Day 7: Repeating Decimals & Equations

Quote of the Day: “You have to think about how you come across to others. Being polite and having good manners are the most important keys. You would think that the first thing people notice about you is your appearance. But generally this is not the case. What others notice more than looks are the positive things that you possess - your smile, your self-confidence, your sincerity, and your attitude.” - Napoleon Hill

Question of the Day (as always from a student): What is the difference between finding the square root and a sign with a 2 in it, a 3 in it, etc.?

Math Objective: Discover patterns for fractions with denominators of 2, 3, 4, 5, 6, 7, 8, and 9.

Lesson Sequence:

1. Number Talk 71 - 29
2. QSSQ
3. Review HW/Pepper
4. Finish the fraction to decimal conversions using a calculator
5. Discuss the various patterns

Regrets: None of the classes have discovered why the denominators of 3, 6, 7, and 9 repeat when the other denominators do not. They have at least brought up the fact that several of these repeat though.

I also think that we need to get more voices to speak up with the number talks. Students are failing to connect ideas from the previous lessons (which also centered on subtraction problems) to this lesson. We are going to do a number talk reflection to being classes next week to see if students can set goals around building off the ideas of others and also getting their voice to be part of the conversation.

Glass Half-Full: It was powerful to see students verbalize their patterns and also stop doing the standard algorithm when they recognized certain patterns - the ninths in particular.

Honors Math Objective: Find a value for a variable in a two step equation

Lesson Sequence:

1. Number Talk 71 - 29
2. QSSQ
3. In each group of 4, students were in charge of one problem. The solution from that one problem was then applied to the second problem. All students worked this out on marker boards.

Regrets: I completely messed up the directions and did not realize it for about eight minutes. I thought that all four students could work on their problems simultaneously. As a result, students were stranded because they had two variables in their problems and had no idea how to find the value of both variables (it would have been impossible as a matter of fact). I finally clarified for them that the solution from Problem 1 must be applied to Problem 2, and that gets applied to problem 3. Indirectly, this was not that big of a deal since the students could in fact begin distributing and move numbers from one side of the equals sign to the other before they knew the values of either variable. If I had explained the directions the way I was supposed to, we would have never recognized this (myself included).

Glass Half-Full: After my blunder, the students had blunders of their own. They had trouble combining a "+30 and -1" to become a 29 (many students wrote 31). They also were not bringing the sign with them when distributing. At the end of class I asked students to raise a fist of five. Five being very frustrated and one being very calm. The entire class had 4s and 5s in the air. I then asked them how fast time went by. Five being fast and 1 being slow. Again there were mostly 4s and 5s in the air. My point: we embraced the struggle.

Day 6: Repeating Decimals (Again)

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 of the Day (As always from a student): Typically when the word combine is used, it means to add. Why is it that with 6b - b it is still considered "combining" like terms?

Regular Math Objective: Locate repeating decimals on a number line

Regular Math Sequence:

1. Number Talk 65 - 28. Again today students were using the standard algorithm, but there were other strategies that were shared that I found encouraging. Some students would add three to 65 and then take off three from their solution (65 + 3) - 28 = 40 - 3 = 37. Other students were taking away the tens digits (60 - 20) = 40 and then subtracting the ones digits (5 - 8) = -3 and then combining those together
2. QSSQ
3. Review the homework and pepper
4. Review the exit ticket
5. Students began to work on the fraction to decimal practice.
6. Exit ticket that mirrored the previous day's exit ticket. 
7. Pass out the homework

Regrets: Students had struggled the previous day, so I wanted to see what type of growth both the written and oral feedback had on students. There was some growth. Students that did not put the fraction into a decimal the day before were generally doing that in the exit ticket this time around. The problem was that these same students were now having difficulty locating the point on the number line because they had not even approached that the day before. For students that struggled with the number line, there was some improvement but still some flaws. 

All that said, this topic is mighty boring if you're not into the whole math thing, so I'm moving on. Students will get the exit tickets back with feedback in the next class. We will also try to fit this into a 2-4-2 homework in the future and of course include it on the study guide before the quiz. By then hopefully students have a grasp on what they need to do. 

Glass Half-Full: Pepper is helping students wrap their heads around what a rational number is versus what it is not. Students were thinking that repeating decimals are not rational, but as their exit tickets have been indicating all rational numbers can be placed on the number line. 

