Day 128: Find Your Own Grade

6th Grade Math Standards: 6.SP.5 Summarize numerical data sets in relation to their context, such as by: a. Reporting the number of observations. b. Describing the nature of the attribute under investigation, including how it was measured and its units of measurement. c. Giving quantitative measures of center (median and/or mean) and variability (interquartile range and/or mean absolute deviation), as well as describing any overall pattern and any striking deviations from the overall pattern with reference to the context in which the data were gathered. d. Relating the choice of measures of center and variability to the shape of the data distribution and the context in which the data were gathered.

6.SP.3 Recognize that a measure of center for a numerical data set summarizes all of its values with a single number, while a measure of variation describes how its values vary with a single number.
The Learning Objective: Discover how your test average is calculated. Rationalize what will make your test average grow and fall.

Quote of the Day: “Two young women told us that they have just returned from Iraq after having their HET truck disabled when an insurgent’s bullet went through the engine block. ‘It was a lucky shot,’ one of them said. Even if the insurgent had been aiming there, it would be highly unlikely to get that result.’ When asked what they did when the truck was disabled they said ‘We were taking fire and just hunkered down, protected the vehicle, and waited for reinforcements to arrive. We’re not just going to leave it there.’ As we parted, they thanked us for being there. I told them we should be profusely thanking them, and we did. Then one of them said, ‘you don’t have to be here. We do.’I could not believe how committed and courageous those two soldiers were. In our time at Camp Arifjan we found that was the rule rather than the exception.” - Jay Bilas

Agenda:

1. MCAS Math March Madness
2. Vocabulary pepper: The four terms were central tendency, mean, median, and mode. With some students I asked questions such as "What data value will increase a mean of 36?" Or "When will we have to average two numbers to find a median?"
3. Review the homework
4. Calculate your own grade.
5. Pass out the next night's homework and work on that in class

The Assessment: The MCAS Madness results were 68% on the question regarding the number line, 68% on the question regarding how many questions were on the quiz, 100% on the question regarding what the algebraic expression was worth, and 95% on the question that gave the diameter and asked for the radius.

I told students we needed to get 5 correct consecutively in order to quit playing pepper. It took ten minutes in one class when I measured it.

Students were able to calculate their own grade with differing levels of cuing and scaffolding. Some students met greater challenges when I told them to find what score on their next quiz (which was printed on the sheet) if they wanted to raise their average by two points. A couple students met this challenge successfully. I put together yet another problem from here for students that involved finding a median.

Glass Half-Full: I liked the idea of having students calculate their test grades. It gave a practical and motivating reason for why finding a mean is a pertinent skill in their lives. It also notified them of their current performance in class. Students who owed me a quiz became worried that their grade was so low. Students that had done poorly begged to retake the next quiz. Students asked why the April 2nd quiz was missing (today's date of course is March 31st). The connections to our objective were great but curiosity only grew as a result of the customization of the question.

One Regret: On the homework I want to try to change the second question so that the axes are already labeled and the graph has a title. There are enough challenges for most of the class on this worksheet that this little "spoon feed" would serve me better to get students to remain focused with their questions as they pertain to central tendency.

Link of the Day: More engaging mean questions. Would you rather carry 80 bars of soap or 30 bath towels?

Day 127: Mean, Median & Mode

6th Grade Math Standards: 6.SP.5 Summarize numerical data sets in relation to their context, such as by: a. Reporting the number of observations. b. Describing the nature of the attribute under investigation, including how it was measured and its units of measurement. c. Giving quantitative measures of center (median and/or mean) and variability (interquartile range and/or mean absolute deviation), as well as describing any overall pattern and any striking deviations from the overall pattern with reference to the context in which the data were gathered. d. Relating the choice of measures of center and variability to the shape of the data distribution and the context in which the data were gathered.

6.SP.3 Recognize that a measure of center for a numerical data set summarizes all of its values with a single number, while a measure of variation describes how its values vary with a single number.
The Learning Objective: Determine the central tendency of a data set with three different measures.

Agenda:

1. Students work on their weekly quiz
2. Have the students silently group themselves by birthday from youngest to oldest
3. Notes on central tendency (mean, median and mode)
4. Classwork examples
5. Pass out and start the homework

The Assessment: I went around the room to check how students did with the classwork. The students also got back last week's weekly quiz. I'm convinced that deciding what deal is better if you have two closely linked rates is a problem that you can spend an entire week on. Here's one sold answer from a student.

Glass Half-Full Take: Today was a lighter day for me as I only actually taught one class. I took a half-personal day after flying in from Chicago and not landing until 2:00 this morning. That said, the one class I taught I felt pretty refreshed and will likely have more energy in the coming days with the opportunity to get away.

One Regret: I wish that I had time to go over last week's weekly quiz. I didn't really build that into today's lesson and this week's weekly quiz is somewhat connected to that.

Day 126 Manipulating the Mean

6th Grade Math Standards: 6.SP.5 Summarize numerical data sets in relation to their context, such as by: a. Reporting the number of observations. b. Describing the nature of the attribute under investigation, including how it was measured and its units of measurement. c. Giving quantitative measures of center (median and/or mean) and variability (interquartile range and/or mean absolute deviation), as well as describing any overall pattern and any striking deviations from the overall pattern with reference to the context in which the data were gathered. d. Relating the choice of measures of center and variability to the shape of the data distribution and the context in which the data were gathered.

6.SP.3 Recognize that a measure of center for a numerical data set summarizes all of its values with a single number, while a measure of variation describes how its values vary with a single number.

