As a group we want to use our textbook and not feel like we need to recreate every lesson in order for it to be of high quality. We assessed the strengths and weaknesses of our textbook and picked a few areas where we saw a need for improvement.

The first area we picked was in chapter 2, the percent chapter—specifically we picked percent change problems. We built the entire unit around an introduction using bar models and a visual concept of 10%. Much of our discussion has been around the proper uses of bar models in helping students understand integers, percent’s, equations and ratios. Bar models are not a one and done concept. We are trying to find ways to use them throughout the year to aid in a visual understanding of many of the 7th grade concepts, such as understanding positive and negative value, including absolute value. Also, the bar models are excellent for helping to understand ratios and proportions. There is real power in learning using the Bar Models. We first heard about their use from the Singapore Math concepts found in the Singapore elementary books. Now we are using those ideas on a middle school level.

Our unit was built around this first lesson that starts with this question:

Everything we taught in the unit was linked back to the idea of visual, mental and 10% 

Next we used the ideas to start doing routine percent problems like:

 
This was all in preparation for teaching our original purpose: Percent Change. With the bar model approach, students could now logically, pictorially and algebraically do many kinds of percent change problems. 

In chapter one, students learned about ratios and used cross products to solve them, relating it to equal fractions. In chapter two students used the visual model to relate the visual to the proportion. It is hard to convey the power of the bar models until you see it in action and see how students can visualize and do problems that were inaccessible before. Even simple things like, “Why do you do 9 lines in a box when you want 10 spaces?” That simple concept alone helps students understand parts of a fraction, ratios and proportions.


About the author: Wayne Smith teaches math at American Fork Junior High School in the Alpine School District. He is the team leader for the 7th grade math teachers, wsmith@alpinedistrict.org

Author: Jennifer Boyer-Thurgood 
Reblogged from: Utah State Elementary Mathematics Teachers Academy


Do your students know you LOVE math? Do they think you even LIKE math? All too often negative attitudes about mathematics start with the negative feelings teachers (and other influential adults) pass on to their young students. As a mathematics methods teacher at a university I am well aware of how pervasive and even debilitating these attitudes can be. Students who enter my classroom as college seniors are not afraid to express their fear of a course syllabus that includes division of fractions and alternative algorithms. In fact, my own fear of mathematics as a college student and an elementary classroom teacher is what eventually led me to my field of study. I was determined that my students would have a more positive experience with math than I did. Learning to love math as a teacher, and in turn teaching your students to love math, often means rethinking math instruction as you’ve always seen it. Students pick-up on and often imitate your attitude. Make sure they are learning that you LOVE (or are at least learning to LOVE) math by eliminating these clues from your curriculum.

5 CLUES THAT LET YOUR STUDENTS KNOW YOU DON’T LIKE MATH:

“I’m Not a Math Person”: I always get a mixed response when I tell strangers what I do. The phrase, “I teach math,” is usually met with the same facial expression as if I’d explained, “I taste test dog food.” In fact, being the math teacher and researcher I am, I have kept track of these responses. Out of the strangers I’ve talked to in my community, on airplane flights, and even educational settings within the past 18 months, only 1 out of dozens has responded positively to my profession. Why? My belief is that we as a society have accepted a dislike towards mathematics. “I’m just not a math person,” is something I frequently hear from others. Consider what would happen if you, as a teacher, said this about reading. “I’m just not really a reading person,” sounds absurd and would likely be met with fierce criticism. Even if you don’t enjoy a good math problem like you do a good book, don’t devalue mathematics, especially in front of your students. Make sure that your students know you are a math person by pointing out the many times we use math during the day. Your class might read a thermometer, count how many days until Friday, or even calculate how many minutes until recess. These are all important mathematical tasks.

Spending Less Time on Math: Spending less time on mathematics than on other subjects is a sure way to let your students know that you don’t value mathematics. If math is something you have to “get through” each day students get the message loud and clear that math isn’t important. Be sure to carve an appropriate amount of time out of your day for mathematics study. Once you’ve worked on making this time interesting and valuable it may even start to take up more of your day than you’ve planned for.

