Day 38: Double Number Line

6th Grade Math Standards: Use ratio and rate reasoning to solve real-world and mathematical problems, e.g., by reasoning about tables of equivalent ratios, tape diagrams, double number line diagrams, or equations. a. Make tables of equivalent ratios relating quantities with whole-number measurements, find missing values in the tables, and plot the pairs of values on the coordinate plane. Use tables to compare ratios. b. Solve unit rate problems, including those involving unit pricing and constant speed. For example, if it took 7 hours to mow 4 lawns, then, at that rate, how many lawns could be mowed in 35 hours? At what rate were lawns being mowed? c. Find a percent of a quantity as a rate per 100 (e.g., 30% of a quantity means 30/100 times the quantity); solve problems involving finding the whole, given a part and the percent. d. Use ratio reasoning to convert measurement units; manipulate and transform units appropriately when multiplying or dividing quantities. MA.3.e. Solve problems that relate the mass of an object to its volume.

The Learning Objective: Use a double number line diagram to describe a relationship between units

Quote of the Day: "If everyone has the attitude that someone else will do it, then nobody will do it." - Camfel Productions

Question from Yesterday (as always from a student): "Is 3.2 a different number then 3.20?"

Assessment: Circumventing the room to see that students were putting the correct numbers on the double number line; checking the homework

Agenda:

1. Jumpstart
2. Review the homework
3. Ask a question upon seeing this video
4. Create a double number line between mini bears and grams 
5. Create a double number line between regular bears and grams 
6. Answer the question how many mini bears are inside the super bear?
7. How many regular bears are inside the super bear?

Glass-Half Full: This concept is on the easier side of tape diagrams which we did yesterday. The hardest part about this topic for me is finding resources for it. I have created my own stuff for it, but enjoy the Dan Meyer Gummy Bear activity much more for engaging the students. I'll show them my stuff on Monday as a review.

Regrets: Today was a shortened day because of an assembly we had in the morning. We rushed the answer to how many mini and regular bears would equal a super bear and about half of the class was probably clueless as to how we discovered the answers here because of time pressure.

Link of the Day: Freakonomics always informs in interesting ways, but my ears perk up when I know that it's about education. Listen to the piece on gaming in schools.

Day 37: Tape Diagrams

6th Grade Math Standards: 6.RP.3 Use ratio and rate reasoning to solve real-world and mathematical problems, e.g., by reasoning about tables of equivalent ratios, tape diagrams, double number line diagrams, or equations.
a. Make tables of equivalent ratios relating quantities with whole-number measurements, find missing values in the tables, and plot the pairs of values on the coordinate plane. Use tables to compare ratios.
b. Solve unit rate problems, including those involving unit pricing and constant speed. For example, if it took 7 hours to mow 4 lawns, then, at that rate, how many lawns could be mowed in 35 hours? At what rate were lawns being mowed?
c. Find a percent of a quantity as a rate per 100 (e.g., 30% of a quantity means 30/100 times the quantity); solve problems involving finding the whole, given a part and the percent.
d. Use ratio reasoning to convert measurement units; manipulate and transform units appropriately when multiplying or dividing quantities.
MA.3.e. Solve problems that relate the mass of an object to its volume.

The Learning Objective: Solve ratio problems using a tape diagram

Quote of the Day: “Curt Carlson was founder of multibillion-dollar Carlson Companies. That today includes Radisson Hotels and the restaurant T.G.I. Friday’s. He said the first five days of the week, Monday through Friday, are when you work to keep up with the competition. It’s on Saturdays and Sundays that you get ahead of them. To a lot of people, Carlson was a workaholic. Of course, he didn’t think so; to him, work wasn’t work.” – Harvey MacKay

Question from Yesterday (as always from a student): Several students were asking about this question from the weekly quiz:

We raised $10,000 simply by selling sweaters. Sweaters were $40 apiece. How many sweaters were sold?

Assessment: Students stood up when they were done the third problem; also used a fist of five

Agenda:

1. Self-Assess on the ratios quiz
2. Review ratio quiz
3. Quote, Star Student, revisit the question of the day
4. Tape Diagram Notes
5. Tape Diagram Practice 

Glass-Half Full: I like hearing students say "Can I retake it?" I like the laughs I get when I tell the story about a student that said that he hated me because I make him learn.

Regrets: The activator was absent. I have students simply try the problem without any instruction first, so it helps explain how useful the process is. That said, I'd like to connect the problem to real life somehow or make this into a 3-act problem somehow.

Link of the Day: Robot grading isn't far off it would seem.

Day 35: Ratios Study Guide

6th Grade Math Standards: 6.RP.2 Understand the concept of a unit rate a/b associated with a ratio a:b with b ≠ 0, and use rate language in the context of a ratio relationship. For example, “This recipe has a ratio of 3 cups of flour to 4 cups of sugar, so there is ¾ cup of flour for each cup of sugar.” “We paid $75 for 15 hamburgers, which is a rate of $5 per hamburger.”

The Learning Objective: Find a unit rate

Quote of the Day: “A person’s happiness is related to the happiness of their friends, their friends’ friends, and their friends’ friends’ friends. We found that each additional happy friend increases a person’s probability of being happy by about 9%. By comparison, an extra $5000 in income increased happiness by about 2%.” – Dale Carnegie

Question from Yesterday (as always from a student): "Why would anyone ever want 0.8 of a pencil?" In response to finding a unit rate of 4.8 pencils per $1.

Assessment: Circumventing the room.

