LessonPlan,PythagoreanTheorem

** Pythagorean Theorem || || . || This lesson uses the cognitive learning theory to drive the design of the lesson as it progresses from the simple or previously known information to new and more complex knowledge. Four well known cognitivists’ ideas are employed in this lesson: Gagne’s nine events of instruction teach skills, Paivio’s use of visual and verbal cues, Gartner’s intelligences and finally, Bloom’s Taxonomy. || ||  . || ** The intended grade levels are 6 – 8. || ||  . || 1. Students will be able to draw and identify a right triangle, with its two legs and hypotenuse. 2. Students will be able to independently demonstrate the relationship between squares and a right triangle to prove the theorem using diagram and computer models. 3. Students will be able to find the length of the sides of a right triangle by using the theorem. 4. Students will be able to work cooperatively in pairs: a. to solve problems together b. to search on the web for examples of real-life problems that use the theorem. c. to create a two page PowerPoint with an adapted real-life problem, provide a diagram or picture, solve it, and later share with classmates || || . || · Pythagorean Theorem Activity (handout), one per student · Extra Pythagorean Problems (handout) · Helps for Pythagorean PowerPoint Project – on class website · Pythagorean Theorem PowerPoint –on class website · Homework Sheet –on class website · Scissors · Projector · White board, markers, erasers · Calculators, one per student · Teacher’s computer · Reserve mobile PC computer lab, one computer per two people || || . || · Handouts needs to be printed prior to class · PowerPoint and all materials related to lesson must be loaded onto teacher’s class website prior to class · AUP policy will be reviewed as it pertains to the use of computers, information, and pictures from the WWW for their PowerPoint. · Access to computers will be by a wireless, mobile computer lab which was reserved in advance. · Students needing help in making a PowerPoint will be identified in advance and paired with a student in their cooperative group who is able to facilitate their learning of technology skills || || . || The students’ computer skills were assessed in the beginning of the school year. I also have utilized the technology department to ensure students experience success and confidence in making their PP. We have come up with a PP help sheet that will guide those needing aid so class time can be spent on the new theorem, not PP basics. Differences in student abilities can be compensated by careful matching up of students in cooperative pairs. The partners will take on a role of facilitator. The review of squares, right triangles, squaring and square roots will make all students feel more comfortable tackling a new theorem today. The PP on Pythagorean Theorem as well as websites used today are available in the classes’ website. This is for review and strengthening newly learned skills. I have used 7 of the 8 intelligences in this lesson plan and feel this diverse approach will best meet the varied intelligences of my class. || ||  || Slides 1:  ** For today’s lesson, we will be using an interactive PowerPoint. We will be learning about a new mathematical theorem called the Pythagorean Theorem. About 2,500 years ago, Pythagoras, a Greek mathematician, made an amazing discovery about the relationship between right triangles and squares. It appears the Egyptians and Babylonians had already understood the same relationship about a thousand years earlier, but Pythagoras gets the credit since he is recognized as the first man to have //proven// this relationship. ** YOU **, a mathematical archeologist, are rediscovering the past. This ancient theorem has recently been rediscovered on a fragile clay tablet ( SHOW TABLET on **slide 4** ) and sent to the National Archeological Museum in Athens, Greece. On the tablet are some drawings. The museum curator has asked for your mathematical expertise to uncover the mysteries of the past. A passport is readied and off you go to Greece. Once you arrive you are quickly escorted to a private lab where you are given replicas of images from the tablet. Look carefully at these shapes. Let’s see if you can figure out this relationship on your own based upon your own knowledge of squares and triangles. Slides 5 – 8: Go through slides 5 - 8 to connect with what they just did.** Ask: How many saw the relationship? (Now have students correctly model the Pythagorean theorem using the shapes.) Ask a student or two to verbalize what they just did. ** Like Pythagoras, you have just proven the Pythagorean Theorem! ** Access 2 websites on the whiteboard: 1. [] This **first site** show shows the sum of the two shorter legs/ sides will equal the third and longest side. Ask for student volunteers to work the interactive site on the white board. Ask for a couple students to note their observations. 2. [] This **second site** shows that no matter how the right triangle gets stretched the sum of the two squares equals the third. Again, ask for volunteers to manipulate the shapes on the whiteboard and offer observations. ||
 * Lesson Plan Template (60 minute lesson) **
 * Lesson Topic:
 * Lesson Topic:
 * The instructional design for this lesson plan is informed by the following learning theory: **
 * The instructional design for this lesson plan is informed by the following learning theory: **
 * Grade Level or Intended Audience:
 * Grade Level or Intended Audience:
 * Learning Outcomes: **
 * Learning Outcomes: **
 * Materials and media required: **
 * Materials and media required: **
 * Consideration for materials and media: **
 * Consideration for materials and media: **
 * Learner Analysis: **
 * Learner Analysis: **
 * Introduction of topic/gain attention: (5 min)
 * Introduction of topic/gain attention: (5 min)
 * Students ** will be handed //Pythagorean Theorem Activity Handout// with cut-outs for three-squares, right triangle, and another triangle. Hand out //scissors// as well.
 * Slide 3: **
 * Instructions:** Cut out the four shapes. Arrange the shapes in such a way to see if there is a relationship between side lengths of the of the squares measure up to side lengths of the triangle. (Circulate among students, compliment on effort, but do not show relationships, let them figure it out.)**
 * PA Academic Standards: **


 * Mathematics **2.8.8.D Use concrete objects to model algebraic concepts.

