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```The difference between memorization and understanding is that students quickly forget things they memorized. How can I help them see that?

— Julie (@jreulbach) May 29, 2015

As we are preparing for final exams, I see a lot of this. Today, we were working on reviewing solving systems of linear equations and many of them had forgotten how to graph lines. And as I was walking around, helping students, it dawned on me why. You see, my students have just come off a section on solving quadratic equations. To be honest, it was a little rushed due to time constraints, but not as rushed as last year. Prior to solving quadratic equations by taking square roots, or completing the square, or the quadratic formula, we had spent quite a bit of time on graphing quadratic equations (here is a post on the start of the unit). While learning how to graph the quadratic equations, students used completing the square to rewrite an equation into vertex form. They also solved equations by either factoring or by taking square roots when finding the x-intercepts (depending on whether or not the equation was in standard form or vertex form). So when we got down to solving by these two methods, they had seen them used before and in a context. They did significantly better on these two learning targets than students had in the past, especially the completing the square learning target. When I finished grading their papers, I was pondering what had gone better this time around. Between that and today, it was pretty clear: the connection students had made while using these processes in graphing quadratic equations helped. In fact, the tweet Glenn Waddell responded to Julie with rang through my head today.

@jreulbach My learners are enthusiastic about math this year. I think one reason is they do see the connections. — Glenn Waddell, Jr. (@gwaddellnvhs) May 29, 2015

It is becoming clearer to me that I need to work on helping my students find connections between the doing of the mathematics and where it can be used and/or why it is used. Glenn has already begun to blog about how he does this in his classes and I am looking forward to reading more about it this summer. I think I have found yet another summer project for #summerlist2015.

]]>I can sketch a rough graph using the zeroes of a polynomial and other easily identifiable points such as the y-intercept.

I’m incorporating A-APR.3 (Identify zeroes of polynomials when suitable factorizations are available, and use the zeroes to construct a rough graph of the function defined by the polynomial.) and A-SSE.3a (Choose and produce an equivalent form of an expression to reveal and explain properties of the quantity represented by the expression.^{*} Factor a quadratic expression to reveal the zeroes of the function it defines.) from the Common Core State Standards. We have just finished solving a quadratic equation by factoring, which incorporates A-SSE.3a. This is the first of four learning targets in my unit on Quadratic Functions, which focuses on graphing. I had originally intended to also do transformations in this unit, which I hope to do in the future. I am running out of time in the school year and I want to make sure I teach how to solve quadratic equations that are not factorable.

My intention (since I start this tomorrow), is to do a mixture of students working with what they know how to do and additional instruction to teach the new information. The pages below are set for my students to add into their interactive notebooks, so they are 2 to a page.

(Pardon the formatting – the graphs for some reason don’t come out right when I PDF it. Here is the link to the Word Doc – it didn’t hold my page formatting when viewed in the viewer.)

The first page is actually 2 copies of the same page – students will begin by doing it on their own. The second page is part notes and part examples students will work on their own. The first half is notes where I introduce the terms roots and zeros for solutions and discuss the important parts of the first page, where students were asked to find the x- and y-intercept of a linear equation and then to try to do the same for a quadratic equation. I am connecting this to their prior knowledge from linear equations. The parts the students work tie back to solving quadratic equations by factoring, which we just finished.

The third page continues in this format, asking students to find the y-intercept for each of the equations and introducing the shape of the quadratic graph as well as how the graph can open. By the end of the third page, students should have enough information to start a graph of the quadratic equation.

The fourth page walks students through the sketching process, at least as I saw it. The last page is where they actually would graph it. I did not introduce the term “axis of symmetry” at this point. The next learning targets will introduce vertex form and at that point I will bring in the term. I wanted to use the idea of the fold line and mirror image with the hope that it would help my struggling students understand how the symmetry in a parabola works. The final learning target in this unit is for students to graph a quadratic function, identifying key features such as the intercepts, maximum and/or minimum values, symmetry, increasing and decreasing intervals, and end behavior of the graph.

The examples I chose were all factorable. The first one is fairly straightforward. The second one factors easily and has only one x-intercept. The third and fourth examples have a negative squared term, one with -x^2, the other with -3x^2. They are a little more of a challenge to factor, but can be factored.

