Teaching algebraic thinking and other problem solving skills can help students build math understanding like the concept of equal and other basic algebra thinking concepts. Perfect for grade 3 math, grade 4 math, math workshop, math centers, math intervention

Algebraic Thinking in Elementary School? YES!

One thing that I have discovered over the years is that students can be very trained to "fill in the box".  The more focus we put on our math series and workbooks, the more true this is.  In my mind, it is our obligation as elementary math teachers to develop mathematical thinking in our classrooms--and there are so many ways to nurture this...providing students with many rigorous problem solving opportunities, having in-depth mathematical discussions, doing number talks, providing many different experiences to build numbers sense--and developing algebra thinking in our students.

What?  Algebra?  YES.

In a nutshell, algebra is the area of math that deals with mathematical symbols and the rules that exist for using them.  As students get more sophisticated with their thinking, of course these symbols and rules get more complex.  For me, one of the key symbols we can address in the intermediate grades is the equal sign.

The concepts of “equals” is such an essential mathematical concept—and one that is often not taught explicitly in many math programs.  Consider the following problem:
7 + 5 =  ___ + 3
Students have a good sense of the concept of equals know that “9” goes in the box.  Every year, I am shocked by the number of students who think it is “12”—or that the problem can’t be done at all!  Many students are programmed to think that the “=“ sign means “find the answer” when, in actuality, it is a sign indicating that the two sides balance.  Deep understanding of this helps down the road with algebra and other more advanced math—and also makes a smooth transition to INequalities as well.  Moving our “box” (or “variable”, or a letter, or a line, or a question mark) to different places in equations helps students think about numbers and equality and not just filling in the blank.  So--give it a try!  See where your students are!
Teaching algebraic thinking and other problem solving skills can help students build math understanding like the concept of equal and other basic algebra thinking concepts. Perfect for grade 3 math, grade 4 math, math workshop, math centers, math intervention
After we have a pretty in-depth discussion about this problem, we do a few more similar ones together and then I send my students off in pairs to try a BUNCH more--where they come up with the answer and must defend it by explaining the math.  I want them using language like "In order to make the equation balance, both sides must be worth 13...the right side already is, and the left side needs 6 more to make 13." or something like that!  I walk around and monitor and coach this.  I really play this up...explain that they are doing ALGEBRA and they love it!  I tell them that we can make these problems more and more challenging as the year goes on--and we do! 
Teaching algebraic thinking and other problem solving skills can help students build math understanding like the concept of equal and other basic algebra thinking concepts. Perfect for grade 3 math, grade 4 math, math workshop, math centers, math intervention
 Because we are working with whole numbers right now, I continue to have my students working on this "concept of equals" with harder and harder problems.  Some of my students are even ready for an enrichment group where we learn about order of operations so we can solve some really fun ones!
Teaching algebraic thinking and other problem solving skills can help students build math understanding like the concept of equal and other basic algebra thinking concepts. Perfect for grade 3 math, grade 4 math, math workshop, math centers, math intervention
 As the year progresses, we can even do similar problems with decimals, fractions, measurement concepts and more!

Algebra thinking for math intervention?  Sure!

For some of my students, they need a little reinforcement with this.  I met with a group to do more of the single digit cards where we broke out some manipulatives to model the balance.  I have even pulled out a pan balance in the past to REALLY show this idea of "balance". With intervention, it's always smart to make things as concrete as possible.  Using counters or objects is always the best place to start.  Once my students started getting more confidence, I used this "The Missing Piece" game to start to expand the concept to double digit numbers so students really see how "two parts" can make a "whole".  Even my stronger students love this game because they start to make some predictions about numbers they "need"--it's fun to watch their growing mathematical thinking
Teaching algebraic thinking and other problem solving skills can help students build math understanding like the concept of equal and other basic algebra thinking concepts. Perfect for grade 3 math, grade 4 math, math workshop, math centers, math intervention

Even more algebraic thinking challenges!

Another great way to get student thinking algebraically is to provide students with problems where they have to "play" with numbers a little...like these "balance"-type problems.  Again, our students are SO used to filling in the blanks that this type of problem is a great way to get students thinking in new ways.  Instead of asking "What is 1 + 7?" or even "What are all the ways to make 8?", these ask students to guess and check and keep muliple numbers in their minds.  

