Left-Over Squares

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SWBAT justify statements in a proof about area. Students will understand why the Pythagorean Theorem is about area and how it can be used to find the missing side of a triangle.

Big Idea

Students analyze a transformation proof for rigor and learn that a famous theorem means more than "a squared plus b squared equals c squared".

Lesson Open

8 minutes

The Warm-Up prompt checks to see if students understand the Pythagorean Theorem.  I am on the lookout for students who assume that they can simply add the lengths of the legs to get the length of the hypotenuse.  I ask students to follow our Team Warm-up routine, which involves sharing their responses with the other members of their cooperative learning team.  I choose students at random to write the team's chosen response on the board.  

I use the team answers to highlight misconceptions and focus attention on the correct meaning of the Pythagorean Theorem.  I display a slide in the slide show and use cold calling to check for understanding.  (This is also an opportunity to see if students recall how to find the side of a square given its area.)

Students will no doubt recall that the Pythagorean Theorem is usually used to find the missing side of a triangle.  I agree.  The squares on the hypotenuse of the triangle in the warm-up problem and the check for understanding both have areas that are perfect squares.  I ask students to find the lengths of their sides.

I display the Agenda and Learning Goals Slide for the lesson and briefly review it with the class.

Re-Creating a Proof

15 minutes

In this section, students work in teams to re-create a transformation proof of the Pythagorean Theorem.  The activity uses a Team Jigsaw format. I distribute the handout for the activity and make sure students know where they can obtain scissors and glue. I give the instructions for the activity with the help of a slide in the slide show.

As students get to work, I display a digital timer.  Students will probably enjoy this part of the lesson, but it is important to keep it to a time limit.  The focus should be on using reason to analyze the proof and evaluate its claims for rigor in the next section.  If a team is stumped, I invite them to send out a 'spy' to see what other teams are doing.  After 5 minutes or so, I use a document camera to display student work showing the solution to the first re-assembling task.

Things to Be On the LookOut for (BOLOs):

  • Did the Cutter read the directions and measure the base and height (legs) of the triangles before cutting them out?  It helps if the student writes the lengths of the sides in the interior of each triangle, since they may get smaller when cut.
  • Are students stumped when they have to find the area of the large left-over square (the square on the hypotenuse)?  Since the sides of the large left-over square are at an angle to the grid-lines, students cannot easily use the unit grid to measure the lengths of the sides of this square (however, see below).  The idea is for them to see that, since both the square on the hypotenuse and the squares on the legs are left over when the single large square (filled with the unit grid) is covered by four right triangles, they must have the same total area (MP2).  This is the argument that proves the theorem (MP3).  I say: Both the single square on the hypotenuse and the two smaller squares on the legs represent area left over when the same large square is covered with four triangles.   What does that say about their total areas?
  • Do students use the unit grid to find the length of the hypotenuse of the right triangles and square it to find the area of the large left-over square?   I say: You have used a smart strategy to find the area of the square, but why isn't it suitable for a proof?  Do all right triangles have sides with these lengths? (MP3)  I talk the team onto the argument described above.  I may share their results with the class, however, since a concrete example may actually be more convincing for some students than the logical argument I want them to understand.  


Is It Rigorous?

20 minutes

In this section, students work in pairs to evaluate the rigor of the proof.  The activity uses a Rally Coach format. I begin by reviewing the previous activity with the whole class to ensure that students understand why the demonstration proves the Pythagorean Theorem.  I may invite teams to share some of their learnings or insights.  I use a slide in the slide show, which contains a hyperlink to a website with an animated demonstration.

The animation shows that the transformation works for any right triangles, but--I ask students to be super-skeptical for a minute--how do we know that the left-over regions are really squares?  (MP3)  (What is a square?  How do we know these sides are all the same length?  How do we know these acute angles form a right angle?)  I invite discussion.  (One way I may do this is by using a game called Math Ball.)

If the major points come out in a whole-class discussion, the activity can just be used to summarize and check for understanding, if there is time remaining.

I distribute the handout for the activity and give the instructions with the help of a slide in the slide show.

Things to Be On the LookOut for (BOLOs):

  • Are students referring to their course notes?  If necessary, I will suggest where to look.
  • Are students having trouble showing that adjacent acute angles sum to 90 degrees (or leave a 90 degree angle when the sides of the right triangles are aligned to form a side of the large square)?  I suggest labeling the acute angles of the right triangles with a pair of variablers, such as x and y.  What equations can we write using our knowledge of triangles and angles? (MP2, MP7)


Applying the Theorem

10 minutes

In this section, students work in pairs to apply the Pythagorean Theorem.  The Activity uses a Rally Coach format.  I display a slide with the instructions as students get to work.

I ask students to draw squares on the sides of  the triangles, because I want to reinforce the meaning of the theorem.  This turns out to be good differentiation, and I will often see students drawing squares on right triangles weeks later.

Lesson Close and Homework

5 minutes

The lesson close follows our Individual Size-Up Routine.  The prompt asks students to recap the argument they used to show that the acute angles of a right triangle can be put together to form a right anle into their Learning Journals.  


For  homework, I assign problems #4-6 of Homework Set 1 for this unit.  Problems #4-5 ask students to practice applying the Pythagorean Theorem to find the missing side of a triangle.  Problem #6 asks students to re-call one of the arguements they used to justify the claims in the proof of the Pythagorean Theorem they analyzed in class.  The second part of this problem is a challenge for students who are ready to use algebra in a related proof of the Pythagorean Theorem.