Honors Math Objective: Analyze the differences in how to find decimal equivalent for repeating decimals of one, two and three digits

Honors Math Sequence:

1. Number Talks
2. QSSQ
3. Homework Review
4. What happens to the decimal when we multiply by 100? By 10? This conversation led us to determining how to get rid of the repeating part of the decimal when the sequence is two digits long, three digits long, etc.
5. Students were given marker boards and I assessed them individually to see if they could find fraction equivalents to a few different decimals
6. Homework was a pre-cursor to square roots. 

Glass Half-Full: I was half-expecting that I would have to stand and deliver the lesson for students. By giving the students wait time and getting them to collaborate with their groups as I asked the question in step four I was encouraged by the diversity of answers that I heard. Some students said to multiply the decimal by 10,000 and then subtract by multiplying it by 100. Others said to multiply by 100 and take away 1. I even heard 1000x minus 10x. All of the answers were correct and it led to a much better conversation around the day's objective.

Day 5: Repeating Decimals

Quote of the Day: “Do rewards motivate people? Absolutely. They motivate people to get rewards.” - Daniel Pink

Question of the Day: Why is 9/9 equal to 1 instead of .9 repeating?

Regular Math Objective: Place a fraction that is converted as a repeated decimal on a number line. 8.NS.1

Lesson Sequence:

1. Number talk 43 - 8. It was astonishing how many students used the standard algorithm in their heads instead of other methods. I cannot picture borrowing in my own head. Myself and the other teachers in the room contributed our own strategies after the students had shared theirs to let them know that even the adults think about the problem differently. 
2. Quote, Star Students, Question of the Day
3. Reviewed the homework and did pepper for the vocabulary terms from yesterday
4. Had students locate points on a number line. I kept telling the students to do the problems on the right of the page first since they were all terminating decimals and were much easier to locate on the number line. The students would then stop instinctively before doing the left side of the sheet. All that being said, I need to edit the sheet so that students can put these points on the number line in the order of terminating decimals first and then repeating decimals. Students struggled to find the right place to put the repeating decimals on the number line. The exit ticket also confirmed this. 
5. Exit Ticket. Many students did not finish this since I gave it out with about three or four minutes left in class. 
6. 2-4-2 Homework

Regrets: The exit ticket was revealing. Many students had issues converting fractions into decimals. For the fraction 4/15 students were putting 3.75 meaning that they were dividing 15 by 4. That being said, by the time they looked at number two on the exit ticket and saw the number line they should have recognized their mistake. I think as a result of the lack of time they had to complete the exit ticket they were not able to process all of this.

Glass Half-Full: I spent more time than I usually would giving feedback directly on the exit ticket for students to see in class the next day. My goal in doing this is to let students know that I value their effort on exit tickets and also that they start to find their mistakes on these exit tickets.

Honors Math Objective: Convert repeating decimals to fractions

Lesson Sequence:

1. Number Talk 43 - 8. 
2. Quote, Star Student, Question of the Day
3. Reviewed what the numbers inside a square root symbol mean from the previous night's homework 
4. Repeated decimals worksheet from the Mathematics Assessment Project. 

Glass Half-Full: The lesson really took off when students had to convert .54 repeating to a decimal. I made the mistake initially of helping a student through that problem, but went around to other groups shortly thereafter who were struggling and did not give it away. I focused instead on what was happening when students tried to subtract .45 repeating from .54 repeating if they were trying to do 100x - 10x. We ran out of time just as students were starting to get some clarity. 

Regrets: As a result of running out of time on that one issue, I could not give an exit ticket. The students were working collaboratively and productively in groups, but I would have liked to see what they retained individually at the end of the class. 

Day 4: Rational Numbers

Quote of the Day: “Katie Ledeckey’s coach, Bruce Gemmell, says she’s always relished a challenge. ‘There’s a little video clip that Katie’s parents have of one of her first swim meets,’ Bruce told me. ‘It’s just one lap. She’s six years old. She swims a few strokes and then grabs on to the lane line. She swims a few more strokes and grabs on to the lane line again. Finally, she gets to the end of the pool and gets out of the water. Dad’s filming it, and he asks, ‘Tell me about your first race. How was it?’ She goes, ‘Great!’ A few seconds later, she adds, ‘That was hard!’ And she’s beaming - a smile from ear to ear. That says it all right there. She has that attitude with everything we do.’” -Angela Duckworth

Question of the Day: How do we know if a number is an outlier?