The Learning Objective: Find a missing value in a data set given the mean and other data values

Agenda:

1. March Madness Day 8
2. Students were given this problem which purposely leaves out information such as what numerical values are equivalent to a B+, A-, and C+ so that they could ask questions and have an opportunity for critiquing the reasoning of one another
3. We did another problem in which they had to determine what grade (a 225%) made going from a 60 average to a 93 average possible.
4. If there was time we closed out with one more problem that was similar to the Bethel problem earlier, but this time the problem had several more assessment and involved critiquing around rounding grades (is a 79.8 a C+ or B-?). 

The Assessment: The March Madness was done with TurningPoint software. The top results among the three classes asked for correctness looked like this:

Division of fractions 86%

Percent proportion 81%

Write a ratio 64%

Length of a side in CP 71%

Overall, pretty good results. Students were also assessed by a partner and myself who was circumventing the room during our class.

Glass Half-Full: The students struggled and we needed only one or two problems to make that possible. Students were truly forced to think with this activity. 

One Regret: For some students, I think this lesson comes too fast. I need to do a better job with small numbers of showing how the mean can change when a data value that is lower than the mean is added and how it changes when a data value that is high as added to the data set. 

Day 125: Finding the Mean

6th Grade Math Standards: 6.SP.3 Recognize that a measure of center for a numerical data set summarizes all of its values with a single number, while a measure of variation describes how its values vary with a single number.
6.SP.5 Summarize numerical data sets in relation to their context, such as by: a. Reporting the number of observations. b. Describing the nature of the attribute under investigation, including how it was measured and its units of measurement. c. Giving quantitative measures of center (median and/or mean) and variability (interquartile range and/or mean absolute deviation), as well as describing any overall pattern and any striking deviations from the overall pattern with reference to the context in which the data were gathered. d. Relating the choice of measures of center and variability to the shape of the data distribution and the context in which the data were gathered.

The Learning Objective: Find the mean of a data set

Agenda:

1. Visual example of equal shares in partners (take a stack of 3, 2, and 1 rectangles and make them all have the same number of rectangles). 
2. Derive the formula to find the equal share (introduced now as the mean or average).
3. Work out what happens on a number line and where an average sits relative to the data set (the sum of the distances before and after the average are equal). 
4. We did an original activity in which pairs of students were given 4 index cards. Each index card had one number written on the front and a different number on the back (so 8 numbers in all). Students were instructed they could only use one side of the index card (we used four different colors for each index card and informed students they could only use one color). From there we gave students a scavenger hunt of sorts. They had to find different means by using trial and error (or a better method if they could think of it) to arrive at the means. 

The Assessment: I went around the room as students tried to work out what data set would be appropriate given the mean.

Homework: I gave students six problems in which they had to find the mean.

My Glass Half-Full Take: As I was going around the room to see how students were doing, I got two positives from different students. The first was that one student recognized that by simply multiplying the mean by four he knew what sum to look for. This made it much easier to derive the mean

Another pair of students recognized quickly how to use trial and error, and got the first three means I asked for without any difficulty. I changed the task for these two students. For them I asked them to find how many different combinations of means could we come up with given the rules of this activity. They only could list 9. I went through a method using variables to demonstrate that there will over 9 and asked them to come up with what they thought it could be for homework. I canceled their normal homework because I knew the task was so easy for the two of them. I told them both not to spend more than 15 minutes and that it was ok not to have the answer as long as I could see the effort.

One Regret: I think I might laminate the index cards next year so that we can take out the aspect in which students have to write what numbers go where. It will save 5 minutes and a small tree.

Day 124: Statistical Questions

6th Grade Math Standards: 6.SP.1 Recognize a statistical question as one that anticipates variability in the data related to the question and accounts for it in the answers. For example, “How old am I?” is not a statistical question, but “How old are the students in my school?” is a statistical question because one anticipates variability in students’ ages.

The Learning Objective: Create a statistical question, differentiate between statistical and non-statistical questions

Quote of the Day: "Thinking of yourself as the best is one of the biggest reasons successful people stumble and fail." - Rick Pitino

Agenda:

1. Self-Assessment of inequalities quiz
2. Review quiz by having students ask any pertinent questions
3. MCAS Math March Madness Day 6
4. I gave all students a post it note and asked them to write down how many minutes it usually takes them before they give up. Once the students had the number written, students posted that note to the board. I then asked for three volunteers to organize that information for us into a line plot (although I didn't use that term). As these students organized the information I did vocabulary pepper with the rest of the class.
5. Statistical questions notes
6. Statistical question examples
7. The class surveyed their classmates with a statistical question they had created
8. Ticket to leave 

The Assessment: I went around the classroom and checked all students statistical questions before they surveyed their classmates. In one class I had a co-teacher so I corrected and returned all exit tickets today. I will give feedback to students on the other exit tickets tomorrow.

Students were also assessed with the March Madness. Today our top scores were 43% for the coordinate plane question (the wording wasn't consistent on the Powerpoint and the assignment linked above so it confused many students), 68% for the question regarding percentage of questions, 74% for dividing fractions (this was 37% in one class), and 62% for the order of ratios. 

Students also assessed themselves using the checklist which was passed out to them on inequalities.

Homework: I had students write a statistical question for something that they are learning in science and a non-statistical question for something they are learning in Africa. Students were given plenty of time to do this in class if needed.

My Glass Half-Full Take: There were several movement breaks provided for students to be out of their seats during the course of the class today. Getting clickers and returning their clickers, posting their sticky note to the board, and surveying classmates. It was an appropriate lesson for bouncy adolescents. 

One Regret: The amount of time to cover statistical and non-statistical questions was too much. I felt like I could have done a social or reflective activity with the students. Instead I ended up simply starting mean with 10 minutes to go in class and that was just a jumbled mess. Tomorrow we will cover mean more extensively anyway, so that might not have been the best way to handle today. 

Link of the Day: This is the best way to get professional development. 

Day 123 Inequalities Quiz

6th Grade Math Standards: 6.EE.5 Understand solving an equation or inequality as a process of answering a question: which values from a specified set, if any, make the equation or inequality true? Use substitution to determine whether a given number in a specified set makes an equation or inequality true.

