Day 144 Box Plots

6th Grade Math Standards: 6.SP.4 Display numerical data in plots on a number line, including dot plots, histograms, and box plots. MA.4.a. Read and interpret circle graphs.

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: Analyze a box plot

Agenda:

1. Jumpstart that asks students to find a missing data value when the median, mode, and mean have been given. The second question asks students to find potential errors in a large data set (a data set that consists of 24 values). 
2. Review line plots/dot plots
3. Box plot of Samantha's family
4. Box plot notes
5. Box plot ticket to leave
6. Box plot homework

The Assessment: The ticket to leave, which was actually the same thing as the notes. If students got one box plot correct, we let them skip the second box plot and start the homework.

Glass Half-Full Take: Today we did Samantha's family for a second time. Here is a look back at the first time it was done:

This time I made a couple of small adjustments. First, I assigned the girls girl names and the boys boy names. This avoided silly distractions. Second, I enlisted cooperation before starting from the class. Again to avoid distractions and to ensure that while the activity was fun we were still working toward the objective. Third, I warned the students not to slice the paper towel when it was rolled out to represent the "box" portion of the box plot. That's all we needed. It was much smoother today.

 

One Regret: Finding the range is a tricky concept for students. The initial instinct most students have is to subtract the lowest number on the number line from the highest number on the number line. That is wrong because the number line does not necessarily reflect the data set. In all of my classes, I had students try and fail in finding the quartiles and the median. This does not add much value in my opinion because where they guess and fail or I "spoon feed them" what to do, eventually it sticks in either case. What does not stick is finding the mean. That's where trying and failing will add more value. Otherwise I'm having them try everything and fail, so no initial struggle really sticks out. Where they struggle most, is with finding the range and determining which part of the data is more spread out. The ladder struggle we deal with in greater detail tomorrow when they create box plots.

Link of the Day: This comparison of DreamWorks and Pixar comes from Yummy Math. One of the components to it is a box plot.

Day 143 Dot Plots Day II

6th Grade Math Standards: 6.SP.4 Display numerical data in plots on a number line, including dot plots, histograms, and box plots. MA.4.a. Read and interpret circle graphs.

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: Interpret dot plots to find measures of center and variation

Quote of the Day: "Each student in this class will get an A for the course. However, there is one requirement that you must fulfill to earn this grade: Some time during the next two weeks, you must write me a letter dated next May... and in this letter you are to tell, in as much detail as you can, the story of what will have happened to you by next May that is in line with this extraordinary grade." They [the students] were to discuss insights, milestones, and even competitions won. "I am interested in the attitude, feelings, and worldview of that person who will have done all she wished to do or become everything he wanted to be." - from How to Win Friends & Influence People by Dale Carnegie

Agenda:

1. Jumpstart
2. Review the homework
3. Dot plots notes on central tendency and variation
4. Dot plot notes on central tendency and variation independent practice
5. Work on the weekly quiz

The Assessment: In coordination with the quote of the day, I promised students that if they could prove to me they could find measures of variation and central tendency by looking at a dot plot that we would have no homework. It instantly helped add value to the assessment which was step four in the agenda. I gave the students two problems to do on their own from this step.

In all honesty, students proving their knowledge in the formative assessment and not needing to do homework is generally just good teaching practice. The homework I have typically given this year is done only because I run out of time to more properly give a formative assessment.

Glass Half-Full Take: The ease with which students are able to answer what the purpose of a dot plot is useful is refreshing for me. When they understand the more basic purpose, it makes the rest easy. All the students demonstrated on the formative assessment from the last class that they knew at least two purposes for the dot plot and now some of the students can state as many as nine purposes.

1. Find the mean
2. Find the median
3. Find the mode or peak
4. Find any possible outliers
5. Find the quartiles and interquartile range
6. Find the range
7. Find any possible gaps
8. Find any clusters 
9. Find how many pieces of data are in the data set

One Regret: I only wish that I could have gotten the students out of their seats at some point during class. Perhaps next time I teach this lesson, I will have them go to the recycle bin to throw their paper away once the jumpstart finishes. The way this class was designed, it was essentially 100 minutes of desk work (although in truth if students were optimally focused it's really 50 minutes).