]]>You see, I have been largely absent from the MTBoS this year. I was trying to put my finger on as to why. A lot of it has boiled down to life getting in the way. I’ve been trying to keep up with my kids and keep my daughter in particular on track with her homework (she has ADHD and it is a daily struggle for us). Most times, when she is working on homework, I sit and play some sort of game – the hidden picture games I play primarily, although Candy Crush has crept in at times. Sometimes I am grading papers. But mostly, I am playing some game that keeps my mind occupied but that I can drop quickly to help her with her homework or get back on task. I could be reading stuff on Digg Reader or Twitter, and the excuse I’ve given myself is that it takes too much thought on my part to give it the focus I want to since many times I have to drop what I’m doing to work with my daughter in one way or another. By the time we get the kids in bed, mentally, I just haven’t wanted to go and read stuff related to work. I’ve told myself I just need a mental break and back to the games I head. Or maybe I’m checking Facebook or something else. But heading to blogs or Twitter hasn’t been the first (or the second or third) thing on my mind.

School hasn’t been incredibly more difficult than last year (which was tough!). There’s been the added pressures of the PARCC exams, but I haven’t been overly stressed about it. But, when the time comes that I could read stuff that is math education related, I just haven’t felt like it many times. I thought maybe it was burnout for a while, and maybe that is somewhat true. But as I have gone through some things this weekend, I don’t think that is entirely true. Maybe I was being selfish and just keeping to myself, but I’m not sure that is entirely true either. I didn’t blog much and I felt like I just didn’t have much to say that would contribute to the MTBoS, plus it was tied to work. So I didn’t blog much unless something really jumped out at me.

And life continued to happen. There has been a lot going on in my personal life in the last few months. I don’t want to share all of that here. It’s not the time, nor really the place. But I will share that the last two weeks have been very difficult on many levels for me. My daughter injured her knee while playing with her brother (he tackled her) and has been on crutches for the last two weeks. There have been some family things going on, including the death of a close family friend unexpectedly early last week. I got sick and it was kind of tough to shake it. And life continued to happen.

Yesterday was incredibly full. The funeral for our family friend was in the afternoon and we had a Girl Scout commitment that was shortly afterwards. My brother was here for the weekend and a good friend of mine was here for the day. Plus my husband’s family was here for the funeral. I had to make a choice. After the Girl Scout event, I could either head back and spend time with his family or have dinner with my friend and her son. One kid wanted to hang out with the cousins, the other wanted to head out to dinner with our friend and her son. And I really didn’t know what to do. I was torn. My brother said something that made a whole lot of sense – do what will make you happy because there is no way to make everyone happy. It was a lot longer than that, but that was the basic message behind it. At the funeral, they had talked about our friend who had died and how she was so kind and really cared for others. She always had time to listen to you and rarely talked about whatever was troubling her. She would generally share what was going on if you asked, but she never began with it. She was such a beautiful person and had such a beautiful soul. And as I thought about my decision – head back to visit with my in-laws or spend time having dinner with my friend, I realized that if I followed what my brother said – to do what would make me happy – it would make me happier to spend time with my friend, who needed someone to listen to her. So, after our Girl Scout event, we went out to dinner. I had time to visit with my friend and enjoy time with her. I was able to listen to her and be supportive of her.

Over the course of today, I have exchanged texts with a couple of MTBoS friends. With one, I began the conversation because of a difficulty she was going through. I wanted to connect with her to confirm what I suspected. We had a brief text conversation and I was glad I reached out to her. Later in the day, another MTBoS friend contacted me about TMC15. We had a conversation about booking flights and a little bit about NCTM. I know I’m looking forward to seeing him this summer and I’m glad to know he’ll be heading to LA.