Agenda:

1. Entering the class students worked on the study guide
2. I passed back and collected weekly quiz materials as students worked on numbers 1 and 2.
3. We worked on each problem part by part until the ten questions were completed
4. Either Get to 10 or pepper to close out the class 

Glass-Half Full: The pace of the study guide was slow, but effective. I think it kept the students that were struggling involved better than simply giving the students the opportunity to do ten at a time.

Regrets: I wish there was something for the higher students to work on as they finished in some ways, but also believe that them helping students who were struggling was valuable as well.

Day 36: Ratios Quiz

6th Grade Math Standards: 6.RP.1 Understand the concept of a ratio and use ratio language to describe a ratio relationship between two quantities. For example, “The ratio of wings to beaks in the bird house at the zoo was 2:1, because for every 2 wings there was 1 beak.” “For every vote candidate A received, candidate C received nearly three votes.”

6.RP.2 Understand the concept of a unit rate a/b associated with a ratio a:b with b ≠ 0, and use rate language in the context of a ratio relationship. For example, “This recipe has a ratio of 3 cups of flour to 4 cups of sugar, so there is ¾ cup of flour for each cup of sugar.” “We paid $75 for 15 hamburgers, which is a rate of $5 per hamburger.”

The Learning Objective: Write a ratio three ways; find a ratio and put it in simplest form; find a unit rate in a real-world context

Quote of the Day: “Cooperative behavior requires individuals to understand that by working together they will be able to accomplish something that no one can accomplish on his or her own. Jim Vesterman considered himself a reasonably good team player, yet he learned an indelible lesson in the power of group effort when he joined the Marine Corps. It started on his first day of boot camp at Parris Island as he and his fellow recruits learned to make their beds. His experience went something like this: the men are told that their objective is to have every bed in the platoon made; the drill instructor begins counting, and everyone has three minutes to make his bed (‘hospital corners and the proverbial quarter bounce’); they step back in line when done. So, Jim explains, he made his bed, stepped back in line, and felt ‘pretty proud, because when three minutes were up, there weren’t more than ten men who had finished.’ However, the drill instructor wasn’t handing out any congratulations; rather, he was shouting out that they had all day to get this right, looking at all the beds that were unfinished. Jim ripped off the sheets again… and again, and again. Finally the drill instructor looked him in the eye and pointed out, ‘Your bunkmate isn’t done. What are you doing?’ Apparently Jim had been thinking that he was done while his bunkmate struggled. Finally the light dawned on Jim, and working together with his bunkmate, they made both beds, and much faster than they had each done on his own.” – Barry Posner

Question from Yesterday (as always from a student):   In 2018, Mrs. Buxton expects to sign up for Instagram. Assuming she gets 45 likes 2 minutes after her photo has been posted and that Mr. Brazille has 98 likes after 12 minutes, which photo is getting more likes per minute? 

On this question, students were asking what to round to. We went over the fact that it's fairly obvious who has more likes and once the work has demonstrated who has more, there's no need to even get into the decimals.

Assessment: The quiz; the weekly quiz

Agenda:

1. Quote/Star Student/Question of the Day
2. Take the quiz
3. Work on the WQ #5

Glass-Half Full: I was really happy with the success we had in highlighting ratio words to solve the last two problems on the quiz. In general students were putting the units in the order that was given in the problem, so reading comprehension was strong.

Regrets: It was a hard quiz and the hardest question was problem five. Overall though I was elated with their effort on this quiz and I thought that the time we had to prepare for it and the way in which we prepared to it is something that I would repeat in the future.

Link of the Day: This is from 2006 and was I found it through Dan Meyer's blog. Everything written is still true.

Day 34: Unit Rates Let's Go Shopping!

6th Grade Math Standards: 6.RP.2 Understand the concept of a unit rate a/b associated with a ratio a:b with b ≠ 0, and use rate language in the context of a ratio relationship. For example, “This recipe has a ratio of 3 cups of flour to 4 cups of sugar, so there is ¾ cup of flour for each cup of sugar.” “We paid $75 for 15 hamburgers, which is a rate of $5 per hamburger.”

The Learning Objective: Find a unit rate in a real-world context

Quote of the Day: “There are plenty of teams in every sport that have great players and never win titles. Most of the time, those players aren’t willing to sacrifice for the greater good of the team. The funny thing is, in the end, their unwillingness to sacrifice only makes individual goals more difficult to achieve. One thing I believe to the fullest is that if you think and achieve as a team, the individual accolades will take care of themselves. Talent wins games, but teamwork and intelligence win championships.” - Michael Jordan

Question from Yesterday (as always from a student): "Why not divide the top part of the ratio by itself and the bottom part of the ratio by itself?"

Assessment: I collected the unit rate work that students had made on the trip to Stop & Shop, but as I was circumventing the room I made a new assessment on the fly. In the last 5 minutes of class, I asked the students if we had 31 pounds to $22, would there be more than 1 or less than 1 pound per $1 and why? The reason I asked this was because students were simply guessing on where to put their dividend and divisor. The exit ticket really helped clarify that this was the main struggle students were having putting together a unit rate. I also asked where the dividend and divisor would go. The thinking is clear, so tomorrow part of the agenda will be to review these types of comments from students. I actually posted one of these on Instagram already.

Agenda:

1. Jumpstart with munchkins from Dunkin Donuts. A 25-count costs $4.99. A 50-count costs $7.49. What is the unit rate of each?
2. Quote, Star Student, Question
3. Weekly Quiz recap
4. Review the homework (students went to the board to do problems)
5. Pictures from my trip to Stop & Shop

Glass-Half Full: Students associate the number one with a unit rate without much difficulty. I thought it was also critical that I integrated the exit ticket into each class because this is historically such a large issue.