Link of the Day: Good take on high school preparation for college by Grant Wiggins. A couple of counterpoints from me:

• High school is not college. The maturity level and hormones are not the same, so why should we treat it as such? College classes are longer, but is that really what we should do with high school classes?
• What about those students in high school that are not going to move on to college? In fairness to Mr. Wiggins, he does make this caveat at the end of the article.
• Advocacy from the students is another thing that must happen gradually. In my own experience many times I didn't seek out my professors for extra help, but my peers. 

Day 142 Dot Plots

6th Grade Math Standards: 6.SP.4 Display numerical data in plots on a number line, including dot plots, histograms, and box plots.
MA.4.a.Read and interpret circle graphs.

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 Objectives: Make three observations from a dot plot

Quote of the Day: “More people die Monday morning at 9 A.M. than any other time. Monday at 9 A.M. is when people start their work week. Think about that - people would rather die than go to work. It sounds funny but actually it’s quite sad. People feel like they don’t have a choice. So they give up. But I am here to tell you today that you have a choice. You don’t have to sit passively by like so many other unhappy souls who let life create them. You can take the wheel and choose to create your life, one thought, one belief, one action, one choice at a time. It’s your bus and you’re the driver and you choose where you are going and the kind of ride it’s going to be.” - Jon Gordon

Agenda:

1. Self-Assessment 
2. Review the test on statistics (central tendency and variation)
3. Review fractions to decimals since that is the topic on this week's weekly quiz
4. Start the weekly quiz
5. Create a frequency table
6. Create a line plot based on the frequency table
7. Make various observations (mean, range, etc.) about a line plot
8. Guided practice from the text book about dot plots
9. Exit ticket
10. Independent homework practice

The Assessment: The guided practice was utilized as a ticket to leave as well as the actual exit ticket. I had students do the guided practice knowing that I had to come around and check.

Glass Half-Full Take: I felt very fresh coming off of vacation. I had read a couple books that helped with my attitude and gave me a more positive outlook in how I relate and deal with the students. Every little issue that typically rattles me I dealt with one at a time and had no issues. I even made a point of complimenting the student that I just criticized.

One Regret: Students were given the opportunity to start the weekly quiz in all classes and I had originally ignored this in the agenda. This week's weekly quiz deals with decimals, fractions, and percents. I liked reviewing the topic in one class because it's apparent this is still a gap in student learning.

Speaking of gaps, that word was given as part of the dot plot vocabulary. Gaps, peaks, and clusters were all mentioned today. I don't regret that part, but it will not be for the best if this is not covered again tomorrow and the next day to reiterate the meaning and value of these terms.

Link of the Day: This is a list of books to read for math teachers. I got my two cents in there.

Day 141 Statistics Test

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.

6.SP.2 Understand that a set of data collected to answer a statistical question has a distribution which can be described by its center, spread, and overall shape.

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: Identify three measures of variation; identify three measures of central tendency; find all measures of central tendency given the data set; find all measure of variation given the data set; find a missing value in a data set given all values but one and the mean; differentiate a statistical question from a non-statistical question

Agenda:

1. Collect the Weekly Quizzes
2. Explain the dilemma of how many ways people can sit next to each other if two people are in the room, three people in the room, etc.
3. Pass out and read over the directions for the test
4. Take the test
5. Determine how many combinations of people can sit next to each other if there are 4 people in a group, 5 people in a group, 6 people in a group, and 7 people in a group
6. How many handshakes take place if each person in a group of two shakes hands with the other? A group of 3? 4? 5? 6? 7?
7. If there are three people chosen among 3 people to be allowed to drop out of school and only attend Red Sox games, how many possible groups can be chosen? What about if there are 4 people and only 3 can be chosen? If there are 5 people and only 3 can be chosen? 6? 7?

The Assessment: The test went relatively well. Based on the study guide from the previous class I was not surprised to see students struggle most with finding a missing data value given the mean. I also was not surprised with the students struggling to identify what measure of central tendency was most appropriate given the data set.

Glass Half-Full Take: Today was the day before a vacation, so it was good to have something to go to after the test where students could work independently and be somewhat engaged. I borrowed all three of the items featured in the agenda 5-7 from a book called The Number Devil. I loved the book and thought it was both curriculum and age appropriate.

One Regret: As part of that after test activity, I had the students share their work on Google Docs. We have not used Google Sheets in my class and I'm not sure if the students have had any experience with this. It was not something I went over in great detail with the students beforehand and I decided that students usually are better equipped to handle technology than me and that the details were not necessary. I was wrong.

