6th Grade Math Standards: 6.NS.1 Interpret and compute quotients of fractions, and solve word
problems involving division of fractions by fractions, e.g., by using
visual fraction models and equations to represent the problem. For
example, create a story context for (2/3) ÷ (3/4) and use a visual
fraction model to show the quotient; use the relationship between
multiplication and division to explain that (2/3) ÷ (3/4) = 8/9 because
3/4 of 8/9 is 2/3. (In general, (a/b) ÷ (c/d) = ad/bc .) How much
chocolate will each person get if 3 people share 1/2 lb of chocolate
equally? How many 3/4-cup servings are in 2/3 of a cup of yogurt? How
wide is a rectangular strip of land with length 3/4 mi and area 1/2
square mi?
6.G.2 Find the volume of a right rectangular prism with fractional edge
lengths by packing it with unit cubes of the appropriate unit fraction
edge lengths, and show that the volume is the same as
would be found by multiplying the edge lengths of the prism. Apply the formulas V = lwh and
V = bh to find volumes of right rectangular prisms with fractional edge lengths in the context of
solving real-world and mathematical problems.
The Learning Objective: Apply multiplication of fractions in order to find a limit.
Quote of the Day:“Spend time with people who constantly drain you, pull you in the wrong direction, or try to knock you down, and it will be almost impossible for your talent to take flight.” - John Maxwell
Agenda:
Jumpstart reviewing like denominators, unlike denominators, and multiplying fractions.
Review the homework. The most interesting problems asked students to find 1/4 of 154 and 1/3 of 120 in a word problem and then compare the amounts.
Physical example of a limit using our classroom.
Limits using a 64-square grid and continuously coloring half of the remaining unshaded grids and the questions which corresponded to this.
Starting the homework in class.
The Assessment: I was consistently circumventing the room during number four in the agenda and also checked homework during step one in the agenda. Overall I was encouraged by the homework results.
Homework: Tonight's homework was page 284 #4-11 and page 285 #14-16
My Glass Half-Full Take: I yelled onions twice today. The first time I got excited was when a student got a common denominator for 4, 10, and 7 compared the fractions and determined which fraction was greatest. Everyone else in the room that could solve it (including myself) multiplied one-fourth, three- tenths and, two-sevenths by 152, 160, and 147 respectively to see how many people were in each group. Then we determined that three-tenths of 160 was the biggest group. By finding a common denominator and changing the numerator, we'd get the same answer. It's always nice to have students sharing alternative ways for solving the problem - especially ways I never would have thought.
Another student eventually recognized that if a person continually walks half-way across a room they will mathematically never reach the wall.
One Thing to Do Differently: The activity we did today was done in partners. One of my colleagues did it in groups of six. I believe that would have worked much better. It would have been easier to facilitate giving colored pencils, and it also would have put the learning on the students instead of on me. I ended up being at the board for most of the block as it were and I believe students did not think as much as they should have. I also believe that perhaps ditching time to start on homework is better because the activity is deeper on Bloom's Taxonomy than our homework. Perhaps a simple assessment of three problems for homework will be a more appropriate way to give out homework in the future on this lesson.
Link of the Day: If you're looking for a hot button issue to talk about with Halloween in educational way, perhaps discussing the appropriateness of costumes would suit you.
6th Grade Math Standards: 6.NS.1 Interpret and compute quotients of fractions, and solve word problems involving division of fractions by fractions, e.g., by using visual fraction models and equations to represent the problem. For example, create a story context for (2/3) ÷ (3/4) and use a visual fraction model to show the quotient; use the relationship between multiplication and division to explain that (2/3) ÷ (3/4) = 8/9 because 3/4 of 8/9 is 2/3. (In general, (a/b) ÷ (c/d) = ad/bc .) How much chocolate will each person get if 3 people share 1/2 lb of chocolate equally? How many 3/4-cup servings are in 2/3 of a cup of yogurt? How wide is a rectangular strip of land with length 3/4 mi and area 1/2 square mi?
6.G.2 Find the volume of a right rectangular prism with fractional edge lengths by packing it with unit cubes of the appropriate unit fraction edge lengths, and show that the volume is the same as
would be found by multiplying the edge lengths of the prism. Apply the formulas V = lwh and
V = bh to find volumes of right rectangular prisms with fractional edge lengths in the context of
solving real-world and mathematical problems.
The Learning Objective: Multiply fractions to find a product.
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
Agenda:
Jumpstart on converting mixed numbers to improper fractions
Review the homework on unlike fractions by having students do the problems on the board (4 at a time)
The Assessment: My favorite no and the ticket to leave. Keep in mind the problem was 1/2 times 3/5. While the most popular was 3/10. The three pictures listed below appeared more than once today during the pre-assessment. The post-assessment in the last picture appears great in that picture, but it too was far from perfect - although students demonstrated gains in fractions times fraction (mixed numbers was harder).
Homework: Page 269-270 all problems.
