5 Day Unit Plan Launch, Explore, Summarize€¦ · Pythagorean Triples: The Pythagorean Theorem states taht: In a right triangle, the square length of the hypotenuse is equal to the

Pythagorean Theorem

Students will need to be able to understand and show they know how the Pythagorean

Theorem was developed.

Students need to understand that taking

the square root of a number is the inverse operation of squaring

a number.

Understand and identify what a Right

Triangle. is

Understand the concept of a proof.

Pythagorean Triples:

The Pythagorean Theorem states taht: In a right triangle, the square length of the

hypotenuse is equal to the sum of the squares

of the length of the legs: a2+b2=c2.

Understand the Properties of a Right

Triangle

Students need to know what the 'legs' and

'hypotenus' of a right triangle are.

5 Day Unit Plan

Launch, Explore, Summarize

1. Topic Selection and Rationale:

I chose to create a mathematics unit covering the Pythagorean Theorem Proofs.

2. Essential Content:

Vocabulary:

Hypotenuse:

The side of a right triangle that is opposite the right angle. The hypotenuse is the

longest side of a right triangle. In the triangle shown below the side labeled c is the

hypotenuse.

Legs:

The sides of a right triangle that are adjacent to the right angle.

Perpendicular:

Forming a right angle. For example, the sides of a right triangle that form the right angle

are perpendicular.

Pythagorean Theorem:

A statement about the relationship among the lengths of the sides of a right triangle.

The theorem states that if a and bare the lengths of the legs of a right triangle and c is

the length of the hypotenuse, then a2+b2=c2.

3. Plan Instructional Methods and Procedures using the Launch, Explore, Summarize format:

Day 1:

Subject: Pythagorean Theorem Introduction

Grade: 8th

Timing:

Reference for lesson:

Learning Goals and Objectives:

CCSS 8.G.6. Explain a proof of the Pythagorean Theorem and its converse.

Resources, Materials, and Preparation for Instruction (What facilities, resources, and tools will be

needed and how are they to be used?):

Mrs. Angela Chappell

Real World Algebra by Edward Zaccaro

a. Launch (5-10 minutes)

Teacher Considerations (Before)

Description of Learning Activities

Anticipated Student Responses

Teacher Guidance (During)

Q: How will I engage the students’ prior knowledge?

Ask students to

For the launch activity I am going to be reading a picture book to the students. The name of

“How tall is the flagpole?”

“We cannot find the measure of the

If a student was to ask the question of “How tall is the flagpole?” or “You

brainstorm about the ways they can measure of how long a piece of string would need to be in order to stretch from the top of the flag pole (point a) to a point x feet away from the flag pole (point b)?

A visual representation of this situation will be placed on the Smart Board for all students.

Q: How can I keep from giving away too much of the problem?

I can use guiding questions to keep students thinking about how they could obtain the measure of something they cannot touch. Such as: “Could there be relationships between the measures of the height of the flagpole and the distance away from the flagpole?” and or “If you were given two of the measures could you find the third measure? If so, how?”, etc.

Q: How can I make it personal and relevant to the students?

I can make this part

the picture book is What’s Your Angle, Pythagoras? Through using this text it is introducing students to the concept of the Pythagorean Theorem. Throughout the book it gives a proof of the Pythagorean Theorem, explains what it is and how it came about, and the book also gives some of the history behind the development of the Pythagorean Theorem. By using this picture book as a launch for this activity and unit allows the students to look at the Pythagorean Theorem from a different light and can be used as a reference throughout the remainder of the unit. After we have finished reading the book as a class students’ who were not sure how the find the measure of the string needed to stretch from the top of the flagpole to a point 10 feet away from the flagpole on the ground, should now have an idea as to how they could find the measure of the string.

string needed to stretch from the top of the flagpole to a point on the ground x feet away from the bottom of the flagpole.”

“The measure of the string needed is going to be the same as the height of the flagpole.”

“You would need to know what the measure of the flagpole and the measure of the distance on the ground away from the flagpole in order to find the measure of the string.”

Some students’ may have no idea as to where to start on explaining how to find the measure of the string, so they may not be active participants of the discussion/activity.

Students’ who do not know where to begin may also be disruptive, causing a distraction from the concept/question at hand.

“You can find the measure of the string by taking the measure of the flagpole squared and adding it to the distance away from the flagpole squared. When

need to know the measures of the flagpole and distance away from the flagpole.” your response to the student could be: in this case it we are not concerned about the height of the flagpole or the exact distance from the flagpole because we only want to explain how we could find the measure of the string needed to stretch from point a to point b because it is a measure of something we cannot reach. But if it would help you explain what the measure of the string needs to be you could give the pole a height; you could call it either x or give it a numerical value of your choice. And you could call the distance from the flagpole either x+n or a numerical value of your choice.

If a statement is made from a student or group of students about not being able to find the length of the string needed to stretch from point a to point b or if a

of the lesson relevant to my students’ lives through using relevant and everyday objects the students are familiar with to set up situations and problems. This could also be made relevant to the lives of my students’ by setting up situations and problems through and with activities they are involved and active in, such as sports.

I could as make this relevant and personal to my students’ lives by asking: “ Have any of you ever been in a situation where you needed to find the measure of a length but you only had the measures for two of the sides?” “If you have, how did you go about finding the measurement of what you could not reach?” “And what was the situation?”

Q: What advantages or difficulties can I foresee?

Students not being familiar with how to find or explain how to find the measurements of something that is out of reach.

those add together it gives you the length of the string squared.”

student or group of students does not know where to begin in explaining how to find the measure; you as the teacher could sit down with the student or group of students and draw a visual representation of the problem on a piece of paper in front of them. After drawing the picture you could have the student or group of students refer back to the picture book, What’s Your Angle, Pythagoras? , at pages 24, and 28-29. In these pages the student(s) should grasp an understanding or a general direction as how to explain how to find the measure of the string.

If a student was to give you the correct explanation behind how to find the length of the string, you could ask them to think about why that works and how it works. This allows the student to reach a higher level of thinking.

If you have a student or students who are being disruptive or

Students will be at various stages of their understanding of how to find the measure of something they cannot physically reach.

distractive to other students in the classroom because they are trying to avoid the problem at hand the best thing to try first would be to sit down with the student(s) and talk through the problem with them through asking guiding questions and giving explanations where needed. If this does not work there could be an alternate assignment for those students.

b. Explore (15-45 minutes)


Descriptions of Learning Activities

Anticipated Student Reponses


Q: How will I organize the students to explore this problem? (Individuals? Groups? Pairs?)

Students for this activity could be placed into pairs, where they could read the narrative as a group and then work on the problems attached with the narrative. If students cannot handle pair work they can always

For the explore activity students will be reading a short narrative out of Real World Algebra. This short narrative is about the Pythagorean Theorem. The short narrative sets up a situation where a firefighter is trying to figure out how all a ladder needs to be in order to reach the top of a building. The firefighter cannot figure it out so he goes to the local hardware store

“I do not understand how to apply the Pythagorean Theorem equation.”

“What are the legs and hypotenuse of a right triangle?”

“How do you know if it is a right triangle?”

“I understand how to use the Pythagorean Theorem equation and how to apply it.”

If you were to be met with the statement of “I do not understand how to apply the Pythagorean Theorem equation.” As the teacher you should sit down with the student and go back over the steps shown in the narrative and explain to them what the steps mean and how to

work individually.

With this activity students could work individually as well. If students work individually they are to read the narrative silently and complete the problems attached to the narrative.

If students have questions while working on the attached problems to the narrative have students ask at least one to two students how to answer the problem before asking the teacher.

Q: What materials will students need to encourage diverse thinking and problem-solving?

Students will each need a copy of the narrative about the Pythagorean Theorem and a copy of level one problems that are associated with the narrative.

If each student if to have a copy of the narrative they will be able to use it for a reference throughout the entire unit and that also allows students to make notes that help them recall and remember steps of the

where he happens to find someone who is very good at this kind of problems. The clerk was able to look at his drawing and figure out how long the ladder needs to be. The clerk ‘Algebra Woman’ walks the firefighter through what and how the Pythagorean Theorem works. The clerk walks the firefighter through every step of the Pythagorean Theorem equation. Students are able to see this in a visual graphic, almost cartoon like representation and visualization of the Pythagorean Theorem. During the cartoon like representation students are able to see how to perform and conduct all of the steps to the Pythagorean Theorem equation. With students being given all of the correct steps and examples shown as how to use the equation students can then use the information they have learned and obtained from the short narrative and cartoon representation of the Pythagorean Theorem on the associated level one problems. There are 10 level one problems for students to work on once they have finished reading.

properly use the steps. By sitting down with the student one on one and going back through and explaining the steps to them, may help the student better understand and gain clarification as how to apply the Pythagorean Theorem equation.