Lesson Sequence in Regular Math:

Objective: Convert fractions to decimals and decimals to fractions

1. Number Talk using this dot configuration. In case the link does not work, it is a 3x3 configuration with the dot in the middle missing. All of the students in my classes were getting 8 dots except for one. That class took the way I asked the question a little too literal because they said the color of the dots was different than how I asked in the question. Despite the lack of variety in student answers, it was interesting for the first time to conduct a number talk as students went through the various ways that they chose to count. For more on number talks just Google Jo Boaler Stanford.
2. Quote of the Day, Star Student, Question of the Day
3. My Favorite No: What is bigger 2.375 or the fraction 2 and 3/8? All students were given an index card to show the math out and then we went over it on the board. Typically, I collect the index cards, but for some reason today I did not do it. The picture above was an interesting way of doing this task that I think demonstrates stronger mastery of the standard than someone doing the standard division algorithm.
4. Vocabulary. I am teaching four classes of regular math this year in eighth grade which is a relatively big change from what I have been doing the past two years in sixth grade. I only have the eighth graders for 50 minutes where as in sixth grade the students had 100 minutes of math per day. In my first class, I was having the students write the definitions and the examples as I essentially stood and delivered. Luckily my colleague and I had a break after this class and decided to ditch this. It was boring and put little emphasis on student learning. Instead we just passed out sheets with the definitions and examples already done out. Then we gave students five minutes to practice the words with a partner. I went around the room and saw that the students were not using this time to fool around which made me think this is what we will do moving forward with new vocabulary.
5. Assigned that night's homework.

Lesson Sequence in Honors:

Objective: Analyze patterns between repeating decimals and terminating decimals

1. I did the same number talk to ensure that students got this routine. This particular class took about fifteen minutes to complete the many ways they came up with for counting the 8 dots.
2. Quote of the Day, Star Student, Question of the Day
3. Converting the fractions with denominators of 2, 3, 4, 5, 6, 7, 8, and 9 to Decimals. This worksheet was taken from a longer assignment from the Mathematics Assessment Project teaching the 8.NS.1 standard of getting students to take repeating decimals and turn them into fractions. We will get to the rest of that part of that resource this week. 
4. Discussing the patterns in groups and then as a whole class to what the fractions were. It was really interesting to see the pattern with the 7ths myself. I had never known that all of the digits were the same for 1/7, 2/7, etc. Some students recognized this on their own, but the one thing I did stop and show students was the hexagon that was formed between the numbers 142857 and how the numbers across from each other had a sum of nine. 
5. Assigned that night's homework. 

Regrets: I had way more things written on the agenda than I taught today. I am still adjusting to only having half the time that I'm used to having with the students. 

Glass Half-Full: In all classes I thought the Number Talk routine was kicked off to a successful start. Students were respectful of the time that I was giving them to wait and determine different ways to arrive at the same solution. 

In the honors class, I was really pleased that the students were left with a cliff hanger. They noticed the pattern for ninths was that the numerator was repeated, but they also questioned that 9/9 should be .9 repeated instead of 1. 

Day 3: What Would School Be Like If There Were No Grades?

Quote of the Day: “Benjamin Bloom, in his study of top-ranked young performers in several fields, found this motivation in some of them from their early years: ‘For most of the mathematicians, the joy of discovering a new way of solving a problem was more important than a high test score, receiving a good grade, or getting the teacher’s approval for their work.” - Geoff Colvin

Question of the Day: From yesterday: What is STEM?
From Today: What would school be like if there were no grades? Why grade homework?

Lesson Sequence:

1. Which One Doesn't Belong? As part of proving what one didn't belong among 9, 16, 25, and 43 we discussed what a prime number is and what the square root of each number was. This proved pivotal when we moved to the discussion about grading and incentives. 
2. Quote of the day, question of the day, star students
3. Sticky Talk. Technically the term for this protocol is Chalk Talk, but because we were using sticky notes I changed the name. This was where the lesson got interesting, so more on this below.
4. Passed out the syllabus. Depending on how much time remained after the Chalk Talk, I gave all the students a highlighter and told them to read it as a group, but only to highlight three things on the syllabus (individuals within a group could highlight different things). I have always found that there is too much information in my syllabus, but I also have a hard time cutting out information. By forcing the students to choose three things I am letting them pick out what is most important. 
5. If time, discussed the group work from Day 1. 