Students were simply typing messages to one another into the cells and not being as productive as they could have been in solving the problems. They did not struggle with the technology, but the task I gave them with the spreadsheet was rather vague. What I should have had them do was create a page with their name and their partners name on the page. Then the students could submit that page to me as a sort of bonus answer. Instead I made the page public and left the directions relatively vague. Even students that knew what they were doing had their work erased by other students who could edit the page. In my last class I expressed my disappointment with the students lack of focus. In reality I'm more to blame than any of the students for the lack of planning. I will know better next time at least.

Link of the Day: I was reading this blog post about if we have mathematical ceilings and loved two analogies. "Try asking adults about their math education: They refer to it like some sort of NCAA tournament. Everybody gets eliminated and it's only a question of how long you can stay in the game." And then this, "if you're missing one simple understanding - that these graphs are simply the x-y pairs satisfying the equation - they you're a broken futon. You're missing a piece which future learning will crucially depend."

Day 140 Statistics Study Guide

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.
6.SP.2 Understand that a set of data collected to answer a statistical question has a distribution which can be described by its center, spread, and overall shape.
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: Identify three measures of variation; identify three measures of central tendency; find all measures of central tendency given the data set; find all measure of variation given the data set; find a missing value in a data set given all values but one and the mean; differentiate a statistical question from a non-statistical question

Agenda:

1. Jumpstart using a problem from the book on page 836 #17 and a graphic organizer for identifying terms that go with central tendency and variation. I had students skip steps 1 and 2 on the link because we had done this already. I also projected the problem from the book on the board so students didn't need to rip the page from the textbook.
2. Analyze how we could check the work after ordering the numbers
3. Homework answers shared
4. Study guide page 1 done in partners
5. Start of second block page 2 of study guide passed out and page 1 is finished by students within the first 5 to 10 minutes of class
6. I either went over the study guide with the whole group or individually for smaller classes. If we did it as a whole group, I intentionally made an error at the end (quartile 3) on #18 to ensure students paid attention.
7. After I have checked the study guides students can receive a Chrome Book and try finding the measures of variation and central tendency of this data set from ticket prices to NHL playoff games
8. Students can work on their weekly quiz (students had the option of doing this ahead of step 6)

The Assessment: Today the assessment was through circumvention of the room. This is probably the least favorite of all of my assessments but the most common. I will say this about circumvention though, I virtually never sit at my desk as students work and work on something else. I'm always working with them so it's almost identical to a ticket to leave or a clicker assignment in that the feedback speed is pretty quick. Today in assessing, it was apparent that students still struggle with finding the missing value in a mean, defining the terms central tendency and variation, and have a difficult time with knowing what measure of central tendency to use in different situations.

Homework: Study for the test and work on the weekly quiz.

Glass Half-Full: The NHL playoff tickets was an extremely current way to bring meaning to data. I saw the ticket prices yesterday on my personal Twitter account which is virtually 100% dedicated to sports. The connections between sports is significant here, but looking at the ticket prices we were able to find out a little more about our data set.

We got a little into why the Rangers prices were higher than any other hockey team. Students were quick to point out that they won the Presidents Trophy (awarded to hockey's best regular season team I believe - hockey is my number four sports priority). I acknowledged that but quickly asked what else caused this and we branched off into a conversation about the cost of living in New York versus some of the other city. I informed students that milk would cost about a dollar or two cheaper in New York than a place such as Nashville.

It was enjoyable to get students on the Chrome books too which I haven't done too often this year. They all worked relatively quietly on them - even in my more energetic classes.

One Regret: I wish that I could have given the students some type of ticket to signify that their study guide "passed inspection." Students were rushing through the study guide to get to the chrome books and as a result not doing the best job possible to be prepared for the assessment tomorrow. Even something as simple as a yellow sticky note to mark off that they were all set and could move on to the next part of our agenda.

Link of the Day: Not really a link. Just saw this tweet from a teacher: "The average 4 year old asks 400 questions a day. What is happening with our secondary students?" I'm definitely guilty of trying to belittle the questions of my sixth graders to get on with "my show."

Day 139 Heights of 6th Graders vs. The Celtics

6th Grade Math Standards: 6.SP.2 Understand that a set of data collected to answer a statistical question has a distribution which can be described by its center, spread, and overall shape.