So as I was putzing around today (and my husband was working with my daughter), I decided to open up my Digg Reader. It has the lovely infinity sign, which means I have more than 1000 posts to read. And rather than declare bankruptcy, I chose to start reading. The first blog on my list I came to was Sarah Hagan’s and I had to go all the way back to August, 2014. I began reading. About the 4th or 5th post was titled “On Blogging.” In the post, Sarah talks about why she was struggling with blogging and why it was important to her. And a lot of it rang true for me. But most of all, the last two paragraphs are spot on:

I need to go back and remember why I blog, though. I blog for me. I blog because I process best through written reflection. I blog because I have a terrible memory. How did I teach this topic last year? Let’s go back and read the blog post about it. I blog because I desire community. My blog made me a part of the MTBoS. I blog because I have a desire to share. I blog because I believe that my sharing will lead others to share. I blog because I want my impact to expand beyond the city limits of Drumright, Oklahoma. I blog to connect.

From here on out, I will stop apologizing about what I blog about. I blog for me, not you. I will not feel guilty when I do not blog. My blogging will happen based on what I need. Dan Meyer told us to be selfish. I’m taking his advice.

She is completely right. I have lost sight of why I became involved in the MTBoS and stayed on Twitter, read others blogs, and blogged myself. I, like Sarah, began blogging for myself. It was a place for me to hash out my thoughts and figure out what I was doing right and needed to improve. Participating in Twitter was a way to connect with others and find other like minded teachers. It was a way to find other teachers who taught the same course or who had similar passions for teaching math and bounce ideas off of them. Through Twitter and blogs, to quote Cheesemonkey, I found my tribe. As I read the last paragraph that Sarah wrote, I remembered how Dan talked about being selfish and some of the conversations we had around that at and after TMC14. And I realized I had gotten away from that. Even though it sounds bad to be selfish, when I was being selfish – blogging for myself, engaging in Twitter and blogs, interacting in the community – I was also being self-less. By being self-less (and selfish), I was giving back to the community that has helped me grow so much as a teacher and as a person.

So, I am publicly going to say this (even though I am really only doing this for myself): I am going to work at being more selfish/self-less. It’s going to take some time, but I am going to go back and read the blog posts I have stored in my Digg Reader. I am going to make a better effort to be more active on Twitter. Hopefully, the inspiration to blog more will strike me and I’ll want to post more here. I have more Common Core / PARCC tables that I want to work through at some point. Doing these things does make me happy. I need to get back to that.

And if you made it this far through my selfish blog post, two things: thank you for reading and I hope you are able to figure out what makes you happy and do whatever that is.

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Throughout this unit, I have been talking about how factoring is un-doing the distributive process that we did when multiplying polynomials. We are looking for the factors that we multiplied to get us the answer we are given.

We are not all of the way through, but this is what I have done so far:

Days -2 through 2 – Introduced X-Puzzles (Julie references them) for students to work on finding two numbers that multiply and give one number and add up to another number (example – two numbers that multiply and give 48 and add up to 14: 6 and 8)

Day 1 – Taught how to factor using the GCF.

Day 2 – Taught how to factor four terms using 2 and 2 grouping.

Day 3 – In class practice factoring using the GCF and 2 and 2 grouping.

Day 4 – Taught how to factor x^2 + bx + c. Here’s how I explained it:

Let’s say you have x^2 + 10x + 21. We’re going to set up an x with the multiply number on top and the add number on the bottom, like we have been doing. To get the add number, we are going to take the coefficient (including the sign!) of the x-term, which is 10. We are calling it the add number because to have gotten it, we added. To get the multiply number, we are going to multiply the coefficients of the x^2 term (which is 1) and the constant (which is 21). 1 times 21 is 21. Now solve the x-puzzle – what two numbers multiply to give you 21 and add up to 10? 7 and 3.

We are now going to rewrite the 10x as the sum of two terms with the coefficients we just got. So now we have x^2 + 7x + 3x + 21. Since we have 4 terms, we can use 2 and 2 grouping like we did last week.

x^2 + 7x + 3x + 21 What do the first two terms have in common? (x)

What do the second two terms have in common? (3)

= x(x + 7) + 3(x + 7)

We should now have the same expression in both parentheses – this is our common factor for the two terms I have. It goes in one parentheses and the other parentheses has what is multiplied by each of the (x + 7)s.

= (x + 7)(x + 3)

Then I have them multiply back to check:

x^2 + 3x + 7x + 21 (Oh look – the same four terms we had a moment ago!)

x^2 + 10x + 21 (Combine like terms to get our original problem).

