I start today by putting a drawing on the board of the number 36 in base 10 blocks, and 20 in base 10 blocks. I ask students how much 36 + 20 might be. They record their answers in their math journals. Next I ask for students to share their solutions to the problem. As they talk I create "concrete referents" to their conversation by writing numbers discussed, pointing to the representational drawing I made of the problem, and pointing to either the number line or number grid, or drawing student representational drawings, depending on what they say.
I repeat the process for 36 - 20.
I ask students to think of some of the models that help us solve math problems. Models are a semi-abstract construction of our thinking, while strategies are tools to increase our efficiency. So, a number line is a thinking model of how numbers actually work (2 comes before 3, numbers go on and on), and we can rely on them as a model when we want to use strategies such as “making ten” to compensate when adding. I list their suggestions on the board (number line, number grid, counting, drawings, etc.). I tell students that these are all models that we can make or use to help us when we are trying to figure out a problem. Modeling with mathematics is an important strategy to encourage student to use in solving a problem (MP4).
I remind students that we have looked for the mystery number (the missing number) in a number sentence and we have found the words that help us in a word problem. Now we are going from, "I know what to do" (add, subtract, compare), to figuring out approaches to help us find solutions.
An example of this is, if I see 36 + 20 I know what to do. I have to add, but what or how are our next steps to be figured out, and that's where "thinking" models can help us.
Today we will practice with several ways of modeling what we want to find out so we can get the answer.
To meet 2.NBT.7, students should have ongoing experiences using models (number lines, number grids), representations (pictures, drawings) and concrete materials to support their work. These have to be introduced in context, alongside the strategies derived from place value and properties of operations that are most appropriate for these experiences. Today's objective is to get students thinking about the possible models they can use to help them make sense of math.
I explain to students that today they will rotate through 3 centers where they will work with different types of math models. They will use the models to solve the problems that they find at each center. At the end of the day I will ask them to tell which models they liked and might remember to use again, and which made it harder for them. I tell them that just like reading where we all like different books, in math we may like different models.
Center 1: I introduce a format where students have a square with 3 sections. The top (largest) is labeled total. The two smaller sections underneath are labeled part/part. Students work with adding the prices of two objects, putting one price in each box and then adding with concrete manipulatives or pictures to get an answer to put in the total box.
Center 2: Here students take 3 coins from a pile without looking. They record what they have picked up and then try to find the total. They may choose to use number sentences, count on number grids or number lines, or count the coins directly to get the total.
Center 3: Students will build 2 numbers with base 10 blocks, and then add the tens together, and then the ones together to get an answer. They will be encouraged to trade the ones in for another ten if they count more than 9 ones.
I ask students to look at the paper I give them and to circle the models they find the easiest to help them with addition and subtraction problems. Tools I can Use to Build Math Models.docx They can pick more than one model that they might use in the future when they have a problem they can't do in their head.