6.EE.7 Write an inequality of the form x > c or x < c to represent a constraint or condition in a real-world or mathematical problem. Recognize that inequalities of the form x > c or x < c have infinitely many solutions; represent solutions of such inequalities on number line diagrams.
The Learning Objective: Write an inequality, graph inequalities, decipher what value make a given equality true

Quote of the Day: “Perfection is what you are striving for, but perfection is an impossibility. However, striving for perfection is not an impossibility.” - John Wooden

Agenda:

1. MCAS Math March Madness Day 6
2. Inequality Powerpoint noting when to use a number and when not to use a number
3. Inequality book work problems #1-12 (skip 11) 
4. Inequality Quiz
5. Weekly Quiz 

The Assessment: The quiz results were good, but the quiz itself perhaps could have been harder. Students were not asked to multiply in order to see if given inequalities worked. We also did not include any fractions or division problems to see if inequalities worked. I think there could have been more real-life examples and we could have provoked students to give reasoning rather than just memorize the direction a graph goes.

Homework: Weekly Quiz #22

Glass Half-Full Take: The results from the prime factorization question were solid. We even had a group get a 100% on writing an algebraic expression from the MCAS Math March Madness.

One Thing I Wish I Had Done Differently: Made the students analyze and think deeper on the quiz (and the study guide for that matter).

Day 122 Inequalities Study Guide

6th Grade Math Standards: 6.EE.5 Understand solving an equation or inequality as a process of answering a question: which values from a specified set, if any, make the equation or inequality true? Use substitution to determine whether a given number in a specified set makes an equation or inequality true.

6.EE.7 Write an inequality of the form x > c or x < c to represent a constraint or condition in a real-world or mathematical problem. Recognize that inequalities of the form x > c or x < c have infinitely many solutions; represent solutions of such inequalities on number line diagrams.

The Learning Objective: Determine what value makes an inequality true, graph in equality, write an inequality in a real-world context

Agenda:

1. Powerpoint Pepper using terms such as "at least" to determine if a number is included in the inequality or not.
2. Inequality study guide 
3. Inequality graphing and writing practice
4. Weekly quiz #21 (10 questions about ratios)

The Assessment: The pepper is a great on the spot assessment because it gives students instant feedback for an issue that they clearly had in the previous class. The study guide was available online with answers.

Homework: Weekly Quiz #21 and the next class there will be a quiz

My Glass Half-Full Take: I wasn't in school for this lesson and as I write this post I'm actually awaiting my flight back to Boston from Atlanta where I just completed my first half-marathon. I'm physically exhausted, but mentally excited to get back to work tomorrow. It's the first day off I've had this year and I feel like it comes at a good time (March can be a relatively long month for teachers).

I think that at this point in the year it's possible for students to know the routines independently, but it's also nice that we have a Title I teacher and special education teacher that I co-teach with who can help fill in the gaps without me in the classroom.

One Thing I Wish I Had Done Differently: I'll find out tomorrow, but it could be an issue giving a quiz the day after I was not in school. We'll see what was accomplished after I talk to the co-teachers and if I need to give a quick review before the quiz that can be done since it is only a ten question quiz.

Day 121 Graphing Inequalities

6th Grade Math Standards: 6.EE.5 Understand solving an equation or inequality as a process of answering a question: which values from a specified set, if any, make the equation or inequality true? Use substitution to determine whether a given number in a specified set makes an equation or inequality true.

6.EE.7 Write an inequality of the form x > c or x < c to represent a constraint or condition in a real-world or mathematical problem. Recognize that inequalities of the form x > c or x < c have infinitely many solutions; represent solutions of such inequalities on number line diagrams.

The Learning Objective: Graph inequalities in a mathematical and real-world context

Quote of the Day: "Be on time. Period."

Agenda:

1. March Madness
2. Graphing examples in notes (for example x > 4, x is less than or equal to 8)
3. Writing inequality examples (for example less than or equal to 48 inches to ride the roller coaster)
4. Exit Ticket with my age being described as 14 years old or older
5. Cheers clip 3-Act Math
6. Math Jack (basically Black Jack but renamed to take the gambling idea out of the minds of the students). I ask students to graph the inequality for the game. 

The Assessment: The exit ticket was a struggle for most students. They didn't know how to describe the variable. They said it was a or s, but didn't know what a or s represented (my age). The students also struggled with the term "14 or older." They mistook that to mean I'm greater than 14. Perhaps that was because I obviously am, but still the problem needs to be interpreted based on how it is written.

We called the March Madness "All-Star Friday." It represented our greatest challenge problems from the week in March Madness. I posted at the top of each jumpstart the record percentage in a given class for each question. The questions included surface area, ratios (with a tape diagram template given), the distributive property and decimal subtraction. All four topics were answered with at least 89% proficiency from each class. Perhaps I'll have the students make an inequality of this statement.

Homework: There was no homework over the weekend. Usually we allow students to work on the weekly quiz online, but this weekend I didn't post it because we are going to work on it in class on Monday.

My Glass Half-Full Take: Math Jack was enjoyable. It's nice to hear students say thank you that was fun as I had one student do at the end of class. It was also interesting to see what evolved from watching the Cheers video. It was supposed to help us calculate inequalities, but instead we were doing work with ratios. Students were calculating how much money $50 per month was in terms of weeks. That involved knowing how many days in a year and how many months in a year. It's a gentle reminder to the worksheet lover in me that math only takes one question to be interesting and assess many different skills if needed.

One Thing I Wish I Had Done Differently: I don't think students have a strong foundation in terms of writing inequalities in a real-world context as evidenced by the exit ticket. This has to be one of the first tasks to tackle when I see the students again. I also know that students will forget how to graph despite the fact that they were doing it without difficulty today. It's one of those skills that's easy to memorize in the moment but taking a break from it will make it much more difficult.