We have spent two more days practicing this in class. With practice, they have caught on rather well. We will start ax^2 + bx + c when we get back from Easter Break in a few days. I feel confident that they will catch on to that with practice as well.

I’ve taught with the boxes that Julie referenced. I think the biggest problem I always found is that students get confused on which term goes in which box. With using 2 and 2 grouping as part of the process, it ties to something they have already learned and it makes clearer to the student why they get the answer they get.

]]>As we have been working through the Common Core State Standards this year, I have really been putting a heavier emphasis on my students being able to explain why they are getting the answer they are getting or what something represents. At times, it has felt like I am pulling teeth to get them to do it. But slowly, surely, we have been getting better. I am starting to see more students being willing to contribute an answer. Discussion is improving a little bit at a time.

I am not sure if I can pinpoint exactly why today’s discussion went well. I felt good about the questions I asked. Once I reflected on the day, I felt good about the responses I received from my students. But could I pinpoint exactly why they had a better idea today? Not really. But what I think I can feel confident about is that my students are coming around to applying the Standards of Mathematical Practice. With perseverance on my part in teaching them, they have begun to come around.

So, if you are reading this, I want to offer you some hope. Keep fighting the good fight. Push the students to use the Standards of Mathematical Practice. Post them in your room, whether the actual ones or the student-friendly ones. Make the students recite them, a la Justin Aion (sorry – I can’t find the actual post where he shares that they do this, but he does!) Whatever it is you need to do to get them to understand and, more importantly, use the Standards of Mathematical Practice, DO IT! Will it be easy? No, not at first. But it will get easier and they will improve. As far me, I will keep fighting the good fight. I’m going to enjoy today for a few minutes first.

]]>I teach in Ohio, which ~~is~~ was (as of June 30, 2015) a PARCC Consortia state. When Common Core was first released in 2010, grades K through 8 had their own set of mathematics standards. There are 5 domains of standards at the high school level, but they are not arranged by course. In addition, as you read the standards, there is some overlap. An Appendix (A) was added with a suggested list of which standards should go with which course (Algebra 1, Geometry, Algebra 2 or Math 1, 2, or 3), but that’s as much guidance that has been given.

So Ohio was involved with both consortia at the beginning. Eventually, they decided to go with PARCC. My students will be taking the PARCC exams this winter and spring. As I was looking at my curriculum map this fall, I was trying to figure out the order I was going to put my units in, knowing that my students were going to take the PBA in late February. Once again, I was hunting all over the PARCC website trying to find the information for each test and look through the End of Year test, and getting incredibly frustrated. As I was getting frustrated, I kind of thought about putting together something that had everything in one place – the actual Common Core Standard and all of the table information from PARCC from every document they had on Algebra 1. As I began assembling it, it made sense to me to add in the End of Year Items and Sample Tasks that were already released by PARCC in with the correct standard. While I was waiting on PARCC to release the PBAs, I also did Algebra 2. Since I teach Algebra 1, it made sense to me to do the same for Algebra 2 so I could see where my students were going. The 8th grade standards are next on my list, so that I can clearly see where my students should come from. I will add the 8th grade once I have it finished.

Once I completed these, I shared them with a few people I know who could use them. I wasn’t totally sure what I wanted to do with them. There were some varying opinions, but the bottom line opinion was that they would be useful to other teachers and it was a good idea to share them.

I have copied and pasted the information provided by the Common Core State Standards and PARCC online. At the very end of the document, you will find the links of where I pulled the information from. All I have done is formatted and arranged the information in a way that made sense to me.

So, here they are. I hope they can help you out. Please share the links.

All the best, Lisa

]]>We are starting our gear up for TMC15, which will be at Harvey Mudd College in Claremont, CA (outside of LA – map is here) from July 23-26, 2015. We are looking forward to a great event! Part of what makes TMC special is the wonderful presentations we have from math teachers who are facing the same challenges that we all are.

To get an idea of what the community is interested in hearing about and/or learning about we set up a Google Doc (http://bit.ly/TMC15-1). It’s an open GDoc for people to list their interests and someone who might be good to present that topic. If multiple people were interested in a session idea, he/she added a “+1” after it. The doc is still open for editing, so if you have an idea of what you’d like to see someone else present as you’re writing your own proposal, feel free to add it!

