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. 

Day 38: Intro to Functions

Quote of the Day: “Treat today as if it were your first day. Ask yourself, ‘If I were just starting this job, what would I do differently?’” – Barry Posner

Question of the Day: Does the order matter in terms of where you put numbers in scientific notation? In other words is 2 x 10 to the third the same as 10 to the third x 2

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: This was where it went wrong. I did not really have a sequence going in. I knew the study guide would not consume the whole class. I interspersed QSSQ, reviewing the homework and pepper throughout the lesson. It was bad. Plus the weather was cloudy for the first time ever in New England which had everyone in a slightly worse than normal mood. If I could recreate the day and do it correctly this is what I would have done.

1. The last two problems of the study guide as a warm up.
2. QSSQ including a look back on the exit ticket from yesterday in which kids could not subtract a negative number.
3. Try number six from the study guide which is a division problem since that problem would be fresh in our minds off of QSSQ and yesterday's exit mistakes. 
4. Give six minutes to do problems four and five of the study guide which in my opinion were the hardest on the study guide because they were reading comprehension questions. 
5. Finish the rest of the study guide which students did not really struggle with. 

Honors Math Objective: Analyze if a graph, set of ordered pairs, or t-chart represents a 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: This sequence actually made sense and went better.

1. 100 divided by 0 and 0 divided by 100 were the warm up question. I was surprised at how few students knew this up front. It's really hard to discuss functions without this as a perquisite in my mind. As my colleague and I were previewing what we were about to teach I kept wondering why it was that the vertical line test worked and that each input needed exactly one output. Finally he started to answer and I just blurted out because the slope would be undefined. I haven't been in eighth grade since I was in eighth grade so take it easy on me with the hateful "you didn't know that and you're a math teacher" jibber jabber. 
2. QSSQ 
3. We reviewed the do now and then I asked one more post-warm up question. How do we know a slope? Students mentioned change in y over change in x. 
4. I passed out the vocabulary terms. Students read them for a minute. They then quizzed each other for a minute (I was timing it to the second). Then I asked students do you think it might be good to make flashcards? They agreed. 
5. Review homework and pepper with the vocabulary terms 
6. Students did in explore activity from Big Ideas and ripped out their homework which was seven problems asking about functions, domain, and range. 

Day 37 Dividing Scientific Notation (Again)

Quote of the Day: “Babies need to learn almost everything from experience. If babies didn’t have a strong drive for novelty, they wouldn’t learn as much, and that would make it less likely they’d survive. ‘So, interest - the desire to learn new things, to explore the world, to seek novelty, to be on the lookout for change and variety - it’s a basic drive.” - Angela Duckworth

Question of the Day: Why is scientific notation listed from the numbers 1 to 10?

Regular Math Objective: Divide and multiply numbers in scientific notation

Regular Math Standards: 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 took ten problems from this LearnZillion practice sheet and immediately had students get to work in groups of three. The problem I typically chose was the first one since these problems got progressively more complex. Each group had a mini marker board. The rules required that whoever had the marker could not speak and could only write what he/she was told to write by the other two group members. After the group got the hang of the first one, I put a second problem up on the board. I intentionally hid all the problems from view so that I could have students compete. 
2. I intentionally skipped to a problem (#6) that required students to divide a dividend and divisor that were the exact same number. Despite the fact that this exact problem was there exit ticket in the previous class and that I was handing back that very exit ticket as they worked this problem out, many students continued to make the same error they had the previous class. I showed them all of their errors on the exit ticket after students had an opportunity to work this particular problem through.
3. We did one more problem and then looked at the QSSQ for this day. 
4. We did another problem, but this one finally became more complicated. It was problem eight which forced students to subtract with negatives. After all groups thoroughly struggled with this, we moved onto the next part of our agenda to help strengthen this weakness.
5. I created a really basic PowerPoint of nine or ten one digit integer problems in subtraction and addition. We did a pepper type of format with me asking four to six students stand at a time and answer about twenty questions in total before moving onto the next group. As a side note, for the amount of time that I know students spend on this topic in seventh grade, none of it sticks. Developmentally these students either are not interested in this topic or it just is that hard because I know the teachers spend a great deal of time trying to teach this standard. In any event, without mastery of this it is impossible to ask students to put division problems into scientific notation with proficiency. 
6. The students did problems six, eight, nine, and ten as an exit ticket. The interesting answers are shown below. 

