Teaching Phase: How will the teacher present the concept or skill to students?
Use Pythagorean Theorem PowerPoint.ppt the to teach the lesson. Read the following before using the PowerPoint to help expand upon what is in the slides as you present it. If a projector is unavailable, do the following.
- Start off with an appropriate bell ringer(s) to assess the students' prior knowledge. A bell ringer is a problem that is prominently displayed in the classroom that the students are required to work on once they enter the classroom. This is usually set up at the beginning of the year as a procedure. These bell ringers can be checked as a separate assignment weekly or monthly, or they can be included in a notebook check. The numbers of questions may vary according to timing and the level of your students. Some suggestions of appropriate bell ringers appear in the formative assessment section of this lesson.
- Briefly review the vocabulary below. Ask students to write definitions or examples for each term on their own. This could be included as part of the bell ringers. Have volunteers share their ideas and record/display their definitions. Make them part of a word wall.
- right triangle
- square root
- Have a picture of a 3-4-5 right triangle on the board or display using an overhead with squares on the edges (This can also be done using Pythagorean Theorem PowerPoint.ppt the the picture is on slide 4).
Point out the right triangle to the students and ask them what the length of the bottom edge is. (3 units.)
- If the length of the bottom edge is 3 units, what do the 9 squares on the bottom here represent? (the area of a square with a length of 3, 3 squared)
- What is the length of the left side of the triangle? (4, if they answer 16, point out that 16 is the area of the square, not the edge)
- What is the length of the long side? (5, if they answer 16, point out that 16 is the area of the square, not the edge)
- Is there a special name for this longer side? (hypotenuse)
- Does there appear to be any relationship between the areas of these three squares and the triangle? What about between the squares themselves?
- If I add the squares of the two shorter sides together, what do I get? (25, the area of the largest square connected to the hypoteneuse)
- Why does that sum seem familiar? (it's the same as the square of the length of the hypotenuse.)
Some guy back in about 500 BC proved mathematically that there is a relationship here, and that it is true for all right triangles. This was roughly 800 years before Algebra was established. This guy was the Greek philosopher Pythagoras of Samos. Ever since, his name has been associated with this theorem. Technically, someone else figured this whole thing out hundreds of years before him.
What this theorem states is that the sum of the squares of the two legs of a right triangle is equal to the square of the hypotenuse. a2+b2=c2
- Let's try it with this example. 9 + 16 = 25
- What can be done with this information? (find a missing length)
- What if you wanted to measure the length of a lake?
- Can you stretch a tape measure over the lake? (Students may answer about a laser measuring device, which you may want to point out how expensive something like that is.)
- Could you measure lengths along the shore of that lake? (with a long enough tape measure, or some other device, yes)
- If you could measure two straight lengths that connect at a 90 degree angle and have end points on opposite ends of the lake, what shape would you form? (A picture may be helpful, answer should be right triangle)
Well, sometimes it is easy to measure two lengths of a right triangle, and not so easy to measure the third length. Using the Pythagorean Theorem, we could calculate the third length.
Let's say I wanted to know the length of the diagonal of a standard sheet of loose leaf paper. What are the dimensions of a standard piece of loose leaf? (Some students may point out that many loose leaf manufacturers have produced loose leaf in 8 x 10.5 inch sizes. You could use this as a teaching moment to point out how manufacturers love to make things smaller and pass them off as the same but with a new lower price, or you could just say computer paper instead, which still measures 8.5 x 11 inches.)
Could I use the Pythagorean to find the length of that diagonal? (A picture might be helpful, draw the diagonal in, if they don't see the right triangle connection. Try and get them to give you the sides as the legs.)
If one leg is 8.5 inches and the other leg is 11 inches, what do I do to figure out what the diagonal is? (Some students may incorrectly respond - "Add them together" - Although that would result in a length longer than either of the two legs, I don't think the diagonal is that long)
What did we do with the lengths of the triangle before? (Refer them to the picture with the squares.)
What do we do with those squares? (add them together 72.25 + 121= 193.25 )
Is this the length of the diagonal? (does the paper look like it has a diagonal of 193.25 inches?)
Let's look at the formula. a2+b2=c2
Try and get the students to recognize that they have calculated the left hand side of the formula and that it is equal to c squared, not just c.
How do we figure out what c is, if c squared is 193.25? (divide it by 2? - that would make it smaller, but is 96.625 a reasonable length for the paper's diagonal?)
How do we undo a square? (square root)
Have a student use a calculator to get the answer of 13.901 (rounded to the nearest hundredth)
Go back to the original 3-4-5 triangle.
What if we knew the bottom was a length of 3, and the hypotenuse was a length of 5, but did not know what the other side was?
Let's start with the formula a2+b2=c2
Where do the 3 and the 5 go in this equation? (a and b? - One of them is going to be one of those, but remember the longest side is the hypothesis, c. Looking for a and c or b and c)
Does it matter if the 3 goes in for a or b?(No, the commutative property of addition)
How do we solve from there?
What are the three basic steps to solving a Pythagorean Theorem problem.
Step 1: Write the equation. Most students will start by writing the numbers in the equation, but it is a good idea to have the students get in the habit of writing out formulas first.
Step 2: Substitute the length of the hypotenuse for c and any lengths of the legs for either a or b. You could just say substitute known values, but some additional detail doesn't hurt here. Some students might respond better to using the term "Plug-in" instead of substitute, but you want them to get used to the higher vocabulary.
Step 3: Solve the equation for the missing side.
Some lower level students might appreciate more detailed steps. This is a judgement call, since too many steps often scare the students. However, if you think they would benefit from more detailed steps, you could use the following:
Step 1: Write the equation
Step 2: Identify the hypotenuse
Step 3: Substitute or "Plug-in" the value for the hypotenuse for c and the values for the legs for a and b.
Step 4: Simplify the Exponents
Step 5: Solve the equation for the remaining squared variable
Step 6: Calculate the square root to get the length of the missing side
Technically, this all boils down to what I like to call a simple "plug'n'chug" (Substitute and evaluate an expression or equation for given variable values) operation, but some students won't see it that way.
Students will generally understand how to solve when they are given the legs and need the hypotenuse, but not when they are given the hypotenuse and have to find one of the legs this throws them off. You'll want to start with problems that have the two legs given, and then show them one where the hypotenuse is given and ask the following:
Should I add these two together?
Someone is bound to say yes. Remind them that the numbers are on opposite sides of the equal sign and you are solving for a leg. To keep an equation balanced, whatever you add to one side must be added to the other.
Someone is bound to do a2-c2 instead of c2-a2. Ask students how they can be sure which of these will yield the correct answer. Ask them to try this with the 3-4-5 triangle to see if they get ans answer that makes sence.
Have the students work on Pythagorean Theorem Worksheet.doc the . This can be started in class as either individual work, or group work for accommodations. It can also be used as a homework assignment. The answers could be posted as the following days Bell Ringers with the note "Check your Homework Answers", or can be provided at the end of class for a student self check.