Day 45: Pythagorean Theorem Study Guide

Regular Math Objective: Find unknown triangle side lengths by applying the Pythagorean Theorem; Apply the Pythagorean Theorem in the coordinate plane

Regular Math Standards: 8.G. 6a. Understand the relationship among the sides of a right triangle. b. Analyze and justify the Pythagorean Theorem and its converse using pictures, diagrams, narratives, or models.
8.EE.7 Apply the Pythagorean Theorem to determine unknown side lengths in right triangles in real-world and mathematical problems in two and three dimensions.
8.EE.8 Apply the Pythagorean Theorem to find the distance between two points in a coordinate system.

Regular Math Lesson Sequence:

1. When students entered class we immediately got to work on two warm up problems. One problem asked students to turn 4/15 into a decimal and put it on a decimal number line correctly. The other problem asked students to divide using scientific notation. The denominator in that problem had a negative exponent. Both of these are questions that we have had misconceptions with and just plain old freezing. They will therefore be on a quiz the next day that have nothing to officially due with those topics. Where would we be without spiraling? 
2. QSSQ - During this time I explained that the second study guide could be done for 50 points again on the quiz. I love giving away 50 points. Standards assessment guru Rick Wormeli really thinks we should all be going to a four-point scale anyway (I bet in five years I will be grading on a four-point scale) because the 0 is way too unfair. 
3. I gave students two minutes to do the first two problems on the study guide and then reviewed them on the board.
4. I waited 8 more minutes for students to do problems 3 through 8 and then reviewed them on the board. After about six more minutes we had five minutes to go, so I reviewed the remaining problems at the board and passed out the second study guide. The answers (which are seen in the link) were available online for students to have feedback.

Honors Math Objective: Identify function notation and differentiate between the input and output in function notation

Honors Math Standards: A1.F-IF 2 Use function notation, evaluate functions for inputs in their domains, and interpret statements that use function notation in terms of a context.

Honors Math Lesson Sequence:

1. Pepper/Review HW 
2. QSSQ 
3. Exit Ticket. I did this with plenty of time left in class because I wanted to see what they retained from the homework and to avoid the bell getting in the way of impacting the authenticity of the assessment. It was a good idea. There was clear mastery in parts of the class, some students that were answering one out of two questions, and a few that have no clue. 
4. Students could ask any question that they wanted from the weekly quiz. The only catch was that I would not answer them. I had done way too much talking and spoon-feeding math answers and concepts for one day. So I had two students that already completed the weekly quiz without errors come to the front of the room as I did other teacher stuff. The class was a little less orderly and the students that went to the front could have done a better job communicating. That being said I would prefer to keep this routine in the future. The students struggle to explain even though they got answers correct and often do not recall if they received help when they originally did the assignment. 

Day 44: Pepper Battles

Quote of the Day: “Philip of Macedonia, the father of Alexander the Great, said ‘An army of deer led by a lion is more to be feared than an army of lions led by a deer.’ That may be true, but I’ve come to believe that Philip missed the bigger point: An army of lions led by a lion is to be feared most of all, for it is unstoppable.” - Mark Sanborn

Question of the Day: Where does the term hypotenuse come from? How do we know whether to subtract or add in doing the Pythagorean Theorem (in reference to finding a leg or finding the hypotenuse)?

Regular Math Objective: Given two points on a coordinate plane find the distance between them using the Pythagorean Theorem.

Regular Math Standards: 8.G. 6a. Understand the relationship among the sides of a right triangle. b. Analyze and justify the Pythagorean Theorem and its converse using pictures, diagrams, narratives, or models.

Regular Math Lesson Sequence:

1. My Favorite No. This was really rewarding to teach. I did not give students anything today. I was locked into ask questions and let students prove why mode. The problem I gave was the points (0, 1) and (-2, 1). Too many students right away got four units. 
   
   Exactly like that. No work. No explanation. So I asked how did you get that? Kids came to the board throughout the day and traced over the legs of the triangle. They were missing the fact that you could go directly from one point to another with a straight line. I used the analogy of walking to the trash bucket in the room and it was effective. Then students had the problem of counting the distance diagonally on a square unit as one unit. To confront this, I took a square object in the room and measured it across as well as diagonally. Students saw that diagonally it was longer. At this point on the SMART Board we had two lines tracing the legs and one tracing the hypotenuse. I told students to look at the board for fifteen seconds and not write or say anything. And finally they started writing a formula for the Pythagorean Theorem. 
2. Depending on the results of My Favorite No, we either did one more problem or worked on what would have been the homework if Halloween wasn't a thing. I started the students with problems three and four on the link since they related back to My Favorite No. I also really like problems five and six as standards to revisit because students are not recognizing the difference between approximating and getting exact values for the irrational numbers. 
3. The exit ticket question was a math mistake of a 3, 4, 5 triangle that had a hypotenuse of 25. We were more successful than normal because at this juncture in the unit, students are actually well ahead of the curve on this mistake. It made for easy correcting and a nice ego boost. 