They need to figure out what numbers go in each "bubble" to make the sum on the little bubbles true for ALL three (or four on some cards) true.  This gets students really thinking about how numbers compose and decompose--so important!  Once students start to think more flexibly, I start to do this type of problem with larger numbers.  
Teaching algebraic thinking and other problem solving skills can help students build math understanding like the concept of equal and other basic algebra thinking concepts. Perfect for grade 3 math, grade 4 math, math workshop, math centers, math intervention

Another way to get students thinking algebraically is by extending the students' work with "equal" and reasoning by using balance mobile problems.  These extend my students' thinking about equal, about "halving", and other key number sense concepts.  I should really write a blog post about them because I LOVE the lessons that come out...from perseverance to "making sense of problems" to modeling with math--and more.  I did make a little video with a few details...


So--I'd LOVE for you to give some of these ideas with your students and then let me know how it goes!  If you want a set of task cards that have a whole bunch of problems like the ones at the top of the post (at 4 levels), just click the image below to see what I have been using with my students.  Enjoy--and make sure to tell me what you find! If you are interested in the other resources, just click the images.
Interested in "The Missing Piece"?  It's one of the five addition and subtraction games in this resource:

Want to pin this for later?  Here you go!
Teaching algebraic thinking and other problem solving skills can help students build math understanding like the concept of equal and other basic algebra thinking concepts. Perfect for grade 3 math, grade 4 math, math workshop, math centers, math intervention


Interested in today's post over at Upper Elementary Snapshots?  Just click either image to take you there!  Have a wonderful day--and a very happy Thanksgiving as well.


As I did my planning for math last week, I really spent some time thinking about how we rush, rush, rush to get through our curriculum sometimes.  In fact, I think having a math SERIES makes it so much worse--we start to think of teaching math as teaching lessons and chapters and units instead of teaching math concepts.  "What lesson are you on?" is a question that makes me cringe when exchanged between teachers because it's sending the message that we are on a timeline and we are trying to get our students to learn math in nice and tidy hour-long chunks.

That isn't how it works.

So this week, we have several lessons in our series that are meant to reinforce the relationship between multiplication and division.  I know it was taught in 3rd grade.  I know some (SOME) of my students "know" their facts.  But I also know that many don't--and if I just keep plowing through the book they never will.  I wanted to slow down and really manipulate numbers to get them thinking about groups and sharing.

Last year I played a game with my students that I called "The Herding Game".  I found an open space in my building (last year an empty classroom, this year a big hall space by the elevator) and told my students they were animals.  They, of course, are used to me calling them this (I frequently say things like, "OK, wombats, let's go to music.") so they didn't flinch.  I explained that for this activity they truly would be animals--animals that live in herds.  We brainstormed a list and then they had to do the following:

When I call out the name of an animal, I will also tell you how big your "herd" needs to be.  You need to quickly form "herds", and any animals who can't form a herd will head over to the holding pen (a taped off area on the floor).  My rules?  You can't be in the pen more than once.

So we got started.  We started the activity with 24 students and I called out, "Buffalo, herds of 6!".  Off they roamed to make their herds.  We noticed how we had no animals in the pen, so I wrote the equation 24 / 6 = 4 on a white board and we discussed how nice and evenly it worked out.  We tried several more combinations...we made herds of 5 (leftovers!), herds of 3 (no leftovers!), herds of 10 (leftover!) and so on.,  Each time, I showed them the matching division equation AND the matching multiplication equation.  2 x 10 + 4 = 24.  The wheels were starting to turn...and students were tuned in to how many animals would be in the pen even before they had formed their herds.
After about 12 minutes, we headed back to the classroom and I taught them the pretzel game.  This was a game I used with an intervention group last year, but I was pretty confident that I could really get the students thinking so I played a game against one of my students with the rest of the class watching and we really started to dig in to division concepts, making predictions about how many remainders there would be, and so on.
 The gist of this game is the same as the herding game...we started with 34 pretzels and took turns rolling the dice to determine the number of bowls we could put them in.  On any turn, any leftovers ("remainders") go to that player.  As my opponent and I got going, I LOVED the discussion that the two of us modeled..."I have 34 pretzels and need to sort them into 4 bowls...I know that 4 groups of 8 pretzels gives me 32--so I have 2 leftovers." or "I have 30 pretzels to sort into 2 bowls, that is easy because I know that 2 groups of 15 is 30."