If you were to be asked “What are the legs and hypotenuse of a right triangle?” it is essential you or the student draw a visual representation of a right triangle or use one of the visuals off of the associated problems and label the legs and hypotenuse of the right triangle. It is important for the student to know that the hypotenuse is always going to be directly across from the right angle and that the two legs of the triangle are always perpendicular to one another creating the right angle.

If the question is asked “How do you know it is a right

Theorem and so on.

It is also essential that each student have an individual copy of the associated problems so they can each have record of their work. Through each student having record of their work, when the work is checked they are hopefully able to see where something went wrong and are then able to fix it. Also with each student having an individual copy of associated problems they can also use these as a reference for later applying the Pythagorean Theorem in more complex situations.


A disadvantage to this activity could be students will not fully understand or grasp the essentials for the Pythagorean Theorem from the narrative in order to apply it to the associated problems.

An advantage to this activity could be students will fully grasp and understand the

The problems are very basic and only application based. Students are being asked to apply the basics of the Pythagorean Theorem. With these 10 problems students should be able to use the examples given in the reading and cartoon visualizations to work through the problems on hand. This is a great introductory activity for students to become familiar with what the Pythagorean Theorem is and its equation before they begin learning proofs of the Theorem. During this activity students will also fill out two Frayer vocabulary models on the terms legs and hypotenuse. These can be used as a reference for students and could be kept in their math journal or notebook. On the Frayer model students are asked to provide a definition, examples, facts/characteristics, and non-examples of the word. In order for a student to be able to accurately apply the Pythagorean Theorem equation it is essential for them to know those two vocabulary words. The more familiar students are with the terms leg and

triangle?” you could talk with the student about how the two legs met perpendicularly. Meaning the two legs met together and forms a 90 degrees angle, or a right angle. Any triangle that has one angle measure of 90 degrees is classified as a right triangle. You could also tell and so the student how a right triangle has one 90 degree angle and the how the other two angles have to be acute angles or less than 90 degrees because the total angle measures for a triangle add together to be 180 degrees.

If you were to encounter a student who says “I understand how to use the Pythagorean Theorem and apply it.” You could ask the student to verbally explain and walk through all the steps with you. This will prove whether they really do understand or if they are just saying this to be done with all of their work. If a student is saying

essentials of the Pythagorean Theorem and be able to correctly apply to the associated problems.

Another disadvantage is some students may understand the narrative portion of the activity but not be able to apply it to the associated problems.

Another disadvantage could be students do not understand the narrative or how to apply it to the associated problems.

An advantage to this activity is hopefully students will keep the narrative and associated problems to use as a reference tool for when applying the Pythagorean Theorem in different situations.

hypotenuse the easier applying the equation should be. With exposure to the vocabulary words comes knowledge and mastery of the vocabulary.

this and shows proof they do understand and can accurately apply the equation allow them to move to the level two problems associated with the narrative.

c. Summarize (15-20 minutes)



Anticipated Student Reponses


Q: How can I orchestrate the discussion so the students summarize the thinking in the problem?

For the summarize activity students are being led in a group discussion. Once students finish the explore phase of this

“The Pythagorean Theorem can be used to find the missing side of a right triangle by plugging in the

If you are met with a correct and accurate description of what the Pythagorean Theorem is and

The students can be asked to think-pair-share. In the think-pair-share students to think of something new they learned today about the process of applying the Pythagorean Theorem equation.

Students could also be asked to think about how the Pythagorean Theorem can be used to find the measure of something that is out of reach. (This is the same question asked in the launch part of this lesson) At this stage students should be able to apply what they have learned through the launch and explore parts of this lesson at this point.

There are many ways you could assess student knowledge and have them summarize what they have learned from these activities. Having students pull together as a group, think individually about a question being posed, then share with a neighbor, then

lesson each student should be asked to think about and write down on a piece of paper or math journal how the Pythagorean Theorem can be used to find the missing piece of a right triangle. Once all students have finished thinking and writing down their answer to the question have the students get into groups of two or pairs and have them share their answers. Allow students about 5-10 minutes to share. By students sharing their answers with another student helps them gain confidence and reassurance in what they are saying and doing. Also this is a great time for students to see what other students think. By students sharing they may gain ideas or information from fellow students who might have looked at the lesson in a different light or picked up on something another student did not. By allowing and giving students the chance to share reveals mastery and misconceptions of the concept being taught. It is important for you, as the teacher, while students are talking in their pairs to be walking around the

appropriate numbers for the appropriate variables and solving.”

“You can only find the hypotenuse of a right triangle if it is missing.”

“The a and b in the equation represents the two legs of a right triangle and the c represents the hypotenuse of the right triangle.”

“The Pythagorean Theorem can only be used for a right triangle.”

“The Pythagorean Theorem can be used to find the missing distance of a right triangle in a real life situation.”

“The Pythagorean Theorem can be used for any triangle.”

how it can be used to find a piece of a right triangle that is missing guide and challenge the student by asking them “Why do you think the Pythagorean Theorem and its equation works?” By guiding the student(s) to a higher level of thinking allows them to think and make connections about why they work.

If you are met with a misconception such as “The Pythagorean Theorem works for all triangles.” It is vital you explain to the student why this is not the case and that “The Pythagorean Theorem only works for right triangles.” You could help the student understand this through explaining the vocabulary of the Pythagorean Theorem such as, legs, and hypotenuse.

With any response you are met with it is essential to have the student explain their thinking and reasoning behind

having a class discussion is a great way to see who understands, and grasped the concept and who does not understand, and who still needs more work.

You could ask students a range of questions. Some of those questions could be: “How could you use the Pythagorean Theorem equation in the real world?” “What is the Pythagorean Theorem?” “Can you use the Pythagorean Theorem to find a missing leg of a right triangle?” “Does the Pythagorean Theorem work only for right triangles? And why?” (This last question will lead you into the next few lessons, proofs of the Pythagorean Theorem.)

Q: What generalizations can be made?

At this point students should be able to apply the Pythagorean Theorem equation to basic problems.

Students should also know and

room monitoring what students are sharing. This is a great way for you to informally gauge where your students understanding is at. After students have shared with their partner led students in a class discussion about the importance of the Pythagorean Theorem and equation. During this time let students share what they shared with their partners and what they wrote on their papers during the brainstorming time. While students are engaged in the class discussion it is your job as the teacher to ask students to “Explain what you meant by that” “Why does that work?” and “How did you get that?” By asking students guiding questions like this you are pushing them to think at a higher level, hopefully developing that higher level of understanding. By listening to the responses of the students about what they understand the Pythagorean Theorem to be and how it works is essential in knowing whether or not they really understand the concept. Also by asking guiding questions will show the depth of what the student

their answer given. By understanding where the student is coming from helps you gain a better understanding for their understanding of the content.

If students are not able to carry a productive discussion about the Pythagorean Theorem an individual writing assignment can be given to the students who are not being productive or the whole class.

understand what each of the variables in the Pythagorean Theorem equation stands for and represents.

After these activities students should only know how to solve for the hypotenuse. This is because they are given all of the side or leg lengths in the problems and they are being asked to plug in and solve.

The students should understand and know how and why the Pythagorean Theorem equation can be used to find the measure of something out of reach, i.e. the hypotenuse of a right triangle, in a real world situation.


An advantage I can see to this summarize activity is students could be very active in the discussion and we would be able to end the discussion wondering how to prove the Pythagorean Theorem works and why. If this was to happen students would leave class

understands and what misconceptions they might have. If a student does share a misconception with the group make sure that you address that right there on the spot and help the student understand why that is not the case and what does work. It is important to do this because more than likely they are not the only student in the class who had that misconception. Towards the end of class time the discussion should be wrapping up with students talking about and wondering why the Pythagorean Theorem and its equation works. This way students will leave your class with the “need to know” and they will enter back in the next day ready to learn and gain that “need to know”. As students leave the classroom they should be asked to turn in and “exist slip”. On this piece of paper the students should write how they could use the Pythagorean Theorem to find the measure of a missing side length of a right triangle in a real life situation.

wondering and thinking about how and why the Pythagorean Theorem works, which would be great because over the next few lessons students are going to be exploring different proofs of the Pythagorean Theorem.

A difficulty I could foresee is students not grasping and understand how to apply the Pythagorean Theorem equation; and also not knowing what the variables stand for.

Day 2:

Subject: Pythagorean Theorem Proofs

Grade: 8th

Timing:






Looking for Pythagoras, The Pythagorean Theorem, Connected Mathematics 2 (Teacher’s Guide)

Mathematics Station Activities for Common Core State Standards, Grade 8, Walch Education

Grid paper

Triangle Table Chart

Glue

Scissors


Teacher Considerations

(Before)




Q: How will I engage students’ prior knowledge?