Honors Math Objective: Use function notation to identify graphs and their matching functions

Honors Math Standards: A1.F-IF 2 Use function notation, evaluate functions for inputs in their domains, and interpret statements that use function notation in terms of a context.

Honors Math Lesson Sequence:

1. The entire class went to the classroom of the other 8th grade math teacher for Pepper Battles. We had students stand one minute at a time and asked them vocabulary and quick theory based questions. Within 8 or 9 minutes we were out of there. It was a great break from the norm. 
2. QSSQ
3. We discussed the various meanings of parenthesis. It ranged from multiplication to an ordered pair to probability to least common multiple. We then introduced f(x). I was quick to point out that this is essentially the same thing as y. 
4. The students did an explore activity from Big Ideas. 
5. We discussed what graphs from the explore activity go with what functions. By the time I handed out the homework I realized that the students were going to be overwhelmed by what they were facing. We'll have to revisit it again tomorrow. 

Day 43: Coordinate Plane & Pythagorean Theorem

Quote of the Day: “Playfulness and humor improve the quality of a person’s ideas and creativity, not to mention a person’s health...Studies have shown that laughter reduces stress hormones and increases the level of mood - and health-enhancing chemicals such as endorphins and dopamine.” - Greg Bell

Question of the Day: How can a graph be linear and continuous? If the legs of a right triangle are the same is the hypotenuse always irrational?

Regular Math Objective: Apply the Pythagorean theorem in the coordinate plane

Regular Math Standards: 8.G.8 Apply the Pythagorean Theorem to find the distance between two points in a coordinate system.

Regular Math Lesson Sequence:

1. Student choice either weekly quiz or 11 squared through 20 squared as a warm up as I passed back exit tickets 
2. QSSQ 
3. Exit ticket review. I found the students tuning me out as I was reviewing so I periodically had them get out of their seat and kinesthetically show me where the legs and hypotenuse of a right triangle were. By the last class I just made students stare at my work for 15 seconds and ask a question. Only two kids volunteered questions, but they were good. "Why did you get 7 instead of 49? How did hypotenuse get its name?" 
4. Homework review and pepper. The pepper topics today centered on coordinate plane, right triangles, and rational numbers 
5. Pythagorean and coordinate plane exploration. It was really a great example to go by for exploration, but I completely left off connecting the hypotenuse in the directions. Here is the document. I would add a question nine about connecting the points to create the hypotenuse before students declare that they have just created a right triangle. 

Honors Math Objective: Derive an equation of a graph

Honors Math Standards: A1.F-BF 1 Write linear, quadratic, and exponential functions that describe a relationship between two quantities.

Honors Math Lesson Sequence:

1. 4 minutes of Pepper
2. Review of the graphs and equations from yesterday
3. I passed out four new situations and had the students take 5 minutes to graph them independently. It takes about 2 and a half minutes per graph, so this was not enough time but at some point we need to move forward. 
4. I gave students 8 minutes to debrief in groups.
5. I went over the problems on the board. Students were really angry when they saw the step function that described the first problem. "Is this going to be on the test?" I don't know. It's going to be in this class though right now in this moment. I thought it would be fascinating to discuss if this problem were a function. The kids were still hung up on the chaos of the whole thing compared to what they thought this situation might be (see the picture below). In their defense, this stuff is definitely hard. I think it's really fun though. 
6. I had students match equations to the graphs. 
7. QSSQ

Day 42: Questions to Consider When Graphing

Quote of the Day: “The word compete comes from the Latin competere, which means ‘to seek together; strive in common; coincide.’ True competition means two (or more) rivals are playing the game they love together.” - Jim Murphy

Question of the Day: Wouldn't a squared plus b squared equal c to the fourth power?

Regular Math Objective: Use the Pythagorean Theorem to calculate the leg of a triangle when the hypotenuse and one leg is given.

Regular Math Standards: 8.G. 6a. Understand the relationship among the sides of a right triangle. b. Analyze and justify the Pythagorean Theorem and its converse using pictures, diagrams, narratives, or models.