Sudden Change in Teaching Methods: This is one I have to say I’m guilty of. As a teacher during my first few years in the classroom I knew my math teaching wasn’t up to par, but I didn’t know how to improve it. My reading instruction was woven in throughout all the other subjects I taught. I often brought in outside resources to supplement stories we were reading as a class and my students knew from the way I talked about reading and the books I was reading that I really loved and valued them as readers. Math in turn was usually a series of timed tests and worksheets completed during the set math time each day. The dual teaching personalities I exhibited to my students and my great discomfort with this dichotomy was what drove my search for better methods. In order to strengthen your ability to teach elementary math search out reputable resources and trainings such as the Elementary Mathematics Teachers Academy. Also, search out mathematics experts in your school district and find out what support and resources are available to you.

Different expectations: Without realizing it teachers, even teachers who are conscious about equity issues, treat students differently. Some have expectations that boys are better at math than girls. Others value students who complete math quickly over students who take more time. Others prefer students who follow prescribed algorithms and show their work to students who arrive at correct answers through mental processes. Think about how you respond to students during math, what expectations you have for them, and why you have these expectations. Considering and valuing the mathematical processes of your students will make you a much more insightful math teacher. I often plan my math lessons based on the questions I predict students will ask and the different ways they will each think about the problem.

Belief in the Math Gene: “Our family just isn’t good at math,” was something I heard from more than one parent at parent-teacher-conference as an elementary teacher. I can’t help but cringe every time I hear a student or parent of a struggling math student use this phrase. Please know and let others know that your genetic makeup may limit you from ever having naturally blonde hair, but it does not determine your mathematics abilities. Often helping students learn to LOVE math also means working to dispel their own negative beliefs about their abilities and helping them learn to LOVE themselves and believe in their own potential. Struggling math students are often those students who don’t “see” math in the clean linear way teachers teach them to. Take time to get to know and complement the unique mathematical thinking of each of your students.

All students (and teachers) are capable of doing well in mathematics. Make sure this is the message you are sending your students.

- See more at: http://online.usu.edu/teachmath/blog/post/do-your-students-know-your-love-math#sthash.q3qgx90v.dpuf

Desmos.com is my number one use of technology in my classroom. My students and I use it to create lines, analyze data, and solve problems. The interface is simple and user-friendly.

Here's how Desmos describes their product, "At Desmos, we imagine a world of universal math literacy, where no student thinks that math is too hard or too dull to pursue. We believe the key is learning by doing. When learning becomes a journey of exploration and discovery, anyone can understand – and enjoy! – math.To achieve this vision, we’ve started by building the next generation of the graphing calculator. Using our powerful and blazingly-fast math engine, the calculator can instantly plot any equation, from lines and parabolas up through derivatives and Fourier series. Data tables open up a world of curve-fitting and modeling. Sliders make it a breeze to demonstrate function transformations. As browser-based html5 technology, the graphing calculator works on any computer or tablet without requiring any downloads. It's intuitive, beautiful math. And best of all: it's completely free."

The best way to start learning about Desmos is to launch the calculator and just start playing with it. Make sure that you check out the Just Add Sliders, Tables of Data, and Regressions tutorials on the main page.  These are some of the coolest features of Desmos.

Desmos sliders have transformed my teaching of linear functions. Sliders have helped my students understand how manipulating the slope and y-intercept of an equation changes the graph of the line. What are sliders? Check it out:

Once you have graphed a line in Desmos, you can click on the line to find ordered pairs that lie on the line. If you have two lines graphed, you can click on their point of intersection to find the solution. Adjusting the window is as easy as zooming in or zooming out like you would do in Google Maps.

I also love the artwork on Desmos. Last year, almost half of my 8th grade honors students had used Desmos to figure out how to graph circles so that they could make their graphing art more interesting. One student was even graphing sine waves and animating them with sliders. He figured out how to do this in his spare time at home. In class, I never taught any graphing beyond linear functions. On their own, my students used the art examples on Desmos to learn so much more than that! Students can create an account so that they can log in and save their graphs.

