LESSON SUMMARY
PRIOR KNOWLEDGE
- Understanding of the relationship between square numbers and square roots, including non-perfect squares
- Understanding the connection between squares and square roots to area of a square and side length of a square
- Solving equations of the form x^2=p
- Knowledge of types of triangles
- Understanding the Pythagorean Theorem as a relationship between the side lengths of a right triangle
- Students will be able to interpret, illustrate, solve real-world problems involving the Pythagorean Theorem
Detailed implementation guide for teachers
In the previous two lessons the students learned about how to prove the Pythagorean Theorem as well as how to apply it to determine the length of a missing side of a right triangle. In this lesson they will be applying their skills to creating models and solving word problems.
Lesson 3 is a flipped classroom. Students are expected to go home the night before this lesson and watch the clip below. They are be required to copy down everything that the person in the video writes on the white board. They must also write down one potential misconception they could see their peers having when analyzing this problem. The teacher briefly will check each students notebook the next day in order to ensure that the students watched the video. The teacher may choose to show the video again at the beginning of class in order to help remind students of what they watched the night before. Students will then participate in a discussion about the problem in which the teacher will act as a facilitator and answer any follow up questions the students may have.
Lesson 3 is a flipped classroom. Students are expected to go home the night before this lesson and watch the clip below. They are be required to copy down everything that the person in the video writes on the white board. They must also write down one potential misconception they could see their peers having when analyzing this problem. The teacher briefly will check each students notebook the next day in order to ensure that the students watched the video. The teacher may choose to show the video again at the beginning of class in order to help remind students of what they watched the night before. Students will then participate in a discussion about the problem in which the teacher will act as a facilitator and answer any follow up questions the students may have.
The video can also be found by clicking on the LINK.
The class activity for the day is to work on a mini-project. Students are able to work in pairs or groups of three (at the discretion of the teacher), but each person must turn in their own work to be evaluated. The project is a series of word problems in which students must draw a diagram and solve each problem. Creativity is encouraged. Students
While students are working on these problems the teacher should monitor the activity, work with individual students and groups that may be struggling, ask guided/follow up questions to students working.
If the teacher notices that multiple people have missed the same problem, he/she should stop the class and discuss that problem. A student may be asked to come to the board to demonstrate how to solve the problem correctly.
Questions that a teacher may ask individuals during the project:
Student responses:
Potential student misconceptions:
If there is time left in the period after all the students have finished the problems the teacher can bring students to the board to explain how to solve selected problems. A great problem to discuss as a class is the CHALLENGE problem.
While students are working on these problems the teacher should monitor the activity, work with individual students and groups that may be struggling, ask guided/follow up questions to students working.
If the teacher notices that multiple people have missed the same problem, he/she should stop the class and discuss that problem. A student may be asked to come to the board to demonstrate how to solve the problem correctly.
Questions that a teacher may ask individuals during the project:
- Are you looking for the length of the side or the hypotenuse?
- What in the question tells you that you are looking for the side/hypotenuse?
- Is there a way I can check to make sure that this is the correct answer? If so, how?
- How do you think this diagram would look?
- Do you think we can create a similar problem to this one to helps us solve this?
Student responses:
- I am looking for the side length/hypotenuse.
- The question says that he ran a straight route and the other guy ran horizontally, so I have my two side lengths. I am looking for the hypotenuse. (Using problem 1 as an example).
- I can substitute all three side lengths back into the equation to see if the two sides are equal. I can also substitute the squares back into the equation to check.
- I think that the picture would look like...STUDENT WOULD DRAW A DIAGRAM.
- Yes. We can change the numbers to make them easier to work with or we can change the problem to one that we have already done that is similar.
Potential student misconceptions:
- Mistaking the hypotenuse for a side of the triangle or visa versa. This will cause the students to sum the squares rather than find the difference.
- Forgetting to square the side lengths before solving for the unknown.
- Not taking the square root of a value at the end of the problem.
If there is time left in the period after all the students have finished the problems the teacher can bring students to the board to explain how to solve selected problems. A great problem to discuss as a class is the CHALLENGE problem.
Your browser does not support viewing this document. Click here to download the document.
ANSWER KEY FOR THE MINI-PROJECT
Your browser does not support viewing this document. Click here to download the document.
TICKET OUT THE DOOR
As the students are preparing to leave class the teacher should pose this question to the class. The students should answer this questions on a separate sheet of paper or at the bottom of their project.
The answers to this question will vary depending upon how students felt about the the problems in the Pythagorean Theorem Project. The goal of this Ticket Out the Door is for the students to reflect upon that day's activity. In addition,
As the students are preparing to leave class the teacher should pose this question to the class. The students should answer this questions on a separate sheet of paper or at the bottom of their project.
- Which problem was the most challenging for you? Explain in detail (including the mathematics) why you felt that way. Make your ELA teachers proud and answer in complete sentences.
The answers to this question will vary depending upon how students felt about the the problems in the Pythagorean Theorem Project. The goal of this Ticket Out the Door is for the students to reflect upon that day's activity. In addition,
References
IXL math practice - Pythagorean theorem: word problems (Eighth grade). (n.d.). IXL Learning. Retrieved July 29, 2014, from http://www.ixl.com/math/grade-8/pythagorean-theorem-word-problems
Practice with Pythagorean Theorem. (n.d.). Practice with Pythagorean Theorem. Retrieved July 29, 2014, from http://www.regentsprep.org/Regents/math/ALGEBRA/AT1/PracPyth.htm
Word Problem Exercises: Pythagorean Theorem. (n.d.). Word Problem Exercises: Pythagorean Theorem. Retrieved July 29, 2014, from http://www.algebralab.org/practice/practice.aspx?file=word_pythagoreantheorem.xml
IXL math practice - Pythagorean theorem: word problems (Eighth grade). (n.d.). IXL Learning. Retrieved July 29, 2014, from http://www.ixl.com/math/grade-8/pythagorean-theorem-word-problems
Practice with Pythagorean Theorem. (n.d.). Practice with Pythagorean Theorem. Retrieved July 29, 2014, from http://www.regentsprep.org/Regents/math/ALGEBRA/AT1/PracPyth.htm
Word Problem Exercises: Pythagorean Theorem. (n.d.). Word Problem Exercises: Pythagorean Theorem. Retrieved July 29, 2014, from http://www.algebralab.org/practice/practice.aspx?file=word_pythagoreantheorem.xml