I consider both of these interesting because I think there is a narrative here. Both the coefficients and the exponents are wrong. The mistakes are obvious in both cases. The top student did 2.5 divided by 2 and the other student multiplied (probably because it was something that could be done mentally) instead of dividing. The exponents were added instead of subtracted.

I think what is telling is that there is no work here. No effort to divide each part up individually. The top student tries something, but does not even want me to see. This could be because I have been saying to students do not do too much math in an effort to persuade them away from writing out the numbers in standard form. I need to reallocate our goals to do math, but recognize what is necessary and what is not. Here the students are clearly guessing and do not have logical proof to show why what they are doing makes actual sense. 

Honors Math Objective: Find the domain and range of a graph

Honors Math Standards: A1-F-A1Understand 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. QSSQ
2. Recap of Ketchup Day. 
3. I explained to students how pictionary would work. Since I have twenty-six students in this class, I grouped students according to a deck of cards (ace through king). One card was red and one was black. Ideally I would have had the kids with the red card do half of the graphs and the ones with the black cards do the other half. That was not what I did unfortunately, but confusion was only temporary. 
4. The students and I began to do Pictionary (credit to John Scammel). They used a great deal of vocabulary and it served as a nice refresher in a topic that they probably have not visited since June at the earliest. 
5. Afterwards we broke down this vocabulary and I mentioned for the first time words like translation and domain and range. 
6. I assigned students to work out the domain and range of six different graphs for homework. It would have been helpful to get students to use colored pencils, but I was time-crunched at the end of the lesson.

Day 36 Dividing Scientific Notation

Regular Math Objective: Divide numbers in scientific notation

Regular Math Standards: 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:
The mistakes were numerous. I don't even know where to begin on Monday when we resume. I wish weekends weren't a thing. JK. There are still kids doing too much math with scientific notation. This is why we have scientific notation. Keep the math out of the problem.

At least this was correct.

Probably the most common error was students multiplying the nines. 

Below is the inverse mistake of the one above. Since we spent most of the class dividing, most kids got caught in the routine of dividing all numbers. In this case they divided 5 and 4.

Forgetting to use scientific notation. 

Forgetting that negative exponents don't create negative numbers. 

Honors Math Objective: N/A (see below)

Honors Math Standards: N/A (see below)

Honors Math Lesson Sequence: Today was our first ever catch up day. I had given it quite the hype because all twenty-six students were in support of it when I first introduced the idea only the week before. At the outset of class I read the quote of the day and then got into the importance of recognizing three things for us to be successful.

1. By helping someone else learn something, you are actually helping yourself achieve mastery and solidify your own understanding.
2. In seeking out help from a classmate, there is no shame or embarrassment. School is a place to learn new things not to regurgitate what is already known. 
3. There is no debt involved at the completion of us helping one another. Let's just move on. 

About two-thirds of students elected to retake a quiz. Most of these took inequalities, but not all of them. A few elected to take the absolute value quiz and one student took our first quiz of the year which featured rational numbers and multi-step equations. 

The other third of the class was working on their Chromebook on TenMarks. Part of their role though was that if a classmate came to them for help that they must abandon their TenMarks work and assist that classmate.

I had warned students that I will simply correct quizzes and monitor the group to make sure that there was no cheating. I would be vague in the feedback that I gave after students completed tests and quizzes. The reason for this is that I could get bogged down with one student for ten minutes at a time and if you do the math that means that about 11 students would not be served at all. 

The students absolutely thrived in this environment and under these constraints. I asked for a 3-2-1 (three things that went well, two questions they had, and one item for improvement) feedback in the last five minutes of class. I also asked at the bottom to put whether they would want to do this again. The results were overwhelmingly positive. In fact, I told a third of the class to start with the improvement component, a third to start with the questions, and a third to start with the positives. I figured by doing this that if students could not finish I would at least have a balance of feedback. In the end, many students were stuck on finding improvements, but easily listed positive results. Here is a complete list of what they thought. 

Day 35: Multiplying Scientific Notation (Again)

Regular Math Objective: Multiply numbers in scientific notation

Regular Math Standards: 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. Visual Pattern #72. No students were able to do this, but the chart was helpful for student learning outcomes and when I did a fist of five after working it through as a whole class, the students agreed that they could do something like this if they saw it a second time. 
2. QSSQ. Today's question was about the exit ticket from the day before and explaining what happens between the factors.
3. Review Homework and Pepper
4. Continue with homework practice (which we barely had time for). 

