Lesson 16

This warm-up encourages students to reason algebraically about various computational relationships and patterns. While students may evaluate each side of the equation to determine if it is true or false, encourage students to think about the properties of arithmetic operations in their reasoning.

Seeing the structure that makes the equations true or false develops MP7.

Display one problem at a time. Tell students to give a signal when they have an answer and a strategy. After each problem, give students 1 minute of quiet think time and follow with a whole-class discussion.

In the first question, students may think you can round or adjust numbers in a subtraction problem in the same way as in addition problems. For example, when adding \(62+28\), taking 2 from the 62 and adding it to the 28 does not change the sum. However, using that same strategy when subtracting, the distance between the numbers on the number line changes and the difference does not remain the same.

Ask students to share their strategies for each problem. Record and display their explanations for all to see. Ask students how they decided upon a strategy. To involve more students in the conversation, consider asking:

• Do you agree or disagree? Why?
• Who can restate ___’s reasoning in a different way?
• Does anyone want to add on to _____’s reasoning?

After each true equation, ask students if they could rely on the reasoning used on the given problem to think about or solve other problems that are similar in type. After each false equation, ask students how we could make the equation true.

In the previous lesson, students conjectured that the measures of the interior angles of a triangle add up to 180 degrees. The purpose of this activity is to explain this structure in some cases. Students apply \(180^\circ\) rotations to a triangle in order to calculate the sum of its three angles. They have applied these transformations earlier in the context of building shapes using rigid transformations. Here they exploit the structure of the coordinate grid to see in a particular case that the sum of the three angles in a triangle is 180 degrees. The next activity removes the grid lines and gives an argument that applies to all triangles.

Some students may have trouble with the rotations. If they struggle, remind them of similar work they did in a previous lesson. Help them with the first rotation, and allow them to do the second rotation on their own.

Ask students how the grid lines helped to show that the sum of the angles in this triangle is 180 degrees. Important ideas to bring out include:

• A 180 rotation of a line \(BC\) (with center the midpoint of \(AB\) or the midpoint of \(AC\)) is parallel to line \(BC\) and lays upon the horizontal grid line that point \(A\) is on.
• The grid lines are parallel so the rotated angles lie on the (same) grid line.

Consider asking students, "Is it always true that the sum of the angles in a triangle is \(180^\circ\)?" (Make sure students understand that the argument here does not apply to most triangles, since it relies heavily on the fact that \(BC\) lies on a grid line which means we know the images of \(BC\), \(AE\) and \(DA\), also lie on grid line that point \(A\) is on.)

It turns out that the key to showing the more general result lies in studying the rotations that were used to generate the three triangle picture. This investigation is the subject of the next activity.

The previous activity recalls the \(180^\circ\) rotations used to create an important diagram. This diagram shows that the sum of the angles in a triangle is \(180^\circ\) if the triangle happens to lie on a grid with a horizontal side. The purpose of this activity is to provide a complete argument, not depending on the grid, of why the sum of the three angles in a triangle is \(180^\circ\).

In this activity, rather than building a complex shape from a triangle and its rotations, students begin with a triangle and a line parallel to the base through the opposite vertex:

In this image, lines \(AC\) and \(DE\) are parallel. The advantage to this situation is that we know that points \(D\), \(B\), and \(E\) all lie on a line. In order to calculate angles \(DBA\) and \(EBC\), students can use either the rotation idea of the previous activity or congruence of alternate interior angles of parallel lines cut by a transversal. In either case, they need to analyze the given constraints and decide on a path to pursue to show the congruence of angles. Students then conclude from the fact that \(D\), \(B\), and \(E\) lie on the same line that \(a + b +c = 180\).

There is a subtle distinction in the logic between this lesson and the previous. The previous lesson suggests that the sum of the angles in a triangle is \(180^\circ\) using direct measurements of a triangle on a grid. This activity shows that this is the case by using a generic triangle and reasoning about parallel lines cut by a transversal. But, in order to do so, we needed to know to draw the line parallel to line \(AC\) through \(B\) and that this idea came through experimenting with rotating triangles.

Keep students in the same groups. Tell students they’ll be working on this activity without the geometry toolkit.

In case students have not seen this notation before, explain that \(m \angle DBA\) is shorthand for "the measure of angle \(DBA\)."

Begin with 5 minutes of quiet work time. Give groups time to compare their arguments, then have a whole-class discussion.

Some students may say that \(a, b,\) and \(c\) are the three angles in a triangle, so they add up to 180. Make sure that these students understand that the goal of this activity is to explain why this must be true. Encourage them to use their answer to the first question and think about what they know about different angles in the diagram.

For the last question students may not understand why their work in the previous question only shows \(a + b + c = 180\) for one particular triangle. Consider drawing a different triangle (without the parallel line to one of the bases), labeling the three angle measures \(a, b, c\), and asking the student why \(a + b + c = 180\) for this triangle.

Ask students how this activity differs from the previous one, where \(\triangle ABC\) had a horizontal side lying on a grid line. Emphasize that:

• This argument applies to any triangle \(ABC\).
• The prior argument relies on having grid lines and having the base of the triangle lie on a grid line.
• This argument relies heavily on having the parallel line to \(AC\) through \(B\) drawn, something we can always add to a triangle.

The key inspiration in this activity is putting in the line \(DE\) through \(B\) parallel to \(AC\). Once this line is drawn, previous results about parallel lines cut by a transversal allow us to see why the sum of the angles in a triangle is \(180^\circ\). Tell students that the line \(DE\) is often called an ‘auxiliary construction’ because we are trying to show something about \(\triangle ABC\) and this line happens to be very helpful in achieving that goal. It often takes experience and creativity to hit upon the right auxiliary construction when trying to prove things in mathematics.

This activity revisits a picture that students have seen earlier and that they will see again later when they investigate the Pythagorean Theorem. The four right triangles around the boundary make a quadrilateral inside which happens to be a square. Since the four triangles surrounding the inner quadrilateral are congruent by construction, this makes the inner shape a rhombus. The angle calculations that students make here justify why it is actually a square. The fact that quadrilateral \(ACEG\) is a square is not asked for in the activity, but if any students notice this, encourage them to share their observations and reasoning.

Students are making use of structure (MP7) throughout this activity, most notably by:

• Using the fact that congruent triangles have congruent angles.
• Using the fact that the measures of angles that make a line add up to 180 degrees.

Provide 3 minutes of work time followed by a whole-class discussion.

Display the image of the 4 triangles for all to see. Invite students to share how they calculated one of the other unknown angles in the image, adding to the image until all the unknown angles are filled in.

In wrapping up, note that angles \(ACE\), \(CEG\), \(EGA\), and \(GAC\) are all right angles. It’s not necessary to show or tell students that \(ACEG\) is a square yet. This diagram will be used in establishing the Pythagorean Theorem later in the course, so it’s a good place to end.