My Glass Half-Full Take: I really enjoyed having students come to the board and do the problems from the homework. This sounds like it's teaching 101, but I almost never have students go to the board. It was great because I did it essentially by cold calling and by picking students that I knew had the wrong answer or no clue how to do a problem. The answers we got were very telling. It provoked questions like what difference does it make when the denominators are 6 and 9, and the common denominator is 36 instead of 18? Things I would have never gotten to thinking on my own.
One Thing to Do Differently: I would change the my favorite no to include mixed numbers instead of regular fractions. The majority of students could do the my favorite no, so in a sense it led to over-confidence during the notes. When I did the ticket to leave most students couldn't actually do a mixed number times a mixed number, and that's after I did a problem with a mixed number in the notes.
I would also change the notes so that the third problem was a mixed number times a mixed number instead of a mixed number times a fraction. As part of this, it's important to let students try to solve it on their own first so they have more interest and are invested when I review it at the board.
I also wish that I had done a jumpstart on something as simple as what two whole numbers is 3 and 2/5 between. In doing an estimate of these problems it's amazing how many students have no idea what to say when I ask them what's a whole number that's too low and too high for 3 and 2/5. They all want to go way too low or way too high. Some say numbers that are both too high.
Link of the Day: Inspiration for working hard and seeing things through courtesy of one of my colleagues.
6th Grade Math Standards: 4.NF.3d. Solve word problems involving addition and subtraction of fractions referring to the same whole and having like denominators, e.g., by using visual fraction models and equations to represent the problem.
5.NF.1. Add and subtract fractions with unlike denominators (including mixed numbers) by replacing given fractions with equivalent fractions in such a way as to produce an equivalent sum or
difference of fractions with like denominators. For example, 2/3 + 5/4 = 8/12 + 15/12 = 23/12 .
(In general, a/b + c/d = (ad + bc)
5.NF.2. Solve word problems involving addition and subtraction of fractions referring to the same whole, including cases of unlike denominators, e.g., by using visual fraction models or equations to
represent the problem. Use benchmark fractions and number sense of fractions to estimate
mentally and assess the reasonableness of answers. For example, recognize an incorrect result
2/5 + 1/2 = 3/7, by observing that 3/7 < 1/2 .
The Learning Objective: Add and subtract fractions.
Quote of the Day: “It starts with control of your emotions, but it also extends to having the resolve to resist the easy choice, the expedient solution, and, at times, temptation in its various and alluring forms...Self-Control in little things leads to control of bigger things. For example, the reason I prohibited profanity - a small issue - during practices was because it was usually caused by frustration or anger. A player that can’t control her language when she got upset during a drill or scrimmage would be more likely to lose control in more damaging ways during the heart of a competition - fouling, fighting or making other poor decisions that would almost certainly hurt the team.” - John Wooden
Agenda:
Jumpstart: Stop & Shop sells Coca Cola in packs of 9 cans for $7 and Richdale sells Coca Cola in packs of 12 cans for $10. Which store offers the better deal? Show your work at least two ways.
Weekly Quiz passed back to all students while they worked on the jumpstart
We reviewed the jumpstart
Like denominators notes
Like denominator practice and tickets to leave
Unlike denominators notes
Unlike denominators practice
The Assessment: I collected the students work on like denominators at the end of the first class and had time to grade and return to one of my classes a few hours later. I did not have the opportunity to return it to another class as I ran out of time, but will give it back tomorrow. The main concern I had for students in terms of the feedback I gave them had nothing to do with the fractions. It was their lack of circling the key parts in a word problem. In my six years of teaching, this as much as any calculation issue has proven problematic again and again. The student below did not execute in the second biggest mistake I saw. She tried to mentally subtract 3 and 2/4 from 8. She got 5 and 1/2. Even for students that wrote 8 minus 3 and 2/4 (or 3 and 1/2), they still struggled to arrive at the difference. I think part of the reason for the struggle is what I would call "problem fatigue." In other words, they had to add (do one step) before getting to the next step.
Another assessment I did was check off students as they worked on the homework. In one class I had much more time to do this than another (more on that in things I'd do differently).
A third assessment was reviewing again the rates issue that we are having as a class. I am confident that we are now turning the corner on this type of problem, but the students will see a similar problem in the near future. The students that are successful seem to be gravitating more and more toward making a chart and finding the least common multiple.
Homework: Students did eight problems finding the sum or difference for of unlike fractions and also two to three word problems. They were given time to start this in class.
My Glass Half-Full Take: One thing that I don't plan for (although in a perfect world I really should) is having students get out of their seats. Today as we were doing the notes, I was constantly asking students to get out of their seats and explain to someone in the class how to do the work or simply to check the problem. It's amazing how beneficial this is for everyone in the class. It shrinks the ratio of student to teacher for me and also gives the students who "get it" a more developed idea of how to solve the problems as they view their classmates who might solve it correctly in a different way or are solving it incorrectly and they try to search for the mistake.