If your students are going to be creating art in Desmos, they will want to know how to 'end their lines' by creating domain and range restrictions. (Did you catch my use of the phrase 'want to know' in that last sentence? I'm not kidding. Your students will be begging you to teach them more math once the magic of Desmos gets under their skin.) Here's a quick tutorial on how to do create domain and range restrictions in Desmos:

I'd love to hear your thoughts after you spend some time playing with Desmos.com. Have you used it before? Do you use Desmos or a similar graphing technology in your classroom? How do you incorporate graphing technology into your classroom?

About the author: MaryAnn Moore (@missnarymm) teaches 8th grade math in Davis School District.  She coordinates the UCTM teacher blog and is also a regular contributor to the UCTM teacher blog. Please email her at mmoore@dsdmail.net if you are a Utah teacher interested in contributing to this blog.

When I first learned about math teacher blogs, Dan Meyer's blog was one of the first that I stumbled onto.  I spent about half the day reading his posts, watching videos on his blog, and browsing through his 3-Act Tasks. I was hooked. In one afternoon, I'd gotten more good quality professional development from Dan Meyer than I'd gotten from years of state and district sponsored classes. And it was completely free!  I'm still learning from him and still working on implementing some of the great ideas I've gleaned from him. Recently, Dan Meyer posted a video of a presentation he gave at the 2014 NCTM annual conference. It's about an hour long but entertaining, thought-provoking, and well worth the time.

Here's what Dan Meyer has to say about this presentation: 

"Students generally prefer video games to our math classes and I wanted to know why. So I played a lot of video games and read a bit about video games and drew some conclusions. I also asked my in-laws to play two video games in front of a camera so we could watch their learning process and draw comparisons to our students.
These are the six lessons I learned:

1. Video games get to the point.
2. The real world is overrated.
3. Video games have an open middle.
4. The middle grows more challenging and more interesting at the same time.
5. Instruction is visual, embedded in practice, and only as needed.
6. Video games lower the cost of failure."

About the author: MaryAnn Moore (@missnarymm) teaches 8th grade math in Davis School District.  She coordinates the UCTM teacher blog and is also a regular contributor to the UCTM teacher blog. Please email MaryAnn at mmoore@dsdmail.net if you are interested in contributing to the UCTM blog.

In my last post I discussed some of the things I learned about mathematical modeling at a recent Math for America professional development.  At that same professional development, I had a conversation with a few people about modeling and developing great tasks.  I mentioned that I have to practice looking for great questions, but until I'm able to come up with all of them on my own, I steal (borrow) a lot of great content that people share online.  Over the next few weeks, I will write blog posts to share with you some of my very favorite modeling and number sense resources.  Let's get this party started by discussing my all-time favorite number sense resource:

Estimation 180

Website url: http://www.estimation180.com/

Author: Andrew Stadel @mr_stadel

Description: "Each day of the school year I present my students with an estimation challenge. I love helping students improve both their number sense and problem solving skills. I'd like to share the estimation challenges with you and your students. Michael Fenton and I have collaborated on this handout for your students. Happy Estimating!

Enjoy,
Andrew Stadel"

Why I Love It: 

Honestly, I first started using Estimation 180 because I was looking for an easy time-filler/warm-up for a class of students that was way below grade level. I needed something that I knew everyone could do no matter their mathematical skill, that would keep them busy for a few minutes at the beginning of class while I took attendance. I'd heard some buzz about Estimation 180 on Twitter and on some blogs, so I decided to try it out. I got a whole lot more than I bargained for!  Here's a few reasons why this site is so awesome:

1. Low-Threshold
Everyone can estimate. Everyone!  That kid who says, "I don't know how to do this," every time you call on him?  Yep, he can estimate too.  I have 100% student participation when we do an estimation challenge.  On some days, it might be the only 5 minutes of class that everyone is participating, but at least everybody was thinking mathematically for at least 5 minutes. 