As we went, the other students did exactly what I was hoping they would do--they started predicting.  "Wait--the only number that will give leftovers is 5 because you can divide 24 into 1, 2, 3, 4, and 6 groups!" and so on.  Their minds were really starting to think logically and recognize the connection between multiplication and division.  We finished our game (I won--in case you are interested) and the students BEGGED to play it on their own.

First things first--we need to do some work in our math book--but let me tell you how fast they were able to do the problems!  They needed to solve problems like 34 divided by 5--and I could HEAR students talking to each other in terms of pretzels and bowls!  As they worked, I pulled 3 students who I didn't think were ready and played a round of the game with them to really slow things down and get them doing it themselves.  Next week I'm going to pull them to do my paper version of the herding game to keep modeling this for them.
As students finished, they could choose from a variety of multiplication and division games to work on fluency...and I felt really good that I taught MATH, not lesson 4.9 (even though I did).  Interested in the games and activities?  They are linked below.  Have a great weekend!





If you have  followed me for any length of time, you know I LOVE using concepts sorts to get students talking and for ME to check for misconceptions.  This solids, liquids, and gases sort was no different--except I took a slightly different approach today!

We have NOT started this unit yet--this lesson was the kickoff.  I wanted to hear the students talking about what they know to be true--AND what they BELIEVE to be true!  We started by getting our sort cards prepared...
The students worked in trios and started having discussions about what the items were--and where they believed they belonged.  (Sand?  A liquid?  I had to hear more..."It pours."  Misconception #1)
After the groups took about 10 minutes preparing and sorting their cards, it was time for a gallery walk.  Each team traveled to check out the other groups to see what they had done--and to see if there were any cards they questioned.
I gave each trio 2 sticky flags that they had to judiciously use...they only got two.  They placed them on the two cards they thought we should talk about.  (You'll notice that the "sand" was a big question!  So were pudding, pillow, and steam.)
We then argued (I mean had quality academic discourse) about some of the tricky ones.  I gave no answers (remember, this is day 1!) and I had each group put all of their sort pieces into an envelope to revisit after we have learned a bit more!  

Interested in seeing the resource?  You could EASILY do this on your own, but if you want everything put together in a neat and tidy package with directions and a quick assessment, here's the link.  


As we begin our unit on opinion writing, I wanted my students to realize that there are many different types of opinion statements…from as simple as requiring a yes or no answer (“Do you agree that school uniforms should be required?”) to more numerical/quantifiable opinions (“How many days long should our school year be?”) to more open-ended statements requiring students to provide their own idea (“In your opinion, what is the best brand of pizza?”).  

To get my students thinking, I provided them with 32 topics on cards and asked them to work with a team to sort them into categories and to be ready to give each category a label.  I wasn’t really interested in a “correct” answer—just in getting them to realize that opinion statements come in many shapes and sizes!  Different groups sorted in different ways, and it was a lot of fun to talk about all the different categories they made.  This was our first day of our opinion writing unit—and the students got REALLY excited about some of the topics!  

By the time we finished, we were able to have a great discussion about what an opinion is and how when we write about our opinions, we need to be able to state them clearly and back them up with several strong, reasonable reasons!  Not bad for a day’s work!
After day 1, many of my students were buzzing about topics that they were super interested in writing about...so mission accomplished!  Today, I asked the students to use the cards again to do another sort…but this time the sort became a little more personal.  I asked them to work on their own using the “mini cards” to sort them in a new way…topics they have strong feelings about, topics that they are more neutral about, and topics that they aren’t interested in very much at all.  Once they were sorted, I had them glue their “I totally could see myself writing about this!” cards right into their writing notebook.  I had them talk with their table groups about their choices—and some students actually added some new choices onto their lists. It was fun to see them take opinions that were on the cards and "tweak" them to be more to their liking.

By the time we finished this lesson, students were able to see that the topics that interested them the most were the ones where they had some experiences or many compelling reasons why they felt the way they did.  I reminded them that good writers choose topics they feel “expert” on—or that they can become experts on!  Before we finished, I asked them to pick one topic to write on the planning page so they would be ready to write on day 3!  We will do a few days of "quick writes" on these topics before we really dig into a true "essay" format.

Interested in these cards?  I put everything I use for my first 3 days in one nice and tidy package!  Click the image below if you are interested.