“What kind of triangle have I drawn on the Smart Board?”

“What are the different parts of this triangle?”

“How do you know those are the legs and this is the hypotenuse?” (point each of them out or label them for the students to see)

“How long are the legs on this

For the launch activity I am using an activity out of Looking for Pythagoras, The Pythagorean Theorem, Connected Mathematics 2 (Teacher’s Guide). This activity will be used to review some of the information from lesson one, but it is also going to be used to get the students thinking about the converse and proofs of the Pythagorean Theorem which will be covered throughout the

“I do not see a connection between the areas of the two legs and the area of the hypotenuse.”

“The areas of the two legs are equal to the area of the hypotenuse.”

“I do not understand how the areas of the two legs are equal to the area of the hypotenuse.”

“The areas of the two legs are equal to the area of the hypotenuse

If a student was to think there is no connection between areas of the two legs and the area of the hypotenuse it would be okay at this point in time. Although I would suggest that you do explain to the student that the Pythagorean Theorem equation seems to show there is some kind of relationship between the two given that

triangle?”

“How can we find the length of the hypotenuse of the right triangle?”


I can use guiding questions to keep students thinking about how the sum of the areas from the two legs could equal the area of the hypotenuse. Such as: “Do you think there could be a relationship between the area of the two legs and the area of the hypotenuse?”

Or if a student has figured out and made the connection that there is a relationship between the area of the two legs and the area of the hypotenuse you could ask them “Why do you think the area of the two legs are equal to the area of the hypotenuse?”

You could also guide students to a deeper level of thinking through asking them “If a2+b2=c2 works then does that mean the triangle

remainder of this unit. This launch activity will begin by a right triangle being projected up on the Smart Board. This right triangle will be drawn on dot or grid paper and will have leg measures of each one unit. To review the information covered in yesterdays or the previous lesson ask students to identify the kind of triangle and the different parts of the triangle; the legs and hypotenuse. After students have identified the different parts of the right triangle have them tell you or justify their answer as to why they say the legs are the legs and why the hypotenuse is the hypotenuse. Once students have identified the triangle and justified their answers as to the parts of the right triangle; ask them if they can see a relationship between the two legs and the hypotenuse. This is a question solely for the students to think about and jot an answer or thoughts down in their math journals. Give each student a sheet of grip paper.

because of the Pythagorean Theorem equation saying so. a2+b2=c2”

“The hypotenuse is always the side across from the right angle.”

“The legs connect and form a right angle.”

“I do not remember what they hypotenuse is, I just know it is across from the right angle.”

“The legs are always the two shortest measures.”

“I do not remember which variable represents the hypotenuse.”

“Does it matter which leg I use for a and which leg I use for b?”

“How should I cut the hypotenuse out of the grid paper because it does not fall on a whole number?”

“What do you mean any three measures?”

“Of course any three measures will work for a right triangle.”

“The Pythagorean Theorem equation does not

a2+b2=c2. If the student still is confused or does not believe there is a relationship that is okay because they are only taking your word for the truth at the moment. They have not yet discovered a proof themselves to see that there is a relationship.

If you were to be met with the response that there is a relationship between the areas of the legs and the area of the hypotenuse have the student explain their reasoning and logic behind their thought. This would be a great time to see what this student knows and where there might be gaps in their understanding.

If you were to be met with students who are confused about what the different parts of a right triangle are, sitting down with a student or a group of student one on one and going back over the vocabulary of legs

is a right triangle?” Q: How can I make it personal and relevant to the students?

One way you could make this relevant to the lives of your students is through relating this back to the launch activity from yesterday. Have students draw a triangle representation of the flagpole scenario. Once students have drawn a triangle representation of the flagpole scenario have them draw squares off the legs and the hypotenuse and find the areas of the squares. For this use of the scenario you will have to give the lengths numerical values instead of variables. Once they have found the areas of the squares they should notice the areas of the two legs are equal to the area of the hypotenuse. But if this connection is not made guiding questions might be asked. Such as: “Do you see a relationship between the area

Before guiding the students to proceed with the grid paper ask the students to think about how can we find the length of the hypotenuse of this right triangle? Allow students a few seconds to think about how they could do this, and then ask a few students to share. There may be a few students who know what it is, but guide them to using a2+b2=c2 to find the measure of the hypotenuse. It is essential that students find the measure of the hypotenuse before they continue on to the next step. With the grid paper guide students to cut out the dimensions of the right triangle that is displayed and have the glue these pieces together in their math journal to form a replica of the right triangle on the board. Once students have replicated the right triangle ask them to label each of the parts; legs and hypotenuse. Make sure students have shown all of their work to find the measure of the hypotenuse in their math journals. When students have completed this part of the activity give them

determine whether a triangle is a right triangle or not.”

The Pythagorean Theorem does determine whether a triangle is right or not.”

and hypotenuse. A visual aid in this situation would be a great help for students who are still struggling and whom might be confused by the parts of a right triangle. It is imperative for all students to understand and know the parts of a right triangle.

If you are met with a student or students knowing the different parts of a right triangle asking them to explain and prove their answers will help them clarify and for they themselves to feel reassured about the knowledge they have obtained. This could also be an opportunity for students to catch their own misunderstandings through talking out loud and proving they know something.

If a student was to be confused about how to cut out the hypotenuse of this triangle out of the grid paper you could either show them or ask questions guiding

of the two legs and the area of the hypotenuse?”


Students may have not made the connection between yesterdays lesson about the Pythagorean Theorem equation. Meaning some students may not understand what the variables represent and how to solve for a missing leg or hypotenuse.

Students being at various stages of their understanding of how the Pythagorean Theorem equation works and what it is and represents.

Students making the connection between the Pythagorean Theorem equation and this launch activity.

a question to ponder on that will take them into the mindset of the exploration phase of this whole lesson. A question such as that could look like is “Do you think that any three measures will work to make a right triangle?” or “Does a2+b2=c2 determine whether a triangle is a right triangle or not?” With these questions student s will really have to explore visually and mentally to make sense of what is being asked. Many students will probably not understand or know the answers to those two questions. This is why it is essential for you as the teacher during the next phase of this lesson to make sure students make sense of those questions and understand the logic behind the answers to them. If students have time to explore if any three measures will work allow them to cut different measures out of the grid paper and see whether or not it works.

them towards where and how they should cut. Such as “Well is the decimal closer to 0, ½, or 1?”

If you are to be met with uncertainty or not knowing about the two guiding questions into the explore activity you could guide students to talk to one another and see what they can come up with. It is important you as the teacher are walking around and intently listening to all student conversations at this time so you can address misconceptions and clarify some thoughts of students whom might be on the right track.







Students for this activity need to be working as individuals. By students working individually they are able to see whether the dimensions given make a right triangle or not and whether they work when plugged into the Pythagorean Theorem equation. When students are working individually they are challenged. Through this activity students will be able to see a visual representation of each ‘triangle’ they have constructed themselves. Through each student working on their own guarantees that every student will have made contact with each set of dimensions, and come up with their own conclusions, that will not have been influenced by

For the explore activity students will be filling out a triangle chart. This triangle chart has five columns. The headings of each of those columns are as follows: a, b, c, Is a2+b2=c2 true? and Makes a right triangle? In the first three columns where a, b, and c are listed students will find each of those columns filled in with different measures. Each row makes up the measurements of what could be or may not be a right triangle. In the fourth column, Is a2+b2=c2 true, students will be plugging in each value given for each variable and seeing if each statement is true or not. In this column each student will need to show where they plugged the values in and expanded the numbers. For example, if the given value for a was 1, the given value for b was 2, and the given value for c was 3 students would not only need to show they plugged the numbers in like 12+22=32, they also need to show 1+4 does not equal 9. By

“I do not understand what converse means.”

“How can it be true that the Pythagorean Theorem and its converse say what they do?”

“How does a2+b2=c2 prove a triangle is a right triangle?”

“I do not understand how to use the grid paper to form a triangle in with these given dimensions into a shape.”

“This does not form a right triangle, it forms a straight line.”

“12+12=22”

If you were to be met with a student not understand what the word converse means you could try to talk through the examples that were given with them, this may bring some clarification. If that does not bring clarification you could present the student with another more relevant example to their life of a converse. If neither one of those options are working you could have the student look up the noun definition of converse. But even with the student finding the definition they are still more than likely going to need clarification as to what converse means.

If a student was to ask how could the Pythagorean Theorem and its converse be true, you could have them think about it on their own and see how they could

another student.

Students could also work in pairs. But for each student to get the most that they can out of this activity it would be better for every student to work individually.

Although if a student has questions about the activity have students ask at least one to two students how to do what they are having troubles with before asking you.