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: Compare the interquartile ranges of two data sets using a double stem and leaf plot

Quote of the Day: "Only a life lived for others is a life worthwhile." - Albert Einstein. We touched briefly on what it meant to celebrate Boston One Day on the two year anniversary of the marathon bombings.

Agenda:

1. Jumpstart with stem and leaf plot of temperatures in London
2. Review homework
3. Heights activity in partners
4. Fix weekly quiz mistakes

The Assessment: I collected all weekly quizzes from students in the first block of class and had the opportunity to return weekly quizzes to students with verbal and written feedback today. In looking over homework, I noticed that students struggled more with appropriate measures of central tendency than they did with finding the interquartile range in correcting the homework.

Homework: Students had to finish the height activity papers and work on their weekly quiz.

Glass Half-Full: The students were not capable today of starting the jumpstart on their own because it was the first time they confronted a stem-and-leaf plot. I anticipated this and asked the class up front if they had heard of a stem and leaf plot before. About half the hands went up, but none of the students could tell me how many data values were in the data set when I asked that question as a follow up. We went through the first couple of numbers and from there, the students were able to identify all the numbers. At that point I corrected the homework and collected the weekly quizzes. Disaster averted.

From a classroom management standpoint, I sat at my desk correcting weekly quizzes as students worked. My desk happens to be in the back of the classroom, so I can see the students while the students cannot see me. This was an advantage because I wasted no time telling students that they should not be turned around or talking to anyone besides their partner. And as I corrected weekly quizzes, I called students up with their partner to first review the weekly quiz and then review the classwork. It was the kind of feedback I would like to deliver all of the time, but can't based on what we do in class. We happened to be at a point with the unit we're in right now that students can work mostly independent from the teacher, so I had the opportunity today to do what we did.

A third plus today was the number of students that could actually create a data set with an interquartile range of 15. To me that's a higher level of thinking than simply finding an interquartile range as students need to work backwards in some respects to accomplish this task.

One Regret: One thing I wish was different was the focus in going over the homework. The Lawyer's Salaries worksheet from yesterday was an excellent example in a real-world context of how an outlier can completely alter a mean. The change in the mean was more than $25,000 with and without the outlier whereas the change in the median was only $5,000. Perhaps next time I teach this I will get students to act as lawyers or even change the names of the lawyers in the graph to students in my class. The presentation can be improved to improve focus.

Link of the Day: A colleague sent this link to our staff the other day and I found it interesting. Here's why winners keep winning. Reason number one is the most underrated skill for any teacher to have - a good mood.

Day 138: Appropriate Measures

6th Grade Math Standards: 6.SP.2 Understand that a set of data collected to answer a statistical question has a distribution which can be described by its center, spread, and overall shape. 3.

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 what measure of central tendency is most useful to use under different constraints in a data set; Identify three types of variation; Find the interquartile range of a data set

Agenda:

1. Jumpstart finding a range and interquartile range
2. Kinesthetic description of vocabulary terms range, interquartile range, and outlier
3. Kinesthetic description of how range, interquartile range, and outlier are measures of variation
4. Review homework problems number three and four from the previous class
5. Appropriate measures of variation notes 
6. Exit ticket of describing what central tendency in a data set of 11, 13, 13, 13, 12, 13, 13 is appropriate and also finding the interquartile range of a different data set
7. Students worked on their homework which included Lawyer's Salaries and NFL Penalties

The Assessment: I warned the students that appropriate measures was not the most exciting topic in the world and for some reason focus seemed to increase. What I initially did was activate their minds by asking what central tendency was used to describe their grades. Students struggled with this more than I would have imagined but eventually stumbled on the mean. I then used the example of a student who had nine 100s on a weekly quiz and one 0, and asked the students to describe this person as a math student. I told them honestly that if I were to describe this student I would use the median or even more likely the mode. That kicked started us to get a more narrowly defined idea of when to use which central tendencies.

I summarized it as the book did by stating that:

• Mean gets used with data sets that have no outliers
• Median gets used when there is an outlier and if the data does not have any gaps in the center (I used the analogy of grandparents day at a nursery school and took the ages of people to explain the gaps). 
• Mode gets used when there are many repeated numbers (there is a little more of a broad definition than the other two).

When the students were left to their own devices after we did two examples together they did not have much trouble. They actually convinced me that a data set of 1, 2, 3, 4, 5 can be appropriate and inappropriate for the median at the same time since there are no outliers but also no gaps.

