’t. Our next activity will give us some more insight into this. [Mrs. S refers the students to their student pages, pages 108 & 109 parts a, b, and c]
7. / Mrs. S / Were you able to create a triangle using the segment you drew on your paper and the two segments you drew on the patty paper? [The students indicated that they were not]
Let’s do this activity again. This time each group will be given a different set of criteria which are written on the index cards I am handing out now. Your group’s job is to determine whether a triangle can be constructed using the side lengths given on your group’s index card.
[Mrs. S hands each group an index card which contains one of the criteria shown below]
Group A: Draw a 6 inch line segment on your paper. Draw a 4 inch segment on one piece of patty paper and a 3 inch segment on another.
Group B: Draw a 6 inch line segment on your paper. Draw a 2 inch segment on one piece of patty paper and a 3 inch segment on another.
Group C: Draw a 6 inch line segment on your paper. Draw a 3 inch segment on one piece of patty paper and a 3 inch segment on another.
Group D: Draw a 6 inch line segment on your paper. Draw a 5 inch segment on one piece of patty paper and a 4 inch segment on another.
Group E: Draw a 7 inch line segment on your paper. Draw a 1 inch segment on one piece of patty paper and a 5 inch segment on another.
Group F: Draw a 7 inch line segment on your paper. Draw a 4 inch segment on one piece of patty paper and a 3 inch segment on another.
Group G: Draw a 7 inch line segment on your paper. Draw a inch segment on one piece of patty paper and a inch segment on another.
[Students are given five minutes to complete the constructions. Mrs. S projects the table shown below under the document camera.]
When you have made your determination, write “yes” or “no” in cell that corresponds to your group.
Group / Segment 1
inches / Segment 2
inches / Segment 3
inches / Triangle?
A / 6 / 4 / 3 / yes
B / 6 / 2 / 3 / no
C / 6 / 3 / 3 / no
D / 6 / 5 / 4 / yes
E / 7 / 1 / 5 / no
F / 7 / 4 / 3 / no
G / 7 / / / yes
8. / Mrs. S / [After all groups had added their entry to the table]
Looking at the information on the table, as a group, formulate a conjecture about when three segments can form a triangle. Write your conjecture in your student notes.
[The students discuss the table within their groups and each student copies down the groups conjecture into their notes]
9. / Mrs. S / [Mrs. S pulls an equity stick from the cup]
Elijah, can you please share your group’s conjecture?
10. / Elijah / We said that the small numbers have to add up to be more than the big number to get a triangle.
11. / Mrs. S / Can you restate your conjecture, putting it in terms of the sides of the triangle?
12. / Elijah / I’m not sure what you mean.
13. / Mrs. S / Can anyone else help Elijah out?
[Mrs. S calls on one of the students whose hands are raised]
Albert
14. / Albert / You can only make a triangle if the two shorter sides are longer than the longest side.
15. / Mrs. S / I think we are getting closer. Would anyone else like to add anything?
[Mrs. S calls on a member of Group G, as she had noticed that this group had written a correct conjecture when she was walking around the room during the activity]
Shelby
16. / Shelby / The sum of the lengths of the two shorter segments must be greater than the length of the longest segment in order to make a triangle.
17. / Mrs. S / Albert, how does Shelby’s statement differ what your groups’ statement?
18. / Albert / My group said that the shorter side had to be longer than the longest side. I guess that doesn’t really make sense. We wanted to say that they had to add up to be more but that isn’t what we really said.
19. / Mrs. S / It can be challenging to say exactly what we mean, especially in math class. That is why it is so important to learn the vocabulary and use the correct terminology when we are talking or writing about mathematics.
20. / Mrs. S /
The diagram shows two different pathways one could take to get from point A to point B. Within your groups, discuss these diagrams in the context of today’s activities.
21. / Mrs. S / [After about one minutes, Mrs. S selects and equity stick from the cup]
Maria, what did your group come up with?
22. / Maria / We said that segment AB was the line we drew on our paper. We were group A, so this would have been six inches long. The other two segments were the ones we drew on the patty paper.
23. / Mrs. S / How do the distances traveled compare? Who walks farthest, the person who travels along pathway 1 or the person who travels along pathway 2?
24. / Maria / Pathway 2
25. / Mrs. S / How long was pathway 2 for your group?
26. / Maria / Seven inches
27. / Mrs. S / [To the class]
Was Group A able to create a triangle using their three segments?