The first thing I had the students do with the sticky talk was divide the class in half to answer one of two questions silently at their desks on a sticky note:

1. What would school be like if there were no grades? Or...
2. Why is homework graded?

Next, students got up and put their sticky notes on the board or a piece of large chart paper which were at opposite ends of the classroom. For five minutes they read through the ideas of others. Again it is important to note that this was done silently. Students struggled with this because they are hormonal thirteen year olds and because they wanted to talk about the topic. I also chose to participate by asking questions and probing students thinking (although none of my questions or thoughts are in the responses below). Students had the opportunity to write their own questions and connect ideas to one another. After five minutes, students switched sides of the room and looked at the answers and questions their classmates had of the other question. Again, they had the opportunity to ask questions and I left extra sticky notes in case a new thought popped into their heads. 

Here was some of the dialogue about what school would be like if there were no grades. These first two pictures was the most common theme I saw.

     

I like how a student wrote the comment in the marker instead of me. Here were some of the other ideas that cropped up about having no grades in school:

 

Here were some responses to the question about why homework is graded starting with my favorite. 

Yes. I want you working 100% of the time. No sleeping. No texting. No dating. No snacks. No collecting $200. Just homework baby. Here were some more thoughts:

Regrets: The students did not have much room to read responses. By the fifth time of the day that I was doing this, I called students up by their groups of four instead of having twelve at a time. I think I could either have the question in two different parts of the room or just simply make the viewing space wider. That would curtail some of the mischief and foolery that was taking place when we were supposed to be silent.

In talking with my eighth grade math colleague, we both were finding ourselves asking the students what is the purpose of school in follow up discussions or even within the protocol. Students responded in huge numbers that there would be no purpose to school without grades and no sense in doing homework if it was not graded. Yet when we asked what the purpose of school was, students responded by saying it was to learn. Learning is what grades are ultimately supposed to measure. If I were to really zoom in on why the students would be in such an uproar in a school with no grades, what it really comes down to is their desire for feedback. And on that point I can't really argue - that is at the core of what grades are and why we have schools. How our brains are trained to interpret feedback versus grades - especially at the student level is where the traditional education system has lost its meaning.

Glass Half-Full: The rationale behind this entire lesson was that my colleague and I are going away from counting homework as part of the grading process. Originally we thought students would sell themselves on such an idea. And then as we got closer to this day we saw all of the potential resistance that this decision could bring as well.

The research is out there. And there's a lot of it with many contradictions. And there are many opinions too. See the sticky notes above. Not pictured are the parents and caretakers of the people who wrote on the sticky note. Ultimately for us as the teachers it was a matter of weighing grading homework versus not grading homework. The cons of grading homework were too great. It puts stress on families. It strips kids of play and opportunities to do extracurricular activities. It puts students from disadvantaged backgrounds at an even further disadvantage. As the educational consultant Rick Wormeli would have you know, it makes grades less informative since homework is typically graded on subjective measures like effort and completeness and not the standards. It encourages students (even against all warnings) to copy each other, which when founded leads to negative emotions for all stakeholders and when unfounded leads to inaccurate formative assessments. Homework gives kids more of a negative outlook on school. And for the students that would not do homework if it was not graded, they probably were not going to put a whole lot of sweat into it anyway since they were not intrinsically motivated. Not grading homework will save just a little time for the teachers in terms of putting it in the grade book - time that can be spent in the trenches on helping students get better at math. That is the job of a math teacher after all.

Inevitably we are going to run into problems with this system, but after doing this protocal I am more convinced we are taking the correct path. We are still going to give homework in a 2-4-2 format (see Steve Leinwand: Thoughts on Revising How We Assign and Review Homework), so we can still benefit from the positive outcomes that the students listed above. Students that do homework with good intentions will know what to work on, enhance their short term memory, and be more prepared for the test. The adults will become more concerned with the standard than the grade, will have a more accurate formative assessment, and maybe will lose that reputation for wanting to possess the soul of the student 100% of the time.

I wrote way more than I ever intended. Time for a weekend.