In addition to the appropriate measures exit ticket, students also had to determine an interquartile range as part of the exit ticket. In one class, this took students twenty minutes. Students are struggling as expected because of the number of steps involved. That said, eventually all students finished this task. It was a clear day where having double-blocked math classes was a benefit because I would never allow twenty minutes for an exit ticket a year ago. Given that there was so much time to play with, I only gave subtle hints and suggestions to help students along in getting the interquartile range. The jumpstart, homework and previous day's work made this process a little less painful than it could have been otherwise.

A third assessment came in checking student homework. Students really struggled to find the error that someone else made in calculating the interquartile range of a data set with nine values. They had no trouble with the median but getting the quartile values proved difficult.

Glass Half-Full Take: In my last class it was wonderful having the Title I teacher in the room. He took half the students and I took half. Students struggled but didn't lose focus as they tend to do when they struggle and cannot get teacher assistance. I was in close proximity to all students and only guided them to the next step at most when they needed my assistance. Sometimes I simply referred students back to their notes. As quick as my feedback was, it would not have been possible without a second teacher in the room.

I was encouraged by the ease of which students recognized what an appropriate measure of central tendency was.

Finally I was also satisfied with how I presented to students what a measure of variation was kinesthetically. Our definition as the textbook provides is that a measure of variation "describes the distribution or spread of the data." This definition means nothing to the students as the term "distribution" does not help anything. I have found that changing the definition to "how spread out the numbers are" is helpful. I will likely modify to this definition going forward as students get a clear picture of what this means.

One Regret: As students worked on the Lawyer's Salary data, I kind of regret not giving them a calculator to find the mean. They get lost in adding and dividing six digit and seven digit number respectively. Consequently when it comes time to analyze an appropriate measure without an outlier (today's objective) they are not in class and could bypass the importance of this question.

Link of the Day: The gender pay gap is still relevant today.

Day 137 Interquartile Range

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.

The Learning Objective: Determine the interquartile range of a data set

Quote of the Day: It's about remaining mentally and emotionally balanced all the time, no matter what is taking place around you. You never want to show your opponent a weakness through your words, facial expressions, or body language. No matter what they are saying to you, no matter what the crowd is chanting, if you can show poise you demonstrate to your opponent that they cannot rattle you. - Coach K

Agenda:

1. Jumpstart with Bethel's grade (he took 9 tests, so find what happens if he gets a 100 on his tenth test or a 0 on his 10th test). I also collected the students' Plan-A-Party work
2. Pass out Weekly Quiz #23
3. Refresh students on what the range and outlier are by having them kinesthetically show what the range is.
4. Get students to think about range and outlier as they apply to sports from the weekend (the Knicks and Magic combined for only 15 points in the 2nd quarter - a season low). 
5. Interquartile Range homework models (two)
6. Samantha's Family - students role played as a family went from 5 to 6 members during the entire time (the 94 year-old great grandfather, the 1 year-old baby, etc.) 
7. Gave ten minutes for students to start their homework which was 4 problems that asked students to find range and interquartile range of a data set. 

The Assessment: I checked all students first problem as they started the homework. As students role played I managed to cold-call every student on what the median or quartiles were in a given situation.

Glass Half-Full: Both the homework models and the homework itself help breakdown a complicated process one step at a time. I think it is much easier than our textbook breaks the process down for students. If you ask a college educated adult to find an interquartile range, I'm willing to guess that less than half can do it successfully. I think it's more than fair to outline each step to help the students get used to the multiple step nature of finding an interquartile range.

Students also love role-playing with Samantha's family. Everyone is dying to play a role in the family for whatever reason. It goes along with my theory that anything that deviates from the classroom norms is craved by the students. I really enjoy student enthusiasm as it makes my work much more rewarding. In fact I don't think the work would be rewarding at all without the enthusiasm of the class.

One Regret: Speaking of the Samantha's family activity, I think the classroom management aspect of this could have been better. The students enthusiasm unfortunately dwarfed the mathematical learning experience a little here. I could have done a better job of telling the students up front that they needed to respect my narration of the story and to participate not just as a "family member", but also as math students. This time of the year is always more challenging as the students comfort level with me grows and they become increasingly willing to push boundaries.