One Thing to Do Differently: Everyday there are many more than one. It's just a matter of whether I share it or not. Here's what comes to mind:
The reason my notes aren't linked here are because I ditched them after the first class. I instead handed out a worksheet and just picked out problems I sensed we could struggle with. It worked much better. I then used the word problems from that sheet as the ticket to leave.
These are fourth grade standards and I'm a sixth grade teacher. Just five years ago these exact standards were sixth grade standards, so guess if the students struggled? "The idea that 4 is 32/8 is crazy. It doesn't make any sense." - Student. I wish I knew that before teaching this lesson. I also noticed that more students got 8 minus (1 and 3/8 + 1 and 1/8) wrong than those that did not get it correct. Fractions are tough. It doesn't matter that "they were already taught this." I probably assumed too much going in and could have relied on a pre-assessment better to guide the lesson.
There is a small pack of students that are struggling converting a mixed number to an improper fraction. Since we are moving toward the sixth grade standards in the coming days (multiplying and dividing fractions), I think it's going to be essential to continue to hammer this skill home perhaps in a formative nature.
Notes practice, notes practice. I tried my best with my energy to liven things up, but I feel like there's something that can be done differently here. I am doing a kinesthetic/visual/notes/partner activity in two days so that will be a nice change for everyone.
I gave out two sheets that are essentially the same sheet, but one says unlike fractions and one says like fractions. The one that says unlike fractions is homework. What are the odds that everyone does that assignment tonight and nobody does the other one?
Link of the Day: What works better? Drilling a concept repeatedly in a day or trying to several different concepts in the same day - for several days? According to research, spreading out the concepts works better. The technical term is interleaving. Only the youngest of children benefit more by drilling initially.
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
4.NF.3c Add and subtract fractions mixed numbers with like denominators, e.g. by replacing each mixed number with an equivalent fraction and/or by using properties of operations and the relationship between addition and subtraction.
The Learning Objective:Differentiate between two rates. Turn a mixed fraction into an improper fraction. Turn an improper fraction into a mixed number.
Quote of the Day:“Out of all the applicants from all over the world, my department at Columbia admitted six new graduate students a year. They all had amazing test scores, nearly perfect grades, and rave recommendations from eminent scholars. Moreover, they’d been courted by the top grad schools. It took one day for some of they to feel like compelte imposters. Yesterday they were hotshots; today they’re failures. Here’s what happens. They look at the faculty with our long list of publications. ‘Oh my God, I can’t do that.’ They look at the advanced students who are submitting articles for publication and writing grant proposals. ‘Oh my God, I can’t do that.’ They know how to take tests and get A’s but they don’t how to do this - yet. They forget the yet.” - Carol Dweck
Agenda:
Passed back the ratios quiz and weekly quiz
Students wrote in their journals, graphed their results, and wrote in their journals
Since the topic should be reviewed based on 4th and 5th grade standards, I had students start the homework in their spiral notebooks without me doing any examples. As students had issues, I had student helpers assist these students in teaching the process.
The Assessment: Circumventing the room as students began homework. Student journals were also used.
Homework: Like denominators practice. Four problems with proper fractions, four problems with whole numbers and a fraction, four problems with mixed numbers, and four word problems.
My Glass Half-Full Take: It took a full block to go over the quiz. Not because I was going over every problem, but because the students were asking me to go over different problems. It's nice that they were not insecure about worrying if their classmates cared that they did not know how to do something. It's also nice that they were willing to learn even though the grade had already been recorded. I think this is one of the advantages to giving retakes for everything that we do in class. Students never want to stop learning.
One Thing to Do Differently: I did better with the Hershey Bar activity without giving the students a worksheet. The activity was truly meant to be an experience with manipulatives and I think the students relished the opportunity to not be writing something for a change in math class. I would probably keep the worksheet for myself as a place to ask students questions, but the students could do without it.
Link of the Day: With fractions the topic of discussion, I thought this was a clever way to introduce fraction division.
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 The Learning Objective: Write a ratio three ways. Calculate the unit rate. Find equivalent rates. Complete a ratio table. Find the equivalent rate. Determine the best rate between two options. Create a ratio table and list the coordinate pairs. Graph ratios on a coordinate plane. List ratios in a double number line.
Quote of the Day: “Students with the growth mindset continued to show the same high level of interest even when they found the work very challenging. ‘It’s a lot more difficult for me than I thought it would be, but it’s what I want to do, so that only makes me more determined. When they tell me I can’t, it really gets me going. Children with the growth mindset can’t tear themselves away from the hard problems.” - Carol Dweck, Mindset
The Assessment: The ratios quiz and weekly quizzes were collected, graded, and will be returned to the students.
Homework: Students are going to finish Weekly Quiz #6 for Monday. When they do, I will return it to them the following day and let them know what needs to be fixed.