Regrets: In reviewing the question from yesterday, there was too much of me speaking and not enough of them speaking. I need to ask less questions and get them to think deeper on the few questions that I'm asking. In reviewing the question of the day, I also hit on the ticket to leave question, but I think my commanding the stage as opposed to the students thwarted students from understanding the lesson sooner.

Link of the Day: Four strikes and your out is a great way to bring people who are finished with work together to continue to be productive while others finish up.

Day 33: Unit Rates

6th Grade Math Standards: Understand the concept of a unit rate a/b associated with a ratio a:b with b ≠ 0, and use rate language in the context of a ratio relationship. For example, “This recipe has a ratio of 3 cups of flour to 4 cups of sugar, so there is ¾ cup of flour for each cup of sugar.” “We paid $75 for 15 hamburgers, which is a rate of $5 per hamburger.” 29

The Learning Objective: Find a unit rate; analyze a unit cost to see what the best deal is

Quote of the Day: "When we attempt to use criticism to win an argument, to make a point, or to incite change, we are taking two steps backward. People can be led to change as horses can be led to water, but deprecation will rarely inspire the results you are aiming for.” – Dale Carnegie

Observation from Yesterday (as always from a student): "The price of cans will also change when there's only 5 cans because there is a recycling refund."
"If there is one missing, it should be cheaper than normal because the customer could argue the goods are damaged."

Assessment: Letting students try problems on their own from the notes; circumventing the room during the visual pattern

Agenda:

1. Visual Pattern #7. Write one thing you know. Write another thing you know. Draw the 4th step. Make a chart. How many trees are in Step 43? 
2. Unit Rate Notes
3. Unit Rate Homework

Glass-Half Full: In the first class, the co-teacher I was working with looked like she was going to jump out from the back of the classroom and attack me like a tiger. We were finding a unit rate and dividing 5 by 74. This was the students first experience dividing 5 by 74 I think (the occasion is rare unfortunately because they are two beautiful numbers). So why was it made into a glass half-full situation? It allowed us to use calculators. Students were literally asking how to clear their answers, were asking questions about how many decimals to write, etc.

Regrets: As far as time went, depending on the class I could have left out visual patterns. The notes took a while and in one class, I did not bother to assign homework because I did not fairly assess those students and it would have been an assignment that they potentially did wrong. I also think that it might have been better to start with a much more basic example in the notes (I did verbally give them two birds to one stone and four wheels to one car).

Link of the Day: We have Chromebooks in my school, so if this is you as well this link could be of use. What to do with Chromebooks.

Day 32: Partial Products

6th Grade Math Standards: 6.RP.1 Understand the concept of a ratio and use ratio language to describe a ratio relationship between two quantities. For example, “The ratio of wings to beaks in the bird house at the zoo was 2:1, because for every 2 wings there was 1 beak.” “For every vote candidate A received, candidate C received nearly three votes.”

The Learning Objective: Find the price of a single item in a product with multiple items

Quote of the Day: Born into poverty, Lincoln was faced with defeat throughout his life. He lost eight elections, twice failed in business and suffered a nervous breakdown. He could have quit many times - but he didn't and because he didn't quit, he became one of the greatest presidents in the history of our country.

Here is a sketch of Lincoln's road to the White House:

1816 His family was forced out of their home. He had to work to support them.

1818 His mother died.

1831 Failed in business.

1832 Ran for state legislature - lost.      

l832 Also lost his job - wanted to go to law school but couldn't get in.         

1833 Borrowed some money from a friend to begin a business and by the

end of the year he was bankrupt. He spent the next 17 years of his life

paying off this debt.

1834 Ran for state legislature again - won.         

1835 Was engaged to be married, sweetheart died and his heart was broken.

1836 Had a total nervous breakdown and was in bed for six months.         

1838 Sought to become speaker of the state legislature - defeated.

1840 Sought to become elector - defeated.        

1843 Ran for Congress - lost.  

1846 Ran for Congress again - this time he won - went to Washington and

did a good job.   

1848 Ran for re-election to Congress - lost.       

1849 Sought the job of land officer in his home state - rejected.   

1854 Ran for Senate of the United States - lost.

1856 Sought the Vice-Presidential nomination at his party's national

convention - got less than 100 votes.    

1858 Ran for U.S. Senate again - again he lost.  

1860 Elected president of the United States.

Question from Yesterday (as always from a student): Does the order matter in a ratio? Is 3 Kittens to 5 puppies the same as 5 puppies to 3 kittens?

What would happen to a ratio that was already put in simplest form if we added one more part to either ingredient? So if we had 20 chickens to 10 wolves and that ratio was simplified to 2:1 could it be 2:2 or even 1:1 if we added one more wolf?

Assessment: The homework from the night before was assessed with emphasis on number 18. In one class, I never had time to distribute homework, so I had them do it in class and worked with them on the highlighting aspect of breaking down the ratio problems.

Agenda:

1. Partial Product (from Dan Meyer)
2. Review homework and exit ticket
3. Nana's Paint Mix Up (from Dan Meyer)
4. Journal four questions. What is a ratio? What are three ways to write a ratio? What are the four steps to write a ratio? What is a wolf in ratios?

Glass Half-Full: I really enjoyed walking around the room and seeing students too high and too low responses to the Partial Product problem. I'm happy to see so many students giving logical numbers and following it up with logical reasoning. I'm sure the estimation will result in improvement in test scores, but if I had only a day or week to prepare for the test I wouldn't do estimation based activities. The time it takes to make it work is so much longer, but by taking the long view these students are really building a real world type of number sense. And the best part of all is that it is so much deeper than simply preparing the students for a standardized test. These problems are everyday things that can change the way that they perceive their decision making today.