Q: What materials will students need to encourage diverse thinking and problem solving?

Students will each need a copy of the triangle chart. Or they each need to copy down the triangle chart into their math notebooks. It is essential for each student to have a copy of the triangle chart so they can record their results and show their work. It is also essential that each person have the same chart so when as a class you are going over the results or

students expanding past just plugging in helps students see how the values given for a and b are either equal or not equal to c. In the fifth column, makes a right triangle, students will be using grip paper to cut out dimensions of each triangle. This will help students see how visually either the measures make a right triangle or do not make a right triangle. In this column students are expected to write a yes or no and if they answer no they need to explain why a right triangle is not constructed. Depending upon the dynamic of your class you could fill in between 10-14 rows for students to complete. It could be helpful to students if you were to do the first one in the chart as an example for the students to be able to see what is expected and how to construct the shape with the grid paper. When students are constructing the triangles with the grid paper there are two ways they can go about doing it. The first way it to add up the values for a, b, and c and cut out a strip of grid paper the

figure out it is true or not true. But I would also reassure them that you are glad they are wondering why and how they work and that you are going to be proving how they work throughout this unit. Although you may have a student or two who just have to know and will search the Internet and figure out how and why they work.

If a student does not understand how to use the grid paper to form the dimensions given it is vital you sit down with this student immediately and help them understand how to use the grid paper. Because if a student is not able to have a visual representation they are not going to know whether or not a right triangle is being formed, they can only make assumptions.

If a student tells you their dimensions make a straight line, this could be very true.

answers each student will be on the same page and be able to compare results or answers.

Each student will need a sheet of grid paper, a pair of scissors. By each student having a pair of scissors and a sheet of grid paper they are able to each visually see how the dimensions make or do not make a right triangle. This will also allow student to be able to see what the dimensions make if it is not a right triangle.


A disadvantage to this activity could be students not fully understanding how to use the grid paper to make a right triangle or non right triangle. This is a disadvantage because if students do not understand how to use the grid paper to form whatever shape the dimensions is going to give them,

length of the sum then fold the grid paper into the correct lengths. Or students can cut out each measure in the grid paper and lay them out on their desk as a visual. It is good to give students different ways to approach hands on tasks, because remember not all students are the same and they all learn differently. Once each student has completed the chart as a class go over the answers. As a suggestion to go over the answer and be able to display the data for the whole class to be able to see use a Smart Board if you have access to one. This is also a way for the class to be interactive. Have a copy of the chart up on the board and ask different students to come up and fill in rows. Ask students to show their work so other students can see how they got their answers. While as a class you are going over answers you might have the opportunity to correct students’ answers or misconceptions during this time, take advantage of that and do not leave anyone confused. After you have finished

You could guide the student to figure out why the given dimensions formed a straight line and allow them to share their findings with the class.

If you happen to have a student who says:”12+12=22” you could have them explain to you or the class their logic behind this, but make sure you explain to the student or have the student figure out why this does not work.

If you have a student who is abusing or taking advantage cutting out the grid paper you can have them draw out the dimensions on the grid paper with a pencil instead of cutting them out. This still works but it does not allow students to see the same kind of visual.

the student will not be able to see or have a visual representation to help them understand why and how some is or is not a right triangle.

Another disadvantage to this activity could be students not understanding how to correctly plug the numbers in for their correct variables. If students do not correctly plug in numbers for the correct values the answers or results could be wrong. Also if students do not expand the values once they are plugged into the formula they may think that a2+b2=c2.

An advantage to think activity could be students fully grasp and understand how to use the dimensions given to make either a right triangle or another shape.

Another advantage could be that student understand and know that it is important to expand the values

going over the answers to fill in the chart ask students what they think the Pythagorean Theorem is saying. Allow them to think for a minute then talk with a partner for a few moments, then come back together as a class and discuss. Students should figure out that the Pythagorean Theorem says:”If a triangle is a right triangle, then a2+b2=c2.” Students may have to be guided towards what the Pythagorean Theorem says. When students have figured out what the Pythagorean Theorem says ask them if they anyone knows what converse means. More than likely they may be able to break it down into stems but they are probably not going to know what the word means. To lead students to figuring out the meaning of converse one or both of these statements could help them. “If a circle is round, then if a shape is round it is a circle.” Or “If a horse has four legs and a tail, then if an animal has four legs and a tail it is a horse.” These two statements will really get students thinking and laughing.

plugged into the formula to see if they are equal or not.

Another disadvantage could be that students do not understand that the Pythagorean Theorem says:”If a triangle is a right triangle, then a2+b2=c2.” If a student does not understand this then they are not going to understand the converse of the Pythagorean Theorem.

Another advantage could be that students do understand what the Pythagorean Theorem and its converse says.

But the most important part it that they come to the understanding of what converse means. As a class discuss how each of those statements is an example of converse. Through class discussion you should come to the consensus of what converse it. Then ask students if the Pythagorean Theorem says:”If a triangle is a right triangle, then a2+b2=c2.” Then what does the converse of the Pythagorean Theorem say? Give students a few minutes to think and share with partner is necessary. Come back together as a class and have students share what they think the converse of the Pythagorean Theorem is. More than likely there will be a student or students’ who figure out what the converse is. The converse of the Pythagorean Theorem says:”If a2+b2=c2, then the triangle is a right triangle.” Some students at this point may start questioning why this is the case and others may take your word without any proof. Although throughout this whole activity

students were proving the converse of the Pythagorean theorem. If students are questioning why the Pythagorean Theorem and its converse says what it does then you have done your job. Students should be left wondering and pondering why this is the case. By leaving students wondering is a perfect way to lead into the next lesson which will be covering a proof or proofs of the Pythagorean Theorem. To challenge students have them create dimensions for two right triangles. In doing this have them prove they work with the Pythagorean Theorem and by cutting the dimensions out of grid paper in order to see a visual.






Q: How can I orchestrate the discussion so students summarize the thinking in the problem?

By students being asked to think-pair-share about their reasoning about the triangles

For the summarize activity students are going to work individually to figure out why a triangle is a right triangle and why the other triangle presented is not a right triangle. On the Smart Board two triangles will be drawn. They could

“You told us it was a right triangle, so why do I have to prove it actually is a right triangle?”

“How do I prove this triangle is not a right triangle?”

“What does it mean to prove something?”

If you are met with a student wondering why they have to prove a triangle is right when you just told them it was right, it is important you let them know you are only telling them it is. So with

could help another student see something they did not pick up on or something they had completely overlooked. By allowing students to share with one another is essential for their growth in depth of knowledge.

Students could be asked why they used the Pythagorean Theorem equation to prove whether the triangle was right or not.

Students could also be asked why they used the grid paper, if they or anyone used grid paper, to figure out whether the triangles are right or not.

There are many ways to assess student knowledge from what they have learned throughout these activities. By allowing students to share with others and having a whole class discussion allows students to possibly understand and see ideas from a different light. But it also allows you

each have the shape of a right triangle or one could have the shape of a right triangle and the other could be an isosceles, scalene or equilateral triangle. Above one of the triangles it will say “This is a right triangle.” Measurements placed on this triangle are 12, 5, and 13. Above the other triangle it will say “This is not a right triangle.” The measurements placed on this triangle are 5, 6, and 9. At this time students are taking your word that each triangle is what you are saying it is. But they need to prove to themselves that one of them really is right and the other one is not. Ask student to tell you why this is a right triangle and why this one is not. Students will need to show their work and explain. Students must use the Pythagorean Theorem equation to prove we have a right triangle and a non right triangle. Students could also use the grid paper from the explore activity if they choose also to prove their conclusions. Once every student has had a chance to

“This one is a right triangle because it looks like a right triangle.”

“Do I use a2+b2=c2 to figure out if a triangle is right or not?”

them taking your word that it is a right triangle could be a risk, because you could have told them it is a right triangle when it really is not. Explain to this student that by using mathematics to prove whether the triangle is right or not will help you better understand why the Pythagorean Theorem and its converse say what they say.

If you are met with a student wondering how to prove whether a triangle is right or not it is essential that you ask them to tell you or explain what the Pythagorean Theorem and its converse says. Through a student knowing or explaining what the Pythagorean Theorem and its converse says should help the student figure out how to prove whether a triangle is right or not.

If a student makes the comment that a triangle is right because it has the shape of a right

as the teacher to see how deep of an understanding your students have for this concept and what gaps or misconceptions they may have.

Asking students’ questions is a great way to see their understanding and depth of knowledge. In this case asking a student WHY they use the tools they did to prove something is essential. Their answer to the why question will give you information you need to help or guide the student onto the next level.


At this point students should know what the Pythagorean Theorem and its converse says.

Students should also know how to prove the converse of the Pythagorean Theorem.