My Glass Half-Full Take: The quiz covered many of the ratio standards and was fairly rigorous. Only two problems were not connected to word problems. I had built in enough time for students to do the quiz and work on their weekly quiz, but could not teach another lesson after the quiz, which was the intention a week ago. My colleagues and I had an inkling after this quiz was put together that this was a possibility, so it wasn't a big deal. In correcting the quizzes, the most encouraging thing for me was seeing that two of my students chose to solve number nine by finding the least common multiple.
One Thing to Do Differently: Many of the quizzes had the same mistakes. The first problem of the quiz students incorrectly ordered the units. Instead of 3:5 they said 5:3. I wish that I had forced students to circle these terms from the beginning of the unit. It's really something that I push all of the time, but I don't think I push it hard enough.
Another mistake was the rate problem of Stop & Shop versus Costco. We had this exact problem (technically I changed two numbers) in class just days before, but it was still the most popular wrong answer on the quiz. Interestingly the preceding question about recipes which measured students ability to tell if rates were equivalent went very well. This question on Costco and Stop & Shot used harder numbers though, and more specifically tougher units. Perhaps I could have devoted a full class to understanding the insight of this idea in lieu of just half of a class, but then I would have compromised actually teaching the lesson on unit rates (notes included). Students actually did quite well on question eight which tested the objective of seeing whether or not two rates were equivalent. I always believe that students who put the units of what they are dividing, multiplying, etc. have a better chance of getting a deeper understanding. In this particular problem, that could not have been more apparent. I need to do a better job of calling students out on not writing the units down. Perhaps some annoying buzz sound whenever they don't put units will serve as a reminder.
Link of the Day: Some worksheets from Mathfunbook on each and every standard with a little differentiation worked in as well. That said, I don't think it's the most engaging thing a student has seen. Still I see this as a cheap alternative to a textbook for any district out there that's looking to transition away from texts.
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
The Learning Objective: Graph ratio tables, compare rates to see which is the better buy, and several more. This day was really a summation of the last two weeks of a lessons.
Quote of the Day: “One day you go to a class that is really important to you and that you like a lot. The professor returns the midterm papers to the class. You get a C+. You’re very disappointed. That evening on the way back to your home, you find that you’ve gotten a parking ticket. Being really frustrated, you call your best friend to share your experience but are sort of brushed off. What would you think? What would you feel? The fixed mindset says I’d feel like a reject. I’m a total failure. I’m an idiot. My life is pitiful. Nobody loves me, everybody hates me. Excuse me, was there death and destruction or just a grade, a ticket, and a bad phone call? The growth mindset says I need to try harder in class, be more careful parking the car, and wonder if my friend too had a bad day.” - Carol Dweck, Mindset
Agenda:
Jumpstart with a couple unit rate problems and a couple ratio tables
Review homework
Pass out the study guide and do the study guide in terms of page 1, page 2, page 3. This took about 70 minutes to complete.
The Assessment: Circumventing the room during the study guide and letting students do their work in partners.
Homework: Study for the quiz and finish WQ #5.
My Glass Half-Full Take: Somehow I managed to go over all 17 of the problems we had today. Originally in the agenda I had the sugar packets video from Dan Meyer as part of the lesson, but we did not have time given the depth of some of the questions that were on this study guide. I left the day feeling as though the students were at the very least challenged by what we did.
One Thing to Do Differently: I feel as though my time would have been better spent if I could consistently say to students check the answer page. I did not have an answer page readily available or multiple answer pages readily available. With that at my disposal though students would not be wondering how to do certain problems. Of course the answer page also reenforces a negative of giving students an easy out in their thinking. I just wished I could have been checking in on more students and critiquing more of their papers. I went over four problems every four to seven minutes (I used a timer and put it on the board all class), but in going over the problems I never could capture the attention of the class and my words were going in one ear and out the other in many situations. I found myself constantly cold calling students to verify understanding. Perhaps creating an answer page with one to three wrong answers and telling the students this was the case would be an interesting compromise.
Link of the Day: Speaking of answer pages, there is now an App that will solve any textbook problem you wish to take a picture of. And not all mathematicians hate it. I too don't hate it. My take has someone who does give textbook homework occasionally (lately I've given a lot from the text but this will slow down), is that at least the students are going to put the effort forth to take a picture and communicate with someone or something else about the response. The app also comes with its flaws and this creates for interesting side discussions.
6th Grade Math Standards: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 The Learning Objective: Determine if two ratios measuring the same units are equivalent
Quote of the Day: “It is a new school year and things seem to be going pretty well. Suddenly some popular kids start teasing you and calling you names. At first you brush it off - these things happen. But it continues. Every day they follow you, they taunt you, they make fun of what you’re wearing, they make fun of what you look like, they tell you you’re a loser - in front of everybody. Every day.” The fixed mindset took the incident more personally. ‘I would think I was a nobody. I would think I was stupid and weird and a misfit.’ Then they’d say they wanted violent revenge, saying they’d explode with rage at them, punch their faces in or run them over. They strongly agreed with the statement, ‘My number one goal would be to get revenge.’