Regrets: For the class that did the homework in class, I could not get to Nana's Paint Mix Up in class. I will obviously try to fit it in the future, but if we end up not getting to it, it will be unfortunate. In one class, I just sat with one group of two and worked it through with them in an unassuming manner. The rest of the class worked in partners and from what I could hear genuinely put effort toward finding a solution. The group I worked with ended up using 45 red and 9 white scoops of paint. The other students that correctly answered the problem used 25 red and 5 white scoops. It was great to listen to the answers that I got when I called on a couple students at random after giving ten minutes to try and solve.

Link of the Day: A list of the most innovative countries in the world. Finland is number one, the U.S. is number five.

Day 31: Introduction to Ratios

6th Grade Math Standards: 6.RP.1 Understand the concept of a ratio and use ratio language to describe a ratio relationship between two quantities. For example, “The ratio of wings to beaks in the bird house at the zoo was 2:1, because for every 2 wings there was 1 beak.” “For every vote candidate A received, candidate C received nearly three votes.”

The Learning Objective: Define ratio; give an example of a ratio

Quote of the Day:

Question from Yesterday (as always from a student): What was Jeanine's rate of putting together calculators for one hour?

Assessment: Here is the ticket to leave:

Agenda:

1. Record and reflect on the decimal quiz 
2. Review the decimal quiz
3. Ratio vocabulary, read it, write it, and say it
4. Ratio scavenger around the room (tell people and then share it to people). End of Class 1
5. More depth to the ratio notes (you can write them three ways, highlight the words before and after "to" in a word problem two different colors
6. Exit Ticket. As students work on the exit ticket, pass out the textbooks. 
7. Rip out and start the homework.

Glass-Half Full: As indicated in the picture under the assessment, highlighting was new and it was effective. I don't have exact data, but the question pictured has historically resulted in more losses than victories. Today there were more victories than losses. The highlighting method forces students to think. It's my fourth year teaching ratios, so this is becoming very predictable to see what students will do wrong.

I also joined Instagram. Students are following like woah.

Regrets: There was a lot of chaos toward the end of class between books being passed out, students needing assistance on the exit ticket, and getting weekly quizzes passed back. It's important for students to get a warning on this chaos before it happens to ensure maximum possible cooperation.

Link of the Day: A parent recommended CamScanner app for creating PDFs with the cell phone. Could definitely be useful in a pinch for me.

Day 30: Decimal Quiz

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

The Learning Objective: Use decimal operations in real-world and mathematical problems; give an example of a rate

Question from Yesterday (as always from a student): Is dividing by 0.5 and multiplying two the same thing?

Assessment: The decimal quiz (including the post assessment of the pre-assessment we did back on October 12th.

Agenda:

1. Take the post-assessment 
2. Decimals quiz
3. Work on WQ #4
4. Read if done numbers 1-3
5. In class #2 we worked on the paperclips activity from Dan Meyer

Glass-Half Full: The ratios unit was a weakness for our math team according to summative assessment data from last year. We have altered the way we will approach students recognizing what the labels in problems are. Today we got a head start on that by telling students to write "3 paper clips per 10 seconds" instead of simply writing 3.

Regrets: More than one colleague thinks that we may have rushed the decimal quiz. One colleague suggested that we could have broken up the addition and subtraction sections of decimals instead of teaching both in the same day. I had not given this consideration because lining up the decimal is the essential part of either operation. Upon further review though, there were students that were able to annex zeros in addition and not subtraction.

I also think that the concept of repeating decimals needs its own days after the results we got on the 8th problem from the quiz today. As I had mentioned in yesterday's regrets, there are definitely ways to differentiate this part of the curriculum for students that are thriving with the basics.

Link of the Day: I just found out about Pear Deck. The link here is to the pricing. It would require every student in the class to have a device and Google drive to participate.

Day 29 Decimals Study Guide

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

The Learning Objective: Divide with decimal divisors and decimal dividends

Quote of the Day: “When your attempt rate is high, each individual failure becomes a lot less significant…Accepting failure doesn’t just make risk-taking easier. In a surprising number of instances, it’s the only reliable path to success.” – Ron Friedman

Question from Yesterday (as always from a student): "Why do we say 'slide' the decimal in division problems? Is it possible to get a quotient that is greater than 1 and less than 1?"

Assessment: Marker boards check. In groups of 4 students found the solution to 4 different problems. They had to take the sum of the hundredths place of all four problems on their marker board.

Agenda:

1. Frayer Model of the 4 problems 
2. Quote, Star Student, and Questions of the Day
3. Repeating decimal problems (on marker board) 
4. Study Guide
5. Exit Ticket (1.1 divided by 0.06) 

Glass-Half Full: Something that the pre-assessment this year and my memory from last year served me with was that students struggled with repeating decimals. Looking at the problems we have done to date in this unit, and it is easy to understand why. Today's exit ticket helped me relay to students the idea that they have to continuously bring down zeros and how to express that they do not have to stay in math for the rest of their lives because this problem is not ending.

Regrets: At what point we drifted off because of the questions of the day. The drifting off wasn't a problem. We were breaking down the reason that some quotients were more than 1 and others were less than 1. It's all about the relationship between the dividend and the divisor. I think this investigation would take 90% of the students the entire block to wrap their heads around. If I'm looking for something at the end of the year to return to as part of an investigation, this would be engaging.

Another larger scale problem to tackle is what divisors cause repeated decimals to happen and why?

Link of the Day: Football season has gotten to Dan Meyer.

Day 28 Dividing with Decimal Divisors

6th Grade Math Standards: 6.NS.2 Fluently add, subtract, multiply, and divide multi-digit decimals using the standard algorithm for each operation 

The Learning Objective: Divide numbers when the divisor is a decimal.