Day 136 Plan A Party Part II

6th Grade Math Standards: 6.NS.3 Fluently add, subtract, multiply, and divide multi-digit decimals using the standard algorithm for each operation.

6.NS.4 Find the greatest common factor of two whole numbers less than or equal to 100 and the least common multiple of two whole numbers less than or equal to 12. Use the distributive property to express a sum of two whole numbers 1–100 with a common factor as a multiple of a sum of two whole numbers with no common factor. For example, express 36 + 8 as 4(9 + 2).

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.

The Learning Objective: Use decimal and greatest common factor calculations in a real-world context.

Agenda:

1. Finish floor plans of the plan-a-party worksheet
2. Make the calculations for inviting 30 people to a party (invitations come in packs of 8, plates come in packs of 25, etc.). Students also had to consider the prices of these items
3. Begin Weekly Quiz #23

The Assessment: All Plan-A-Party will be collected by myself of Monday. I circumvented the room today and noticed that students seemed to be understanding the problem well enough.

Glass Half-Full: Today was a half-day and a Friday, so it's always harder to teach. Students are naturally a little more energized and many of them are confused by little issues such as how long class will be and if we will have lunch on a day like today. When we were planning this day about a week ago, none of the math teachers really anticipated the student attitude in our planning. It was not until Wednesday after the students were done MCAS testing that we decided to do a U-turn and try a mini project for this class and yesterday's classes. I was the most resistant to this change as I wanted to get right back in the routine, but today being as hard as it was to transition from class to class it was nice to have students at least know what was going to be asked of them in math class. The level of independence the students had with the class today was much higher than it would have been had we been doing a lesson with interquartile range as we had originally planned.

One Regret: It might not have done much harm if I had had students work on this whole project with a partner and had both partners submit an assignment. As it were, many students were asking questions of one another anyway.

I find this to be a common occurrence in my classroom. Very often we are doing activities that require a partner anyway and the desks are also arranged in groups of two. Over the years I have increasingly stopped fighting the idea of letting students help other students. The reality is that eventually all students are going to have to be capable of working collaboratively (even though in the immediate future they are tested independently).

Link of the Day: This article discusses the relevance of arrests in the classroom and advocates that police can cause more harm than good in schools. On average Massachusetts has about 2 arrests for every 1000 students. That is one of the lowest rates in the country

Day 135 Play a Party

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.NS.3 Fluently add, subtract, multiply, and divide multi-digit decimals using the standard algorithm for each operation.

6.NS.4 Find the greatest common factor of two whole numbers less than or equal to 100 and the least common multiple of two whole numbers less than or equal to 12. Use the distributive property to express a sum of two whole numbers 1–100 with a common factor as a multiple of a sum of two whole numbers with no common factor. For example, express 36 + 8 as 4(9 + 2).

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.

SP.1 Make sense of problems and persevere in solving them.

SP.3 Construct viable arguments and critique the reasoning of others.

The Learning Objective: Find the dimensions of a rectangle given the area and perimeter

Agenda:

1. Jumpstart with football pushups 
2. Plan-A-Party Rough Draft
3. Plan-A-Party Final Draft
4. Plan-A-Party Supplies

The Assessment: I circumvented the room as students worked on the plan-a-party and jumpstart.

Glass Half-Full: In two out of three classes, the Plan-A-Party went very well. Students were able to identify the dimensions of the rectangles.

I was especially happy with having students determine the dimensions of the room. This involved students finding a perimeter of 66 and an area of 270. Even some of the students that typically do not struggle had difficulty finding the solution to this problem. Work organization proved to be essential for students that met this problem with success and without much assistance. As we discussed earlier in the year, listing the factors one at a time and in order is a great way to discover what factors work and having a through understanding of what factors will not work.

To take this another step further for students next year I could even try to show them (or have them show me) how this type of problem can be solved algebraically. I am willing to venture a guess that none of the students could do that if I simply asked "come up with a rule algebraically for the length or width of all rectangles."

On a separate note this is a good jumpstart to have timed after students have learned central tendencies. We had good discussion and good thinking pertaining to the questions about mean and mode.

One Regret: My assessment was not done very thoroughly and this cost me in one class where the entire class was wondering what was wrong. What I wish I rather did was have students find the area and perimeter of one part of the party (such as the Pool Table which had a perimeter of 14 and area or 10). From there I could better diagnose who had troubles and why.