The growth mindset students saw it as a psychological problem of the bullies, a way for bullies to gain status or charge their self-esteem. ‘I’d think that the reason he/she is bothering me is probably that he/she has problems at home or at school with grades...I would want to forgive them eventually. My number one goal would be to help them become better people.’ Now individual children can’t usually stop the bullies, especially when the bullies attract a group of supporters, but the school can.” - Carol Dweck from Mindset
The Assessment: I told students that their ticket to start the homework was me checking off the first homework problem. When seven students stood up simultaneously to be checked, I realized I was going to have a room of chaos shortly with students waiting to be checked and not waiting to do math. Thus, after checking the first correct answer, I gave the student feedback on the fact that she had made the numerator have the same number through multiplication and had her go around and check others. I often use this peer mentor method, and today it proved useful.
The other assessment I had was with the problem described in step three of the agenda. Where I noticed they were lacking was in emphasizing how important units were. Several students said a store was better because the value 3 is cheaper than 3.20, but the unit was really 3 bottles and 3.20 bottles. In effect the idea of getting 3 bottles per dollar was of course worse than getting 3.20 bottles per dollar. It was interesting to see one student solve this problem trying to use the least common multiple too.
Homework: Page 63-65 odds only on equivalent ratios. I especially like the question that asks students to find if $28 in 4 weeks is equivalent to $49 in 49 days.
My Glass Half-Full Take: The problem that I took the picture of motivates students. They enjoy the challenge. And I don't just mean the students that enjoy math. Everyone is enjoying it. It's when I do notes that I start to lose them. Speaking of notes...
One Thing to Do Differently: I came up with the conclusion by the third time I'd done notes that to tell if two rates are equivalent simply multiply or divide a rate or two rates until either the numerator or denominator is the same. In my last class I made the students repeat. "To see if two ratios are equivalent." To see if two ratios are equivalent. "Make the two numerators." Make the two numerators. "Or." Or. "Denominators." Denominators. "The same." The same. It proved to be the most effective way I did it all day, although there was still a good deal of confusion. Yet, when the students had no direction from the notes, they weren't without a clue as referenced by the pictures above (although all of the pictures do have their flaws).
Link of the Day: Tape diagrams have been a problem for parents to understand, so I made a video.
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
6.MP3 Construct viable arguments and critique the reasoning of others
The Learning Objective: Graph the relationship of two different units
Quote of the Day: “All my life I’ve been playing up, meaning I’ve challenged myself with players older, bigger, more skillful, more experienced - in short, better than me.’ First she played with her older brother. Then at ten, she joined the eleven-year -old boys’ team. Then she threw herself into the number one college team in the United States. ‘Each day I attempted to play up to their level...and I was improving faster than I ever dreamed possible.” - Mia Hamm
Students begin their homework (and in one class finish)
The Assessment: The homework, partner practice. I circumvented the room as students were doing the homework.
Homework: Page 51-52 problems 1-10 in the textbook.
My Glass Half-Full Take: This lesson on the surface looks like I did nothing to prep or involve myself. I used the book for notes and for the actual homework. The fact is that I did work quite hard to find the right questions to ask to make sure students were understanding the basics of the objectives and I poked a little harder with the other questions I asked. I list them in the next topic.
I'm not a huge fan of using the book. I think it's boring usually, but we have them for a reason. Teaching graphing without the book involves a good deal more labor from me and the students in terms of writing the graph, labeling the axis, etc. There are more important places to deposit my time during the school day and I think I spent my time worrying about what questions to ask. Perhaps next year I could spice up the actual lesson.
Questions to Provoke Learning: During the jumpstart: What graph do you think starts out making the most paperclips? What graph do you think doesn't change the speed that it adds paperclips to the chain? What graph starts out slow and then gets faster? Why do you think that it's possible in real life to start out fast clipping paperclips and then slow down? Why do you think it's possible in real life to start out slow clipping paperclips and then speed up? What graph is closest to a machine? Why would a machine sometimes not be Graph A?
During the ratio table review: How can we get from 6 to 4 using division and multiplication? Is there more than one way to get there? If 5 DVDs cost $60, how much money does 3 DVDs cost? How does the unit rate help solving this problem?
During the coordinate plane introduction: What axis is the x-axis? Which is the y-axis? How can we remember which is which? How do we find a given point (what comes first the x coordinate or the y coordinate)? How do we remember this? Why do we connect the dots in some graphs and some graphs leave the dots unconnected? How can we compare two different ratios that measure the same units? How can a ratio be described in words?
There's a lot of questions there. And I asked every one of those today. Could every student answer everyone correct tomorrow. I doubt it. In fact I think there isn't a single student that could answer every single one of these questions, but many of them go outside of the objective and were simply asked to ensure that I was differentiating and pushing the thinking of students that were quick to accomplish the objective.