Question from Yesterday (as always from a student): Why does it say to round in the directions? Is it possible to get an answer bigger than one when dividing decimals?

Assessment: Students started the homework in class and got 3 in a row correct.

Agenda: Jumpstart where the students work in groups to find wolves (common mistakes) of the four decimal operations

Review the bottle of water question from the homework

What happens when the dividend stays the same and the divisor gets lower and lower?

Homework model questions (14.4 divided by 1.2 and 15 divided by .03)

Work on the homework (not assigned as homework since today was Friday)

Glass-Half Full: The question about what is 360 divided by 0 is excellent for these students because they all carry the notion that it is 0. In seeing the trend of the divisor getting closer and closer to zero, some students could make sense of the fact that the quotient could not all of a sudden reverse course. This third part of the agenda was done entirely in groups which was effective because there needed to be dialogue even for the students that were comprehending to make sure they were on the right track.

Regrets: I want a ticket to leave that asks students to describe the relationship between the quotient and the divisor when the dividend is constant.

Day 27 Dividing Decimals

6th Grade Math Standards: 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).
MA.4.a. Apply number theory concepts, including prime factorization and relatively prime numbers, to the solution of problems.

The Learning Objective: Multiply decimal numbers

Quote of the Day: “Group A (83% of the people in the study) were embarking on a career chosen for the prospects of making money now in order to do what they wanted later, and Group B (the other 17% of people in the study) had chosen their career path for the reverse reason, they were going to pursue what they wanted to do now and worry about the money later. The data showed some startling revelations: At the end of 20 years, 101 of the 1,500 people in the study had become millionaires. Of the millionaires, all but one - 100 out of 101 - were from Group B, the group that had chosen to pursue what they loved!” - John Maxwell

Question from Yesterday (as always from a student): Not really a question, but more of a statement. "If the factors are both less than one, the product is also less than one." This led to a further conclusion that if both factors are less than one, the product is less than the factors.

Assessment: Checking the homework, my favorite no, circumventing the room as students start tonight's homework, letting groups check each other for understanding of the notes.

Agenda:

1. Visual Pattern #5 
2. Review homework
3. Multiply, add, and subtract worksheet (we ended up doing #1, #8, and #13 because students on the whole that could do #1 could also do the other 6 problems)
4. Dividing decimals my favorite no (4.56 divided by 7)
5. Division of decimals notes. The only things I did were tell students to bring the decimal up immediately when the divisor was a whole number and to annex zeros if the dividend was a whole number (which they got a chance to try on their own during My Favorite No)
6. Division of decimals practice 

Glass-Half Full: The inventive spirit in me came up with the following script for students that needed a challenge.

After annexing zero, will the following always have 0 as a remainder?

• 3
• 4
• 5
• 6
• 7
• 9

I also had students divide 10, 11, 12, and 13 by 9 to see a pattern.

Regrets: I did not get the students out of their seats in the second part of class.

Link of the Day: An interesting slant on a parent's slant of common core math. Something for parents and teachers alike.

Day 26: Multiplying Decimals

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

The Learning Objective: Add decimals to find a sum; subtract decimals to find a difference; multiply decimals to find a product

Quote of the Day: “General William Westmoreland was once reviewing a platoon of paratroopers in Vietnam. As he went down the line, he asked them a question. ‘How do you like jumping, son?’ ‘Love it, sir!’ was the first answer. ‘How do you like jumping?’ he asked the next. ‘The greatest experience in my life, sir!’ exclaimed the paratrooper. ‘How do you like jumping?’ he asked the third. ‘I hate it, sir,’ he replied. ‘Then why do you do it?’ ‘Because I want to be around guys who love to jump.’” – Harvey MacKay

Question from Yesterday (as always from a student): How many meters are in a foot?

Assessment: The students used 2 foot by 2 foot marker boards in groups of four and had to find the sum or difference of four different word problems. Once the solution was found they added all four of their solutions and came up with a bigger sum. That's the number I checked. If it was wrong, they kept trying. And it's amazing how quickly they forget...

I also had students do a my favorite no at the start of the multiplying notes part of class. Students that had no difficulty were given an alternative assignment (the basketball player stats from yesterday) in lieu of doing the notes.

During the notes, I had students try problems on their own.

Agenda:

1. Estimation 180 - Bubble Wrap 
2. Adding and subtracting decimals homework review
3. Markerboards assessment. 
4. My favorite no (start of second class). 0.3 x 0.71
5. Notes on multiplying decimals
6. Starting the homework from the textbook on page 205

Glass-Half Full: The marker boards assessment is amazing. It gets the students working together. It gets the students talking about math. It gives me the chance to see that they understand the math. I just need to find the opportunity to implement them more often.

Regrets: In the last class of the day I had a student tell the class that it would not make sense for the product to be over 1 since neither factor was over 1. It was unfortunate that it took the last class to jump to this conclusion. I wonder if there's a way to get all students to consider this because it provides a why to the "slide the decimal over as many times as there are digits after the decimal" rule. It will certainly be something I go over tomorrow. The one student who recognized this was delighted when I yelled onions in excitement.

Link of the Day: Andrew Stadel doing shooting basketballs at 101 questions. This could lead to equations, variable expressions, ratios, etc.

Day 25: Adding and Subtracting Decimals

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

The Learning Objective: Add and subtract decimal numbers.