Regular Math Lesson Sequence:

1. Pepper. I loved kicking the class off with pepper today. My heavy review topics were the square roots of numbers that are perfect squares up to 400 and defining why rational numbers were rational numbers. The kids were dead - especially in the morning with the sun not rising yet and the weekend still fresh in their minds. 
2. QSSQ - I included a recap of the previous lesson today in QSSQ because we were coming off of a weekend. Well worth it.
3. Marker boards to try two problems of a rational and irrational hypotenuse. Students had all sorts of unique answers such as the hypotenuse is 100 for a triangle that had legs of 6 and 8. I only did this in two out of four classes, but when it came time for the exit ticket this proved to be very helpful. We only did two review problems with the boards, but it makes all the difference. The student in the picture below tried to argue that the side was exactly 7 and 1/7 because she added it 7 times. Her group actually corrected her by saying she was not multiplying the same number by itself but was doing 7 times 7 and 1/7. 
   
4. Exploration of finding the legs of a triangle. I really enjoyed this. Some students were able to do it on their own without my instruction. Other students needed some prompting. It was also interesting to see that no students dared to leave the answer as an irrational in a square root sign because we were taught to find approximations earlier this year. Leaving it in a square root is virtually unheard of for them so I had to introduce that. In some classes I referenced the fact that we write pi as the symbol not 3.14. 
5. Exit ticket. One triangle needed the leg solved and one needed the hypotenuse solved. One was irrational and one was rational.

The student above froze once they saw that it was irrational. The student below just sort of ditched the c squared part of the equation. 

Honors Math Objective: Derive an equation of a graph

Honors Math Standards: A1.F-BF 1 Write linear, quadratic, and exponential functions that describe a relationship between two quantities.

Honors Math Lesson Sequence:

1. I had the students work on a graph for a problem that had the number of workers on the x-axis and the amount of time it takes to paint a bridge as the y-axis.
2. We discussed the problem in small groups and then as a large group.
3. I went over four questions worth considering as we were comparing and contrasting different graphs. Is it linear or nonlinear? Is it discrete or continuous? Is it increasing or decreasing? Where are the intercepts?

 This picture just frustrated me. Then again, it does help the students line up the numbers on certain points. I suppose marker board graph paper (such a product does exist) would be useful.

 It was cool to see the student's explanation in the corner. That said I was wondering why the y-axis was horizontal and the x-axis was vertical.

Here this student is about to erase the continuous function and make it discrete. I did not say anything. I think she knew by virtue of the camera coming out it needed to be tended to. 

Eventually in going over the four questions we needed all of these students seemed to have an uh-huh moment in terms of the fact that it was decreasing, discrete, non-linear and had to avoid the intercepts. This lesson is taking a much longer time than I would have anticipated, but I'm finding it engaging for the students and fun as a teacher. We have been lacking a good deal of real-world connection to this point in the year and so this is a little refreshing. 

Day 41: Pythagorean Theorem Exploration

Quote of the Day: “Mark Twain said, ‘I can live for two months on one good compliment.’ That’s the way most people feel. They are willing to work hard if they receive recognition for their efforts.” - John Maxwell

Regular Math Objective: Use area of squares and triangles to find an unknown side length

Regular Math Standards: 8.G. 6a. Understand the relationship among the sides of a right triangle. b. Analyze and justify the Pythagorean Theorem and its converse using pictures, diagrams, narratives, or models.

Regular Math Lesson Sequence:

1. WODB #28
2. Review the first four questions from the Pythagorean theorem exploration. I got the idea for this exploration from Math is Fun. I modified it a little. Instead of using a and b for the legs of the right triangle, I used 3 and 4 (color pic in word) to make the problem less abstract. I feel like students at this age are still having a hard time with variables. I kept one side a variable so that we were discovering something in this problem. As we were uncovering what c was, I would give students three minutes to discuss in groups and go over problems in groups of two every two to three minutes. As students were exploring I circumvented to classroom manage, but also to direct the conversation where necessary. It was really cool when they got to the final questions.
3. In classes where time allowed, I had students cut out the four triangles so we could see what it looked like to subtract visually. 
4. After the exploration, I used the term hypotenuse. I had students write it at the bottom of the exploration as a picture. I then let the cat out of the bag and put the a-squared plus b-squared equals c-squared jargon on the board. I felt bad, but I also remember taking the test to become a math teacher and finding the origins of this theory to be baffling as a college graduate, so I wanted students to see it. After seeing it we could break it down again. 
5. For an exit ticket (again where time allowed) I had groups find the hypotenuse of a right triangle with 6 and 8 as the legs. 

