Day 48: Adding and Subtracting Unlike Fractions

6th Grade Math Standards: 4.NF.3 Understand a fraction a/b with a > 1 as a sum of fractions 1/b. a. Understand addition and subtraction of fractions as joining and separating parts referring to the same whole. b. Decompose a fraction into a sum of fractions with the same denominator in more than one way, recording each decomposition by an equation. Justify decompositions, e.g., by using a visual fraction model. Examples: 3/8 = 1/8 + 1/8 + 1/8 ; 3/8 = 1/8 + 2/8 ; 21/8 = 1 + 1 + 1/8 = 8/8 + 8/8 + 1/8. c. Add and subtract 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. d. 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) /bd.)
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 .

Quote of the Day: "If I had an hour to solve a problem, I'd spend the 55 minutes thinking about it and the last 5 minutes solving it." - Albert Einstein

Question of the Day (from students): Is it possible to have a mixed improper fraction?

Objective: Add and subtract fractions

Agenda:

1. Visual Pattern #23
2. QSSQ 
3. Review Homework (#6, #8, #11, #16). These problems either involved mixed improper fractions or whole numbers subtracted by a fraction
4. My favorite no 11 and 3/5 minus 6 and 4/5.
5. Notes on my favorite no
6. My favorite no 11 and 4/5 minus 6 and 3/4
7. Notes on my favorite no
8. Marker boards to practice other problems that forced students to find common denominators

Assessment: The marker boards was an effective assessment today. It is much quicker for me to assess if students are writing on marker boards because the writing is bigger. The students are also a tick more engaged.

The my favorite no problems were also assessed as every student showed me their index cards. Most of them tried to say that the first problem's solution was 5 and 1/5 even though 3/5 minus 4/5 is negative. They gave a thumbs up to notify me that they were able to recognize this mistake when we reviewed.

In the second MFN problem, students for the most part knew to get a common denominator. The most common mistake I saw was that they failed to change the numerator after the denominators were changed.

On the homework, I consistently saw students leave answers as 2 and 9/8 or when they were asked to subtract 1 and 2/7 from 2 they got 1 and 2/7 for a difference instead of 5/7.

Glass Half-Full: It was great to diagnose the mistakes and why they were mistakes today. Having said that, I'm on a personal day on Monday and have a long weekend with Veteran's Day before we meet again as a class. Old habits and ways of thinking have a hard time of dying quickly, so it will be interesting to see what students carry over from today until we meet again in five days.

The other great part of the class today was the connection between the quote of the day and the Visual Pattern. Students were unable to get the answer immediately in all of my classes, but some of them did eventually crack the code.

To keep this in perspective, this picture was taken from a student that two months ago would have a blank stare at a visual pattern problem. It is unbelievable to see some of the transformations in the problem solving processes with these students - even if they are not all solving the problems.

Regrets: We rushed reviewing the homework problems and I still believe most of the students who could not do 2 - 1 and 2/7 for homework still cannot do it now. This will undoubtedly be part of my lesson on Tuesday.

Link: I liked this Tweet from Mark Chubb who asks what is a better question. What is closer to one on a number 4/5 or 5/4? Or 4/5 is closer than 5/4 than 1. Show why using a number line.

Day 47: Fraction Introduction

6th Grade Math Standards: 3.NF.3 Explain equivalence of fractions in special cases, and compare fractions by reasoning about their size. a. Understand two fractions as equivalent (equal) if they are the same size, or the same point on a number line. b. Recognize and generate simple equivalent fractions, e.g., 1/2 = 2/4, 4/6 = 2/3. Explain why the fractions are equivalent, e.g., by using a visual fraction model. c. Express whole numbers as fractions, and recognize fractions that are equivalent to whole numbers. Examples: Express 3 in the form 3 = 3/1; recognize that 6/1 = 6; locate 4/4 and 1 at the same point of a number line diagram. d. Compare two fractions with the same numerator or the same denominator by reasoning about their size. Recognize that comparisons are valid only when the two fractions refer to the same whole. Record the results of comparisons with the symbols >, =, or <, and justify the conclusions, e.g., by using a visual fraction model

Objective: Convert improper and mixed fractions

Agenda:

1. Self Assessment checklist from the test
2. Review the Ratio Test
3. QSSQ 
4. Fraction Notes
5. Hershey Bar
6. Adding and subtracting fractions with like denominators

Assessment: I tried to flip the classroom the day before this lesson by having students complete a Google Form after watching my video. About half of my students completed it.

Glass Half-Full: We defined the different types of fractions and what exactly a fraction is. Getting these core pieces is critical to making estimates and checking the work in future lessons with the four operations.

Regrets: This was a very easy lesson to create a pre-assessment for and I essentially did it with the Google Drive, but then I never bothered to differentiate the lesson. As the standards indicate above, this is a third grade skill. That being said 35% of students did not know how to convert between mixed and improper fractions, so it was worth the time to review for that crew, but the rest of the class could have been held to higher standards or given a harder task than the notes which they really already master.