Students should be able to use the Pythagorean Theorem equation to figure out whether a triangle is a right triangle or

make their own conclusions and explanations give students a few minutes to share with one another. Pull back together as a class and have students share their answers. Each student should now know how and why the right triangle is a right triangle and why the non right triangle is not right. Once again this shows students know how to prove whether a triangle is right or not, meaning they can prove the converse of the Pythagorean Theorem, but do they really know why is works?

triangle, is not acceptable. Again just as before when the student was asking why they should prove the triangle is right; the same thing will work here. A picture can be misleading, it may not be drawn to scale; if this was the case the triangle could look right but really not be. So it is better to have proved through mathematics that it really is a right triangle rather than just relying on the picture.

not.

Student should know what the legs and hypotenuse of a right triangle are.


An advantage to I can see to this summarize activity is it allows for student confidence to increase by proving something through mathematics. Meaning, by students sharing their ideas with one another and having to show this is how they proved this is a right triangle and this one is not gives the student the ability to be and stand confident in what they have done.

Another advantage I see with this activity is students being able to use different ways to prove whether the triangle is right or not.

A disadvantage I could foresee is students not grasping how to prove whether a triangle is right or not through use of the Pythagorean Theorem equation.

Day 3:


Grade: 8th

Timing:






Each student will need two 8x10 pieces of paper or a square piece of paper

Each student will need a baggie that has 7 cut out pieces, when put together correctly it forms a

right triangle with squares on the legs and hypotenuse

Each student will need 5-6 colored pencils (each a different color)

Each student will need a pair of scissors for the second station and possibly the first, depends on if

you are using an 8x10 piece of paper

Each student will need a ruler, this is to help draw straight line, optional

Each student will also need a copy of the proof foldable they will be cutting out and coloring (there

are two copies on each page, they will need both)

Mrs. Angela Chappell



(Before)





You could ask students if by just looking at the figure whether or not they think shapes 1-5 are going to fit into the larger square.

If students were to figure out how to place shapes 1-5

For this launch activity students are each going to receive a baggie that has 7 cut out pieces. These 7 pieces range from different sizes of triangles and squares. With these 7 pieces students will be asked to reconstruct the original figure before it was cut apart. On the Smart Board

“These shapes do not fit into this larger square.”

“Why are we doing this?”

“I got all of the shapes to fit in here, but I do not know why they were all able to fit.”

“I got shapes 1-5 to fit into the larger square. I

If a student was to say shapes 1-5 do not fit into the larger square encourage the student to keep trying. You could also ask a student who has already figured out how shapes 1-5 fit into the square to help guide the student as to how the

into the larger square they could be asked to think about, jot down, or share with a partner why shapes 1-5 fit into the larger square.

This is review, but it would not hurt to make sure that all students still know and remember what the different parts of a right triangle are. With this being said once students have reconstructed the original figure just quickly ask the students to describe the legs and the hypotenuse of this right triangle.


It would be very easy to give away what is going to be covered throughout the exploration phase of this lesson right here in the launch. It is very important you be careful as the teacher not to give away what is happening.

By using guiding questions such as:”Why do you think shapes 1-5 fit into the larger

the original figure will be displayed for each student to reconstruct. The original figure has a right triangle in the center and a square adjoining each of the legs and the hypotenuse of this triangle. Once each student has reconstructed the original figure direct the students to see if they can fit shapes 1-5 into the larger square adjoining the hypotenuse. Shapes 1-5 are the figures that make up the squares off of the two legs of the right triangle in the center of all three squares. Some students may be able to fit shapes 1-5 easily into the larger square and for other they may not be able to. If a student was to figure out how to place shapes 1-5 into the larger square have them jot down on a piece of paper why they think they were able to fit shapes 1-5 into the larger square. You should only allow students about 5 minutes to try to fit shapes 1-5 into the larger square after that students should be ready to move on to the exploration

think they were able to fit because of something to do with area.”

“I am not doing this, I do not understand it, and I cannot do it.”

shapes it. Make sure the student helping is not telling the other student how to place the shapes but rather assisting with guiding suggestions, such as: “Have you thought about turning that shape this way?”

If a student was to ask why we are doing this activity I would make sure you explain to them it is to help us be prepared for what we are going to be doing next. By students participating in this activity they are proving the Pythagorean Theorem without knowing it.

If a student is able to fit all shapes inside the larger square and they do not know why guiding question are more than likely the key to getting this student to try to understand. You could suggest that they might should look back at what the Pythagorean Theorem says and its equation.

If a student has figured out how all

square?” or “What kind of triangle is this?” or “What does the Pythagorean Theorem equation say?”

Question such as those can get students thinking in the right direction.


I could make this activity relevant to the students’ lives by relating this activity to a puzzle. If you really think about that is what they are trying to do is solve a puzzle, getting shapes 1-5 to fit into a larger square. The only difference here is there is a mathematical reason for why shapes 1-5 are able to fit into the larger square.


A difficulty I can foresee with this activity is a student possibly getting frustrated with not being able to fit shapes 1-5 into the larger square. If this was to happen this

phase. In the exploration phase students are going to learn why and how shapes 1-5 fit into the larger square.

shapes fit into the square and have an idea as to why shapes 1-5 fit into the square have them find other students in the class who think they know why and have them share with one another. By allowing students to share with one another it allows them to see something from a different point of view or maybe pick up on something they might have overlooked.

If you are meet with a student who is done trying, does not understand, and who has shut down sit down with this student. By taking time to sit down with this student shows them that you care. Try to figure out why the student it frustrated and is done trying. By trying to figure out what is going on and causing frustrations to the student will help you as the teacher know where to start in helping the student succeed in

student could shut down, worst case scenario, or completely give up.

An advantage I see to this activity could be it is easy for students to figure out how shapes 1-5 fit into the larger square. If this is the case students might have ideas as to why this works.

this activity.







For this activity it is crucial that students work individually.

Although throughout this whole activity the teacher will be directing students step by step as how to fold the paper and what to do next after they have completed this.

If students were to be working in groups or partners students would

For the explore activity is dealing with a proof of the Pythagorean Theorem. Once students have completed this proof they should know a proof or explanation of the Pythagorean Theorem. Students are going to need either an 8x10 piece of paper or a square piece of paper. If students are using an 8x10 piece of paper direct them as how to fold the paper in order to get a square and cut it out. Once each student has a square piece of paper direct the students to fold it in half so that a

“I missed a fold. What did I do wrong?”

“What do you mean the creases of the paper?”

“All of the triangles are congruent to one another.”

“All of the triangles are right triangles.”

“Where is the square that adjoins this leg?”

“How do the areas of the two triangles on the legs equal the area of the square on the hypotenuse?”

“That is really cool how you can see

If a student is to miss a fold or is not sure what they have done it is essential that you help them get caught up and back on the right track. One way you could do this is go ahead and direct the class as how to perform the next fold. While the class is folding go over and help the student who is in need of help.

Also depending upon your class it might be a good idea to make sure that every student

become distracted and would not fully understand how this is a proof of the Pythagorean Theorem.

It is also essential that each student work as an individual so they can each have their own proof foldable to place in their math journals to use as a later reference tool.


Each student will need a piece of 8x10 paper or a square piece of paper.

Each student will need 5-6 different colored pencils. This will help the students be able to distinguish different parts of the triangle and squares.

Each student may also need a ruler in order to create sharp and exact outlines of the creases, where the paper was folded, in order for their visual representation to be as accurate as possible.

Q: What advantages or

triangle if formed. Then have them fold the triangle in half, then in half again, and then in half once more. Then have students open the square. Inside the square at this time have students stop and count how many triangles they have. At this point they should have 16 squares. Once each student gives the okay that they have 16 squares direct them to now take their square and fold top half in half down to the center line then have the fold the bottom half in half up to the center line. After students have done this step have them open the square and they should be able to count 24 triangles inside the square. Next have students fold the right and left sides in half towards the center line. When students open the square this time they should be able to count 32 triangles inside the square. After students have finished folding their paper have each student pick 5-6 different colored pencils and get a ruler. Once students have done this direct the students to take one of

how the areas of the squares on the legs are equal to the area of the square on the hypotenuse.”

has completed the current fold before moving on to the next one.

If a student is confused by what outlining the creases of the paper is make sure you tell them it is where you folded the paper. Describe to them they will need to outline or trace every fold they created in the paper.

If a student is to notice all of the triangles in the square are congruent that is great. You could have this student share their findings with the class.

If a student was to be confused about where the square is located that adjoins a leg or hypotenuse of the triangle, show them. By showing the student where the square is located or how the squares in this case are made up of two or more right triangles could help them throughout the remainder of this activity.

If a student is confused as to

difficulties can I foresee?