Link of the Day: The Fast Clapper lesson courtesy of Nathan Kraft.

Day 133 & Day 134 MCAS Testing

I did not have the opportunity to teach math lessons for the past two school days. On Day 133 the students took the MCAS test so late into the day that we actually had to push lunch back. After the students were done this it was only fair to them to have a "fun class." I played Scattegories with them. I also fought every instinct within me to move seats and pass out boring worksheets to the class as they were consistently talking over me. I never did discipline the students as I thought they had a long day as it were and deep down I knew better than to care about how well we did in going over our answers to the game.

On Day 134, I went over the directions faster than the previous day. We started about 15 minutes earlier than we had in the previous day. The students on this day had their encore classes at the end of the day, so again I did not have the opportunity to teach a lesson which is typical for this time of the year.

This is probably the last year of MCAS as we make the switch over to PARCC for both ELA and math. I'm not looking forward to the switch in math as I think the PARCC test is designed to be read by an adult more than a child.

Day 132 Measures of Variation Introduction

6th Grade Math Standards: 6.SP.2 Understand that a set of data collected to answer a statistical question has a distribution which can be described by its center, spread, and overall shape.

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 the range of a data set

Quote of the Day: "Learning is defined as a change in behavior. You haven't learned a thing until you can take action and use it." - Don Shula

Agenda:

1. Self-Assessment
2. Review Quiz
3. Get the central tendencies of Derek and Michael in partners 
4. Ask what the difference is in the data between the two after students agree that the central tendencies for each student is identical
5. Ask students to define range or tell students if nobody knows
6. Do the rest of the problem with Derek and Michael
7. Give examples of outliers (my age, months hockey players were born in in Canada, home runs in a baseball season by Sammy Sosa from 1993-1998, etc.)
8. Homework models (could serve as tickets to leave additionally)

The Assessment: I circumvented the room as students worked on the central tendencies in Step 3 of the directions. Off-hand I would say about 60% of the students recognized that Michael needed a 0 on the quiz he had on division. I pointed this out to students. They seemed to remember what mean, median, and mode were after I gave them a little time to work this out. It was a perfect sheet to have when most students had not been in class since last Thursday (three days ago).

I also asked students to think of a range that would give them an answer of 20 and to come up with their own outlier examples to ensure that they knew enough about the definition to identify how these numbers could fit into a problem.

Glass Half-Full: In one class today, I asked why they believed our mean was about 6 or 7 points away from the other two classes. I did not do it with a threatening or sarcastic tone. I simply asked it curiously to evoke some of their ideas. One student brought up paying attention and others said we did not have enough practice. It was good feedback. My personal opinion is that in terms of classroom management that particular group has a harder time working in partners as we did on the study guide the day before the quiz. I had their attention though after we as a class came to this realization and it helped in going over the quiz. Hopefully that makes up for the 6 or 7 percentage point drop between the two classes as they seemed to pay attention more thoroughly when I reviewed the quiz.

Overall today was a classic example of mood winning over the students. I was not in a joking mood, but I was extremely tolerant of the needs of needy students. I had students go to their lockers, but only at logical points in the course of the class. I used please and thank you to get students to work. My listen to lecture ratio was a little more tilted in the side of listening than usual. More than anything what helped with this was the students previous learning experiences with the three central tendencies. It was a well-timed lesson.

One Regret: So many steps, so little time. I read Malcolm Gladwell's book Outliers over the summer. There were interesting anecdotes from this book that analyze why outliers came to be and I wish I had more time to share these nuggets because it's a great example of math telling a story.

Link of the Day: Great article on Emojis although it does talk a little too much about alcohol for my liking.

Day 131 Good Friday

Today I only had 40% of my homeroom students. Rather than teach an actual lesson, I used the day to let students decorate our door for the upcoming MCAS test. I had students brainstorm ideas individually then share them collectively. I then put together two supervisors and before picking the volunteers for this role I asked that these students be better listeners than they were bosses. It proved to be effective in getting cooperation.

The students also got an opportunity for a diversion when a guest speaker came to speak to the entire sixth grade about animal adaptations.

At the end of the day I worked with students one on one who happened to struggle on the quiz the prior day. These students were not resistant and I enjoyed throwing the starfish back in the ocean for the basic (and not so basic) skills of statistics.