Link: I follow Eric Sheninger on Twitter and I came across a nice Tweet from something he wrote back in August about a solution to negative people. Don't complain and if you do complain present two solutions to solve your problem or issue. He stole it from Jon Gordon's book No Complaining Rule. It's on my short list of must reads as I've also read The Energy Bus and Training Camp by the same author and gotten things out of those books.

Day 46 Ratios Test

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.” 29
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

Objective: Write a ratio three ways in a word problem context; find equivalent ratios by getting a unit rate; solve problems with ratio reasoning; apply a tape diagram to solve for an equivalent ratio; apply a double number line to solve ratio problems; find the least common multiple; compare two different prices

Agenda:

1. Quote, Star Student, Question
2. Take the Test
3. Retake any previous assessments from this year
4. 5, 4, 3, 2, 1 challenge (get to 20 through 30)
5. Read a book

Assessment: The test results were about what I would expect. This was a test that featured many concepts and covered almost all of the ratio common core standards, so there were definitely students with gaps. Class averages were in the high 70s. What will be interesting to see is how these gaps get filled moving forward. In general, ratios do not always correlate with future topics such as algebra and statistics, so I may explore getting these concepts in.

Here is an example of students getting two different answers (bottom is correct, top is incorrect) by doing the problem two different ways. I have used this problem for three years now and like the depth it forces students to use when they solve it multiple ways. 

Glass Half-Full: On part three of the agenda, students were allowed to retake any test or quiz from earlier this year. Students have folders in my room which show me which questions that they have answered wrong on any prior test or quiz we have had at this point in the year. It's not always the most efficient way to differentiate (I would love to rely on software instead), but it does give students the opportunity to increase their grade and me the opportunity to reteach and motivate students to continue learning the skills that they lack. Students were retaking quizzes from September and October (not the exact questions, but similar ones) and answering them well. This was encouraging to me because I do not think there's anyway that these skills can be classified as simply memory activating their brains. They were truly capable of performing the skill and this is a sign of mastery.

Regrets: Many students approach me during a test to sneak an answer out of me. I very rarely will comply. I wonder what research says about providing assistance during tests. My personal view which was influenced by Rick Wormeli is that mastery is not mastery until it is independent.

Link: That quiz was a resource I just heard about on Twitter. Not very fancy, but seems easy to use and a good place for remediation, pre-assessment, and even homework.

Day 45: Ratios Study Guide

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.” 29
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

Objective: Write a ratio three ways in a word problem context; find equivalent ratios by getting a unit rate; solve problems with ratio reasoning; apply a tape diagram to solve for an equivalent ratio; apply a double number line to solve ratio problems; find the least common multiple; compare two different prices

Agenda:

1. Which one doesn't belong (Number 11 from Mrs. Morgan)
2. QSSQ 
3. Collect WQs and Review HW 
4. Start the study guides 
5. Review the study guides

Assessment: Circumventing the room as students worked on the study guide. The study guide is definitely longer than a full class. In the future I can cut problems out of the study guide. Most notably numbers five, six and any two of numbers fourteen through sixteen. It is a busy test with almost the entire ratio standard incorporated (with the exception of percentages using proportions).

Glass Half-Full: A colleague who teaches ELA came to observe my classroom last block and he was very complimentary of the way in which I have the students use highlighters to find the information that is important in a word problem. As he put it, treating word problems as close reading activities helps reinforce literacy skills and problem solving skills.

We also tried something spontaneous that the ELA teacher liked. All of the students were encouraged to send me a picture of them studying. The test was coming after a weekend and the study guide was given on a Friday. Experience has taught me that testing on a Monday inevitably leads to worse results. There are a number of factors that are likely to cause this - most notably that memory is not as strong so this was something worth a shot.

Regrets: Aside from what was already mentioned with eliminating problems, I feel as though students would also benefit from an answer key readily available to them on this study guide due to the high number of problems that they were asked to do. I also firmly believe students do not take advantage of this accommodation regularly because they are unaware of it and have never been taught with answer keys before. The security of knowing that they did the right process and got the right answer does not have to be exclusive to me walking around. Unfortunately we do not have Chromebooks on a 1:1 ratio (no pun intended given the content we were focused on) and I do not like to have to photocopy this many answer keys. I could have students check their phones, but my room gets awful service. Excuses are lovely but I can certainly navigate around all of these excuses.

Link: Warren Buffet drove people in a trolley to get out to vote. It was great to see the unified spirit of the country to just vote in so many instances (without actually following that with vote for ____). Having said that, Buffet of course did openly endorse Hilary.

Day 44: Equivalent Ratios

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.

Objective: Determine if two ratios are equivalent

Question of the Day: Would the points on a graph always be connected?