Quote of the Day: “You can have the courage to be positive as you get up in the morning to face the day. You can have the courage to be gracious in defeat. You can have the courage to apologize when you hurt someone or make a mistake. You can have the courage to try something new - any small thing. Each time you display bravery of any kind, you make an investment in your courage. Do that long enough, and you will begin to live a lifestyle of courage. And when the bigger risks come, they will seem much smaller to you because you will have become much larger.” - John Maxwell

Assessment: The checklist from the quiz (to check for understanding), students doing problems and standing up when finished during the notes, and starting the homework

Agenda:

1. Self Assessment 
2. Review the quiz
3. Decimal Notes
4. Start the homework

Glass-Half Full: Since a few students did so well on the pre-assessment I asked them if they wanted the differentiated assignment which I told them was more of a challenge, but I tried to sell it to them by saying the work that was in the notes was something that would bore them since they knew it already. They all agreed. In all of my classes. I gave them this assignment on the greatest basketball players from Yummy Math.

Regrets: I keep insisting that students come after school or at lunch to do retakes, but I'm not advertising it well enough. Not enough students are taking advantages, or to put it another way, students are resisting my efforts to make them learn.

I will start to make phone calls, place emails, and put the times and days I'm available on the board for retakes.

Link of the Day: $3 Booritos from Chipotle.

Day 24: Prime Factorization Quiz

6th Grade Math Standards: 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).
MA.4.a. Apply number theory concepts, including prime factorization and relatively prime numbers, to the solution of problems.

The Learning Objective: Find the greatest common factor of two or more numbers using prime factorization; determine if two numbers are relatively prime

Quote of the Day: “My mother was the greatest person in my life. She had an eighth grade education and cleaned floors at the Chicago Athletic Club for a living. Some of the greatest people in the world clean floors for a living. When you value everyone and treat everyone with respect, you may just be amazed at how they can make you better. Too often people will miss out on meaningful relationships with amazing people because they quickly pass judgment based on what that person does for a living, the clothes they wear, what kind of car they drive.” - Coach K

Question from Yesterday (as always from a student): "Why do we pronounce a number like two to the third power as two cubed sometimes?"

Assessment: The Quiz, the weekly quiz, and the decimals pre-assessment. I spent a lot of time correcting because what do teachers enjoy more?

Agenda:

1. Collect WQ #3 
2. Write no homework in the agenda books
3. Take the quiz
4. Sieve of Erastothenes when the students are finished
5. Decimals pre-assessment
6. Place value song 
7. Destination Elimination (if time) 

Glass-Half Full: After the pre-test, I know there are a handful of students that have the 4 operations down and can be challenged as a result. This will help let me differentiate so that these students are being pushed, but mainly it helps other students not get as rushed because I already know going in that they are going to need more time to comprehend.

Regrets: I wish that leading up to today's assessment more time was put into showing students examples of numbers that were relatively prime. I think it could have even been introduced the first day of greatest common factor. There were not enough examples and application, and too much emphasis on the definition.

Day 23: Prime Factorization Study Guide

6th Grade Math Standards: 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). MA.4.a. Apply number theory concepts, including prime factorization and relatively prime numbers, to the solution of problems.

The Learning Objective: Define relatively prime; identify if two numbers are relatively prime or not; find the greatest common factor using prime factorization

Quote of the Day: "Don't tell me how tough the situation is. Show me how tough you are facing the situation." - Jay Bilas

Question from Yesterday (as always from a student): Why is it that if we're finding the greatest common factor of 16 and 28 that we circle 4 two's that the numbers have in common and we don't multiply 2 x 2 x 2 x 2?

Assessment: I had the students define relatively prime for me on the study guide and also had students answer if 11 and 44 were relatively prime. I used the student check method in which I checked off a couple students and used those students to then verify the answers of other students.

Agenda: Shortened day because we had the school fundraiser in which the students went around the schools on wheels called the Rollathon.

1. Visual Pattern #4
2. Reviewed the homework by having students do three problems on the board as the rest of the class worked on Visual Pattern #4
3. Pepper
4. Study guide 

Glass-Half Full: With the shortened day, the four activities were well-timed. I did not need extra time and still got an assessment and students to think in it. I also loved sharing what one of the students said yesterday about relatively prime numbers. "If you look at any two numbers that are next to each other on a number line, they are always going to be relatively prime."

Regrets: My first class of the day had difficulty getting started on the jumpstart. I think it was excitement with the Rollathon. I addressed it pretty quickly though and we were off and running from there.

Link of the Day: Foxy Fives from Fawn Nguyen. I'm literally creating an entire challenging math morning with materials I've learned about from Fawn.

Day 22: Prime Factorization to Find Greatest Common Factor

6th Grade Math Standards: 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).
MA.4.a. Apply number theory concepts, including prime factorization and relatively prime numbers, to the solution of problems.

The Learning Objective: Find the greatest common factor using prime factorization

Quote of the Day: “When the work is piled high on your desk, think about how thankful you are to even have a job while so many are unemployed. When work is driving you crazy, think about the fact that you are healthy enough to work. When you are sitting in traffic, be thankful you can drive a car while so many have to walk miles just to get clean water. When the restaurant messes up your meal, think about how many unfed mouths there are in the world. And as I told my father a number of years ago when he lost the love of his life - my mother ‘You had the kind of love for so many years that many people spend a lifetime searching for and never find. Let’s be thankful for that. Where there is a negative there is always a positive. Where there is a dark cloud, there is always a sun shining behind it.” - Jon Gordon

Question from yesterday: "If there is more weight added to an elevator will that slow the elevator down?" "If we use the birthday cake method for prime factorization, will there be a couple different ways to do it like there are with factor trees?"

Assessment: Exit ticket using the marker boards on the problem 36 and 54

Agenda:

1. Jumpstart & weekly quiz feedback
2. Review the homework
3. Pepper
4. Greatest common factor & prime factorization notes
5. Start the homework 

Glass-Half Full: The questions from yesterday brought forth great ideas and debate from across the room. It's been such a positive influence on my class to do this everyday so far this year.