Day 2: Baseline Exam

Quote of the Day:

Question of the Day:

Lesson Sequence:

1. I had the students find their seats which were in rows (yesterday they were in groups)
2. Agenda books to write down what supplies we need for class
3. Quote, Star Student, Question of the Day
4. Pass out the materials for the Benchmark
5. Illustrative Mathematics problems about what fractions will be repeating and what will be terminating decimals 

Regrets: The baseline was pulled from all eighth grade standards using released items from the past couple of MCAS tests. As a result the students performance was awful across the board. One of the primary goals of this test though is to show growth and to verify that we will be teaching concepts that the students have not yet learned. Clearly that is the case in this instance.

Glass Half-Full: I had one conversation with a student who was concerned that he did not know virtually any question. In his words it was the worst he had felt about anything he was tested on in the last two years. I asked him why he came to school and he quickly replied to learn. I then said, good this test is the best thing that has happened to you in the last two years because you will be learning this information.

Link of the Day: This Ontario educator has some realistic viewpoints that seem like common sense, but in reality I've heard arguments like this my whole career. Problems such as finding the volume of a triangular prism are not things that 99% of us encounter everyday. Nor is it fair to expect every student to know how many players are in a starting lineup in basketball. I'm glad that America is not alone in the quest toward common sense in education.

Day 1: What Does Good Group Work Look Like?

Quote of the Day: “I’ve noticed an interesting thing. When some star players are interviewed after a game, they say we. They are part of the team and think of themselves that way. When others are interviewed, they say say I and they refer to their teammates as something apart from themselves - as people who are privileged to participate in their greatness.” - Carol Dweck

Question of the Day: What does good group work look and sound like?

Lesson Sequence: Full credit to this lesson really goes to Sara VanDerWerf (definitely worth a follow on Twitter). Here was Sara's original post about what good group work looks and sounds like, which has had some great tweaks put in by other teachers. Here's what I did:

1. I introduced myself. It was the first day of school after all.
2. Discussed the importance of me saying less in order for us to learn more. 
3. Read the quote of the day.
4. Read the question of the day. 
5. Told students to all take out a pencil.
6. Told students to number themselves from 1 to 4. For groups that had three people, I was kind of disappointed because the pattern was harder to find. If possible keep groups in only 2 or 4. That could mean the teacher participates in odd number classes. 
7. I passed out the sheets that are numbered from 1 to 100 (see the linked post above) face down. 
8. I explained the rules for the 1 to 100 sheets. I always stopped in my directions and asked so when we get to number five who will circle that? 
9. The students had three minutes to circle numbers. The engagement could be summed up by many photos that I took. 

1. They discussed their strategy for a couple minutes and I collected and threw out their sheets from round 1. As I took their sheets I made sure they knew what number they had to beat in Round 2. I then passed out the sheets for Round 2 as well as a set of crayons.
2. I instructed students that Round 2 would be the same as Round 1 except now they would be using colored pencils and got each group focused on beating their record from the previous round. 
3. After Round 2, students again discussed strategies and I asked them to look at their sheets as part of this strategy. If groups correctly followed the directions, the pattern could not be missed. Although they did not use the word quadrant so I had to make them swallow that. 
4. We did one more round and students set their records across all classes. 
5. Students wrote an exit ticket that answered our question of the day.  

As the students were circling numbers, I took pictures the first round and then wrote down what I heard in the second round verbatim. The students really enjoyed hearing their exact phrases after the round. 

Regrets: Initially I had students write one thing that demonstrated what good group work looked or sounded like. That led to one word answers. My last two classes I asked for three after I consulted my colleauge about this during lunch. While I still only ended up with three words in some cases, I could also call it a 200% increase. 

Glass Half-Full: This is the first time I have not gone over the procedures and rules in my room on the first day of school. It's also my first day ever teaching eighth grade. I felt as though I needed to change up from the norm because the kids would be bored out of their minds from procedures and protocols all day. In sixth grade they need that sort of thing because the building is brand new, but in eighth grade I thought it was a little harder to do that. 

Despite my "dangerous plunge" we still invested in the future of the classroom and kept the students engaged. We have established norms for group work. The words I kept reading again and again on exit tickets were everyone, help, and teamwork. I also got deeper thoughts such as "Nobody is left out." "No one saying negative things to each other." "Using new ideas learned from our past mistakes."