2. High Ceiling
Not only can every kid DO the Estimation Challenges, but every kid can LEARN from the estimation challenges. I find that the best learning often happens when I ask kids the 'why' behind their estimates.  Other good teaching moments come from having students calculate their error.  For example, I had no idea how difficult it would be for my students to calculate their errors of estimating Mr. Stadel's height and the height(s) of his family and friends.  First we had a discussion about all the different ways to correctly write out his height (6 feet 4 inches) and why it couldn't just be written as 6.4.  Then we had some interesting discussions on how to calculate error if someone had estimated that Mr Stadel was 5 ft 9 in.  Why can't we just do 6.4 minus 5.9?  Students brainstormed several strategies to determine the error, including counting up and regrouping to write his height as 5 feet 16 inches.  Was this part of my lesson plan?  No way!  Is it in the eighth grade core that I teach? Not exactly. Do my students need it and does it help them improve their number sense? Absolutely!  I consider it to be time very well spent.

3. Kids Love It
I wish I could record for you that moment that I 'reveal' the correct answer!  There is often shouting, occasionally there are fist pumps. There are sometimes exclamations of disbelief! "What?! There's no way that is only eighteen napkins!"  When a student's estimate is perfect, it's as though he or she had just won The Price Is Right?  I usually try to take a moment to acknowledge the person with the smallest error.  That's all they get - no candy, no extra credit - just a moment of verbal praise, and the knowledge that for that moment they had been more successful than every other person in the room.  You wouldn't believe how motivating that is.

4. Estimating and Problem Solving
As a math student and, until fairly recently, as a math teacher, I always thought that teaching estimating was a waste of time. You know what I'm talking about - those worksheets that teachers gave kids where they had to use rounding to find their answers before they learned the actual algorithm.  Students' answers are invariably different from those in the textbook because they'd rounded to a different place value than the textbook had used.  In all honesty, I still don't see any value in the 'kill-and-drill' form of teaching estimating.  However, I see a lot of value in teaching the Estimation 180 form of estimating.  When I give my students a task like Noah's Ark or Styrofoam Cups, I now begin the task by asking my students to tell me something that is too high, something too low, and then an estimate.  This starts them thinking about what the answer should be.  For my students, choosing the appropriate mathematical operation is often the most difficult part of a problem.  If they already have a good estimate in their head before they start to problem solve, the students are able to figure out much more quickly if their strategy is leading them down a wrong path.

Have you ever found a great resource on the web that you would like to use to teach your students but ended up spending hours turning it into something you could use because the file was in PDF or some other non-adjustable format?  Are you the type of teacher who likes sharing lesson content ideas with other professionals?  Would you like to have easy access to lesson materials in their original file format so that you can customize them to your teaching style in the format that you have chosen?  Then the Weber School District (WSD) Math-3 (honors) Professional Learning Community (PLC) file sharing site is for you.  

I created the WSD Math3H PLC file sharing site as my Math for America (MfA) Master Teacher project. My vision for the file sharing site is an outgrowth of the The Weber School District file sharing site called "Weber Tube" which allows Weber School District teachers to upload their teaching files that they can then embed into their blogs.  Unfortunately, the files on WeberTube all end up being PDF and are not customizable.  

The WeberTube file sharing site gets quite a bit of traffic.  In fact, my files have been viewed over 197,000 times!

I wanted access to original content and felt that sharing my own materials was an "even trade."  I have created my own file sharing site for uploading files in their original file format as a means of bringing together member lesson ideas so that everyone can benefit.  The link to my site is:  http://math3h.pbworks.com/ which takes you to the FrontPage.

The file organization format follows a schedule of topics that was developed based upon the Weber School District Math-3 Scope and Sequence Document (a suggested order of topics created by a team of Weber School District teachers).  Each PLC member takes the lead responsibility for a given week on an alternating basis.  The lead person selects in advance the lesson they are responsible for and other members provide inputs to that lesson.   Lesson inputs can be used by the lead lesson preparer but are not required to be used.  The goal is having original content available.  For the first iteration of this process, one finalized lesson per week is the goal. 