Day 120 Inequalities Introduction

6th Grade Math Standards: 6.EE.5 Understand solving an equation or inequality as a process of answering a question: which values from a specified set, if any, make the equation or inequality true? Use substitution to determine whether a given number in a specified set makes an equation or inequality true.

The Learning Objective: Identify a value or values that make an inequality true.

Quote of the Day: “After seven experiments with hundreds of children, we had some of the clearest findings I’ve ever seen: Praising children’s intelligence harms their motivation and it harms their performance.” - Carol Dweck

Agenda:

1. Self-Assessment
2. Review the Functions Test
3. March Madness
4. Inequality frayer model and notes
5. Inequality exit ticket
6. Inequality practice

The Assessment: The exit ticket and March Madness. The exit ticket results were solid overall. In March Madness I had two classes get less than 25% on the ratio question. I showed the students how to use a tape diagram. Tomorrow I will run a question like this again and hope that showing them a chart as well as algebra to solve it.

Homework: Students did a worksheet with 14 inequality questions. I eventually took problems off for students that were being productive in class and demonstrated the skill. There was no need for them to keep doing the work once they showed me they knew what they were doing.

My Glass Half-Full Take: While going over the test it was good to hear a variety of voices asking questions. I take pride in students being continually committed to mastery. So many students will just quit the moment after they see the number at the top of their quiz or test, and that isn't the true purpose of a test or quiz.

One Thing I Wish I Had Done Differently: I make students write the inequality, substitute a number for the variable, and solve the inequality. The process is multiplied by three when the students need to test for three different values of the variable as they did on the homework and exit ticket. I don't think this is asking too much of the students. It would take me about 8 minutes (if that). I'm not worried about whether it's too much, I'm worried about whether it helps students solve the problem or not. Most of them can recognize what values work and don't work immediately. Is that mental math good for them and should I encourage them not to show work in this instance? It's worth considering for next year.

Day 119 Functions Study Guide & Quiz

6th Grade Math Standards: 6.EE.9 Use variables to represent two quantities in a real-world problem that change in relationship to one another; write an equation to express one quantity, thought of as the dependent variable, in terms of the other quantity, thought of as the independent variable. Analyze the relationship between the dependent and independent variables using graphs and tables, and relate these to the equation. For example, in a problem involving motion at constant speed, list and graph ordered pairs of distances and times, and write the equation d = 65t to represent the relationship between distance and time.

6.EE.1 Write and evaluate numerical expressions involving whole-number exponents.

6.EE.3 Evaluate expressions at specific values of their variables. Include expressions that arise from formulas used in real-world problems. Perform arithmetic operations, including those involving whole-number exponents, in the conventional order when there are no parentheses to specify a particular order (Order of Operations). For example, use the formulas V = s3 and A = 6s2 to find the volume and surface area of a cube with sides of length s = ½.

The Learning Objective: Find an input based on an output and a function rule, determine the rule for a table that shows two different variables, graph a function in a coordinate plane, create a table of values based on a function rule.

Quote of the Day: “I received an email from a gentleman who wrote that during his ten years in the military, whenever something went wrong, the only acceptable response was, ‘No excuses!’ He accepted it, he believed it, and he lived it. When he returned to civilian life, he started working as a territory manager for a large firm in the food industry. He wasn’t doing as well as his company expected, and he wasn’t pleased with his own performance either so he started asking his manager questions like these: Why don’t I get more of your time? Why don’t you coach me more? Why isn’t our pricing competitive? Why doesn’t marketing support us? If this man after living and breathing ten years of “No excuses” can slip into this victim thinking, no wonder the rest of us can too.” - Jon Miller

Agenda:

1. MCAS Math March Madness
2. Study Guide page 1 (4 minutes)and review 
3. Study Guide page 2 (4 minutes) and review 
4. Study Guide page 3 (4 minutes) and review 
5. Study Guide page 4 (4 minutes) and review
6. Study Guide page 5 (4 minutes) and review
7. Take the test on functions
8. Work on the weekly quiz
9. Fill out an NCAA tournament bracket if the weekly quiz is perfect

The Assessment: Performance on the test was very good overall. As far as the March Madness went, students struggled most with surface area and writing ratios. One class only had 24% get the ratio question (keep in mind it was multiple choice so a monkey would do better). The ratio topic was one that I was concerned with, so we will be reviewing that topic on next week's weekly quiz. The top score all day on the surface area question was 57%.

Homework: There was no homework other then continuing to work on the weekly quiz.

My Glass Half-Full Take: I really liked timing everything on the study guide today. I knew going into class that the agenda was jam packed if I let the students do March Madness. Since I wanted to keep this routine going, I decided to ditch using partners for the study guide and to break each section into chunks. I put the timer on the board so that students could be reminded to stay focused and I circumvented the room. There wasn't much difficulty with this from a classroom management standpoint. After doing the study guide in the first block, I was relatively pleased with the results on the test in the second block.

One Thing I Wish I Had Done Differently: The test asks for students to show work a certain way and I'm not really a fan of it now. I think that I will give less restrictions as to how the answers get on the paper the next time around as it only adds to the confusion and for the most part is an impediment toward mastery of the objectives.

Day 118 Multiple Representations of Functions

6th Grade Math Standards: 6.EE.9 Use variables to represent two quantities in a real-world problem that change in relationship to one another; write an equation to express one quantity, thought of as the dependent variable, in terms of the other quantity, thought of as the independent variable. Analyze the relationship between the dependent and independent variables using graphs and tables, and relate these to the equation. For example, in a problem involving motion at constant speed, list and graph ordered pairs of distances and times, and write the equation d = 65t to represent the relationship between distance and time.

