It's October, so it's still relatively early in the school year. During this month's Module, you will be helping students to work with addition and subtraction equations. Did you notice, though, that there is an emphasis on using a variety of structures?
Problem Structures are a very important part of understanding addition and subtraction equations and learning to be flexible with the numbers. In later years, when our students begin their Pre-Algebra and Algebra courses, the ability to understand Problem Structures becomes the foundation of solving algebraic equations.
So before we dive in, let's take a few moments to study the different types of Problem Structures. As you look at these examples, notice the importance of understanding where the missing part will occur in the problem. If we are not purposeful, we can limit our students' understanding by overusing a single type of structure, such as the very popular "Result Unknown" structure. By being mindful of the various Problem Structures, we can work progressively toward including a wide variety of Problem Structures over the course of the school year.
Phase I: CONCRETE
Remember... this is simply ONE idea among a million possibilities. When beginning to work with problem structures, we always want students to experience new concepts using concrete materials. For this example, we will use small paper plates and counters.
To give more meaning and power to the number sentences, try to always include context (a story) for the numbers with each problem.
Tell your students that today we are going to "build" math equations.
Tell your students that today we are going to "build" math equations.
Give each student a collection of cubes (or other counters), 3 small containers (such as a simple paper plates), and some mathematical symbol cards [TIP: create two cards for each student – the first card has an addition symbol (+) on one side and a subtraction symbol (–) on the other side. The second card has an equal symbol (=) on both sides].
SAY: Today, boys and girls, we are going to "build" some math equations. On your desks, you each have some counters, a small paper plate, and some math symbol cards.
Listen carefully to this story: When I was taking a walk yesterday, I noticed there were 4 ducks on the pond. As I watched the ducks swim around, some more ducks flew down and joined them. When I counted all of the ducks, I saw that there were now 9 ducks swimming on the pond.
Can you help me create a Number Sentence for the story I just told you? The kind of number sentence we will talk about today will have three parts: A beginning, a middle, and an end. We will have a number that is added or subtracted with another and then we will have an answer. It will look like this: Use the paper plates to demonstrate where the numbers will go. Flip the +/– symbol over to show that they might need the addition symbol or they may need the subtraction symbol depending on what happens in the story.
- Read the story again: When I was taking a walk yesterday, I noticed there were 4 ducks on the pond. As I watched the ducks swim around, some more ducks flew down and joined them. When I counted all of the ducks, I saw that there were now 9 ducks swimming on the pond.
- What do we know from the story? As students begin to retell the story, use counters to represent the ducks.
- Did I know how many ducks were on the pond when I first arrived? Yes, there were 4 ducks. Let's put four counters at the beginning of our equation to represent those four ducks.
- What happened next? Yes, some more ducks flew in. If more ducks flew in and joined the first four ducks, are we adding more ducks to the pond or are we taking away some ducks? Yes, we are adding ducks, so let's put the addition symbol here next to the four counters to show that we are going to add some more ducks.
- Does the story tell us how many ducks we added? Correct. The story just says that "some more flew in". Let's we have to leave that plate empty until we know how many were added.
- Think about the end of the story. Do we know how many ducks were on the pond at the end of the story? Yes, there were 9 ducks on the pond, so let's place 9 counters at the end of our number sentence to represent all nine ducks.
At this point, we will begin a process of thinking through the possibilities. This is an important step for many students, so you don't want to skip it.
- Okay, so let's imagine that ONE more duck flew in. What number would we add? Yes, a 1. Put one counter on the second plate to represent one more duck.
- Let's count up all of the counters. Count or say "4" and then add on the 1 more to make 5.
- Ask, were there 5 ducks at the end of the story? No, there were 9 ducks. Is 9 more or less than 5? (more) So we know that MORE than just ONE duck flew in. Let's try 2.
- Repeat this process with 2 counters. Add up the 4 plus the addtional 2. Does it add up to 9?
- Repeat with 3 and 4 taking the time to count each time and checking to see if it is the total of 9 that you need.
- Finally, try it by adding 5. This one will work.
Repeat this process with a new story that uses the same or perhaps a different story structure:
Jessie had some crayons on the table. Her brother Mark borrowed 4 of the crayons to make a drawing. Jessie only has 2 crayons left. Can we use the counters and paper plates to make a model of this question? Guide students through each phase by asking about the beginning, the operation needed, the part added/subtracted, and the ending.
Phase II: REPRESENTATIONAL
Once students have a deep understanding, you can continue this process by drawing simple dots or tallies to represent each counter. Dry erase boards, a template inside of a plastic sleeve that allows students to write on and wipe off, or the Smart Board are all good tools to use during this Phase allowing students to easily draw and adjust their representations as they work.
Phase III: ABSTRACT
The Abstract Phase of this process uses numbers in a typical equation. An excellent way to smoothly transition into the Abstract Phase is to pair the numbers with the counters during the Concrete Phase and then pair the numbers again with the drawings during the Representational Phase. Eventually working toward using just the numbers and mathematical symbols to represent the story.
