# How to Subtract and Regroup Mixed Numbers with Visual Models

Flashback to 4th grade when my math teacher asked me to subtract mixed numbers, my palms were sweaty, and I felt like I was looking at the demonstration cross eyed WITH brain fog. If you are a loyal follower of my blog, you might know about my past traumatic experience with fractions. It was not pretty, and the last thing I want is for any student of mine to feel this way.

I created the video below to try to show regrouping with visual models. I can’t stress enough how visual models have changed and revolutionized my teaching. It almost never fails that when students struggle, I’ve forgotten to include a visual to help them “see” the math. I am not alone, check out Dr. Jo Boaler, Berkeley Everett and many others who truly understand how we can deeply teach mathematics through visual representations.

The video below will speak for itself, but if you don’t have time to watch it (it’s less than five minutes), the idea is that you can regroup whole numbers into fractions just like students do with base ten models. It’s SO MUCH FUN when students get to learn this way with you. Please check it out and let me know what you think!

# Start Geometry Vocabulary Early!

I love to be sneaky with using vocabulary in the classroom. One of the ways that I like to embed it is through game playing. For example, instead of using two color counters, we pull out the pattern blocks. Instead of saying “I want blue!”, I tell the students that I would like them to sound more like a fancy mathematician. Instead you must say, “I would like to play with the trapezoid, please.” It causes a laugh with the kids, but even better it starts to get them to use those words more regularly and with purpose. Plus they love to say the word trapezoid! There is a nice benefit to getting these words in their mind early, they’ll start to connect that vocabulary to real world examples of lines, points, segments and other geometry terms.

There is a natural reason to determine the difference between the two tools in the photo below, and to examine how the rhombus and trapezoid are different so that they know which one to pick. This can start as early as the primary levels! Most pattern block sets come with hexagons, triangles, rhombuses, parallelograms, and trapezoids.

# I’m Taking a Big Risk!

I used to hate teaching math when I began my education career. It was scary! I didn’t know what I was doing and had ZERO confidence. 18 years later, you’ll find me traveling across the country teaching other teachers the best ways to teach math. My number one goal in starting this blog is to help YOU love to teach math. However, blogging isn’t the main form of learning these days. I find myself going to social media to learn so many things. So…I am (very nervously) trying a new idea where I do things way beyond my skill and comfort level:

1. I have started an instagram account (a professional one), you’ll find it by clicking here: @beyondtmath
2. My goal is to release as many teaching tips as I can using reels or video posts.
3. I want every video to be under 60 seconds, and something you can try immediately.
4. I plan to try to do these weekly.

It feels worrisome to put myself out there, but I’m going to be posting all those tips that I tell teachers and I hear back: “I wish someone had told me this!” or “I love that idea!” or “I am SO trying that!” or “That finally makes sense to me!”

My first post is about how to develop number sense with cubes. I hope you’ll venture over and find me! (Please, please, please!)

# How to Stop Meaningless Problem Solving

Today while working with some third graders in my intervention group, I had that moment that I know you all have as a teacher. When you pose a story to be solved (I’m tired of calling them problems…is it ok to call them stories to be solved?), and the students just start panic crunching numbers. Like throwing numbers around and adding up every number they hear in the story.

Today it was simple story, one that I know these three could probably picture in their minds, or at the very least draw:

“There are two trees, each one has 15 cherries. How many cherries are there altogether?”

I’ve been reading about the benefits of using vertical white board surfaces, to get students to do more thinking. One student wrote 2 + 15 = 17 with her partner. She was CONVINCED (and there was nothing I could say to change her mind), that because there were two trees and 15 cherries, those two numbers must be added together. The other student knew to add 15 and 15, but used an inefficient strategy of drawing each individual cherry on the two trees. I will address the inefficient strategy issue in another post, but today I really want to address the idea of students just crunching numbers without thinking.

I think for too long, students who struggle with math have been led down this road of not have the proper exposure to tools, drawings and then the abstract notation of mathematics. I see tools disappearing as early as first grade in classrooms, which we all KNOW are critical in child development. Not only are tools disappearing, but I see math programs that encourage abstract notation before connecting tools and drawings to that notation.

Take for example, 3 x 14 or 14 x 3. This student was able to build it with tiles (showing an understanding of area), then was able to take a shortcut and show an area model (without the tiles inside), and eventually use partial product notation to solve.

Somewhere along the line, students who struggle to problem solve missed these connections. They look at you with panic in their eyes as they try to figure out what you are asking and what they have to do. Then in pure desperation instead of solving, they just start mindlessly crunching numbers, not making any connections whatsoever.