Overall I loved this lesson. Students were asking me questions like does it matter if it's a right angle and why a squared plus b squared did not equal c to the fourth power. I also had a student try to make sense of all it algebraically (which we will revisit on Monday). 

I also saw many rich mistakes with units and square root calculation. 

Honors Math Objective: Graph everyday situations

Honors Math Standards: A1.A-SSE Interpret expressions that represent a quantity in terms of its context.

A1.F-LLE Distinguish between situations that can be modeled with linear functions and with exponential functions.

Honors Math Lesson Sequence:

1. WODB (same as above). 
2. Revisiting the exit ticket from yesterday. Today I let students talk by discussing what they wrote on the exit ticket. I think the conversation was vastly improved because students had the opportunity to try it on their own and then could compare. As we discussed a candle burning over time, one student mentioned that she had thought she was graphing how many inches had been burnt over time and not the height of the candle. The conversations were really powerful. 

Ultimately as I was taking part in the conversations with students, I realized it boiled down to four questions. Is the graph increasing or decreasing? Is it discrete or continuous? Is it linear or nonlinear? What should the intercepts look like? We only really broke down two graphs, but that is how rich this content is for where these students are in their learning. Next class, we can continue the conversation with how long it will take to paint a bridge. 

Day 40: Linear Functions

Quote of the Day: “I don’t have that much self-discipline. But if I’m surrounded by people who are writing articles and giving lectures and working hard, I tend to fall in line. If I’m in a crowd of people doing things a certain way, I follow along.” - Dan Chambliss

Regular Math Objective: Divide exponents that have negative and positive integers; find how many times a number in scientific notation is relative to another number in scientific notation

Regular Math Standards: 8.EE.3 Use numbers expressed in the form of a single digit multiplied by an integer power of 10 to estimate very large or very small quantities, and express how many times as much one is than the other. For example, estimate the population of the United States as 3  108 and the population of the world as 7  109 , and determine that the world population is more than 20 times larger.

8.EE.4 Perform operations with numbers expressed in scientific notation, including problems where both decimal and scientific notation are used. Use scientific notation and choose units of appropriate size for measurements of very large or very small quantities (e.g., use millimeters per year for seafloor spreading). Interpret scientific notation that has been generated by technology.

Regular Math Lesson Sequence:

1. Students got a self-assessment and their quizzes returned to them.
2. Students shared a pathway to get an answer with a partner. This process is painful for me. I tell the kids that all they have to do is put one problem that they got wrong on a marker board by asking a person in their group who got that question correctly. Yet, way too many of the marker boards are left blank. Maybe I need to have it modeled and let all the students see it done. 
3. After the self-assessment and quiz review in groups, I asked if there were any remaining questions and if students thought that they could do better on the quiz at that moment than they had the prior class. All hands went up. 
4. I gave the students two problems to try on the marker board. First I had them simplify ten to the fifth divided by ten to the negative first. Next I had them find how many times greater 6 x 10 to the fourth is than 2 x 10 to the third power. 
5. We discussed a picture that modeled the Pythagorean Theorem although I left out the term 'Pythagorean theorem' intentionally (this pic is Microsoft Word but in color). Then I had the students answer four questions pertaining to the formulas for area of a triangle and area of a square. I did not want to assume anything when it came to their prior knowledge even though I had many of these students in sixth grade and knew that my colleagues and I hammered the formulas into their brains. Not literally. We'd get fired, but we did everything but that. These four questions were passed in as an exit ticket. That way I could see what students wrote, but more importantly they wouldn't lose this sheet between this class and the next class. 

Honors Math Objective: Determine if a function is a linear function;

Honors Math Standards: A1.F-IF A1 Understand that a function from one set (called the domain) to another set (called the range) assigns to each element of the domain exactly one element of the range. If f is a function and x is an element of its domain, then f(x) denotes the output (range) of f corresponding to the input x. The graph of f is the graph of the equation y = f(x).

Honors Math Lesson Sequence:

1. As a jumpstart students found the domain and determined if graphs from the back of the Domain and Range Activity. 
2. I had students do the tables and graphs from the linear functions explore that we did for homework on the board. Assigning an explore for homework was a bad idea and I didn't help myself by giving away every copy I had to students. 
3. We reviewed the exit ticket. Students were having trouble writing the notation for domain and range. They had a stronger grasp on whether it was a function or not. 
4. I passed out the linear functions homework
5. I had them start the Functions and Everyday Situations by making them graph the first four problems from that activity. This was great as I gave them seven minutes to do it, but they wouldn't shut up. "Should we add numbers?" "It's too hard." "How am I supposed to do it?" I told them I wanted them to struggle up front and said that they would be working with a partner on this tomorrow, but they still wouldn't listen. Despite the frustration I did get what I ultimately wanted. I saw what kids thought. And they thought that virtually every graph was continuous and linear. And almost every graph was increasing. 

