Monday, November 2, 2015

Math Tip Monday: Operations & Algebraic Thinking

Welcome back to Math Tip Monday, a monthly blog hop filled with wonderful ideas and resources for teaching Mathematics.  This month we're focusing on Operations & Algebraic Thinking.

I remember seeing the Common Core State Standards for the first time a few years ago. When I saw that standard, I shook my head in disbelief. "Algebraic thinking," I thought incredulously, "but these are first and second graders!?" I don't think I realized then how much my instruction formed the foundation for what my students would be expected to do years in the future. I tend to overlook the label now and focus on making sure my students understand these important math concepts.  For the purposes of this post, I am providing tips for the three second grade standards included in this thread.

Represent and solve problems involving addition and subtraction.

Story problems can be so challenging for students at all grade levels. To me, the most important way I can help my students to solve them is to help them to understand the problem. I don't believe in teaching key words, because sometimes those words can be misleading in a story problem. Instead, I help students use reading comprehension strategies to understand the problem before attempting to solve it.

During our math talk time, I introduce a story problem. We read and reread the problem. I ask students to restate the problem in their own words. We visualize the problem or we act out the problem, both physically and by using math tools (counters, cubes, etc.).  Only then do I ask students to tell me what operation would help to solve the problem. I always ask them to explain why or how they know that's the correct operation.

It seems like a lot of work to solve one problem, but I think these strategies are crucial to helping my students become successful mathematicians.  So many times, I've seen excellent math students bomb a test because they added the numbers in subtraction problems or vice versa. Emphasizing comprehension of story problems helps prevent this type of error. Here's a poster I made to remind my students of the steps to solving story problems.

Add and subtract within 20.

Teacher confession: I hate the traditional use of flash cards and timed fact tests. I think they are tedious and, for students who have difficulty with memorizing (like myself, in third and fourth grade), they make math seem more difficult than it is.  The key to this standard is teaching students effective and efficient mental math strategies and then providing them with ample opportunities to practice using those strategies. Here are three ways I help students with fact fluency.

1) Make Mental Math Visible

As part of our math talks or my math mini-lesson, I often help students to "show their thinking." I'll ask them to explain how they got the answer to a problem. I'll then draw an empty number line, break down the equation, or jot a quick picture to show that thinking.

Here are some ways we might show 8 + 7.
In this example, the student used a known doubles fact to help solve the problem. I would rewrite the problem as above to help students to visualize the strategy.

I often use empty number lines to show student thinking. This is one way I might show how a student used the Making 10 strategy to solve the problem quickly and accurately.

2) Help students to see and use number patterns. 

When we teach fact fluency, we start with the Plus 1 facts. Then we move on to the Plus 2 facts. Next, we focus on doubles. Then, we introduce near doubles.  One of the ways I help students to see the patterns for each of these facts is by giving them time to explore the answers on their own. We use math tools, such as cubes, counters, number lines, and the one hundred twenty chart. I allow my students time to find the answers to the equations on their own. Next, I make an organized list of the facts.

I ask students to look for patterns in the list. What do they notice? How does that help them to solve this type of problem? I love it when the kids get excited at discovering the patterns on their own!

3) Use math games to practice!

I shared this idea last month. Check out my post for some great games to use for practicing fact fluency.

Work with equal groups of objects to gain foundations for multiplication.

Patterns, patterns, patterns! Whether you're working on even versus odd numbers, repeated addition, partitioning, or rectangular arrays, helping students see and understand number patterns is crucial to building algebraic thinking.
Just as I use an organized list to help students to find number patterns and increase fact fluency, I often have students look for patterns as they work on this standard. We list even numbers and odd numbers, comparing them and looking for clues to identify each. We use objects and draw pictures to demonstrate why a number is even or why it is odd. Students have to justify their thinking to demonstrate understanding. Here's one way to help your students see patterns during a math talk.
Write on chart paper or create a slide with a question similar to this one.

Some of my students can tell you the answer to the first question in two seconds. Because of this, I always put more emphasis on the second question than the first. We draw pictures and write equations (often students will use doubles facts to explain) to "prove" that the number is odd. By focusing on the "why" or the "how do you know" part of the question, you are encouraging your students to share their strategies with their peers. You are building conceptual understanding, rather than a surface recall of whether a number is odd or even.
During math talks for our multiplication unit, I also encourage students to discover patterns. We build arrays using cubes or counters. We draw them on grid paper. We record repeated addition equations. We look for patterns and draw connections to what we already know. I love to give my students 24 color tiles and ask them to make rectangles. 
We draw our solutions on grid paper. Then we record the repeated addition sentences.
I help students to see that this array represents 4 + 4 + 4 + 4 + 4 + 4 =  24. Often students notice that it can also represent 6 + 6 + 6 + 6 = 24. In this way, my students are already internalizing the commutative property of multiplication. They are building connections that will help them when they begin multiplication in earnest in third grade.

So these are just some of the ideas that I use in my classroom to help students build a foundation in Operations and Algebraic Thinking. I hope you find these tips helpful. You may also want to check out the variety of math products I have available in my TPT store
In the meantime, I hope you'll hop to the next blog to find more tips about this important math topic.

An InLinkz Link-up

Until next time, 

1 comment:

  1. You have a very good collection of activities to help with understanding addition, subtraction, and odd/even. So glad you have the kids record the addition sentences. there is the part of the standards for addition/subtraction in second grade that says these single digit facts must be memorized. Writing the facts will certainly help them reach that goal!