This conference is by teachers, for teachers. That means we need you to present. Yes, you! In the past everyone who submitted on time was accepted, so we really, honestly and truly **need** you to submit/present! What can you share that you do in your classroom that others can learn from? Presentations can be anything from a strategy you use to how you organize your entire curriculum. Anything someone has ever asked you about is something worth sharing. And that thing that no one has asked about but you wish they would? That’s worth sharing too. Once you’ve decided on a topic, come up with a title and description and submit the form. The description you submit now is the one that will go into the program, so make sure it is clear and enticing.

If you have an idea for something short (between 5 and 15 minutes) to share, plan on doing a My Favorite. Those will be submitted at a later date.

The deadline for submitting your TMC Speaker Proposal is January 19, 2015 at 11:59 pm Eastern time. This is a firm deadline since we will reserve spots for all presenters before we begin to open registration on February 1st.

Thank you for your interest!

Team TMC – Lisa Henry, Lead Organizer, Mary Bourassa, Tina Cardone, James Cleveland, Cortni Kemlage, Jami Packer, Max Ray, Glenn Waddell, and Darryl Yong

]]>After recent talk about notice&wonder, sense-making, T said, “Sounds great, but I have to finish Ch4 by end of Nov or else…(1/2)

— Annie Fetter (@MFAnnie) November 24, 2014

…I don’t have time for that ‘habits of mind’ stuff because I have to cover too much content.” What say you? (2/2)

— Annie Fetter (@MFAnnie) November 24, 2014

I didn’t think I could reply in 140 characters (or even 280), so here goes:

I used to think the same way. I have to cover x, y, and z by the end of the year and I wouldn’t have enough time to do the Math Forum Problems of the Week or problem solving or (fill in the blank of your favorite thing we don’t get to). This year I decided that I was going to do the Algebra Problems of the Week with my Algebra 1 classes. Every 2 weeks, the new AlgPoW is released and on the first day of the week, I project the scenario for my students. Our current routine is that I read them the scenario, students list their noticings and wonderings as I read and for a moment or so afterwards. They then get about 2-3 minutes to share their noticings and wonderings with a partner. I then ask each group to share one notice, which I compile. Even if all of their noticings are up there, they have to tell me which notice they had and I add a * to indicate that more than one group had it. After each group has had a chance to contribute, I ask for any additional noticings, which I add to the list. We repeat this for their wonderings. This whole process takes about 15 minutes total.

This was the list of my students’ noticings and wonderings the first week of school:

This was the list of my students’ noticings and wonderings today:

My students have made incredible strides just in their noticings and wonderings over the course of the first 12 weeks or so of school. The quality of their noticings and wonderings is far superior to where they were at the beginning of the year. Now they are looking more for the mathematics and their wonderings are more mathematical than they were at the beginning of the year.

Do we solve every problem in class? No. What will happen next is I will give them the problem of the week that has the question. This week, they will get the PoW tomorrow. They are expected to work on it outside of class. Right now, I am working with them to focus on attempting and revising a solution. When we have work time in class, they can be working on the PoW. I have some students who diligently use their class time to work on the PoW so that they can bounce ideas off of other students or ask me for some direction. Other students will work on submitting it online. Out of my approximately 70 Algebra 1 students, about 20 of them consistently submit at least one draft to a PoW. This is probably the biggest area I am struggling with right now. I would like this number to be higher. Since it is my first year doing this, I am just kind of going with the flow right now.

What is my goal with giving my students the PoW? I want them to be exposed to mathematical situations. I want them to be able to find the pertinent mathematics in a problem situation and be able to use it to solve a problem. My hope is that by the end of the year, my students are more confident in attempting these types of problems because from what I have seen on the PARCC exam, these are skills they will need to be successful on them. But most importantly, they are skills that they will need to be successful in solving any type of problem in life. Will they encounter quadratic equations in their daily lives? Probably not. But will they encounter problems? Yes. Being a good problem solver is an important life skill. If I don’t cover all of the material in my course, yes, they’ll be lacking a little bit when heading to the next course. But if they are good problem solvers, they’ll be able to figure it out and apply what mathematics they do know to the situation. The mathematics will come. Meanwhile, I will keep plugging away at helping my students be better problem solvers. I know it is time well-spent. I can see the improvement in my own students in just over 12 weeks (we are doing our 7th Noticing and Wondering / Problem of the Week).