The Learning Objective: Represent the same function using a table, a graph, an equation, and words.

Quote of the Day: “Try to picture Thomas Edison as vividly as you can. Thank about where he is and what he’s doing. Is he alone?...Edison was not a loner. For the invention of the lightbuld, he had thirty assistants, including well-trained scientists, often working around the clock in a corporate-funded state-of-the-art laboratory. It did not happen suddenly. The lightbulb has become the symbol for that single moment when the brilliant solution strikes, but there was no single moment of invention." - Carol Dweck

Agenda:

1. Jumpstart - Math March Madness
2. Review Homework
3. Notes on representing functions on page 606 and 607 of the textbook
4. Multiple Representation of Functions Practice
5. Work on the weekly quiz

The Assessment: On the link jumpstart above, students were given clickers from TurningPoint Technology to submit their answers. I do this everyday typically for the entire month of March, but this year the software has not been cooperative with my new Windows upgrade. I had the district technology guru look at it and even he was stumped. Finally I got it to work by closing out of my Epson projector software. Needless to say March Madness will follow in the footsteps of the college basketball tournament and drift into April a little. The great news is, the kids don't care.

I throw out the incentive of raffle prizes for right answers, points on tests for any class that gets 100% correct, and points on tests for any class that has the highest percentage of correct answers on any given day. Today, classes were getting over 80% correct answers on the area of a triangle skill and the volume of a rectangular prism (without fractional edges). We got roughly 40% correct on surface area and 50% correct on the distributive property. Clearly two holes that I will try to cover up in the coming days and weeks.

Additionally I was able to collect and provide feedback on the weekly quiz for all students. I also took a look at all homework from the previous class.

Homework: For one class, I gave them the multiple representation of functions worksheet for homework because we did not get through it in class. That class was at the end of the day and I spent more time than I would like with classroom management. The other two classes finished the assignment in class and just have to study for the test we have tomorrow. I put the study guide and answers online for all classes.

My Glass Half-Full Take: I was happy to see that classes went as high as 94% understanding of the volume question. Granted, this could be considered a fifth grade skill, but numbers this high serve notice to myself and the other math teachers that fractional edge volume problems (sixth grade standard) are a necessity to increase the challenge for these students. The students solid understanding of volume can likely be correlated with the weekly quizzes which have featured volume questions for the past two weeks.

One Thing I Wish I Did Differently: On Day 1 of March Madness I always let students take their time to find an answer as I go through the Powerpoint with the questions because they are completely engaged. I wish I did not do this though. The trouble is that students have access to the Madness questions well before the Powerpoint goes up. When they come into class they get a worksheet of all the problems, which they can work on as I grade homework. This time is not fully taken advantage of and if I speed through the Powerpoint students will have a higher sense of urgency and we can use the time that would be wasted on the Powerpoint on something else.

Link of the Day: Just took a quick glance at #slowmathchat on Twitter, but I'm liking what I see.

Day 117: Functions & Equations

6th Grade Math Standards: 6.EE.8 Use variables to represent two quantities in a real-world problem that change in relationship to one another; write an equation to express one quantity, thought of as the dependent variable, in terms of the other quantity, thought of as the independent variable. Analyze the relationship between the dependent and independent variables using graphs and tables, and relate these to the equation. For example, in a problem involving motion at constant speed, list and graph ordered pairs of distances and times, and write the equation d = 65t to represent the relationship between distance and time.

The Learning Objective: Describe the relationship between the independent and dependent variable.

Quote of the Day: “Optimists are normally cheerful and happy, and therefore popular; they are resilient in adapting to failures and hardships, their chances of clinical depression are reduced, their immune system is stronger, they take better care of their health, they feel healthier than others and are in fact likely to live longer.” – Daniel Kahneman

Agenda:

1. Journal on things to focus on in trimester 3: attendance, homework, notes, weekly quizzes, class participation, studying, peer collaboration, and my favorite: math for the sake of math.
2. Independent vs. Dependent Variables 
3. Notes on Functions & Equations (from the textbook) 
4. Functions & Equations Practice (from the textbook)
5. Start the weekly quiz

The Assessment: I had the students start the homework by doing number one on the guided practice. If I checked that problem off they did problem number four on the independent practice which was a two-step rule that required them to multiply by 10 and add 30. Not something that they would be able to get right away if they were not finding the change in y over the change in x.

Homework: The weekly quiz and the functions homework are due the next class, but thanks to the magic of double block math, it is not nearly as overwhelming as it would typically be. Students were finishing the textbook homework in class which left them just the four question weekly quiz for homework.

My Glass Half-Full Take: I liked two things I did today. First of all, on the homework I had the students put an E, M, or H next to each problem. It stood for easy, medium, and hard and gave them forewarning about what sort of effort the problem was going to require. I also liked the analogy of using cause and effect to explain independent and dependent variables. I had learned this from a colleague and did it for the first time this year. I felt like it added to the number of students who were able to differentiate between the two variables.

One Thing I Wish I Had Done Differently: In my last class, I wish that I had students check the work of other students. It got to the point where five students were raising their hand at the same exact time. I could get away with it in my other classes because they were co-taught and class size was smaller.

Link of the Day: I heard in a Ted Talk the other day the reason for why X is "the" variable instead of other letters. Thought it was pretty interesting and also enjoyed how the speaker referenced that we don't call the movie "B-Men" we call it "X-Men."

Day 116: Pi Day

6th Grade Math Standards: 6.G.1 Find the area of right triangles, other triangles, special quadrilaterals, and polygons by composing into rectangles or decomposing into triangles and other shapes; apply these techniques in the context of solving real-world and mathematical problems.
MA.1.a.Use the relationships among radius, diameter, and center of a circle to find its circumference and area.
MA.1.b. Solve real-world and mathematical problems involving the measurements of circles.

