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."