One difficulty I can foresee with this activity is a student missing a fold or getting lost in the directions as how to fold the paper. If a student was to get lost they could then miss other folds and their visual would not turn out correctly.

An advantage I could see with this activity is more than likely you will have your students’ attention because they will want to make sure they have their paper folded correctly and make sure they have not missed a step in the process.

Another difficulty I could foresee with this activity is students not seeing how the areas of the squares on legs are equal to the area of the square on the hypotenuse.

Another advantage I could see to this activity is students picking up on and understanding how the areas of the squares on the legs are equal to the area of the square

their colors and outline all of the creases in this square. There are 12 creases throughout the entire square that students should outline. It would be helpful to the students if they would use a ruler when outlining the creases in the square so they will have straight lines. When students have finished outlining the creases ask students if they notice anything about the size or appearances of the triangles. Students should notice all of the triangles are the same size, congruent to one another and they should also notice they all appear to be right triangles. Next draw the students’ attention to the center square inside the square that is made up of 8 right triangles. Here each student should choose one of those 8 right triangles and color it in. To color this one in they should use a different colored pencil. When students have finished coloring in their right triangle have them label each part of the right triangle. They should label the triangle with a, b, and c. When students have properly

how the areas of the squares of the legs are equal to the area of the square on the hypotenuse, explain. There are many different ways you could go about explaining or helping a student understand this. A possible strategy would be to have students pull into groups and explain what they understand about this model to one another and how it works. This could help a student understand or possibly clarify questions they had. You could also sit down with the student and have them explain to you what they understand or what they think they know about the model. From there you would have a better understanding of where the students’ confusion is coming from and what needs to be clarified.

If a student is to understand the model completely you could have them help other

on the hypotenuse. labeled the triangle direct the students to choose one of the legs, either a or b. On the leg they have chosen they now need to color in the square that adjoins the leg. This square should consist of two right triangles that form a square. Next have them do the same thing on the next leg. After a square has been drawn on each leg direct the students’ attention to the hypotenuse. This part of the activity could be tricky for some students. The square that adjoins the hypotenuse is larger than the two squares that adjoin the legs individually. It is important you point out to all students that the square adjoining the hypotenuse they are going to color in has four triangles making up the square instead of just two. Each of the squares adjoining the right triangle should be represented in a different color. When every student has these steps complete is it time to explain and make sense of what it is we just did and created. Draw the students attention back to the

students who are struggling with understanding. Or you could have them think about another way they could prove the Pythagorean Theorem.

If you were to have a student in your class with a learning disability you could already have the paper folded for this student and have already outlined all of the creases. If this was already done all the student now needs to do is color in the triangle and squares.

If you were to have a student who could not participate in this activity because of a learning disability you could make the foldable for them, but make sure you explain how and why to the student in depth to where they understand what is happening.

beginning were they labeled the sides of the triangle a, b, and c. Then as an example you could isolate one of the squares adjoining the leg a. Guide the students with questions from here. What do you all know about a square? Students should discuss how all side measures are equal. Once that has been discussed you could ask:”So if we know one side measure is a then what are the other side measures?” Students’ should respond with the other side measures being a as well. When this has been established you could ask the students’:”How do you find the area of a square?” Every student should know you find the area of a square either by bh or lw. Then you could ask each student to find the area of the square you are working with. Some students might get an answer that is (a)(a) or a2. The answer you want to encourage students to use in this case is a2. When students have come to the conclusion a2 is the area of the square; ask them:”Why is the area a2?” Students should

know area of an object is represented by square units. Have students label this entire square a2. Repeat the steps you see necessary for students to understand the other square adjoining the other leg of the triangle, b, is going to be represented by b2. Then also repeat the steps you see necessary for students to understand the square adjoining the hypotenuse is going to be represented by c2. At this point some students will probably see how this proof work and where the formula a2+b2=c2 came from, but others may not. At this point you could prompt the student to examine their model they have constructed. Ask them if they see any connections between the areas of the squares. As a class come back together and discuss how the areas of the legs squared is equal to the area of the hypotenuse squared. It may help some students to see this in their model by taking the colors they used for the squares on the legs and taking those colors and outlining

those two squares in the square of the hypotenuse. This will help them be able to see how the areas of the legs squared are equal to the area of the hypotenuse squared. At the top of this foldable direct students to write Pythagorean Theorem Proof, a2+b2=c2.



(Before)





Students can be asked to think-pair-share. In the think-pair-share student will be able to share their understandings of this proof of the Pythagorean Theorem to one another and with the class.

Students can also be asked to use the model from the launch activity to show a proof of the Pythagorean Theorem.

By students thinking, pairing,

For this summarize activity students are going to be led in a class discussion. Once students have finished the explore phase of this lesson each student should be asked to write down on a piece of paper or in their math journal how we proved the Pythagorean Theorem works. Once all students have finished thinking and writing down their answers, have students get into groups of two or three and have them share their answers. Allow somewhere between 3-5 minutes for students to share with one another in their small groups. By students sharing their

“This proof showed us that the area of the square adjoining leg a plus the area of the square adjoining leg b is equal to the area of the square adjoining c, the hypotenuse.”

“a2+b2=c2”

“Are the two legs always going to be the same size?”

“I am still not sure how this shows the Pythagorean Theorem works.”

“The area of two legs is equal to the area of the hypotenuse.”

If you are met with a correct answer as to how this proof proves how the Pythagorean Theorem works and where its equation comes from have this student draw a visual on the Smart Board. When the visual is ready have the student explain how it works.

You could also challenge a student who gave a correct answer to use the model from the launch activity to explain the proof of the Pythagorean Theorem.

If a student gives the answer

and sharing is a great way to see who has grasped the concept and who still needs more clarification or work.

You could ask students guiding questions to see what the depth of their understanding is. Some of those questions could be: “How did you come to that conclusion?” “How did the visual model represent that for you?” or “Did the model we used for the launch activity show you a proof of the Pythagorean Theorem?”


At this point students should be able to explain how the proof in the exploration phase of this lesson proves the Pythagorean Theorem.

Students should be able to prove, see, and understand where a2+b2=c2 comes from.

After these activities students should have made the connection

thoughts with one another helps them gain confidence and reassurance in what they are saying and doing. This is also a great time for students to see what other students think about the proof of the Pythagorean Theorem. By students sharing they may gain idea or information from fellow students who might have looked at the lesson in a different light or picked up on something another student did not. By allowing and giving students the chance to share reveals mastery and misconceptions of the concept being taught. It is important for you, as the teacher, while students are talking in their groups to be walking around the room monitoring what students are sharing. This is a great way for you to informally gauge where your student understanding is at. When students have finished sharing with their small groups lead the entire in class in a discussion about why this proof works for the Pythagorean Theorem. During this time it would be key you to let students

a2+b2=c2, ask them how they know this is true. From there the student will need to both verbally and visually explain their answer. The answer is correct but vague, missing vital information.

If a student at this point is still not sure how this proof shows how the Pythagorean Theorem works you may need to take some time after this discussion wraps up to help the student. Depending on what the student understands or does not understand at this point is when you as the teacher will need to make a decision as to address the student at the moment or wait and help them after class, one on one.

If a student is to ask if the legs of a right triangle are always going to be the same length you need to let the student know that for this example yes. But in most cases the legs are

that both the launch and exploration phases of this lesson were proofs of the Pythagorean Theorem.


An advantage to this summarize activity I could see is students could be very active in the discussion. This discussion could also end with students wondering if there are other proofs to show how the Pythagorean Theorem works. Which would be great because students because in the next lesson students are going to be discovering another proof of the Pythagorean Theorem.

A difficulty I could foresee is students’ not fully grasping and understanding how either proof shows how and why the Pythagorean Theorem works.

share what they shared with their partners and what they wrote on their papers during brainstorming time. While students are engaged in the class discussion it is your job as the teacher to ask students to “How did you come to that understanding” “What steps did you take to get there” or “Please explain what you mean by that” By asking students guiding questions like this you are pushing them to think at a higher level, hopefully developing a higher level of understanding. By listening to the responses of students about what they understand of the proof of the Pythagorean Theorem is essential in knowing whether or not they really understand this concept. Also with asking guiding questions and the responses by the students will show their depth of content knowledge and what misconceptions they might have. Remember that a misconception one student might have, other students probably have. It is important to address those misconceptions.

going to differ in length. By giving the student an example or guiding them back to previous days in their math journals they will see examples of right triangles whose legs are not the same lengths.

If a student was to say the areas of the two legs are equal to the area of the hypotenuse, they are revealing a misconception. You need to make sure that students understand that it is the areas of the squares on the legs that are equal to the area of the square on the hypotenuse.

