Missing a Leg (Day 1)

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SWBAT use the Pythagorean theorem algebraically to find missing leg lengths.

Big Idea

The Pythagorean equation is skillfully used to find missing sides that are not the hypotenuse.

Launch: Square Roots and Squaring Task

20 minutes

The purpose of today's Launch is to review both squaring and square roots of numbers. I want to make sure that my students have sufficient fluency to successfully solve problems with the Pythagorean TheoremI start the lesson projecting this table on the board and asking my students to number a piece of paper from 1 to 7 for each question in the task. (An alternate way of starting the lesson would be to print Launch table with questions and hand one copy to each small group of students, preferably pairs of elbow partners.) 

I plan to ask each small group to identify and list as many pairs of expressions that satisfy each question as they can find in the table. To add a little spark, I tell the class that students who find ALL possible answer-pairs will be rewarded (there are 12 possible answers). I try have a bag of lollipops prepared for this lesson. 

As my students work I walk around checking to see if a group has found all the solutions without mentioning how many there are. If I see any mistakes, I will ask the group to correct a mistake, but I will not help them find it.

I make sure that the responses to each of the 7 questions are shared with the whole class and briefly explained on the board. I usually call on volunteers to go up to the board for this. When this process is complete, I give a lollipop to all the groups that obtain the 12 correct solutions. 

The content in this exercise should not be new to students, but if I find that certain students struggle though this, I take note and ask that they remain for enrichment after class or after school. It is very important that students become fluent with handling square roots and squaring of numbers to be successful with solving right triangles.  



Activity 1: Beginning to Solve

20 minutes

The Beginning to Solve Activity works well when students work in small groups, preferably pairs with similar ability levels. Question 1 revisits the geometric concept of the Pythagorean Theorem based on a popular area proof. Options b, c, and d portray the theorem correctly.

In Question 2 students engage in algebraic work to apply the Theorem to different expressions. In this question, I want my students to see that options "a" and "e" can be used to solve for legs "z" and "x", respectively. Some students will always use option c, which is the classic way of writing the Pythagorean equation, (a2  + b2  = c2, where a and b are the legs) to find any side of the triangle.

a)    y2  –   x2  = z2

b)    y  –  x  =  z2

c)     x2  + z2  = y2 

d)    z2  +  y2  = x2

e)    y2  – z2  = x2

In Question 3, I want my students to recognize option 2 and 5 as equations solved for the legs of the given triangle. Option 6, again can be used to find any side but it's solved for the hypotenuse.

  1. 62  + 102  = 82
  2. 102  –  62  = 82
  3. 82  – 6=  102
  4. 82  – 6=  10
  5. 102 – 82 = 62
  6. 62  +  82  = 102

The previous 3 questions lead up to Question 4. Students should analyze each process and recognize a and d, as the correct ones leading to the correct side length, 12. Once again I show that the "classic" Pythagorean Equation, a2  + b2  = c2 can be used to find any side, including the leg lengths. 

Activity 2 and Closure

20 minutes

For Activity 2 I will ask pairs of students to form groups of four. With each foursome, each student will still be work with their former partner. I hand each group a set of Task Cards and ask that they analyze and discuss the problem that they are given. To hold each student accountable, I tell the students that each should have the work in their notebooks. I will only be collecting one sheet with both problems worked out for each group.

Since I plan to reuse these cards, I ask students not to write on them. I also tell the students that I will be calling on two students at random to walk up to the board and explain the problem to the class.

In both problems, the length of a leg must be found. Each problem includes a Pythagorean Triple. I will try very hard to have enough time for explanation and demonstration of the problems, so if I see that time is running out, I ask that each group finish one of the tasks only.


As usual, I give an appropriate amount of Homework.  Today's homework gives students practice with solving problems with Pythagorean Triples.