Here are some ways that I intervene:

1. Make tools available, have them act it out first, then draw what they made. Don’t worry at this point if the notation is correct, can they make sense of the problem with tools? At first it might be helpful to provide the tool, then the next time have a few different ones to choose from. Start watching if they are choosing an appropriate tool.
2. Write a hint in a different color. Explain that the hint will lead them in a different direction, and that they need to “unthink” some part of what they did. In the case of the cherry trees, I drew an additional tree next to the one they had already drawn and wrote 15 under it. That little hint made all the difference.
3. Look for places they are actually making connections and compliment them on that. The student who wrote 2+15 was actually thinking about multiplication! I told her that, but then asked the two of them to back up and think about what that would mean if they multiplied those two numbers. Find SOMETHING that you like about what they did. Those compliments stick, and they remember it again next time and will use it.
4. Don’t give up on them, and have high expectations. I hear this said over and over again, “My ‘low’ students have a really hard time problem solving.” When you call them low, you will expect nothing but low performance.
5. Try not to give the answer at the end. Explain that problem solving is about thinking, not the answer. This will enrage them at first, but eventually they will start thinking instead of spitting out an answer. They will also see that they need to persevere while solving instead of waiting for the teacher to give the strategy and answer in the end.
6. Have them compare two strategies next to each other. Have the students look for similarities and difference.
7. Choose random partners whenever possible. This allows students to learn different strategies from others, and it keeps your groups fresh.

Students who struggle in mathematics can be excellent problem solvers. It’s so critical that we believe this, and that we continue to challenge them at high levels with the right supports.

# Small Steps for Differentiation: Same Task – Different Entry Points

When thinking of differentiation in the classroom, it is easy to fall into the trap of putting pressure on ourselves to perfectly level activities for every student. My mind goes to having a rotated set of groups and centers all perfectly ready to go. In this scenario you never run out of time, every student is exactly where they need to be, AND they are accountable, focused and staying on task the entire time.

YEAH RIGHT!

Don’t get me wrong, math workshops are a beautiful thing, but it doesn’t always work as smoothly as we’d like. It is okay to differentiate in small ways, taking small steps to be sure that we are meeting the needs of all children without going crazy ourselves.

This past summer I was lucky enough to read work by Timothy Kanold, and then I was able to work with him in a workshop as well. He proposes that instead of coming up with different activities for every student, we have the same task, but with different entry points. So what does this look like exactly?

Here is an example for a second grade classroom where the learning target would be “I can count money up to a specific amount.”(CCSS 2.MD.C.8):

Task: How many ways can you build 58 cents? Build it and record it.

All of the students in this group are given the same task, but usually all of the students have different levels of knowledge surrounding this task. So instead of coming up with 25 different activities you have only one.  As the activity begins and students begin to work, two things will happen which we all can predict every time. Some students will struggle, and others will fly.  This is when you strategically give certain students more.

For the students who struggle in this case you would lay down another task next to it, where the number is more accessible, and you may also consider telling them the value of all of the coins.

Instead of 58 cents, the students who struggle are working with a more accessible number, side by side with the other students.

In this case for the students that are excelling there are many options: ask them to find the solution if you eliminate one of the types of coins, ask them to show their thinking algebraically using a table, give a different amount and have them predict the number of solutions they may find before solving, or ask them to write a story in which you may need to come up with 58 cents worth of change.

The main thing is that you have to truly be walking your room, listening to your students and conferring with them as they solve. The BEST part of this method, is students are working together and hearing one another’s thinking, elevating the learning for all in the room.

# What Does Success Look Like?

Pretend with me for a moment, that you have never seen an apple before in your life. Now pretend that someone has asked you to peel it, but you’ve also never peeled anything before in your life.  How do you know what to do to be successful? How will you know when you have been successful?

That in a nutshell is what “success criteria” is.  It’s all about letting students know what success looks like, and how they will know that they have met the learning target.

In my mission to examine learning targets and communicate them, I learned that it wasn’t enough to simply display them for students. It STILL wasn’t enough for students to write them in their math journals.  My students needed to see the learning target, write it, and then have some sort of interaction with it.  This is where I combine this idea of success criteria (Hattie 2012) with Marzano’s Levels of Understanding.

Here is an example of what this can look like.  You begin your content lesson by reading the learning target out loud, allowing students time to write it down in a math journal (or some other place to take notes). Notice the learning target starts with “I can”.  (It would be just like saying, I can peel an apple.)

I can read and write numbers up to 1,000.

Now some students will have prior knowledge about the learning target, so allowing them a moment to interact and think about this learning target is essential.  You can show them (or you can do this orally) what the different levels of understanding look like. Here is what it could be for this particular learning target:

Using success criteria in the classroom can help students understand the outcome of their learning.

I use Marzano’s Levels of Understanding to anchor my thinking (which I blew up and posted on the wall-this is a free resource by the way!) because the students connect easily to the language.  After I show them what each level looks like, I have the students rate themselves on the current target. Their goal is always the same every day, get to the next level, get higher and get better.

After this mini intro, I teach the lesson. We practice with tiered examples so that everyone is challenged, we talk it through with each other, we help each other come to an understanding.  We break into independent practice work where I can catch the students who still feel like a 1 or 2.  Then we close the lesson with an exit slip or an assignment, rating ourselves once again to see where we fall on the scale.  I take a look at what they wrote for their final rating and catch those students during the review, intervention block or some recess time the next day.