Regrets: I never went over two of the problems from last night's homework. They were good problems, but we never covered them in class today. I thought the jumpstart was horrible. There were five problems and all but one I could have covered in Pepper. It was disengaging - a boring worksheet. I would rather ask only one question that made the students think or use something from Visual Patterns, 7 Puzzles, or Estimation 180.

Link of the Day: 3 Act math for circles.

Day 21: Prime Factorization

6th Grade Math Standards: 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). MA.4.a. Apply number theory concepts, including prime factorization and relatively prime numbers, to the solution of problems.

The Learning Objective: Factor a composite number to its prime factors

Quote of the Day: “Some think it’s their job to change people. Sorry. You can coach, counsel, teach, and guide, but no one changes another person. Change only comes from the inside as a result of decisions made by the individual.” - Jimmy Miller

Question from Yesterday (as always from a student): Are there any two numbers for which there is no least common multiple?

Assessment: Students posted their work on various composite numbers I gave them to try on a marker board.

The kids were in love with the concept after we looked back at the prime factors and got the product. I've never had a group be more enthusiastic about prime factorization, but to me it's art so I'm glad we saw eye to eye.

I also took literally two hours to review their weekly quizzes. There has to be an easier way, but I guess I can safely say it was assessed. Good to see students applying what we have learned from least common multiple. At the same time bad to see them not making charts or drawing pictures to organize their work.

Agenda:

1. Estimation 180 the Elevators (Day 75 through 77)
2. Review of the homework. Is 121 prime or composite? Apparently if the sixth grade were polled it would be an overwhelming victory for prime. Politics. 
3. Repeating the words prime factorization is the same as product of prime numbers ten times with people around us
4. Prime Factorization Notes
5. Prime factorization practice (see the assessment pictures)
6. Prime factorization homework practice started in class

Glass-Half Full: I loved the questions that I got and the thinking that was taking place during Estimation 180. Students asking if elevators go slower if there are more people on it, students sharing that it looks like three rows of people can fit on an elevator based on the picture, etc.

I also enjoyed the use of the marker boards again. I skipped having students do a bunch of problems in the notebook and had them just do the minimum in the notebook. The novelty helps me and the kids as I can see their answers from further distances and assess quicker.

Regrets: I wish I gave them a ticket to leave and simply asked them the question what concept did you learn today. Students refer to this as birthday cake and factor trees when the technical term is really prime factorization or writing the product of prime numbers. The sun will come out tomorrow. And so will this question.

Link of the Day: Prime Climb has gotten some good reviews on Twitter. It's tough to say I will pay the money though because one set will never be enough.

Day 20 Test Review, Open Responses & Prime vs. Composite

6th Grade Math Standards: 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). MA.4.a. Apply number theory concepts, including prime factorization and relatively prime numbers, to the solution of problems.

The Learning Objective: Find the least common multiple of two or more numbers in a real-world context

Quote of the Day: “As much as our culture talks about individual effort and self-improvement, deep down we revere the naturals. We like to think of our champions and idols as superheroes who were born different from us. We don’t like to think of them as relatively ordinary people who made themselves extraordinary. Why not? To me that is so much more amazing.” - Malcolm Gladwell

Assessment: The weekly quiz will be passed in tomorrow and checked by me. In circumventing the room today, students needed prompting in many cases.

Agenda:

1. Self-Assessment of the test
2. Journaling about the test
3. Review of the test 
4. Distribute the weekly quiz and review the directions
5. Let the students start the weekly quiz
6. Prime & Composite Number notes
7. Prime & Composite Number practice/homework
8. Continue working on the weekly quiz

Glass Half-Full Take: Repeating a factor always starts with one and itself several times. This is a concept that students will get at some point if they have not yet because of this slogan.

One Regret: It was hard to estimate the timing of everything today. I took almost a full block to review the quiz by the time I was through relating how the quote relates to the lesson, students had filled out their checklists, I showed pictures of common mistakes, etc. In the next block of class, I allotted 15 minutes to the weekly quiz being started. I could have allotted and 25 minutes and still had no issue working on prime and composite numbers.

I also had meant to reintroduce (or for many it would be a first introduction) the concept of numbers being relatively prime and forgot to do that today.

Homework: Prime & Composite number practice and the weekly quiz is due tomorrow as a rough draft since the students had time today in class.

Link of the Day: mmmm large cheeseburgers (courtesy of Robert Kaplinsky)

Day 19: Test on Factors and Multiples

6th Grade Math Standards: 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). MA.4.a. Apply number theory concepts, including prime factorization and relatively prime numbers, to the solution of problems.

The Learning Objective: Find the least common multiple of two or more numbers, identify all the factors in a number, find the greatest common factor of two numbers, memorize the divisibility rules

Quote of the Day: "A carpenter was fixing to retire and tells the company he works for he was through building houses. They ask him to build one more before he retires. He agrees and through the process of building this last house he begins to cut corners, uses cheaper materials, doesn’t use the same precision in which he used to and basically built an inferior home. When he’s done he gives the keys to his boss and says here you go I’m done. The boss returns the key and says thanks for all your hard work and dedication to this company. This home is yours and we appreciate you. Obviously, had the carpenter known he was building his own house he would not have cut corners. You are building your house every day and you don’t even know it.”

Question of the Day (as always from the mouth of a student): How do we tell on a word problem whether to find the least common multiple or greatest common factor?