Members upload files and put them into the folder corresponding to the week that topic is listed on the schedule.

If you do not teach Math-3 (honors), I envision the project expanding to include Math-2 and Math-1.  If you would like to be involved in the project by uploading content for your own use and sharing your own materials, go to the file sharing site: math3h.pbworks.com and click on the link requesting to join.  If that does not work, please send a request to jelong@wsd.net (subject: math3h.pbworks.com).  

We can all benefit from a file sharing site like this.  Please consider joining. 

About the author: Jeff Long teaches math at Roy High School in the Weber School District. He coordinates the Weber School District Math-3 (honors) Professional learning community whose file sharing site is math3h.pbworks.com.  His school bog is at http://blog.wsd.net/jelong/ .

Last weekend I attended the Math for America Utah Fall Conference.  The topic of the conference was Mathematical Modeling and the presenter was Kara Imm, Co-Director of Math in the City.  My mind is still spinning a little from some of the ideas I received, but I wanted to share one or two in particular that have already had an impact on my teaching practice.

Less is More

Dan Meyer has been talking recently about how "You can always add. You can't subtract."  Usually a mediocre math task or even a good math task can be made much better by removing some of the information.  For example, in this task called Styrofoam Cups by Andrew Stadel, rather than presenting students will all of the information they need to solve the problem (how many cups will stack to reach the height of the door), Andrew Stadel asks students "What information would be useful to know here and how would you get it?"  Since we're not in Andrew's classroom and can't actually measure the height of the door and the height of the cup, he provides us with links to pictures of that information which can be given to students when they request it.  By leaving that information off to begin with, students are given the opportunity to think about the problem and why that information would be useful.  

I find that sometimes when students are given a problem that gives them every piece of information they need, they have a hard time then knowing how to process the information.  In contrast, when I allow students to think about and request information, they have a much better idea of what to do with it because they've thought about why the information would be important and how it relates to the situation.


Processing the Information

As students ask me for information, I ask them "Why do you think that would be important to know?" After that student gives their reasoning, I like to ask a few other students if there is any other reason they would want to know that information.  I don't tell them how to use the information they request.  They tell me! 

Choosing the Model

Each of these ideas about mathematical modeling (Collecting and Selecting Information, Processing the Information, and Selecting the Model) can be represented along continuum.  Some tasks are high in one area and low in another. Sometimes all a task needs is a small tweak to make it fall a little higher on the modeling continuum.

A Final Thought

In the four classes that have done the Styrofoam Cups task, my students were able to determine the information that they needed rather quickly.  However, all of those classes were honors classes.  This has me wondering if I perhaps ought to make some changes before I run this with my regular ed classes next week.  One thing that really got me thinking was this blog post by Joe Schwartz from Exit10A.  Joe talks about working on his own to solve this swingwraps task, also written by Andrew Stadel.  What fascinated me most about Joe's thinking was this image that he shared:

To help him think about how many times a swing would wrap around a pole, he grabbed a poster tube and a chain, and wrapped the chain around the tube.  What a beautifully simple model!  I would have never thought of doing that!

In my last post, I mentioned that I read a ton of math teaching blogs and am constantly collecting ideas from them.  Great math tasks are just one of the many things I glean from reading math blogs.  Since I'm attending a conference this weekend and have been asked to bring one of my favorite tasks, I thought I'd take this opportunity to tell you about it too. 

Noah's Ark by Fawn Nguyen 
(click on the link above to check out Fawn's original post about this task and download a Word Doc of the task)






Mr. Noah wants his Ark to sail along on an even keel.  The ark is divided down the middle, and on each deck the animals on the left exactly balance those on the right — all but the third deck.  Can you figure out how many seals are needed in place of the question mark so that they (and the bear) will exactly balance the six zebras?