Near the end of class time the discussion should be wrapping up with students talking about why this proof of the Pythagorean Theorem works. Students might also wrap up the discussion wondering if there are other proofs for the Pythagorean Theorem. You could also have each student pull out the baggie of shapes they used in the launch activity and show them how shapes 1-5 fit into the square. If you were to do this have student glue this proof into their math journals.

Day 4:


Grade: 8th

Timing:



CCS 8.G.6. Explain a proof of the Pythagorean Theorem and its converse.

CCS 8.G.7. Apply the Pythagorean Theorem to determine unknown side lengths in right triangles in real-

world and mathematical problems in two and three dimensions.

CCS 8.G.8. Apply the Pythagorean Theorem to find the distance between two points in a coordinate

system.



There will need to be enough scissors for each student at one station

Each student will need a hard copy of each station

Station Activities for Common Core Mathematics, Grade 8

There will need to be a basket full of colored pencils or markers at one station

There will need to be glue or tape at one station, glue preferably

http://mathfoldables.blogspot.com/2012/04/pythagoras-o-pythagoras.html







I can ask students before they begin the task at hand: “How do you all think TV manufactures determine how large a TV screen is?” If I was to ask students this they would have to think about the shape a TV is and what the

For the launch activity a picture is going to be displayed and students are going to be asked to answer two questions by using the picture. The picture being displayed on the Smart Board is a flat screen TV. This flat screen TV though has a diagonal line running through the TV from one corner to its opposite corner. It

“How are we supposed to do this when the shape of a TV is a rectangle?”

“Do we use the Pythagorean Theorem to solve these questions?”

“What is diagonal?”

“This or any TV that has the shape of a rectangle can be broken into two right

If you were to be asked “How are we supposed to do this when the shape of a TV is a rectangle?” ask the student to describe the picture to you. Make sure that they notice that where the diagonal line cuts across turns the rectangle into two right triangles.


characteristics are they know about that shape.

I could also ask students how they would find the measure of a TV screen if they were asked to. By asking this question will get students minds thinking.


I can use guiding questions to keep students thinking about how they could find the measure of a TV screen. Such as: “What kinds of tools would you all use to assist you in finding the measure of the TV screen?” “What dimensions of the TV would you need to know in order to find out how big the TV screen is?” and “Have any of you ever had to figure out how big a TV screen was?”


I can make this part of the lesson personal and relevant to my students’ lives through changing the context of the problem. For

is important to note to the students that the size of a TV screen is measured by the length of the screens diagonal. Have students copy this picture down into their math journals. Next display the two questions each student is being asked to complete. The first question is “If a TV screen measures 24 inches high and 18 inches wide, what size TV is it?” The second question is “Susan told Johnny that she wanted a 35-inch TV. The height of the screen is 21 inches. What is the width?” Through this activity students have to think of the TV as two right triangles instead of one rectangle. This may through some students off but others will be able to figure out the problems quickly. This activity will help students with one of activities for station one in the explore phase of this lesson.

triangles.”

“Now we can use the Pythagorean Theorem to figure out how large a TV screen is.”

From here the student should understand how they can go about solving the problems.

If a student is to ask if the Pythagorean Theorem is going to be used to solve these problems make sure you ask them why they think it is going to be used. By asking a student to justify why they think something is going to be used can help them understand why it is going to be used and why it is effective to use it.

You may be met with a student wondering what a diagonal is. If a student is to wonder this explaining it to them using the picture giving would be the best possible thing to do. That way the student knows what the diagonal is in this scenario and can apply it properly.

If a student is to notice that when a rectangle is diagonally split apart it forms two right triangles this

example I could say we are getting a new TV for our classroom and I need their help figuring out how large the screen is going to be on the TV. From there students could answer guiding questions and answer the two questions.

This could also be made relevant to the students’ lives through placing the students in this situation outside of the school environment.


I difficulty I can foresee is students being thrown off by the fact that a TV is a rectangle not a right triangle.

Another difficulty I could foresee with this activity that goes hand in hand with the one above it getting students to realize a rectangle is two right triangles.

An advantage to this activity that I could foresee is it will help student understand part of station one in the explore phase of the lesson.

is great. Make sure you ask the student how they came to this conclusion.

If students are to realize they could use this method outside of school to apply it to real-world situations, wonderful! This could mean the student is grasping what is going on and beginning to understand how to apply their knowledge.

Another advantage I could foresee is students grasping onto this idea and being able to use it later in real world situations.







For this activity students will be placed into small groups of three to four students, all depending on the size of your class.

Students will be going through three stations. Through these station students are asked to work together to solve problems and get through tasks.

If a student though cannot handle group work these stations can be done individually if necessary.


Students will each need a hard copy of

For the explore activity students will be doing station activities on the Pythagorean Theorem. The station activities were taken from Station Activities for Common Core Mathematics, Grade 8 and from http://mathfoldables.blogspot.com/2012/04/pythagoras-o-pythagoras.html. The template being followed for this activity was taken from Station Activities for Common Core Mathematics, Grade 8 have been modified given the information that has been covered in previous lessons. Some of the stations are going to be skipped or supplemented with a different activity. For station one students will completing what is

“I do not want to work with this person.”

“What does diagonally mean?”

“Which plot of land is the oldest brother’s property?”

“Does the term ‘square feet’ mean area?”

“This diagram for station two is confusing.”

“Are all of those triangles the same size?”

“What is the formula for finding area of a triangle?”

“If I found the area of one triangle, can I just multiply it by four to get the area of all four triangles?”

“This is an interesting proof of the Pythagorean Theorem, I had not thought about it that way.”

If you were to be approached or hear that a student does not want to work with the ones they are working with there are several different things you could do. One of the first things you could do is talk to the student and ask them why they do not want to work with their group. If a serious and legit reason is given you could give the student an opportunity to switch groups. If a student does just not want to work in a group they need to tough it out. But if this student becomes a distraction to the rest of the group you can have them work separately from the group.

If you are





each of the stations. It is not necessary for students to have all of the hard copies before they begin the stations. They can collect a hard copy at each of the stations; this way other students cannot begin to work ahead and leave others in their group behind.

At the third station there will need to be enough copies of the figure for each student. Students are going to be using to demonstrate another proof of the Pythagorean Theorem. Also at this station there will need to be scissors, glue or tape, and makers or colored pencils.

Again for station three it is essential that each student has their math journal or a piece of loose leaf paper they can place into their math journal later.


A difficulty I can foresee with this activity is some students not taking each station

station two from Station Activities for Common Core Mathematics, Grade 8. At this station students are being asked to discuss and answer the following questions as a group. Students are to read the scenario given. The scenario is about three brothers who own land around a park, the shape of the park is in a right triangle. A measurement of two of the brothers’ land is given and the other brothers land measurements are not given. There is a visual representation for this problem displayed on the hard copy of the station activity that each student will have. Students are being asked to find the area of the third brothers’ land. Students are asked to explain their reasoning, logic, and a strategy behind how they found what the area was of his land was. Students are also being asked to find out how far the oldest brother would walk if he walked diagonally across his land. For this station students are being asked to apply the Pythagorean Theorem to a real-

“I do not understand how to cut out this figures for station 4.”

“How are both of these squares supposed to fit into this one big square?”

“This proof of the Pythagorean Theorem really helps me understand what is happening.”

“Was I supposed to color these two squares the same color?”

approached with “What does diagonal mean?” while a student is working on station one you could give the student an example of something that is diagonal to something else. From those examples the student should be able to figure out from context clues what diagonal means. If the student still does not understand you could have the students in the group help guide the student in the right direction.

If you are met with students asking questions such as: “Which plot of land is the oldest brother’s?” or “Does the term square feet mean area?” For the first question you should direct the students back to the reading, because in the reading it clearly states which piece of land is the oldest brother’s. The second question is one most students should know. If a student is really

seriously and relying on other students in the group to do their work for them.

An advantage I can foresee with this activity is an increase in student interaction, students helping one another.

Another disadvantage I could foresee is some students not being willing to help other students in their group who might be struggling.

Another advantage to this activity I could foresee is it allows the teacher to be able to roam in between stations. Being able to roam in between stations allows the teacher to be an active monitor, but it also allows them to listen to conversations and informally assess students knowledge.

world mathematical situation. For the second station students will be completing station three out of Station Activities for Common Core Mathematics, Grade 8. In this station students are being asked use the diagram and the area of the diagram to prove the Pythagorean Theorem. There is a diagram on the page that is made up of four right triangles placed together. Below the diagram is a single right triangle, labeled, that will help students be able to make sense of and understand how the diagram works. Once students have closely examined the diagram they are then guided to label the sides of the four triangles in the diagram with a, b, and c. After students have labeled the diagram they then need to answer the following questions: “What is the area of the small square?” “What is the area of one of the triangles?” “What is the area of the four triangles?” and “What is the area of the large square?” Once students have completed this station

struggling with this you might want to say something to them like “What is area measured in?” This should hopefully clear up confusions and clarify your answer.