2.3.8.E Describe how a change in linear dimension of an object affects its perimeter, area and volume. || || . || · Having just discovered the proof for Pythagoras’ theorem, you will be to solve modern, real-life problems using a calculator and mental math · We will use what you already learned about right triangles, squares, and square roots to solve the Pythagorean Theorem. · All throughout the lesson, I will be observing your effort individually and in small group tasks. Points can be earned towards tomorrow’s assessments final score. Effort is rewarded by **2 extra points** when you work individually on your cutouts. Effort is rewarded by **2 extra points** when you work well in your cooperative groups. · At the end of today’s lesson, you and your cooperative partner will begin a short PowerPoint project. You will make up your own real-life problem that uses this theorem and later share it with the class. || ||  || area of “a” + area of “b” = area of “c” OR  a2 + b2 = c2    N.B. Finding the hypotenuse’s value is not simply a + b = c. It must the sum of the “square” of the legs and then finding the square root of c2. N.B. Also, leg a and leg b are interchangeable, “a” doesn’t have to be the shortest leg; it just can’t be the hypotenuse. The same goes for “b”. Simplify and expand algebraic expressions using exponential forms. || || || For a few minutes, let’s go back to your paper squares and right triangle. On white board write: a2 + b2 = c2   Ask What happens if you know the values of squares “c” and “b” and don’t know square “a”? It should be c2 – b2 = a2. How do you find the value of “b”? (Ask for student input.) c2 – a2 = b2 In the same way you can solve for any of the three sides as long as two other sides of a triangle are known. Ask: Are there any questions? ** Make sure cooperative partners work together at this point. ** Here is our first real-life problem. We will be solving for “c” or hypotenuse. Ask: Do you see any relationship in a baseball diamond to a right triangle? Visualize the baseball diamond as a square with two triangles within a square. (Read the problem and accompanying notes. Have students work the problem with their cooperative partner.) Ask: How did you do? Check for correctness or needed help. Pythagorean Problems (handout) to work on triples: [6,8,10] [5,12,13] [8,15,17] [12,16,20] [7,24,25]  [10,24,26] [20,21,29] [16,30,34] [9,40,41] (Wait till all class is done before asking for answer.) Here is the answer to baseball problem. Compare this answer to yours. Any questions? Ask: Why are some answers are more understandable using a calculator than with mental math answers? Answers that include a radical sign may cause confusion, but a decimal answer helps when measuring distances in feet. Moving van problem. Read slide notes for problem. Answer. This next real-life problem is now a sample of your PP project. For this problem I chose to solve for one of the legs. Go over real-life word problem. Solve together as a class, calling on students to answer. || Mathematics 2.4.8.B **Combine numeric relationships to reach a conclusion. 2.10.8.A** Solve problems requiring indirect measurement for side lengths of triangles. || || . || · I will compliment students who are on the right track and offer constructive suggestions to redirect thought processes where needed. · I can praise their efforts when using previous knowledge to solve problems · I can offer positive input on group effort when they work together solving the two real-life problems again or offer suggestions for cooperative learning where needed. · The practice Quia test is a form of feedback for students to see what they learned. · As they begin their work on the PP project, I plan to circulate the class with words of encouragements and suggestions of what to do if they seem to be floundering. · The classes’ website with supplemental materials is a form of feedback and should ease student’s fears if uncertain how to proceed. || ||  . ||
 * Statement of learning outcomes/objectives: ** **(5 min)**
 * Statement of learning outcomes/objectives: ** **(5 min)**
 * ssessment of prior knowledge/stimulate recall of prior learning: (10 minutes) **
 * Slide 9: ** Let’s **review** some terms to understand this theorem.
 * Right triangle**, click the link for “right triangle” and have a student come to white board to manipulate the shape. Ask: What do you notice about the shape no matter what the student does? Right, it always keeps one angle the same shape, the right or 90 degree angle. Angles were called right because they were the true angle needed for building a wall perpendicular to the ground. Ask: What is the symbol to show a corner is a right or 90 degree angle? Yes, a small square. **//This theorem only uses right triangles!//**
 * Square**, click link for “square” and have a student come to the white board and manipulate the shape. Ask: What do you notice about the shape? Right, no matter how it is manipulated the sides remain equal. How do we solve for the area of a square? Look at the squares on your desk, yes, we multiply on side by the other s x s = s2 or area of the square.
 * Hypotenuse** means “line stretching under right angle”. The longest edge of the right triangle is the hypotenuse. Ask: What angle is opposite to the hypotenuse? Yes, the right or 90 degree angle.
 * Slide 10: ** This slide now ties everything together of what we have learned so far into a single, simple mathematical statement.
 * Turn off projector** and show how simple the equation really is. On the **white board** write and draw:
 * Turn off projector** and show how simple the equation really is. On the **white board** write and draw:
 * Pass out calculators**.
 * RELEVANCE:** Recently we learned two mathematical skills called **squaring** and **square roots.** They are used to solve the Pythagorean Theorem.
 * Slide 11: ** Ask: Who remembers how to find the area of a square? This is called squaring. Practice as a class a few squaring problems, call on student randomly for answers. Quickly move on if concept is mastered.
 * Slide 12: ** We just learned about square roots in our last lesson. Let’s use the calculator to practice. Again call on students randomly to check for mastery. Quickly move on if concept is mastered.
 * Slide 13:** Here are some games you can try out on your own.
 * Slide 14:** YouTube review of mental math for square roots. Practice a few on whiteboard: perfect squares like 4, 9, 16, 25 but also ones that come out with decimal answers 2,5,12, etc. Have students compare answers for accuracy . ||
 * PA Academic Standards for Mathematics **
 * 2.3.8.A** Develop formulas and procedures for determining measurements (e.g., area, volume, distance).
 * 2.1.8.E **
 * 2.1.8.G **Use the inverse relationships between addition, subtraction, multiplication, division, exponentiation and root extraction to determine unknown quantities in equations.
 * Present and conduct the guided learning activity/instruction ** : **(15 minutes)**
 * Present and conduct the guided learning activity/instruction ** : **(15 minutes)**
 * Slide 15: **
 * Tell class:** If you have completed this word problem ahead of time, I want you to work on triples. These are right triangles whose lengths are in whole numbers, not square roots or decimals.
 * Slide 16: **
 * Slide 17: **
 * Slide 18: **
 * Slides 22 & 23: **
 * PA Academic Standards for
 * 2.5.8.D **Determine pertinent information in problems and whether any more info is needed for solution. **2.10.8.A** Compute measures of sides and angles using proportions, the Pythagorean Theorem and right triangle relationships.**
 * Provide Feedback: ** (Feedback will be given throughout the lesson.)
 * Provide Feedback: ** (Feedback will be given throughout the lesson.)
 * Assess Performance and Achievement of Learning Outcomes: (10 min)
 * Assess Performance and Achievement of Learning Outcomes: (10 min)