Agenda:

1. Estimation 180 Day 54 & Day 55
2. Quote, Star Students, Question of the Day
3. Review the homework or coordinate plane and ratios
4. Pepper
5. "One chocolate chip recipe makes 48 cookies and call for two cups of flour. A different recipe calls for 3 cups of flour and makes 60 cookies. Are these rates equivalent? Explain why or why not."
6. We looked at student answers to this question
7. Practice using the textbook on page 43 and 45. 
8. Interleaving homework assignment started in partners

Assessment: There was actually a nice question in the textbook that asked students to determine if a rate per day and a rate per week was equivalent. About 90 percent of the class got it wrong, and quickly did the "ohhhh" when the students that got it right explained. I checked homework and the chocolate chip problem.

Glass Half-Full: I really enjoyed walking around the room during the problem that students did on the chocolate chip cookies. In the past, I have just given notes and showed students different methods. The methods were always either scaling or unit rate, but by letting students work they showed me ratio tables, double number lines, and least common multiple as alternatives to solving this problem.

Regrets: I do not think students fully grasp the concept of scaling yet and if they do they are not using the term scaling because I am leaving it out. I think this terms will be helpful to them in years to come. I also think this is the most frequently use way to compare ratios in the real world because it is the easiest. This is partially a flaw of the common core system. There are so many tools in the toolbox, but some tools are only necessary once in a blue moon while others require more wear and tear.

Link: 71 - 58 in the eyes of a second grade. When you listen to them it's a little better than when you look for how you want to do it.

Day 43: Graphing Ratios

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?

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

Objective: Graph ratios; list ratios into ordered pairs; describe the relationship in a graph of a ratio relationship

Question of the Day: Does it matter which unit goes on the top or bottom of a double number line?

Agenda:

1. Correct the mistake (see Double Number Line mistake described here)
2. QSSQ 
3. Review the homework and pepper
4. Graphing review in partners as I passed back corrected weekly quizzes for students
5. Graphing kinesthetic exercise 
6. Book Notes
7. Book Homework

Assessment:

Glass Half-Full: I really enjoyed my new touch to the kinesthetic exercise. We did so much vocabulary today that it was especially helpful. Coordinate plane, x-axis, y-axis, origin, ordered pair, independent variable, dependent variable, and linear function were all said for the first time today this year. That's a ton. The kinesthetic helped, but there's no doubt we will need to work on this some more.

Regrets: The book problems are always kind of lame, but they do not take near the effort it takes to re-create other problems that are more engaging and more relevant.

Day 42: Double Number Line & Ratio Tables

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?

Objective: Apply a double number line to solve ratio problems; Create ratio tables to solve problems in which one ratio is given and only one of the other units is given; Use scaling to find equivalent ratios

Agenda:

1. Visual Pattern #18
2. QSSQ 
3. Resuming the double number line gummy bears lesson
4. Double number line guided practice
5. My favorite no. If I mow 8 lawns in 14 hours, at that rate how many lawns could I mow in 49 hours? 
6. Notes
7. Practice/Homework

Assessment: It was a busy day with a lot of content. I mostly circumvented to see how students were doing and gave light instruction at the board, but mostly let them try and fail. Here are a couple of the more common things I saw.

This first picture is from my favorite no. Only about 10 percent of students could do this, so most students required my instruction for how to find equivalent ratios. The student pictured here got much closer than most of the students. They demonstrated the ability to find a unit rate which would lead to an easy solution if they flipped it around to find hours per lawn.

Even during the notes, I was still trying to let them fail a little. Here the student continues to have a flawed definition of what a ratio is by leaving a difference of 10 between the two currencies. The analogy of if I had 10 Canadian dollars being worthless in American dollars goes over their heads so I try to use a wheels to bike analogy to explain away the error. 

This last one is similar to the mistake above. I had students stand and repeat after me. Multiply to find equivalent ratios. Divide to find equivalent ratios. Don't add. Don't add. Don't add. Don't subtract. Don't subtract. Don't subtract. Unfortunately, the students and the co-teacher then bring out the argument that as long as your adding by two wheels on top and one bike on the bottom you can add (or subtract) just not by the same number. Their right. That's so much more complicated to think about though. Let's just call it multiplying.

Glass Half-Full: I had the students complete a Google Form for the homework. First time I've done it all year. It was great for the few students that completed My Favorite No correctly because they got to take a Chromebook and answer these questions in class instead of doing the notes.

Regrets: It was bad for the majority of students who completed the notes because they did not have time to use the Chromebooks in class and I never explained how to do the assignment. Oh well. There was so much going on in one day. Usually I split these lessons into two separate lessons, but with Election Day and Veteran's Day and assembly after assembly I did everything that everyone tells me not to do and rushed it.

Link: Socratic Seminar courtesy of a colleague who I got to see try this today in his 7th Grade ELA class. Always fun to try new things, but more fun when you don't have to be there.