Day 39: Scientific Notation Quiz

Quote of the Day: “Because our brains are programmed to habituate quickly to our circumstances, we tend to tune out events that happen repeatedly, no matter how positive.” – Ron Friedman

Question of the Day: What happens to the exponent when it is converted to scientific notation in the expression 12 x 10 to the negative second?

Regular Math Objective: Write numbers in scientific notation; multiply and divide numbers in scientific notation find cube roots/square roots of negative exponents and perfect cubes/squares

Regular Math Standards: 8.EE.2 Use square root and cube root symbols to represent solutions to equations of the form x 2 = p and x 3 = p, where p is a positive rational number. Evaluate square roots of small perfect squares and cube roots of small perfect cubes. Know that is irrational.

8.EE.3 Use numbers expressed in the form of a single digit multiplied by an integer power of 10 to estimate very large or very small quantities, and express how many times as much one is than the other. For example, estimate the population of the United States as 3  108 and the population of the world as 7  109 , and determine that the world population is more than 20 times larger.

8.EE.4 Perform operations with numbers expressed in scientific notation, including problems where both decimal and scientific notation are used. Use scientific notation and choose units of appropriate size for measurements of very large or very small quantities (e.g., use millimeters per year for seafloor spreading). Interpret scientific notation that has been generated by technology.

Regular Math Lesson Sequence:

1. I collected the study guides. I have never tried this before, but today the study guides were worth 50 points. I put the answers up online for all kids to check their work. What I asked the students to do was try it on their own and check the answers against mine with a different colored pen and write in the corrections. The two examples above were what I was looking for. Some students understood it clearly while others just passed in the study guide we had done from class or tried the online study guide without checking the answers. Overall I think more people studied than they typically would, but for this process to be worth it students needed to check their answers because otherwise they may have just been reenforcing a bad habit. 
2. I read the directions of the quiz and then gave students one minute to ask each other anything they wanted. Again, I have not tried this before. Some students were silent. Others took advantage. I left out the directions of not looking at your quiz while this was taking place and it was a mistake because I had no control over not telling each other the answer to number one is D or whatever. 
3. They took the quiz.
4. They worked on their TenMarks weekly quiz. 
5. They did any necessary retakes. 

I love what this student did on Problem 1. She is not being apathetic. She wrote that none of the options were scientific notation. She was not nickel grabbing to bring her grade up. And I acknowledged her courage even though her answer was ultimately wrong.

The hardest problems on the quiz were problems four and five (straight out of MCAS and the common core standards) as well as number six. Here's what I saw:

This answer to number five was about as solid as I saw. I liked it because the student used words in combination with the math. Number four is also done correctly here. 

The top answer is a misconception I saw a decent amount of. The students were multiplying instead of dividing to show how many times greater something is. The student made things worse by putting a negative seventeen for some reason. The bottom of the two pictures above is a correct answer, but I think it is more of a guess based on the lack of evidence to support the answer.

No problem got more red X's than number six. Students used the exponent 1 more than anything else. They simply do not know how to subtract integers yet. I think there is also a lack of willingness to persevere when it comes to recognizing the meaning behind a negative exponent in a denominator. 

Honors Math Objective: Explain why a function exists or does not exit.

Honors Math Standards: A1.F-IF A1 Understand that a function from one set (called the domain) to another set (called the range) assigns to each element of the domain exactly one element of the range. If f is a function and x is an element of its domain, then f(x) denotes the output (range) of f corresponding to the input x. The graph of f is the graph of the equation y = f(x).

Honors Math Lesson Sequence:

1. Exponents Jumpstart. I still think that there are students that are failing to see why a negative exponent creates fractions. Since this was a high point of emphasis with regular math and an eighth grade standard I thought it was worth the time to create a chart with the base ten. 
2. QSSQ
3. Review the homework. This was extremely helpful in my opinion for defining what tables and graphs were functions and what were not. We also continued to hammer home the definitions for range, domain and function. Today I introduced discrete and continuous functions as well in pepper. 
4. Domain, range, and function exit tickets
5. Linear functions exploration (got cut off due to time constraints)

Here are some mistakes that were evident on the exit ticket. The students are struggling much more with domain and range than what is and isn't a function. That said, here's a look at what they're doing with the function definitions. The one above is good the one below is good except the one on the right.