The children have become proficient in building an array to "prove" their work, but I realize the amount of time it takes to construct one, that represents a larger product, is taking too long and is more likely to hold mistakes.
So today's lesson opens with examples of "quick arrays" that can be used for multiplication and division. A quick array is essentially the first row and first column of an array. It sets up the factors and helps the students move away from counting one by one.
Boys and girls, I have been noticing that you are doing a wonderful job building arrays to show your work with multiplication. I was thinking that maybe you would like to see a strategy of how to represent those larger multiplication equations.
Show the students these quick arrays and allow them a moment to study them. Ask them to describe what they notice and how these arrays might work and be helpful.
You may be wondering why an image of a Desert Hedgehog is representative of this lesson. To find out, check what a group of hedgehogs is called.
I made a decision with this lesson to have the students work in pairs to create posters to show what they know about multiplication. My thinking in this was to determine if the work we are doing in class is helping the children grow deeper in their understanding of what multiplication is and how it is related to addition and division.
Mathematicians, I have a problem up here on the board. It is 3 x 7 = n. Let's look at all the ways we can tell the "story" of 3 x 7 = n. I know I could use an array. I will draw it here. Can you turn and talk with your partner about other ways to explain 3 x 7 = n?
If you have not introduced variables, then simply use a "?" or even a blank line following the equal sign. The goal of this lesson is to have students represent their multiplicative thinking.
Following their discussions, add their ideas to the board. Guide students, if they don't mention it, to think of quick arrays and addition problems. When considering the Common Core and what children need to be able to do with multiplication, I think this discussion is important. Multiplication requires that students need to think in terms of groups of things, rather than individual things. This repetitive addition and array representation will help them see this concept and guide you to what is missing in their understanding (MP7).
Okay, mathematicians, I am going to send you off with a partner and an equation. I would like you to create a poster of the story of your equation. What you will teach us. In 20 minutes, we will share our posters and then put them in the hallway to show everyone your smart thinking.
As students work in pairs, roam the room, guiding them to think in terms of showing and telling what they know. Prompt students, as they speak, to use vocabulary associated with the representations and watch for misunderstandings. Also, ask them what their strategies are and what they are doing. Sometimes things that look correct, when explained are founded on or include misconceptions.
Watch for misunderstandings while students are working. Many students will draw the opposite array or group model for the equation. This is the misconception we need to correct. Mistakes arise from simple mis-counts or being unorganized. Really watch for the equal groupings.
After the students have had enough time to put down several ideas, I ask them to stand silently. I explain that we are going to do a silent "poster tour" in order to gain more ideas. This is the beginning of creating a community of learners that can critique another's work and learn from peers. Students will be looking for more ideas of how to represent their own thinking, but some may even begin to critique other's work.
Students, it has occurred to me that everyone is doing such smart thinking that maybe we shouldn't just wait until our sharing time. I am going to set the timer for one minute. During that time, please take a silent poster tour. Notice what the other mathematicians are doing to explain their thinking. When the timer sounds, please go back to your poster and continue your work.
As the students share their posters, ask clarifying questions. It is also a good time to begin working with the audience on asking questions, which is tricky for third graders at this time. They always can add information, but asking clarifying questions is a challenge.
These students are learning to share their work at the board. Everything at the beginning of the year is a process, even how to show are work to others and find the right vocabulary (notice the word "cool" in the share)!