This seems like a lot of work, and I won’t lie that at first it was for me. It was a different way of thinking.  But soon after I started to do this, I noticed that it was easier and easier to think about what a 0-4 looks like.  If I ever skipped the rating part, my students would actually shout at me “What does a 3 look like?!” They wanted to know what it would take to be successful! It was very powerful.  You may not have time to write it out like this for every lesson, but you can do it orally while referring to the levels on the wall.

This tweak to my instruction was a total game changer.  Thank you John Hattie and Robert Marzano for your inspiration!

# Real Life Examples of Geometry

The number of terms that students are expected to learn in geometry is a little crazy.  We counted 30 different new vocabulary words at the end of four days of instruction.  So I checked out an iPad cart and decided to have the students find real life examples of geometry in the world around them. After introducing the symbols, and describing each term’s features…they solidified their understanding of each new word with photos. (We pulled out some of the trickier ones from our minds as well.)

We recorded the findings on a giant chart!

Students captured real life examples of: point, line segment, line, ray, intersecting lines, perpendicular lines, and parallel lines with iPads.

It was both motivating and fun to use technology, as well as promote math talk in the classroom.

# Open Ended Math Problems Promote Reading, Writing AND Math

Last spring I had the opportunity to take a practice version of our new state assessment (the Smarter Balanced Assessment). In some states in the U.S. the PARCC is the new assessment which is similar in nature.

Talk about a jaw dropping, sweat on my forehead, instant anxiety through my whole body moment.

What the students are being asked to do is way more than a few math problems. They are expected to read, write and use appropriate grade level math in VERY complex ways. I realized that I needed to add some deep problem solving to my math instruction.  So I began to make open ended problem solving problems to introduce regularly into the classroom.

I decided to create Doggy Dilemma, a free problem for anyone to try out.  It is a highly motivating, real world problem in which students must read through information to decide what dog they must adopt. They draw a diagram of the dog pen, calculate the cost of the fencing, and write a letter to their parents explaining why they made the choices they did.

My third graders have gone crazy over it.  They love it!  There are two full pages of reading involved which mimics the new assessments.  I have enjoyed creating it and want to make it available to anyone who teaches elementary math so that you can give your students the experience they need before the real assessments begin. You can get it by clicking on the picture below:

I’d love to hear how other teachers are encouraging this type of thinking in their classrooms. Please feel free to share in the comments!

I am happy to link up here:

# Spotlight: 4 Really Cool Math Educators

I’ve really always been a learner, everything I ever do I just try to read and read and soak in every bit of information I can.  It is both a blessing and a curse!

If you are like me (you also have the drive and passion for getting better at teaching math), you have this never ending quest to read about math.  I wanted to introduce you to some great teachers I’ve been virtually meeting along the way.  They are just full of great ideas and also have a lot of interesting things to say on their own blogs as well. Check them out!

Evil Math Wizard: She is the first person I met virtually when I got started blogging.  We definitely think alike and have the same goals for our students in math!

The Elementary Math Maniac:  I also have noticed that we have similar beliefs in math. She also connects technology really well to learning, and reviews websites and apps on her blog.

The Research Based Classroom:  You won’t find just math here at her site, but other subjects as well.  She is focused on those very young learners, and believes in research based teaching methods. Outstanding!

Mr. Elementary Math: Full of great ideas, you can find Greg blogging about all sorts of classroom activities with lots of bright, vivid photos.

Enjoy meeting these fine folks!

# Should We Ignore Them? (Tips for When Problem Solving Gets Tough)

Sometimes I feel like a magnet, with a trail of students behind me as I walk around to conference/help during work time.  We are working on Open Ended Word Problem Challenges right now (I have gone through set one in the first quarter, and we are beginning set two.) These problems include a lot of reading, are many steps, and are open ended.  There can be more than one right answer.

So they hit the panic button right away!

Right now, I am in the middle of training my students to trust themselves, to be okay with feeling a little uncomfortable. I want them to seek the answers to their problem WITHOUT me.  This is very hard for them, especially when we are working on challenging math concepts.

Here is what one of those problems looks like!

Here are some ways that I try to raise rigor, and to help students persevere:

1. Ignore them! (What? Are you kidding? How horrible!) Of course the kind of ignoring I am talking about, is the kind where they ask for your help without trying the problem first.  There is nothing worse than when you pass out a tough problem, and the hands go up immediately. This leads to my next tip, a very simple tip.

2.  Make sure the students read the problem three times. Read it once to get familiar, read it a second time to zoom in to what you need to do, then read it even closer a third time to circle key details. The answer to their question is almost always in the problem. Most times I’ll read it out loud!

3.  Encourage students to do what they can in the problem while they wait for help. Sitting there with a hand up, or following the teacher around, trains students that they must rely on the teacher to continue on. When I approach students my first question is always: “What parts did you understand?” They realize that they can do much more than they originally thought.

4.  I teach routines when solving problems. For example, my students cannot actually get up and follow me, rather they wait as I circulate so that everyone gets equal time. Sometimes I’ll have a schedule posted where I meet with small groups.  Knowing that they will all get equal time with me makes everyone relax (including me!).

Teach the students that an “I can do it!” attitude is the most powerful problem solving strategy!