So, Teacher, wherever you are, give it a try. And I don’t just mean give it one try. One is not enough. This is where my students were after 2 PoWs. This is where my students were after 4 PoWs. And now you see where my students are today. 15 minutes of class time to do Noticing and Wondering every 2 weeks plus another 10 or so minutes of class time to pass out the problem and give some further direction (such as sharing all of the noticings and wonderings or having students look back at their own lists) isn’t a lot of time. You don’t need to give up a whole day of instruction. A little bit at a time will help your students out. If you have a PoW membership, you can find PoWs that will work with what you are teaching right now, which will let you work with two things at once – reinforcing your content and teaching problem solving skills. It took me three years to finally get there and I don’t regret doing it at all with my students. Just try it. I don’t think you’ll regret it.

]]>Most of my students could determine if a solution worked in an equation. I did goof and the second one did not work out as I had thought it would. What is most interesting to me is that 64% of my students could find the correct value for x, but would not use inverse operations to solve the problem. For example, on #3, rather than showing:

x + 572 = 893

– 572 – 572

x = 321

they showed:

x + 572 = 893

893 – 572 = 321

x = 321

In other words, they knew what operation to do, but they did not recognize that they needed to do the same operation to each side of the equation.

Any suggestions on how to approach instructing the students? Although what they did was correct (as far as getting the answer), I am concerned that without understanding the ideas of what an equation is and using inverse operations to solve them that they will struggle as we get to more difficult equations. Thanks in advance for any suggestions.

]]>- I can use and interpret units when solving formulas.
- I can perform unit conversions.

I found some resources online I was happy with for actually doing the unit conversions, however, I did not find anything I was real happy with for using and interpreting units when solving formulas. So I developed this:

My intent was to introduce the idea, put the students into groups of four and have each student work through 2 problems, comparing with other students in his or her group who would have done the problems to see if they got reasonable answers. When they were satisfied they had good answers and units, they would get another half sheet. We did number 1 together on the SMART Board so they would see what I wanted to see as far as work went, showing all units throughout the process. I then directed them to work through another problem on that page and then compare notes, asking for the next sheet when they were ready.

Originally, my thought was to start them off on different half sheets (so they wouldn’t go to another group to get answers possibly) and try to rotate which sheets they got. Then I had the thought that students would not understand what I wanted them to show work-wise, so I opted to start all students on the same sheet. In my first class, I rotated who got what sheet next. After watching that in action, I decided that was not my best option and I revised for the next two classes what order I wanted to go in. The present order is in my document above.

We only got through 2 half sheets, which also surprised me. I guess I expected them to work quicker on their own. Part of our hold up today was getting the Unit 2 Learning Target page and divider page set up in our Interactive Notebooks and that took more time than I realized. I was surprised how many of them could not substitute into the formula even when given the variable and its value (such as D = 400 miles). It was not a huge amount, but it was enough that it bothered me. I had hoped to spend one day on this learning target, but after the first group of students, it was obvious to me that we needed to continue tomorrow. We didn’t get anywhere near enough practice. Many of the students who had completed problems on their own did not follow directions, not using units throughout the entire problem. This is one of my larger concerns. A couple of students commented that it was “harder” to do the problems this way – why couldn’t they just do it the way they were used to? I shared with them the idea of them understanding why the units for the final answer ended up whatever they are, but I don’t think they’re buying that.

My intent was to have students practicing the skill and not be lecturing at the SMART Board, walking them through step by step on example problems. I can tell that my students are not accustomed to this kind of work, both in groups as well as having to be a little more self-reliant. When they are looking at something they have seen before that they know well or are working with something they have just been taught, they are mostly good to go. However, put something in front of them that is different and enough don’t have an idea of what to do and the questions pile in.

As I am writing this, I am still pondering how I want to change things tomorrow. My goal is to have them get through 4-5 more of the half sheets. (I’m not sure if I want to put the BMI sheet in front of them or not.) I also would like for them to work through more of the problems on their own, including the units as I directed them to. Hopefully something will click in my head between now and class time tomorrow.

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