Many of the questions that could be asked for station two are question students should be able to figure out on their own or either in groups. Some struggles students might have with this station could be understanding the diagram and understanding that all of the triangles are the same. To help students understand this you could have a tangible version of the diagram that comes apart. This would be helpful for students to be able to see how it works and that all of the triangles are the same size. Another thing students might have some confusion about could be finding the area of either the triangles or the squares. Here

they have added another proof to their ‘library’ of the Pythagorean Theorem. At station three students will be doing a combination of station four from Station Activities for Common Core Mathematics and from the site listed above. At this station students are going to answer the questions from station four but they are going recreate the model from the website in their math journals. At this station there should be a copy of the diagram for each student. This diagram is a right triangle with grid squares on the legs and hypotenuse. On each copy there are two copies of this diagram; students will be using both of them. Student will cut out one of the copies around the perimeter of the figure and the other diagram will need to be cut so that the largest square is still intact with the right triangle, the other two squares need to be loose. Once they have cut out the diagrams they will be instructed to label the dimensions of the right triangle

students are working with variables and not numbers, which can easily throw them off. If this is the case working through one question with that group of students or student could greatly impact and help their understanding.

In station three students could very easily have questions about how they are supposed to cut each of the figures out and how they are supposed to either glue or tape them down. If these questions are to arise have students reread the directions they have been given. If they can still not make sense of what they need to do a visual demonstration could and more than likely would be helpful.

Also at station three you may have some students who know that the two squares on the legs are supposed to fit into the square off the hypotenuse but

and then label the area of the squares on the diagram where they cut around the perimeter of the figure. The square off the legs of the hypotenuse need to be colored in, each in a different color. Next students will be directed to glue or tape down the right triangle that has largest square intact into their math journals. Once students have done this they will then be directed to take the figure that was cut out around the perimeter and glue or tape down the right triangle and the two smaller squares on top of the other right triangle already on the page. When glued or taped down the two figures should align, the large square should be underneath the large square, but the top large square should not be glued or taped down. Students will then be directed to color in the two squares that were cut off to match the two squares on the other diagram. When the coloring is completed students will then be directed to place the larger of the two squares onto the

they are not sure how to go about it. If this is the case remind them to place the larger of the two squares inside the largest square and then that they can cut apart the small square to make it fit into the largest square.

bottom larger square. Students will need to place this square in one of the four corners and glue or tape it down. Students should then be left with the smaller square. It is there challenge to figure out how that square will fit into the larger square. Students will be told they can cut the square apart to make it fit. When students figure out how it fits they need to glue the pieces of that square into the bottom larger square. When students are done with this part of the station they should have a neat proof of the Pythagorean Theorem. This is a useful visual for students to be able to see how the areas of the squares on the legs are equal to the area of the square on the hypotenuse. Next students need to answer the following questions: “How many squares are there?” “Count the boxes in the square that is attached to the hypotenuse; how many total squares are there?” “What do you notice?” and “How does this

represent the Pythagorean Theorem?” When this station is completed student should have another neat and interactive proof to add to their ‘library’ of proofs for the Pythagorean Theorem. When students have completed all stations they should have applied the Pythagorean Theorem to a real-world situation, found the distance between two points, and explained two proofs of the Pythagorean Theorem.







In this particular activity students are not going to be discussing with one another unless there is some leftover time near the end of class.

Although to orchestrate students summarizing their thinking a serious of open-ended

For this summarize activity students are going to be reflective writing. Reflective writing allows the students to look back on their learning over this lesson to see what they know, how they think, and where we need to go next. In particular students will be reflecting on their comfort level with the content, their personal reactions and feelings, and how they can make use of their new learning.

“I appreciate how we covered several different proofs of the Pythagorean Theorem. This way I was able to see how and why it really works.”

“I now understand how to apply the Pythagorean Theorem to real-world situations.”

“I am still not sure about how the Pythagorean Theorem works.”

“I do not want to

Because students will be writing and not actually talking it is vital that you as the teacher be walking around the room. While walking around the room be observing what students are writing. Remember most students are being honest about their reflections. Although you are the one who knows your students best. So

thoughts can be posed for them to answer. Such as: “Today, I learned…” “I appreciate how we…” “Now, I understand…” “I still wonder about…” and “I can apply this to…” By students reflecting back on those and honestly giving answers will help them be able to summarize what was covered in the lesson. This is also a great way for students to tell how comfortable or uncomfortable they are with certain areas of the content.


At this point in time each student should understand and be able to explain several different proofs of the Pythagorean Theorem.

They should also be able to describe the converse of the Pythagorean Theorem as well.

Students should be able to find the measure of a diagonal drawn across a rectangle or square, distance.

Students should

Ask students to turn to the next clean page in their math journals. Here students will write about their comfort level with the content, through guided suggestion starters. Such as: “Today, I learned…” “Now, I understand…” “I still wonder about…” “I appreciate how we…” and “I can apply this to…” As students are reflective writing in their math journals it is key that you as the teacher be walking around observing what students are writing. This is a great way for you to be able to see what your students understand, what their comfort level is with the content, and what they may still be struggling with. If students wrap up their reflective writing with some time leftover at the end you could ask students who feel comfortable in sharing to share something they wrote about in their reflective writing.

write any of this down, I have it all in my head.”

you should be able to judge pretty quickly who is being serious and who is not.

If you catch a student not being serious about what they are writing ask them a question on the spot about something they have written and have them explain to you why they feel that way or think that.

While observing students if you notice a student a written down something in regards to struggling with a piece of the content take this as an opportunity to help this student. This does not mean stop everything you and the student are currently doing, but rather you could arrange a time later to meet and discuss what is going on and help them better understand the content.

know how to apply the Pythagorean Theorem to real-world situations.


A difficulty I could foresee with this activity is some students not taking the reflective writing seriously. If a student does this, it is not accurately conveying how well they know the content.

An advantage I can foresee with this activity is those students who excel in writing might be able to express what they understand better through writing it down rather than translating it orally.

Another advantage I can foresee with this activity is students being given the opportunity to be honest and real about what they understand and what they might still be having difficulties with. This would be great for a shy or timid student.

Day 4:


Grade: 8th

Timing:



CCS 8.G.6. Explain a proof of the Pythagorean Theorem and its converse.

CCS 8.G.7. Apply the Pythagorean Theorem to determine unknown side lengths in right triangles in real-

world and mathematical problems in two and three dimensions.

CCS 8.G.8. Apply the Pythagorean Theorem to find the distance between two points in a coordinate

system.



Enough construction paper for each student

Markers

Colored pencils

Scissors

Glue

Tape






Q: How will I engage students’ prior knowledge? Q: How can I keep from giving away too much of the problem? Q: How can I make it personal and relevant to the students? Q: What advantages or difficulties can I foresee?






Q: How will I organize the students to explore this problem? (Individuals? Groups? Pairs?) will students need to encourage diverse thinking and problem solving? Q: What advantages or difficulties can I foresee?






Q: How can I orchestrate the discussion so students summarize the thinking in the problem? Q: What generalizations can be made? Q: What advantages or difficulties can I foresee?

For this summarize activity students will be creating a memory box of the Pythagorean Theorem. In this memory box students are going to be writing down everything they know about the Pythagorean Theorem, it converse, the equation, proofs, and explanations of proofs. Have students each get a few pieces of construction paper. With one piece of the construction paper have each student fold it in half, hamburger style. Once students have done this allow them

to decorate the outside of their memory box. You might suggest to students to decorate the outside of their memory box with objects that relate to the Pythagorean Theorem. Allow students a few minutes to do this. Most students will not be finished with decorating their box, but that is not the important part of this activity. Allow students some more time at the beginning of class or near the end of class another day to finish decorating their boxes. On the inside students should begin writing down everything they have learned about the Pythagorean Theorem. Allow students to use their notes and their notebooks so they will have something to pull from. Students should be writing about the proofs they have learned about and explain how they work, characteristics and parts of a right triangle, how to apply the Pythagorean Theorem to real-world situations, what the converse is and how they can prove it, and

anything else the student found interesting about the unit. Give students about 10 minutes to complete this portion of the task, they can always go back later and add more to their memory box. When the 10 minutes is up have students think about what the most important thing is that they wrote down in their memory box, this should be something that if they did not understand then they would not understand what the Pythagorean Theorem is or how it works. Have them move outside of their memory box onto a separate sheet of paper and write down what they chose and then explain why they chose that in detail.

Documents

5 Day Unit Plan Launch, Explore, Summarize€¦ · Pythagorean Triples: The Pythagorean Theorem states taht: In a right triangle, the square length of the hypotenuse is equal to the