Student computers will be set up and turned on. ** 1. A short on-line, in-class quiz will assess what students retained and understood from the lesson. There is no formal score, but if students receive a score lower than 80% they are to review and then retake it as home work. [] 2. The class was already informed that student performance/effort in class is rewarded by added points to tomorrow’s assessment grade: · Effort is rewarded when students work individually on their cutouts (2 pts.) · Effort is rewarded when students work in their cooperative groups. (2 pts.) 3. Student’s PP is also a way to assess student successful internalization of the Pythagorean Theorem. || ||  . || 1. Students, log onto the classes’ webpage and access the Pythagorean PP: Slides 22 & 23: Let’s look over these 2 slides. This is what your 2 page PP should look like. You’ll need a simply stated, real-life word problem using the Pythagorean Theorem and solving for a, b, or c. Slides 19, 20, & 21: These three slides show project requirements.
 * Closure to Summarize and Enhance Retention: ** **(10 min)**
 * Closure to Summarize and Enhance Retention: ** **(10 min)**

2. Access “Helps for PowerPoint Project” page. Go over other aspects of project. Tomorrows’ class will be a lab where you will complete your PP. Ask: Do you have any questions about PP project?

3. For **homework tonight**, log onto our classes’ website: · Find “Homework Worksheet” going over required aspects. · Ask if any elements not clear. Ask: · Can students identify a right triangle, with its two legs and hypotenuse? · Can students prove, define, and use the Pythagorean Theorem from memory? · Can students master the in-class quiz with 80% accuracy? · Can students cooperatively make their own PP, make up their own real-life word problem? || ||  . || Revision may be needed if: · Students are taking longer to grasp the theorem, media, and materials due to questions or extensive explanations. · Students did not appear engaged or respond to the lesson · Students of certain learning styles did not understand the lesson. · The lesson was not worth the effort. · Teacher did not appear to be able to communicate the theorem to the extent that they internalized the concept and were able to successfully solve problems that required the Pythagorean Theorem.
 * 4. Students will be given a chance to reflect on their own learning experiences and if the outcomes were achieved. **
 * Evaluate and Revise: **
 * Evaluate and Revise: **
 * What can I do to improve this lesson? ** ||