One Thing to Do Differently: The question that asked the students to describe the pattern of a given table or ratio was answered with vagueness. I wanted students to use units in their answer. Many would simply state, "The pattern is going up by 4 each time." I would follow up by asking 4 what? It's imperative that students state 4 minutes per page rather than 4 pages per minute, so that they would not specify at all was wrong to me. In one class I even forewarned students I was going to ask the same question three times (this particular question about describing the pattern). I ended up asking it five because the students asked the question the third and fourth times couldn't answer after hearing valid answers two times before. It's essential to make sure all students are as detailed as possible in giving a description of a pattern.
Link of the Day: America needs to encourage discovery in learning. Harvard is. There are days where I believe I am lacking here. Today might have been one although I really think the jumpstart got students thinking before I taught them anything.
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 The Learning Objective: First part of class: Construct a double number line to estimate the solutions of problems. Second part of class: Find an equivalent ratio through scaling.
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
Agenda:
Students worked on a division of decimals jumpstart as I passed back the Weekly Quiz and in one class the progress reports. I also collected Weekly Quiz #5 which I required to have done today.
I cold-called a student to tell me why 1.44 divded by 0.4 was not 0.36.
We did I do, we do, you do of a double number line that served as a way to check Weekly Quiz #4. Really cool way to show students a new topic while also reviewing their homework.
I reviewed the answers to the ticket to leave questions with the students and will do so again tomorrow when I hand the exit tickets back.
Before we began looking at ratio tables and more specifically scaling, I had the students try it independently.
We did some tables in notes. First they were problems that were easy to get an equivalent rate, but eventually they involved either decimals or two steps. There were two such problems.
The Assessment: Step 6 of the agenda was a primary assessment for today. "If 14 lawns get mowed in 8 hours, how many lawns can go mowed in 49 hours?" Their pictures are shown below. Many were demonstrating solid ability with the double number line and others were not far away from finding the solution through scaling before we started the activity. None of the three pictures shown actually were able to give an answer, but it let me know students were knocking on the door to this process without any instruction. I also gave the ticket to leave in step 4 and checked the Weekly Quiz in Step 1.
Homework: Page 43-44 in the textbook problems 1-10 and skipping number five.
My Glass Half-Full Take: I was pleasantly surprised at the connection one of my students made as we transitioned to ratio tables. He asked me if ratio tables and the double number line were really two ways of solving the same type of problem. I said there's more than one way to skin a cat (yes).
One Thing to Do Differently: I'm reading Teach Like a Champion right now, and the author uses the analogy of giving students at bats. I don't think students had quite enough at bats with scaling. I think I could have modeled more problems and they could have attempted more. I don't think doing one problem for the class and having them do one independently was enough given that this was my objective for the fifty minute block of time.
Link of the Day: Interesting way to bring students together to communicate about problems in math class. The site/app is called Math Chat and I haven't tried it yet, but I think I'm going to mention it at Parent Conferences this week. Two years ago I'd hesitate and say this won't teach kids anything since they will just be scrounging around for an answer. Today I'd say in our world this is the way in which we add to our knowledge pool and a healthy way to not only learn but have some students teach to demonstrate mastery of a subject.
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 The Learning Objective: First class: Construct a double number line to compare two units. Second Class: Find unit rates of items in a grocery store Quote of the Day: "At 211 degrees, water is hot. At 212 degrees, it boils. And with boiling water comes steam. And steam can power a locomotive. One extra degree makes all the difference." Agenda:
Use a double number line to discover how many gummy bears fit in the larger gummy bear
In the second part of class, we discovered the answer for the gummy bear activity
The students were given a sheet to go shopping for different items I had placed around the classroom. I explained the directions that I would clap my hands when it was time for the students to switch sections of the classroom and try new problems. As students worked, I circumvented the room.
The Assessment: Student work on the unit rate and decimal division problems. Homework: The students were given Weekly Quiz #5 to do over the weekend for a homework grade. My Glass Half-Full Take: As students got into their groups, I overheard the conversation of two students. One student was asking another student what his grades were. The student shot back that he did not want to share his grades. He was basically handling it in the exact way that I have been preaching the students to handle that issue. One Thing to Do Differently: One? How about three today. First of all, I wish the Jets and Patriots game didn't last as it late as it did. I got about ninety minutes less sleep than I usually do and was grumpy at the end of the day. Second, in the first half of class I really didn't have an assessment. Typically I enjoy Dan Meyer's work, but the numbers were big and intimidating which detracted from putting together the double number line. The numbers we used were the same as the numbers he used. I could have simplified the numbers for the kids by telling them to round. It would shorten the length of time on the activity and not compromise the objective. If I could do this over again, I would probably give the students this as notes and perhaps give them Dan Meyer's problem in the second part of class. As it is I'm doing the linked assignment next class because the students were never assessed. This was a classic case of doing placing the activity ahead of the assessment, and hopefully I don't run into the issue again this year. Third, I taught unit rates toward the end of class. We did unit rates the class before, but it would have helped to do this lesson in sync with the other unit rates lesson and would have worked out time wise (we could simply take out the tape diagram lesson and insert it in today's lesson). The activity was a good way to get students out of their seats, but students were doing straight decimal division in many cases in lieu of showing the units and making sense of the real world application of each problem. I think if I either modeled one problem for the students or simply had them recall their notes from the prior class it would have solidified unit rates for them in a deeper way than we did in the last two days of school. Link of the Day: 20 minutes well spent. Admiral William H. McCraven's words to the Class of 2014 at the University of Texas at Austin.
