LESSON SUMMARY
PRIOR KNOWLEDGE
- Understanding the connection between squares and square roots to area of a square and side length of a square
- Solve an equation for a specific unknown
- Knowledge of types of triangles
- Proof of the Pythagorean Theorem
- Knowledge of the equation a^2 + b^2 = c^2
- Students will be able to apply the previous days lesson to determine the missing side of right triangle.
Detailed implementation guide for teachers
STARTER Students will investigate a right triangle with squares using a Geogebra applet. Students should be given 7 minutes to complete this task. Teachers will facilitate a discussion involving the questions below.
The following directions should be included and used to guide students with the applet:
1. Start by moving point F. What do you notice about the relationship between the three side lengths? What about the squares?
2. Now move point D. Although the orientation of the figure may change, do the relationships you found in part 1 still hold true? How can you tell?
Expected student responses:
The following directions should be included and used to guide students with the applet:
1. Start by moving point F. What do you notice about the relationship between the three side lengths? What about the squares?
2. Now move point D. Although the orientation of the figure may change, do the relationships you found in part 1 still hold true? How can you tell?
Expected student responses:
- the sum of the two side lengths squared is equal to the hypotenuse squared; the green square plus the orange square is equal to the blue
- yes, it doesn't matter how the figure looks the sum of the two sides squared is equal to the hypotenuse squared
- by using the formula we can see that the relationship is the same as before
In order to bridge the gap between investigating these types of problems with squares and determining the measure of a missing side without squares we can use these examples.
Using this slideshow teachers should walk students through the first problem on each of the first two slides and demonstrate how to find the missing side length. The values given in each figure represent the side lengths not the area of the squares.
The class should begin each problem by writing the Pythagorean Theorem and then fill in the corresponding side lengths. This allows students to get into the habit of using the formula and will help them remember it more quickly.
Questions to pose to students:
The class should begin each problem by writing the Pythagorean Theorem and then fill in the corresponding side lengths. This allows students to get into the habit of using the formula and will help them remember it more quickly.
Questions to pose to students:
- What is the area of the squares with known side lengths?
- How will knowing the area of these squares help us find the missing side length/hypotenuse?
- What can use you to determine that missing side length/hypotenuse?
- The area of the squares is the side length squared or times itself (e.g. Side length = 4 , so the area of the square is 4^2 = 4 * 4 = 16).
- The Pythagorean Theorem formula is a^2 + b^2 = c^2. This means that the sum of the two side lengths squared is equal to the hypotenuse squared.
- We can use the Pythagorean Theorem to help us solve this. Substitute the given values into the formula and solve the equation for the unknown.
Once students have examined Pythagorean Theorem problems with squares they should transition into solving these types of problems without squares. The instructor will take students to an interactive website that allows them to solve problems and check their answers. Students will be given a brief tutorial of the site in order to prevent confusion during the work period of the class.
Directions for the activity:
Potential student misconceptions:
Directions for the activity:
- Each person must copy down the triangle and show all work on a separate sheet of paper. Calculators are allowed.
- Start on level 1. Answer 6 questions correctly in order to advance to level 2.
- In level 2 students only need to get 4 correct in order to advance to level 3.
- For level 3, students only need to answer 2 answers correct.
- Students may choose to continue working on any of the levels once they have answered 12 answers correct.
Potential student misconceptions:
- Mistake 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.
- Forget to square the side lengths before solving for the unknown.
- Forget to take the square root of a value at the end of the problem.
TICKET OUT THE DOOR
For the ticket out the door, students can choose to work any 3 of the the 10 problems to turn in as they depart the classroom. Students must select either number 3 or 7 as one of their problems. These are the only two problems that require students to determine the hypotenuse given the measures of the two sides. Students are allowed to use calculators to help them find the square roots of numbers, but they must show all of their work.
Below is the answer key to the Ticket Out the Door.
Below is the answer key to the Ticket Out the Door.
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HOMEWORK
That night for homework students will be required to go home and watch this video demonstrating how to solve a multi-step Pythagorean Theorem word problem. They are 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 will check each students notebook the next day in order to ensure that the students watched the video.
That night for homework students will be required to go home and watch this video demonstrating how to solve a multi-step Pythagorean Theorem word problem. They are 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 will check each students notebook the next day in order to ensure that the students watched the video.
REFERENCES
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
Pythagorean Explorer. (n.d.). Interactivate:. Retrieved July 29, 2014, from http://www.shodor.org/interactivate/activities/PythagoreanExplorer/
Pythagorean Theorem Practice. (n.d.). . Retrieved July 29, 2014, from http://www.cpm.org/pdfs/skillBuilders/GC/GC_Extra_Practice_Section5.pdf
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
Pythagorean Explorer. (n.d.). Interactivate:. Retrieved July 29, 2014, from http://www.shodor.org/interactivate/activities/PythagoreanExplorer/
Pythagorean Theorem Practice. (n.d.). . Retrieved July 29, 2014, from http://www.cpm.org/pdfs/skillBuilders/GC/GC_Extra_Practice_Section5.pdf