Assessment: The test went well. I really do not care about the average. Thank god for sabremetrics in baseball. I think it can change the way that tests are analyzed. Without looking at the actual mistakes students make, it's really easy to make a quick judgment about the grade, but look at the mistake below. The first student has done several things well on problem number twelve below. First, they seem to know what a factor is. Second, they list every factor in pairs. Third, the student knows to only list 6 once. The red x can be distracting because it does indicates the concept has not been mastered, but this student is very close. My only job left is to make sure the student knows that and this problem isn't a reason to shout how math is hard (which I can live with) or even worse that "I'm not good at math."

The number got caught off, but it's 30. And again the student had to list the factors, and again the vase is three-quarters full. Just need a little flower.

This next mistake on problems 15 and 16 is another example of a student being very close to mastery. The red x has not appeared yet in the picture because I am going to show it to the students without the red x and see if they have a problem with it. That said, the student seems to understand what a multiple is. The math is a little murky as numbers get higher with 15 and the student also fails to list the first multiple for each number, but this is the best way for the student to learn that fact. 

Here, there is a clear indication that a student is well on her way to mastery of listing factors. Getting that 3 x 17 is 51 is especially significant since so many students mistaken that number to be a prime number. That said in the word problem below (which I should really push that we change because of the wording), the student fails to list the number six after listing 2 and 3 as factors. Oddly enough in the same test the student successfully stated that for a number to be divisible by 6, 2 and 3 also must go into that number. The student failed to apply that rule here, but this mistake is correctable. The 3 x 29 math is also an issue, but again if the student knows 2 x 39 is 78 we are on our way to determining what number times 3 will get 78. 

The mistake on number 21 happened a handful of times. Students mixed up greatest common factor and least common multiple. We need to keep drilling for them that factors start with 1 and itself. Factors start with 1 and itself. Factors start with 1 and itself. And then hopefully as the math is being done here and the student sees one is written, they recognize that they are listing factors instead of multiples.

Factors start with 1 and itself. Factors start with 1 and itself. Finkle and Einhorn. Einhorn and Finkle.

Agenda:

1. Agenda Books
2. Collect the Weekly Quizzes
3. Take the test
4. Do the wicked hard and totally unfair bonus
5. Super Bowl Commercials from Yummy Math

Glass Half-Full Take: Much of this test was about memorization. Divisibility rules and definitions of vocabulary specifically. I'm not a fan of teaching students (or myself for that matter) to memorize, but sometimes it is necessary. This overachieving answer to what a factor is brought a smile to my face.

If you can't see it, a student defined factor (which was what the question asked) and then proceeded to define multiple too. 

Another good part of today was the effort that students put forth with the bonus. So many students are starting by writing down what they know and not giving up without a fight. It's nice to see persistence and struggling even though not a single one left that problem thinking they had solved it. 

One Regret: The time after a test is such a land of opportunity for learning, but it's very difficult to manage since some students are done with the test and some are still working. Obviously the priority of the day was the test, but I chose the Super Bowl commercial activity based on where the students had struggled in the fifth grade (numbers and the base 10). I would have loved to be more active in explaining the worksheet, but did not want to take away from the students still working on the test. What I am going to do next year is explain it the sheet at the start of the second lesson.

Link of the Day: The problem that 1,000 math teachers can't solve from Dan Meyer.

Day 18 Study Guide for LCM, GCF, and Divisibility

6th Grade Math Standards: 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).
MA.4.a. Apply number theory concepts, including prime factorization and relatively prime numbers, to the solution of problems.

The Learning Objective: Apply the rules of divisibility for 2, 3, 4, 5, 6, 9 and 10; find the greatest common factor of two or more numbers; find the least common multiple of two or more numbers

Quote of the Day: “Care is so important to a team because if you want to change limits, there are going to be times when members of the team make mistakes. When you make a mistake, and you know it, you become very vulnerable. The immediate responses of those on your team, those you trust the most, will determine how you perceive your mistake. It can make you feel fearful of making that mistake again. Or you can feel that you put yourself on the line and even though you did not succeed, you know that your teammates care about you and you will not hesitate to step up again.” - Coach K

Assessment: At the end of class, we used the clickers and asked these five questions:

1. How many factors does 36 have?
2. What is the greatest common factor of 18 and 30
3. What number does not go into 240 out of 2, 3, 4, and 9
4. 20 Skittles. 12 Nerds. How many goodie packages?
5. 20 Skittles, 12 Nerds. 4 Goodie packages. How many Skittles? How many nerds?

Agenda:

1. Estimation 180 
2. Study guide 
3. Gizmos & Gadgets Problem (from imlem.org) At the Gadgets and Gizmos factory workers complete a gadget every 588 minutes and a gizmo every 882 minutes. The factory is open 24 hours a day, seven days a week, and the workers always complete the gadgets and gizmos on schedule. If it happens that a gadget and a gizmo are completed at the same time at 3:00 P.M. on Thursday November 29th, how many additional times will a gadget and a gizmo be completed at the same time before 3:00 PM on Thursday December 6th?  
4. Clickers (Exit) 
5. Work on Weekly Quiz #2

Glass Half-Full Take: I utilized the individual marker boards for the first time this year and in my career as part of the exit ticket. Historically I have had poor luck with getting students to show their work when we used clickers although they always have enjoyed using the clickers and the clickers have been a great way for me to get feedback. The marker boards improved the process immensely for me to assess student errors and also for the students to actually want to show the work. Yay marker boards.

One Regret: I really did not anticipate the study guide taking as long as it did. It could take up a whole block and then some, if I had wanted it to. In the future, I might eliminate Estimation 180 and just do the study guide right from the start of class. In the event that students finish early, the gizmos and gadget problem was not solved all day to my knowledge, so it's something for students to work on.

Homework: Study for the test and finish the weekly quiz.

Link of the Day: A lack of play is one reason to blame for a rise in ADHD.