6th Grade Math Standards: 6.RP.1 Understand the concept of a ratio and use ratio language to describe the relationship between two quantities. For example, "The ratio of wings to beaks at the birdhouse in the zoo was 2:1, because for every 2 wings there was one beak." "For every vote candidate A received, candidate C received nearly 3 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.”
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
The Learning Objective: Find the unit rate of a product to determine the value of that product for any given number of units. Find an equivalent ratio given an original ratio and the sum value of the two units.
Quote of the Day: "I believe the most important thing in any endeavor is hope. You cannot believe it is hopeless because if you do, it is." - Dr. Henry Friendman
Agenda:
Upon entering the room the students simplified 4 ratios that had a greatest common factor bigger than 1.
We reviewed the homework particularly the question that I had liked yesterday about 6 students having a baseball glove in a class of 24. What was the ratio of students that had a glove to the students that did not have a glove?
We reviewed the exit ticket from yesterday. In one class in particular, more than 70% of students made a mistake on at least one of the two problems. We discussed circling what the ratio is asking for since most students ignored the word total.
I gave the students this picture and asked them to consider a price that they would think is too high to pay for the apple juice, a price that is too low, and a price that seems just right. It was amazing.
We took notes on unit rate
We revisited the 6-pack turned 5-pack issue described in step four and found the unit rate of that to make sense of what was a fair price.
In the second part of class, I gave students the following prompt: The Patriots are going to outscore the Jets tonight by a ratio of 3:2. If there are 85 total points, what is the final score?
We took notes on tape diagrams. I was shocked by how much they enjoyed it. I think they enjoyed it so much though because of the struggle they had before starting the unit.
The students answered the question described in Step 7
The students did the tape diagram homework
The students did select unit rate problems for homework on a different sheet
The Assessment: There were a few:
Last night's homework. Every students but about ten (out of sixty-five) answered the baseball glove question wrong.
I cold called students to help me answer the jumpstart questions.
I went around the room on the Dan Meyer Apple Juice pre-assessment to see what students were writing. They were really thinking today on this one. Many students wrote a too high of $3.29 and other students were discovering the unit cost and multiplying by five. I was particularly pleased with who was putting $3.29 for a too high answer because these students are showing growth in the early part of the year.
On the pre-assessment to tape diagrams, one out of sixty-five students could tell me the score was going to be 51 to 34. After taking notes on three problems, more than half the class was able to do this. The students that were left confused were helped by their peers.
I did start to look at homework answers for tonight as well.
Homework: Tape diagram practice as well as the unit rate worksheet (not scanned here) and of course the Weekly Quiz which is due tomorrow
My Glass Half-Full Take: I loved throwing the challenging problem at the students before we did the notes today. It was such a helpful way to get them to buy into what we were doing.
One Thing to Do Differently: As we were doing the notes on unit rate in the first part of class one of my students said "Mr. Schruender I'm bored." I put my ego aside because quite frankly I was too. The problem with the five cans came to life partially because it was a picture but mainly because it was a problem worth solving. The bored student was one of the students who was actually finding the unit rate and multiplying by five to find a price that seemed fair. In his defense my notes were far more basic than this type of problem. I was putting him in fourth grade when he was ready for grade 6 and a half. The notes could have been differentiated. There was one problem that gave the students ounces and prices. I know now that if I had asked those students who tested out of the can problem that they could have been asked to determine two different unit rates in the same problem ($1.36 per ounce of ice cream and 0.72 ounces of ice cream per $1 - that sort of thing).
As a a side note, I'm really glad that student said he was bored. I actually came up with the Patriots vs. Jets problem on the spot as a way to say, "ok class I dare you to get this without me actually teaching you how to do a thing." The kids relished the challenge. And I relished the fact that the kids were relishing the challenge.
Link of the Day: Seth Godin discusses why some people just aren't good at math. I agree with everything he writes except for the fact that he thinks standardized tests don't measure anything. Although I think their value is inflated in our society, they can inform us of a student's fluency. I also think they can do a lot more of bad than good when we spend 100 percent of our time preparing students that never do well on these tests to prepare to take yet another one. If it's broke for these students, shouldn't we fix it?
6th Grade Math Standards: 6.RP.1 Understand the concept of a ratio and use ratio language to describe the relationship between two quantities. For example, "The ratio of wings to beaks at the birdhouse in the zoo was 2:1, because for every 2 wings there was one beak." "For every vote candidate A received, candidate C received nearly 3 votes."