This task is pure gold. It's problem solving; it's balancing equations; it's systems of equations.  And there are no numbers.  A sixth grader could do this task (it was written by a 6th grade teacher), but it is engaging enough for a calculus student. Here are a few snippets of dialogue from my students as they were working on this task last year.

"So one bear equals three zebras." 
"How many seals does the elephant equal?" 
"Can you have half a kangaroo?" 
"Get a smaller kangaroo!"

Student: So there are 6 zebras on that side, so then we take zebra out.
Me: Wait, you can't just take a zebra off one side. Your ark isn't balanced any more.
Student: Well, we're not really taking it off. We're just saying 'Sit down, Zebra. Wait for us to catch up.'

This all sounds completely crazy, but believe me, it's very fun.  Last year, I gave the task right before Thanksgiving. Many of my students figured it out in class, but for those who didn't, I informed them that it was NOT homework and they didn't have to do it over the break.  The following Monday, a few of those students complained to me that I had ruined their holiday.  They spent hours working on that task because they just had to figure it out.

Actually, Noah's Ark sort of took over my own family Thanksgiving last year too.  My oldest nephew, Max, was in eighth grade at the time, so I printed out a copy of Noah's Ark and handed it to him before dinner.  Intrigued by Max's intense frustration/determination, one by one, each of the adults in my family found a copy of the puzzle and started working on it.  In case you think this sort of thing is a regular occurrence in my family, believe me when I tell you that I'm the mathy oddball in my family.  We don't sit down and do math together. Ever. It was one of the most strangely awesome holidays I've ever spent with my family.  

"If you're the sharpest crayon in the box, find a bigger box." I heard this sentence at a summer conference I attended just over a year ago and it changed my career.  Even in a school with six other math teachers, I still feel isolated sometimes.  I certainly don't feel I'm the sharpest crayon in the math-teaching box at my school. But I learned at a very young age that a box of 120 crayons is always better than a box of eight.

I have found this bigger box by participating in the MathTwitterBlogoSphere.  This crazy sounding word is abbreviated as #MTBoS (pronounced 'mit-boss') and describes a group of teachers from all over the world (but mostly North America) who communicate with each other through blogs and twitter. I'm a moderate twitter user, but for me the real magic comes from reading blogs.  On the first afternoon that I found the MTBoS list of blogs and tweeps, I signed up for a whole bunch of blogs and suddenly found myself sitting in an enormous crayon box of math teachers!  Hardly a day goes by when I don't get some great idea from a blog written by a teacher who lives on the other side of the country in some place I've never visited.  It's amazing! I've even started writing my own blog at msmooremath.wordpress.com so that I can be a more active participant in this community. As a result of my participation in #MTBoS, I feel connected, energized, and am happier in my career than I ever have been.

What's missing in this online community of math teachers?  Utah!  I interact with a lot of teachers online, but rarely are they from Utah.  I'm pretty certain that Utah has some of the best math teachers in the country, but the only time we ever hear from them is at our annual UCTM conference each November.  Even then, I'm always disappointed that I can't hear from everyone who is presenting.

All that is about to change.  Welcome to the UCTM Collaborative Blog!  I will be maintaining this blog as part of my work for the Utah chapter of Math for America.  I will also be a regular contributor to this blog.  My vision is for this blog to become a space where Utah math teachers share some of their best ideas, resources, teaching strategies and lessons with each other. You can write one, two, or ten posts - whatever you feel you have to share.  Blog posts will be reviewed before posting to ensure that each meets the goals of UCTM.  

If you are reading this, and you are a Utah math teacher, I want to hear from you. I want to learn what is that you are doing in your classes that is amazing.  What is your favorite task? What is your favorite resource? What are you learning from your students? What is it that you know about teaching that I don't know? I won't be satisfied until I learn it too.  I'm very selfish that way.  How do you sign up to share your amazing ideas? I thought you'd never ask. Just sent me an email to mmoore@dsdmail.net and I will send you an email ASAP to figure out a date for your post.  Let's get this party started!