[The majority of the class answers in the affirmative]
28. / Mrs. S / How do segments AC and CB compare with segment AB in the diagram?
[Mrs. S chooses an equity stick]
Xavier
29. / Xavier / They are shorter.
30. / Mrs. S / Anything else?
31. / Xavier / I’m not sure.
32. / Mrs. S / [to the class]
Is there any other important relationship between these segments and segment AB?
[Maria raises her hand] Maria
33. / Maria / They add up to be more than AB. That is why the second path is longer.
34. / Mrs. S / Very well done. Has anyone ever heard the saying “The shortest distance between two points is a straight line?” [Some students nod their heads]Can you see this saying in action here? The path that is the shortest is represented by segment AB. The other path requires the traveler to first walk along segment AC and then along segment CB making the total trip longer than if they had taken the direct route from A to B.
35. / Mrs. S / Let’s do something a little bit different. So far you have been given three segment lengths and you were asked to determine whether you could construct a triangle using these segments. Now I’m going to give you the lengths of two segments and ask you to find the conditions for the length of the third side that will allow us to construct a triangle.
[Mrs. S refers the students to Exercise 1 on page 110]
36. / Mrs. S / [As the students grapple with this task, Mrs. S walks from group to group stopping periodically to ask questions such as: What do you remember about the conditions under which we can construct a triangle?, Do you know which side is the longest? Do you know which side is not the longest? Could the unknown side be the longest? When Mrs. S is confident that all groups have come to the correct conclusion she pulls the students back together.]
36. / Mrs. S / What are the whole number lengths for the remaining side?
[Mrs. S pulls an equity stick from the cup]
Tobias
37. / Tobias / 4, 5, 6, 7, 8, 9, 10, 11, 12
38. / Mrs. S / Why couldn’t we choose a whole number greater than twelve?
[Mrs. S pulls an equity stick from the cup]
Elizabeth
39. / Elizabeth / Because five plus eight is thirteen and this sum has to be greater than the remaining side.
40. / Mrs. S / Why couldn’t we choose a whole number less than four?
[Mrs. S pulls an equity stick from the cup]
Samantha
41. / Samantha / Because it would be too small.
42. / Mrs. S / That is a true statement. Can you explain why it would be too small?
43. / Samantha / Because it wouldn’t make a triangle.
44. / Mrs. S / True again. But why won’t it make a triangle. Why can’t we construct a triangle with segments of length 3cm, 5cm, and 8cm?[Mrs. Sutton writes 3cm, 5cm, 8cm” on the SmartBoard]
45. / Samantha / Oh! Because three and five adds up to eight and it needs to be more than eight to make a triangle.
46. / Mrs. S / Exactly.
47. / Mrs. S / Is it possible for triangles to have side lengths that aren’t whole numbers?
[All members of the class indicate that it is possible]
48. / Mrs. S / How can we restate the restrictions on the third side in this example assuming the side lengths are not limited to whole numbers?
In this example, when we assumed the largest side was 8 cm long, how long could the unknown side be? Turn and tell your partner. If you both agree, raise your hands.
[Mrs. S calls on a student whose hand is raised]
Elijah
49. / Elijah / It has to be bigger than three because if it were equal to three then they two short sides wouldn’t add up to more than the long side.
50. / Mrs. S / Excellent. How can we write this? What notation do we use to indicate greater than?
[Mrs. S selects a student whose hand is raised]
Noah, what variable would you like to use to represent the length of the third side?
51. / Noah / Um, I guess x.
52. / Mrs. S / Using the variable x, please write what we know about the length of the third side using the appropriate notation.
[Noah writes on the board]
53. / Mrs. S / Do we all agree? [Students indicate they do by giving thumbs up]
Thank you Noah[Noah returns to his seat]
54. / Mrs. S / What if the longest side of the triangle were the unknown side, how long could this be?
[Mrs. S calls on a student whose hand is raised] Tiffany
55. / Tiffany / It can’t be more than thirteen.
56. / Mrs. S / Does this mean that it can be equal to thirteen?
57. / Tiffany / No. What I meant to say is that it has to be less than thirteen.
58 / Mrs. S / Can you write this restriction on the board using the appropriate notation?
58. / Tiffany / Yes. [Tiffany walks to the board writes ]
59. / Mrs. S / Do we all agree? [Students give thumbs up]
Can we write both conditions as one statement?
[As the students write the condition in their notes, Mrs. S walks from group to group]