Download poster problems

POSTER PROBLEMS
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.
The way this works: one lesson in six phases
• 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 Conventions 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 Constraints. The main point is that two
triangles are the same if they have the same size and shape.
Another 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.
Common Core State Standards for Mathematics:
7.G.A.2
Teacher Tune Up:
See Triangles and Constraints
See Triangle Conventions and Mechanics
LAUNCH
Teachers set the stage by leading an introductory discussion that
orients students to the context of the problem.
POSE A PROBLEM
Day 1
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.
FLEXIBLE
Challenges
POST, SHARE, COMMENT
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).
STRATEGIC TEACHER-LED DISCUSSION
Day 2
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. By doing this, teachers
emphasize standards-aligned mathematics using studentgenerated examples.
FOCUS PROBLEM: SAME CONCEPT IN A NEW CONTEXT
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. LAUNCH
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. An animator would color the
triangles and use more of them to make smoother, more realistic
heads.
Ask, “What do you need to know to determine a triangle? After
all, rectangles are easy: if you know the height and the width, you
have the rectangle. But what do you need to know for a
triangle?”
POSTER PROBLEMS - T RIANGLES T O ORDER SLIDE #1
Slide #1
Show Slide #2:
Ask: what are all the numbers on this diagram? What do
they mean?
• 5 cm is the length of AB and AC.
• 7.1 cm is the approximate length of BC.
• 45° is the measure of ∠B or ∠C.
• 90° is the measure of ∠A.
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. (A “clue” is one of the numbers on
the slide.)
Slide #3
SERP 2014!
!
!
Triangles To Order - Seventh Grade Poster Problem !!
!
!
2
2. POSE A PROBLEM
Directions for teacher:
Make sure students have rulers, protractors, scissors,
scratch paper.
The Form
• 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.
Cut these out
• 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 AB, AC,
or BC. Those are side lengths.
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.
AC
? cm
? cm
A
?º
?º
BC
B
? cm
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
Angles (with °) go where you see ∠A, ∠B, or ∠C.
•
• The group must agree about whether the specifications tell
you what you need to make the triangle.
1
?º
AB
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.”
•
Triangle Conditions Handout
C
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.)
•
Student Name: ____________________________________________________
Triangles to Order
2014 http://math.serpmedia.org/diagnostic_teaching/
Handout #1
Student Name: ____________________________________________________
Triangles to Order
Triangle Record Sheet Handout
2
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.
C
AC
BC
B
A
AB
With these specifications, we could make
❏ no triangles ❏ two triangles
❏ one triangle ❏ many triangles
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.
C
AC
BC
B
A
AB
With these specifications, we could make
❏ no triangles ❏ two triangles
❏ one triangle ❏ many triangles
2014 http://math.serpmedia.org/diagnostic_teaching/
Handout #2
SERP 2014!
!
!
Triangles To Order - Seventh Grade Poster Problem !!
!
!
3
3. W ORKSHOP
Directions for teacher:
Arrange 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?
As students report, be alert for misunderstandings:
• Be sure that students always fill three cells out of the six
available on the form.
• Be sure sides and angles go in the right slots.
• Be 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.
And 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 SSA situation—one with two sides
and the angle that is not between the two sides, e.g., AB, AC,
and ∠B. 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
Student Name: ____________________________________________________
Workshop & Poster Handout
3
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. And
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?
2014 http://math.serpmedia.org/diagnostic_teaching/
Handout #3
SERP 2014!
!
!
Triangles To Order - Seventh Grade Poster Problem !!
!
!
4
4. POST, SHARE, COMMENT
Directions for teacher:
A
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 A, 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.”
But 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 B 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.
B
C
Poster C 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 “SAS,” the situation with two sides
and the included angle.
SERP 2014!
!
!
Triangles To Order - Seventh Grade Poster Problem !!
!
!
5
5. STRATEGIC TEACHER-LED DISCUSSION
Directions for teacher:
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. But 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. (“Any time you have side
AB with angle A and angle B [and A + B is less than 180°], just
draw side AB first, then make the two angles. Where the anglelines meet is point C. Connect them up to make your triangle.”)
Students may explain situations with two or more triangles.
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.
• Angles too large: the sum of two angles exceeds 180°.
• Swing-and-a-miss: suppose you have AB = 3 cm, BC = 4
cm, and angle C is 90°. If you make BC first, then angle C,
it’s clear that AB can’t “reach” to the other side.
And 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 (SAS). Always 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 (ASA). Works if the angles
aren’t too big. Draw the side first.
• Two angles and a different side (AAS). Hard unless you know
about 180°; guess and check.
Three angles: the miracle of 180°
Level 5: Students recognize that it’s not the specific points and
labels that matter, but rather the relationship between them. (“Any
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.
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. All the triangles will be the
same shape, but of any size. They’re “similar” instead of
“congruent.”
Supporting Generalization
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. Another, 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.
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, “Are these specifications
the same sort as the others we’ve seen, or are they different?
Could 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.
How you get two possibilities
One way is on Handout #4, problem #5. See the answers on
the next page.
Connecting across groups
As 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. FOCUS PROBLEM : SAME CONCEPT IN A N EW CONTEXT
Directions for teacher:
Pass out Handout #4 for this problem (“Zero, One, Two, Many”)
Student Name: ____________________________________________________
Triangles to Order
Zero, One, Two, Many Handout
4
Instructions
Answers
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. And if 0, explain why it’s
impossible to make such a triangle (a diagram can help!).
1.
One triangle (ASA).
2.
One triangle (SAS).
3.
One triangle (Hypotenuse-leg; angle C is 90°).
1.
2.
C
C
AC
BC
45 º
A
AB
4.
One triangle (Two angles give a third—60°—and you have
ASA. This result is similar to #3).
90 º
6.
Two triangles, at last! This is SSA. See the illustration
below. If you make side AB (6 cm) and angle A (15°), you
now have to figure out side BC (3 cm). Imagine it swinging
from point B. It can rest its end on the line coming out of
A in two places, marked C1 and C2. Either of these could
be the third vertex of the triangle.
No triangles. Sum of angles is too great.
7.
Many triangles, all similar (Sum of angles is 180°).
8.
No triangles, because of the triangle inequality: the two 2cm sides are not long enough to meet opposite the 50cm side.
B
60 º
A
2 cm
AB
3 cm
C
30°
4.
C
3 cm BC
AC
5.
B
3.
4 cm BC
AC
A
B
30 º
AB
AC
BC
6 cm
AB
5.
B
30º
90°
A
5 cm
6.
C
C
3 cm BC
AC
A
AC
B
15º
AB
6 cm
C
75°
A
BC
120º
6 cm
AB
7.
B
90º
8.
C
AC
A
BC
60°
45°
B
AC
2 cm
BC
2 cm
B
A
AB
AB
5 cm
2014 http://math.serpmedia.org/diagnostic_teaching/
Handout #4
More about Answer #5:
SERP 2014!
!
!
Triangles To Order - Seventh Grade Poster Problem !!
!
!
7