The Learning Objective: Write a ratio in three different ways. Differentiate part to part and part to whole ratios. Define ratio. Give a real world example of a ratio.
Quote of the Day: "When kids play video games, they fail 80% of the time. They look at failure there as an opportunity to learn. However students can find school mistakes humiliating." - David Dockterman, Harvard Graduate School of Education
Agenda:
Jumpstart with divisibility, listing factors, and prime factorization. As students worked I collected their completed weekly quizzes from the previous class.
Ratio discussion (I listed several ratios and then cold called kids to give me ratio examples)
Students got started on the homework which was page 25 in the textbook as well as a couple problems on page 26.
The Assessment: The exit ticket went pretty well. Students were given a problem about 4 different items and told to write a ratio of fruit to the total number of items. Most of them were able to do this in the correct order and also simplified the ratio into a unit rate (even though that term wasn't introduced yet). The second question on the exit ticket (there were only 2 questions) involved more thinking than the first as the students had to again find a total in a question that was unnecessarily verbose. Again I was surprised with the success they had - although more students struggled here.
I also was able to assess all weekly quizzes (although this was done somewhat yesterday too) in class and return them to students as the students worked on Nana's Paint Mix Up and also during lunch.
Pepper helped me infiltrate their brains and insert the vocabulary term ratio. I feel like infiltrate is the right word.
Homework: From the math book page 25 and part of page 26. I really liked one question that asked students if 6 out of 24 students in a class had a baseball glove, write the number that have a baseball glove to the ones that don't in the class. Among the papers that I saw as students were working on the homework, more than half were writing the ratio as 6:24.
My Glass Half-Full Take: I had a student say I'm not good at math today. The same student was correctly answering questions in class and tested pretty well on my assessments today. I like that the student can have that attitude, but can come back fighting in class. I told the student it's my mission this year to create confidence in her ability.
Another positive from today was the quote. All three classes cited peer influence and the fear of embarrassment as the most lopsided reason why there is a difference between failure in school and video games. I wholeheartedly agreed that making mistakes can be embarrassing. I shared with them a personal example from a class I took this past summer with about forty other math teachers. The professor saw the work I had (and knew I was close but wrong) and asked me to show my work on the board. It's easy to be embarrassed about this situation and I felt like maybe my peers were judging me. Today I asked my students how many of those forty teachers were thinking about that moment today. The students agreed that none of the teachers were. In fact, I had grown in this moment. I was humbled. I realized for the one billionth time that I need to keep learning and that my status as a teacher doesn't change that in the least. Making a mistake is normal and it is also forgettable.
One Thing to Do Differently: I spoke with a colleague at the end of the day who was arguing that the students do not care as much as they used to. I got the sense that she was not saying this to vent or in frustration, but it was a matter-of-fact position that she had reached. She compared her own childhood to the childhood of the students we have. She said that when she got home from school her siblings and her would do their work "in numbers." Meaning that they would sit around the kitchen table (they had no desks in her home) and their mother wouldn't let them leave until they were done. It was discipline 101. She mentioned how her mother would check their work and even call the Boston Public Library (this is before Google remember) to learn things that they were curious about. She said this same discipline and this same curiosity is rare in the students we teach. Perhaps it has always been the case. Or perhaps we're not letting them find their curiosity as we are teaching right now. This video from Sarah Almeida argues strongly that schools are factories that make it hard to care, be creative, and be curious.
Comparing eras in baseball is always hard because there are so many variables in play. Steroids, the size of the pitchers mound, the strike zone, defensive shifts, situational lefties, etc. I think the same can be said with students today versus when we were their age. Today ADHD is more commonly diagnosed - not necessarily more common. Is this leading to better teaching practices for these students or are these teaching practices being used as a crutch by the students? The internet is essentially making any deep question such as can a number be divided by 0 a quick google type away rather than really investigating. Is having information a positive or is taking away critical thinking a negative? Grades can be seen instantaneously with grade books online. Is this causing the students and parents to view the grades instead of the learning more now than ever or is it a good thing that there instant communication that there is a learning gap or high achievement?
The answers can be debated obviously and I'm not going to be able to answer one way or another. What I do know is this. What I wish I'd done differently is thank my grandmother for instilling my work ethic on homework as a child. How did she do it? Simple. She didn't threaten me by taking away the television if it wasn't done or reward me with cookies if I did problems right. She simply praised my effort. When I was done she complimented me. Occasionally she complimented me in front of my parents. That was it. She didn't even have to ask me to ever take out my homework or check it. The environment of seeing her quiet happiness was the only motivation I needed. Perhaps the same can be said of my colleague's mother's phone calls to the library.
Link of the Day: Tomorrow we are using the Partial Product lesson from Dan Meyer. I'm getting a lot of his lessons right now, and the kids are doing a lot of thinking as a result.