6.EE.9 Use variables to represent two quantities in a real-world problem that change in relationship to one another; write an equation to express one quantity, thought of as the dependent variable, in terms of the other quantity, thought of as the independent variable. Analyze the relationship between the dependent and independent variables using graphs and tables, and relate these to the equation. For example, in a problem involving motion at constant speed, list and graph ordered pairs of distances and times, and write the equation d = 65t to represent the relationship between distance and time.

The Learning Objective: Discover a function rule based on an chart or picture; Define pi as an irrational number that is the ratio of the circumference to the diameter

Agenda:

1. Jumpstart with function rules and sequences
2. Review the functions homework from the previous class
3. Visual Patterns # 7, #1, and #22 (they got progressively more difficult)
4. Read the story of Sir Cumference and the First Round Table by Cindy Neuschwander and Wayne Geehan
5.  Pi Chains
6. Pass out WQ #20 for next week

The Assessment: I asked students to come up with one sentence that would describe how they would start solving visual pattern #7. A few of them said they would start with a chart. When we then proceeded to work on Visual Pattern #1 more students were using a chart immediately. I also went around the room looking at homework. Students had difficulty with the sequence 5, __, 10, 12 and 1/2...

Homework: Weekly Quiz #20 will be due not next class, but the following class.

My Glass Half-Full Take: We assigned a color to each of the ten digits in the number system and gave students 10 different colors of construction paper. From there, students were asked to make a "pi chain." We left the method to how they could go about doing this entirely up to the students. I enjoyed that aspect of the project very much as it gave the students the opportunity to think critically about division of labor and other strategies that would optimize the size of their chains. Students left class stating that it was an enjoyable math class. 

One Thing I Wish I Had Done Differently: I wish at the end of class I had done an index card sized formative assessment (exit ticket) and asked students what type of number pi was and if they could describe what ratio pi represented. I don't think the second objective was fully measured even though the chain is an exaggerated way of emphasizing pi is much more than 3.14. 

Day 115: Finding a Rule

6th Grade Math Standards: 6.NS.9 Use variables to represent two quantities in a real-world problem that change in relationship to one another; write an equation to express one quantity, thought of as the dependent variable, in terms of the other quantity, thought of as the independent variable. Analyze the relationship between the dependent and independent variables using graphs and tables, and relate these to the equation. For example, in a problem involving motion at constant speed, list and graph ordered pairs of distances and times, and write the equation d = 65t to represent the relationship between distance and time.

6.EE.6 Use variables to represent numbers and write expressions when solving a real-world or mathematical problem; understand that a variable can represent an unknown number, or, depending on the purpose at hand, any number in a specified set.

The Learning Objective: Discover a function rule based on a chart or picture

Agenda:

1. Page 587 match the vocab with an example
2. Notes on functions and practice
3. Distribute and start homework
4. Visual Patterns 
5. Visual Patterns Challenge (finding a triangular number)

The Assessment: Today in class students had time to finish homework, but we also had time to try some visual patterns. Students worked in partners on these two parts of class and seemed to understand the learning objective. It was difficult at first for students with the two step rules, but soon they began to crave these problems as they discovered the rules.

Homework: Finding the rule practice practice (done in class).

My Glass Half-Full Take: I loved giving the challenge question. One class in particular really took to it. Nobody got it, but obviously that was not the point. Interestingly, giving the challenge enabled me to work with students that are attention seekers or need more wait time to accomplish tasks, so it served as a nice way to manage the classroom.

One Thing I Wish I had Done Differently: This is not a topic that I even need to utter the term homework with. I want students to learn to love math for its own sake. I wish today students did not feel compelled to rush through something just to get it done. I visited an eighth grade classroom today and confronted a former student who was averse to math in my classroom, but today he simply was resistant to my help. I don't want students to ever feel this way, so I need to occasionally have days where the students are solving problems for its own sake. There are enough tricks out there to motivate students beyond grades and I need to utilize them better.

Day 114: Graphing Functions

6th Grade Math Standards: 6.NS.8 Solve real-world and mathematical problems by graphing points in all four quadrants of the coordinate plane. Include use of coordinates and absolute value to find distances between points with the same first coordinate or the same second coordinate.

6.NS.9 Use variables to represent two quantities in a real-world problem that change in relationship to one another; write an equation to express one quantity, thought of as the dependent variable, in terms of the other quantity, thought of as the independent variable. Analyze the relationship between the dependent and independent variables using graphs and tables, and relate these to the equation. For example, in a problem involving motion at constant speed, list and graph ordered pairs of distances and times, and write the equation d = 65t to represent the relationship between distance and time.

The Learning Objective: Graph functions based on an input and output table.

Agenda:

1. Equation word problem jumpstart
2. Homework review
3. My favorite no
4. Graphing functions notes
5. Graphing functions homework
6. McSquare & McTriangle (find the rule based on a pattern)

The Assessment: My favorite no made a connection to the homework in one class and to finding a rule in the other two classes. The one class I made the homework connection with because their work was subpar.

The other assessment came with the homework which was essentially done in the class.

Homework: Finding the rule practice based on the notes.

My Glass Half-Full Take: Graphing is easy once students differentiate in a word problem the output and the input. For students that have grown used to circling what they know, putting the input and output in the right place in a table is relatively simple. For students that are not used to it, they automatically put it in the input column. I am glad that I have hammered into students minds at this point that they should be circling what they know and underlining what they do not know.

One Thing I Wish I had Done Differently: The McSquare and McTriangle activity is nice, but for many students it wasn't enough of a challenge. I am going to have visual patterns ready in the event of this next year and give students a problem that involves a quadratic if we need a further challenge.

Day 113: Functions Introduction

6th Grade Math Standards: 8.F.1 Understand that a function is a rule that assigns exactly one input to each output.