Honors Math Objective: Multiply numbers in scientific notation

Honors Math Standards: 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. 

Honors Math Lesson Sequence: We again did something similar to regular math, but we had extra time so I had the students work out problems on the marker boards that came from the homework that was supposed to be that night. Students went around and helped others if they finished early.

Day 34: Multiplying with Scientific Notation

Quote of the Day: “Some people, when they do someone a favor, are always looking for a chance to call it in. And some aren’t, but they’re still aware of it - still regard it as a debt. But others don’t even do that. They’re like a vine that produces grapes without looking for anything in return...after helping others...They just go on to something else...We should be like that.” - Marcus Aurelius, Roman emperor

Question of the Day: Why is it that 40 times 10 to the 6th power is 40,000,000 and why is this incorrect with scientific notation?

Regular Math Objective: Multiply using scientific notation

Regular Math Standards: 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. QSSQ. I had the students answer the question which was why is it that 40 x 10 to the 6th power equals 40,000,000 is correct, but also incorrect. It was great conversation around scientific notation. 
2. Robert Kaplinsky lesson on How Many Stars in the Universe. It was fun to notice when I asked students to multiply the numbers in the video that they all defaulted to standard form. 
   
3. We did a chart of four different multiplication of exponents problems. In it students used two methods. If I could have this back, I would focus entirely on the method of having students multiply coefficients. It goes against the theory of letting students discover it for themselves, but in having the students do it both ways, they completely missed the convenience of multiplying coefficients and more importantly the base ten numbers. They also lost track of keeping the answers in scientific notation. 
4. The exit ticket came fast today. The majority of students could not complete the chart. 

Not surprisingly when my lack of directions on the ticket to leave lack directions the students calculate the answer in standard form. Note to next year: change the directions so that they say write in scientific notation. Despite that, the student above did use scientific notation on the second problem. Just had some issues with the decimal where the student was trying to slide the decimal over two spots with the base ten and in reality the number is actually less than one. Pretty normal error all things considered. 

This error of getting a coefficient greater than ten continues to haunt us. Past screaming and throwing things, I don't know how else to reteach except to say the same stuff again and again. I like this photo cause you can see me in the shadow taking the picture. Behind the magic if you will. 

The student in this answer does manage to get the coefficient between 1 and 10, but fails to change the exponent. What I am going to do to help clarify this issue is take the factors in a problem like 4 x 6 and divide one of the factors and ask students if I would still get the same answer if the other factor remained the same. 

Honors Math Objective: Multiply numbers in scientific notation

Honors Math Standards: 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. 

Honors Math Lesson Sequence: It was very similar to regular math in terms of the objective and standard, but we got there a different way. First, since students had such a rough time on the exit ticket the previous day, I did something called "Wildfire." I had the students go throughout the classroom and ask around for help on their exit tickets. Virtually every student had something right and something wrong on this exit ticket so it was an appropriate way of trying to get students to understand their mistakes and teach what they thought they knew. I simply acted as another student so that my answers could be cloned for other students to teach. The activity took longer than I would have liked, but based on the dialogue I was having and the students were having it was necessary.

Next, I used this sheet which was required more independence and bridging connections between what they knew already and what they did not know. Students also took note of how the exponent changed when the number was greater than ten, but remained the same if it was less than ten. Unfortunately, with everything that we accomplished in this explore activity we did not have time for an exit assessment.

Day 33: Scientific Notation Mistakes

Quote of the Day: “Provide employees with what I call ‘Goldilocks tasks’ - challenges that are not too hot and not too cold, neither overly difficult nor overly simple. When what they must do exceeds their capabilities, the result is anxiety. When what they must do falls short of their capabilities, the result is boredom. But when the match is just right, the results can be glorious.” - Daniel Pink

Question of the Day: Why is 12 x 10 to the third power not scientific notation?

Regular Math Objective: Write numbers in scientific notation; determine why a number is not in scientific notation and change it so that it is in proper 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. 

Regular Math Lesson Sequence:

1. Visual Pattern #32
2. QSSQ
3. Review Exit Tickets (so many students put 12 instead of 1.2)
4. Correct the mistakes. I got this idea from the Math Assessment Project

Here were the four problems:

3 times 10 to the negative 3 = .0001

This was designed to get kids to stop thinking that since the exponent is three we put three zeroes.