Day 130 Central Tendency Quiz

6th Grade Math Standards: 6.SP.2 Understand that a set of data collected to answer a statistical question has a distribution which can be described by its center, spread, and overall shape.

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 Objective: Determine a median, mode, and mean of a data set. Analyze mistakes in finding a mean. Determine a missing value from a data set given the mean and almost all data values.

Agenda: 

1. Collect the Weekly Quiz while students do the grades of Elvis vs. Elvida
2. Take the central tendency quiz
3. Make necessary changes to the weekly quiz 
4. Retake any previous math assessment (inequalities or the functions test)
5. Read silently

The Assessment: The quiz and weekly quiz. As far as the quiz goes, question 8 was difficult for students because they left Friday out of the data set. Question 9 was difficult because for lack of a better phrase it was difficult. Students actually did better on question 10 which makes sense because there was a concrete data set. I was surprised by the amount of students that knew Taylor's mistake in question 11. 

Glass Half-Full: The first seven questions of the quiz were handled well by the class. If a mistake was made, it was not as if the student was guessing aimlessly. The pepper practice we had with the vocabulary proved worthwhile and perhaps it also helped students to have a very concrete example of what median is as we reviewed it. Going forward we are going to need these skills as fundamentals in studying dot plots, box and whisker plots, and measures of variation such as interquartile range and mean absolute deviation. It is comforting to know that the students shouldn't be lost with the prerequisite steps toward all of these topics. 

One Regret: I have two regrets today. The first regret is that I wish that items three through five were not a part of the agenda. I overestimated the amount of time that the quiz would take. The reason that I overestimated can likely be attributed to the anticipated difficulty of questions eight through twelve. These questions were difficult, but the amount of time spent on them by the students did not change which led to students having an entire block of time to work on potential retakes and the weekly quiz. The trouble with retakes and the weekly quiz is that both of these items are heavily reliant on getting assistance from me. Students would not be retaking a quiz or fixing a weekly quiz if they understood the material in the first place. I also have to coach students on the process for what problems they need to fix, remind them how poorly they did, and stop for a moment to cue the students that are being disruptive to get back to task. Overall the process of an entire class spending an entire block on fixing holes in their learning sounds utopian in theory, but in execution it's extremely stressful and partially chaotic. 

The other issue of course is with those problems eight through eleven that students struggled more with. On the first day of statistics I asked students how long they would spend before giving up on a problem. Perhaps that is what I lead off with when reviewing this quiz. Students were not nearly as persistent as I would like with the problems in which they had to finding a missing data value. 

I learned today that even though on the surface students should be motivated by the prospect of being able to retake a quiz and bring their grade up, that students could care less. I kind of already knew this, but it was safely confirmed for me today. Students crave diversions. They do not want to do more of the same. For all the complaining I do about students being over-tested I was guilty of adding to the problem today. 

Day 129: Study Guide for Central Tendency Quiz

6th Grade Math Standards: 6.SP.2 Understand that a set of data collected to answer a statistical question has a distribution which can be described by its center, spread, and overall shape.

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 Objective: Find a mean, median, and mode. Find a missing value in a data set given the mean. Manipulate the mean to get bigger or lower when a new data value comes into a data set.

Quote of the Day: “I don’t want a bunch of guys with their shoulders down and sighing when something goes wrong. I don’t ever let body language go. Ever.” - Bill Self

Agenda: 

1. MCAS Math March Madness. 
2. Review the homework
3. Study guide in partners
4. Stats 2 on 2 game. Using a deck of cards and taking out the face cards (less to explain to students), the students were given 7 cards each and then told to find the mean, median, and mode of their data sets. The partnership with the highest combined mean, median, and mode wins. 

The Assessment: Using the clickers, about 85% of students were able to answer the question finding the mean, 74% were able to correctly determine how many questions were on a test given the number correct and the percentage, 56% were able to determine what fraction is between 10/3 and 11/3, and 83% were able to find the volume of a cube given an edge length. 

I circumvented the room as students worked on the study guide. 

Glass Half-Full: As I went over the study guide, I made students that didn't do the homework take notes on what I was reviewing with the class. I warned them that if they did not do a good job I would have to have them come at lunch to do the work. This was a good enough incentive to help them pay attention. I still collected the homework sheets from these students to be sure that they were focused. 

One Regret: The study guide takes a long time to do, which left less time to play the game. As a result, I'm going to allow students to play tomorrow if they are finished with the quiz early.