Solving the Pythagorean Theorem Algebraically
Lesson 4 of 10
Objective: Understand the structure of the Pythagorean Theorem algebraically in order to solve for an unknown side length as an equation.
Bellringer – Lesson Opener
As a review of the relationship between radicals and exponents, place the following expressions on the board with the directions for students to simplify each expression including estimating the radicals without a calculator.
42 √24 √100 23 √55 (3x)2 √x2
While students are working to simplify each expression, move about the room formatively assessing students, providing feedback to struggling students, and choosing correct work for students to share on the board to each question. If possible, choose students to present who wrote or thought about the expressions in different ways in order to view a variety of thinking. Have each student present their own work and explain their thinking.
Another great method of assessing every student, is to have students use dry erase markers and marker boards to show their work. You could show the expressions one at a time and put one minute or thirty seconds even on a clock and allow only that much time to simplify then hold up their boards when the time was up. This way, you can quickly see everyone’s thinking without needing to walk about the room so much. The marker boards are faster and you can keep a scratch sheet of paper to list the names of all students who seem to struggle through the bellringer as your formative assessment.
Again, because the activity today is focused around connecting the Pythagorean Theorem to solving equations, this connection to radicals and exponents as inverse operations is important to establish again as the lesson opener. Make sure during your whole group discussion time, you ask explicitly, “What is the relationship between radicals such as a square root and exponents, such as squaring a number?”
Beginning the Activity
Click here to watch this video: Solving Pythagorean Theorem Algebraically, that explains the rational for this lesson and why I have designed the opening of the lesson as evaluating previous student work.
Clarifying and Sharing Learning Goals
Always begin by clarifying for the students what it is they will be learning from the activity today.
The learning question for today is: “How do I think about using the Pythagorean Theorem algebraically to solve for missing side lengths?” I want to push as many students as possible into thinking about the Pythagorean Theorem algebraically and connecting the solving for a missing side length to solving equations, even liner equations as in previous unit. However, I always have some students who are just more successful with visual strategies. I do not force my students to write and solve the Pythagorean Theorem algebraically. Many visual learners like to draw the squares on the diagram and still consider area, moving area of squares, and then finding the side lengths of a square to find the missing sides of the right triangle. I allow these students to continue using this strategy, which is why I have allowed so much exterior space around each diagram. It is my hope that most students will begin to transition into solving algebraically as an equation, but I do still allow choice in solving. Later when trying to develop the distance formula from the Pythagorean Theorem, a sound algebraic understanding of the Pythagorean Theorem is very useful. In high school geometry, students will be required to use the distance formula algebraically because it is also directly connected to the equation of a circle as I mentioned in the lesson rational video.
Starting the Activity
Once your expectations and learning goals are clear, allow students about three to five minutes at the most to work within their cooperative groups to answer questions one and two. Students must decide which student has worked the problem correctly and then describe in full sentences the exact mistakes of the other two students. It is important for students to write in full sentences and coherent thoughts what the specific mistakes are for the two incorrect students. Again, a goal of this activity is to bring to light all the most common mistakes students make, so that your current students will not repeat these incorrect actions.
As students are working within groups to complete questions one and two, move about the room providing feedback that moves learning forward. It is common for some groups to say, “I don’t know who is right. They all look the same.” I usually ask these groups what is the correct answer to the question? Have you worked out the problem your way yet to consider who might be correct?” Usually these groups have not begun to solve the problem for themselves and this suggestion is enough to get them started in a productive direction.
Also, as you move about the room from group to group, select students or groups for presentation during the mini-wrap-up session. Choose student work that is correct and possibly worked or explained in different ways.
After allowing about five minutes for groups to working through questions one and two bring the whole class back together and hold a mini wrap up for students to share what they have learned. This mini wrap up should be student lead and student focused so students feel that they own their own learning and can discuss and defend it to others.
I call this time of students presenting their work a “mini wrap-up” because I do not spend long periods of time closing a lesson at the end of the class period. We use small lesson closers after a small chunk of material has been completed.
Throughout the mini-wrap up, either the student groups or you should be scripting important ideas from the presentation onto the whiteboard for preservation. Many students zone out or cannot follow verbal explanations because there are visual learners. Scripting important ideas on the board for all students to use a guide map throughout the class period is very important. For further explanation of scripting, click on the follow link to watch a short video about how and why I believe scripting is vital to student success.
Students zone out or are visual learners. Helps students to see a map of suggestions as they work – a resource for one another strategy. No excuses when say can’t get started. Good to take picture of it and post to teacher website for the day’s notes.
Working in Cooperative Groups to Make Connections Continued
Once you conclude a mini wrap up of student work, allow students about five minutes to complete questions three, four, and five. Prepare students for these questions by saying, “The following two questions are scaffolded questions designed to help you ease into setting up and solving the Pythagorean Theorem algebraically. The blank lines are missing values in each step of solving the Pythagorean Theorem as an equation. Students should work together to solve these equations as you also move about the room providing feedback. If you are unsure of why I allow students to work together or how students are expected to work in groups, then click on the following link to watch a short video explaining this strategy.
After allowing students time to finish questions three, four, and five pull three student groups to the board to present their work as a mini wrap up for the class. Make sure either the students or you script the entire solution onto the white board for notes and reference as a map throughout class. This scripting can lead to a great discussion question right about now, “Is the Pythagorean Theorem equation a linear equation? If it is not linear, what type of equation is it?” (Quadratic equation because of x squared). “What is different about the solving process for this non-linear equation vs. solving linear equations?” This is a good connection to the main theme of eighth grade math, linear algebra. If you run short on time to fit this discussion in properly, you can always begin class tomorrow with one example and this series of questions for class discussion.
Where to End the Lesson
I try to make it to question six which contains practice problems solving for the hypotenuse, with about five to ten minutes left in class so that students can begin to work together to solve for the missing hypotenuse. Homework is to finish all of the practice problems in question six. I again allowed extra white space around each diagram for students who plan to draw the three squares on each side in order to solve. I encourage the algebraic equation method but do not force it. For my classes with technology at home, I often require students to take pictures of their homework answers to question six and post these images through their Edmodo account to the rest of the class. I then require students to logo into Edmodo and also post at least two comments to another student that night in Edmodo.com.
Here are a few images of sample work produced by my students throughout this school year.