40 times ten to the sixth = 40,000,000

This was put on there to revisit our question of the day. I screamed at students in a ridiculous monologue upfront on the question of the day that they would fail to execute on this exact idea later in class. Sure enough they forgot even though I screamed at them. The great news is that now they are really registering this idea. 

0.5 times ten to the negative second = 0.005

Again students were unable to recognize that the number that is the coefficient needs to be under 1. I failed to call it a coefficient today so we need to drill that in tomorrow.

4 x 10 to the negative second < 8 x 10 to the negative fourth

Students were actually able to quickly point out that the inequality should be flipped. I know. That's pretty much the only way to fix the mistake (well not the only way but you know what I mean). What was cool though was that they got the why in there too.

Honors Math Objective: Write numbers in scientific notation; determine why a number is not in scientific notation and change it so that it is in proper scientific notation

Honors 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. 

Honors Math Lesson Sequence:

1. Exponents Exploration. The link is a Google Doc. I messed up question 1 as I wrote the numbers backward, but other than that it was pretty effective way to show students that they just multiply whatever the coefficients are and then multiply the differences in the exponents as a base ten. The harder part was when there was a coefficient that was larger and an exponent that was smaller. Something like 7 x 10 to the 3 versus 4 x 10 to the 5. I had students just do the  math all the way out in standard form, some estimated, and others use division. I honestly was just hoping that they could do the estimation and know to take the base 10s and divide by the number that separated the coefficients. 
2. Exponents Exploration II. 
3. Exit Ticket from MARS. See the pictures below. The results were less than desirable. Tomorrow we will just have to build off of this. 

Day 32: Scientific Notation Introduction

Regular Math Objective: Write numbers in scientific notation with positive and negative exponents; convert numbers in decimal notation to scientific form.

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. 

Regular Math Lesson Sequence:

1. Pass out the weekly quizzes and have students work on those
2. Scientific Notation Pre-Assessment
3. Scientific Notation Matching 
4. Scientific Notation Post-Assessment

I wanted to have students work on the weekly quiz when I was out on Friday, but that message was poorly communicated to the substitute, so the students worked on something else Friday. Thus, I wanted them to have some attempt at this week's weekly quiz and that's what we did to start class. 

The next three items on the agenda all came from the Math Assessment Project. I modified it to be just one lesson so I left out the objects that are shown in that lesson and focused just on the numbers. The Pre-Assessment pictures below are indications that students did have some working knowledge of scientific notation from their work with exponents, but really struggled with the concept of proper scientific notation. 

As part of the lesson, I did go to the board and show students that the value being multiplied by the base ten had to be greater than or equal to 1 and less than 10. That did not stick however as indicated by the post-assessment. I think I need to scream from a mountain to get this point through. 

I really enjoyed the matching however. The part of the lesson that incorporated a blank number really solidified for me if students understood the lesson or not because it incorporated more complicated exponents to the zero power and a negative power. 

Honors Math Objective: Solve absolute value inequalities; solve compound inequalities; solve multiple step inequalities

Honors Math Standards: A1-A-REI B3 Solve linear equations and inequalities in one variable; including equations with coefficients represented by letters

Honors Math Lesson Sequence:

1. Return quizzes to students
2. Have students fill out self-assessment checklists
3. Have students review one problem that they got wrong with a partner
4. Discuss student answers to the self-assessment 

The students checklists were filled with valuable information for me to work with them on going forward. Many students had nothing but positive things to say, which is encouraging to me. That said, I like these because I know they are being honest - at least in the moment with how they truly feel. 

I have very little doubt when I say that this is the hardest class that these students have taken to date in their academic lives. And I have mixed feelings saying that. I do believe that a challenge is a great thing, but it might be that these students are either over-challenged (which means I need to slow down) or that these students have always been under-challenged. I find the ladder to be hard to believe. When I asked the students about these things aloud, the students that were pictured above expressed themselves as well as others. Some of the takeaways...

• "I want notes."
• "I like this class because it's exciting and we don't take notes like every other class." 
• "I am willing to help other people on catch up days."
• "I would not want to go to a class where I felt overly challenged and automatically got an A+." 
• "I didn't do the study guide a second time, but will now." 
• "This is the toughest class I've ever had, but I like it because I'm not bored." 