6.NS.6 Use variables to represent numbers and write expressions when solving a real-world or mathematical problem; understand that a variable can represent an unknown number, or, depending on the purpose at hand, any number in a specified set.

The Learning Objective: Determine the input given a function rule and the output in a table. Determine the output given the function rule and the input in a table.

Agenda:

1. Self-Assessment from the Equations Quiz
2. Review the quiz as a whole class
3. Functions My Favorite No
4. Function Notes
5. Function Practice

The Assessment: The My Favorite No question gave students a chart where the rule was 2x, but most students filled in a number. Only four out of the 60 students in attendance wrote 2x, so for most students determining a function rule based on the input and output will be a new experience. For those students that already meet this concept, I will look to use quadratic, cubic, and multiple step rules to push their thinking.

Homework: Several students completed all work in class, but students were asked to finish the function worksheets for homework. The greatest difficulty for them was working backward when they had the output and had to derive the input based on the function rule and the output.

My Glass Half-Full Take: I think this is a relatively low-stress topic to teach as students level of confusion is dwarfed by the fact that we have already gone into great detail regarding variables over the past couple of weeks. This is where I happen to like the common core because it feels like we are able to dig deeper into a concept and the students can solidify holes in their learning on prior assessments as we continue to see variables.

One Thing I Wish I Did Differently: In my last class, I went ahead and did a little chant with motion stating that a function is a rule that assigns exactly one input to each output. I did not realize that this was an eighth grade standard until looking it up just now. I could have shared this information with the class and also done this chant with all classes. It's an opportunity to teach and have students up out of their seats in the action.

Day 112 Equations Quiz & Weekly Quiz Work

6th Grade Math Standards: Solve real-world and mathematical problems by writing and solving equations of the form x + p = q and px = q for cases in which p, q, and x are all nonnegative rational numbers.

The Learning Objective: Isolate a variable in order to solve an equation. Create an equation in a real-world context.

Agenda:

1. Pass back Weekly Quiz #18
2. Quiz on Equations
3. Pass out and start Weekly Quiz #19

The Assessment: The equations quiz

Homework: Finish Weekly Quiz #19

My Glass Half-Full Take: We intentionally left the Weekly Quiz off the internet over the weekend. I like the idea that the students did not have to worry about that over the weekend and could really get their work done in class. As I was reminded by one of the other math teachers earlier today, "These are 11 and 12 year olds after all." To some degree we are batting them over the heads with worksheets and quizzes and to a larger degree that's not who I want to be as a teacher.

One Thing to Do Differently: I don't think that students had a clear expectation of what the equations word problems expectations were as far as what I was assessing. I wish that I had told them up front that if the equation or the solution to the equation is wrong, the whole problem is wrong.

I also would have liked to put a timer on the board as students worked on the weekly quiz and told them that they should be completing at least two problems every five minutes. While this is obviously counter-productive in terms of putting time pressure on them and forcing errors upon them, I have a hard time keeping all students productive in a quiet room when I have to deal one on one with students' questions. One of the other math teachers told me that after students got done problems 1 and 2 they had to get checked by her. I liked this idea as long as I had a checklist to keep track of what students had not seen me. I often find that students can do nothing until a couple minutes left in class and by then it's too late to act in a way to help them.

Day 111 Equations Study Guide

6th Grade Math Standards: Solve real-world and mathematical problems by writing and solving equations of the form x + p = q and px = q for cases in which p, q, and x are all nonnegative rational numbers.

The Learning Objective: Isolate a variable by using the inverse operation. Compare an equation to an expression.

Agenda:

1. Equation balancing with pictures (MCAS Sample Problems Jumpstart)
2. Gave students a giant marker board to try the division word problems from the homework
3. Gave students the study guides to do in groups.
4. Reviewed study guides as a whole class.
5. Gave students challenge problems to work out as groups. One problem asked students to use the numbers 2, 0, 1, and 5 (for the year 2015) and come up with a solution using the four arithmetical operations ranging from 0 to 15 for all but two numbers. Another problem was a visual pattern from Fawn Ngyuen, and a third problem asked students to find a median (they haven't done this yet). 
6. Gave students a game of "Equation Wars." Basically I threw up a variable and two numbers. Each group had to put together four equations with the four major operations (group members picked the role of everyone in the group). For the first round, I gave 3 minutes for students to put the best effort into their problems. This included them checking the work and even explaining the work. 

The Assessment: Checking student marker boards, circumventing the classroom during the jumpstart, and in two classes circumventing the classroom for the study guide.

Homework: Study for the quiz which will be on Monday.

My Glass Half-Full Take: In one of my classes, I sat on my computer the entire time as students worked on the study guide. Why is this a good thing? Because I never do this. I always go on the computer to enter grades and homework assignments after school. This means I work later. That in turn means I have less time to read books, go for runs, watch sports, and have a life outside of school. I also should mention that I still listened to groups and was pleasantly surprised at the level of focus and teamwork in the groups of four. I went over the study guide with the entire class afterwards. Unfortunately I didn't try this in my other two classes, and I'm not sure if I could given the makeup of the classes.

One Thing to Do Differently: Equation Wars was so effective because I gave out munchkins to winning groups. I'm always listening to Freakonomics and the value of incentives. I hate having to bribe students to do work, but it did create a spike in student focus. I would like to try this game without giving students the physical reward (and if it fails try the physical reward again).

Link of the Day: Imprisonment can cause a vicious cycle for students. Only 1% of students with mothers in prison graduate college and 13% of students with fathers in college graduate. This study also said students with a parent in prison have a higher rate of attention deficit than students who miss a parent through divorce or death.

Day 110 Division Equations

6th Grade Math Standards: Solve real-world and mathematical problems by writing and solving equations of the form x + p = q and px = q for cases in which p, q, and x are all nonnegative rational numbers.