Day 30: Inequality Study Guide

Regular Math Objective: Find square roots and cube roots of rational numbers

Regular Math Standards: 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.

Regular Math Lesson Sequence: Classes were extremely short today with one exception because we had a school fundraiser that shortened blocks. With the time we had I was able to do another visual pattern similar to the one the day before with a cubed and squared theme to the rule. We also went over the homework.

Honors Math Objective: Solve multiple step absolute value and compound inequalities

Honors Math Standards: A1-A-REI B3 Solve linear equations and inequalities in one variable; including equations with coefficients represented by letters

Honors Math Lesson Sequence: Students were limited with time in this class as well, so I had them work on our study guide for the quiz that was taking place the following day. I got a fascininating question regarding how to solve the inequality 5 | b + 8 | - 7 < 13 in which the student put the inequality 5 | b + 8 | - 7 > -13 and couldn't get the second solution as a result. She and I were baffled for a long time before I finally stumble on the - 7 needing to be multiplied by negative one.

Day 29: Day 2 of Cubed Roots

Quote of the Day: “Part of our problem is that we think about ourselves way too much. The more we obsess about our problems, our weaknesses, and our deficiencies, the more we perpetuate them. It’s ironic but true.” - Judah Smith

Question of the Day: "Why did you get 21 for the square root of 1/441?" In reference to my own mistake

Regular Math Objective: Find square roots and cube roots of rational numbers

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.

Regular Math Lesson Sequence:

1. Visual Pattern #23
2. Review HW 
3. Day two of Engage New York lessons (what was originally the Engage New York Day 3 Lesson HW) 
4. Exit Ticket
5. Assign new homework

Honors Math Objective: Solve multiple step absolute value and compound inequalities

Honors Math Standards: A1-A-REI B3 Solve linear equations and inequalities in one variable; including equations with coefficients represented by letters

Honors Math Lesson Sequence: I wasn't in class this class because I had a TenMarks professional development. Thus I left the students 7 problems that could be done in their notebooks and checked on the Chromebook plus homework. It would have been great to be there too because students continue to struggle with absolute value, but we're making serious progress.

Day 28: Absolute Value Inequalities Are Stifling

Quote of the Day: “Csikszentmihalyi reports that people are happier when they are at work than they’re caring for a child, and researchers observe that happiness increases when children leave home. Yet many people also say that bringing up their children was the best experience of their life. Happiness is not simply the aggregate of happy moments.” – John Kay

Regular Math Objective: Find the cubed root and square root of low numbers

Regular Math Standard: 8.EE.3

Regular Math Lesson Sequence:

1. Jumpstart via Engage New York. This included two charts. One chart was x to the second power and the other was x to the third (or m and p respectively as the chart chose those variables instead of the world famous x). I really thought that this was engaging and a goldilocks task as students struggled, but did not quit in doing this. They were especially perplexed about how to express the relationship when it was a variable and not a constant that they needed to find a rule fo
2. QSSQ
3. Partner Work via Engage New York. I had to reteach how to solve basic equations and explain the merits behind that in terms of multiple step equations. Students were very ignorant of doing something to both sides of the equations. 
4. Exit Ticket. This was really difficult to fit in because students were unable to complete more than five of the nine questions in most classes which left them stranded on negative exponents which was a component of this exit ticket. Instead I verbally went over negative exponents in the last two minutes of class, which is not typically what I like to do. 

Honors Math Objective: Solve absolute value inequalities

Honors Math Standard: A1-A-REI B3 Solve linear equations and inequalities in one variable; including equations with coefficients represented by letters 

Honors Math Lesson Sequence: 

1. QSSQ 
2. Basic absolute value inequalities |x| > 2, |x| < 3 and two that were similar. Students were able to  recognize the pattern with absolute value that started out as greater as being "or" compound inequality and less than to mean that they were and compound inequalities. 
3. Advance toward higher problems that involve moving constants around such as 5 |x - 4| - 3 > 7 

Students were able to conquer the very basic inequalities such as the ones listed in problem two. As soon as we start to introduce the questions with more meat on them, students have a phobia. We could not get to a third problem in this day and completely go over it. Obviously this is where we will enter tomorrow. It's amazing to me how much they understood absolute value and yet could not muster a solid answer on a multiple step absolute value inequality.