The Learning Objective: Isolate a variable in a division problem by isolating the variable.

Agenda:

1. Weight balance jumpstart challenge
2. Review the homework
3. Division equations my favorite no 
4. Division equation notes
5. Multiplication and division combined homework worksheet (word problems included) 

The Assessment: If I was teaching this for the first time, I'd assume all the students knew to cancel out whatever number is with the operation by doing the inverse operation. By year six of teaching equations, I know though that despite practicing with the other three operations there is still a lot of this:

It's what makes My Favorite No so useful as a pre-assessment. Kids see their own mistakes and can actually learn before taking notes.

Homework: Instead of focusing on giving just division, we combined the division and subtraction equations. It could have been harder if we completely mixed the problems but instead just division was on one side and just multiplication equations on the other. That said, word problems with division are still a great struggle. It's like in baseball when you tell a hitter a fastball is coming, that doesn't mean they can hit it.

My Glass Half-Full Take: I enjoy my classes enthusiasm and competitive edge in my favorite no. I always set out a random number such as "10 students in this class will make an error on this problem." It captivates them and gets them to think. When I display the wrong answer students are so quick to want to tell me what is wrong, but I always ask what is right first and it leads to great conversation.

One Thing to Do Differently: In terms of the more basic problems, I'd like to get the two operations mixed up. I also think that doing division word problems as a whole class is a good idea because the division word problems are a headache for many. So often students ask why they can't just multiply to solve. It's a fair question. I always say that they need to learn how to set up and solve one step equations because bigger things await.

Link of the Day: For math teachers who want to know more about teachers who blog and tweet.

Day 109 Multiplication Equations

6th Grade Math Standards: Solve real-world and mathematical problems by writing and solving equations of the form x + p = q and px = q for cases in which p, q, and x are all nonnegative rational numbers.

The Learning Objective: Isolate a variable in a multiplication equation by isolating the variable.

Quote of the Day: “Failing doesn’t make you a failure. Failing makes you a competitor. Every competitor fails. If you lay it on the line, you will come up short at times. Failure is a part of competing, and embracing that fact is an important component of toughness. Tough people fail, but tough people are not failures. The only failures are those who give up, or give in.” - Jay Bilas

Agenda:

1. Jumpstart
2. Subtraction Equations reviewed
3. Addition and subtraction equations combined practice
4. Multiplication equations notes
5. Multiplication equations practice
6. Equations Challenge problem (if time)

The Assessment: The addition and subtraction combined practice was done with a partner and the teacher circumventing the classroom. The multiplication equation practice was also checked by myself.

Homework: Page 555 #1-7, 556 # 11, 13-15, 557 #17-28 & The weekly quiz

My Glass Half-Full Take: When the equations challenge problem was done I had much more success when I read the quote of the day right before. I only used this problem in two classes, but the students really enjoyed it. I have gotten away from the quote of the day of late, but it does add value as I saw today. I also think it was key that the students got a challenge problem because the week has been dry in general.

One Thing to Do Differently: The combined addition and subtraction practice should focus one hundred percent on word problems instead of drill and kill types of problems (the sheet as it is constructed now has both word problems and drill and kill types). At most students should be asked to do three of these types of problems. The rest are a waste of time.

Day 108 Subtraction Equations

6th Grade Math Standards: Solve real-world and mathematical problems by writing and solving equations of the form x + p = q and px = q for cases in which p, q, and x are all nonnegative rational numbers.

The Learning Objective: Isolate a variable in an addition or subtraction problem by isolating the variable.

Agenda:

1. Self-Assessment 
2. Review Quizzes on distributive property and combining like terms
3. Addition Equation homework review
4. Subtraction Equation Notes
5. Subtraction Equation Practice 
6. Start homework

The Assessment: After the notes were done, students were asked to try a couple problems from the book that were not part of the homework and were checked before going onto the homework. The self-assessment as always was kept in the classroom and served as a way for students to verify what they need more exposure to in the future.

Homework: Page 539 #1-9 and #15-17

My Glass Half-Full Take: It was apparent in looking at homework from addition equations that students were not properly isolating the variable in one class particularly more than the others. In doing the same drill as the previous day, students were able to take a step in the right direction and begin showing how to cancel the work.

One Thing to Do Differently: Overall this was a very boring lesson. I say that not only for the kids, but for myself. I have no voice to begin with today, but I also wasn't helped by the lack of flare this lesson had. I need a song, a dance, a challenge - something to get enthusiastic about it myself.

Day 107: Equivalent Expressions Quiz & Introducing Equations

6th Grade Math Standards: 6.EE.7 Solve real-world and mathematical problems by writing and solving equations of the form x + p = q and px = q for cases in which p, x, and q are all rational nonnegative numbers.

The Learning Objective: Combine like terms to find equivalent expressions. Use the distributive property to find equivalent expressions. Solve an addition equation using the inverse operation to both sides of the equation.

Agenda:

1. Pass back last week's weekly quiz and hand out this week's weekly quiz
2. Take the quiz
3. Work on the weekly quiz
4. Basketball team - I had students pick teams and then said that one player got injured. If a player is taken off one team, what should the other team do to be fair? 
5. Addition equation notes
6. Addition equation practice and homework

The Assessment: The quiz and problems #20 and 21 out of the textbook as well as the homework problems.

Homework: Page 529 out of the Glencoe book problems #1-19 and 22 as well as the weekly quiz being due tomorrow

My Glass Half-Full Take: I think that thanks to having more time and longer classes these students understand the distributive property and combining like terms much better than the classes the past two years have.

One Thing to Do Differently: I need to give much more examples of the formatting for adding equations in the notes. Students should do about 8 until they have the process down. I also liked showing the last class the problem 5n + 8 = n - 12 to show them why the process is so important (even though that problem was completely over their heads).