Circle and Stars: Grouping Compared to Arrays

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SWBAT compare how to represent a product of two factors by grouping or by using arrays and decide which is a better tool to represent factor pairs.

Big Idea

Students explain the difference between arrays and grouping to represent multiplication.

Circle and Stars Game

15 minutes

I opened up the lesson today with an activity that was the base for the whole lesson. Circle and Stars is a grouping game. While students have mastered the concept of using arrays, another model, grouping, had been taught in the third grade to introduce multiplication. Rationale: I thought it was a good idea to compare the two models and  decide which model is the best tool to help us with identifying factor pairs. They need to see that the neatness and simplicity of arrays is easier. I also what them to understand the wording of grouping in an equation in relationship to what the grouping should look like. Example: 4 groups of 3 will be 4 circles with three stars. This reinforces the language part of a factor pair in a different model.

I brought the Smart Board up and tried to use the interactive dice, thinking that would be the best way for students to participate. It didn't work! The dice were just a black box on my page. Time for an upgrade or something! Time to think of something quick!

I resorted to borrowing my colleague's big dice. I was given two large regular dice and one geodesic die with larger numbers on it. I rolled one die on the floor and produced the number 1. Not exciting, but it served as the 1 group of ...and I rolled the second die, which produced a 5. I drew 5 stars in my circle. ( Thus the name: Circle and Stars). 

I tossed a die to one student ( they love to have things thrown to them unexpectedly) and another the opposite direction. I asked the first student to toss the die. It rolled my way producing a 4. The next student rolled the second die. It was a three. I realized that the big dice produced an element of excitement that the interactive Smart Board dice do not. 

I drew 12 stars on the board and asked a volunteer to group them as I wrote 4 groups of 3 or 4x3 on the Smart Board page. A student who really struggles volunteered so eagerly, I called on him. As he worked, he struggled, and as he proceeded, I struggled with myself to let him make the mistakes he was making. Students chimed in and I hushed them. I decided to let him go and rephrase my directions somehow to help him see the grouping.Student struggles. You can see the process on this Circle and Stars Lesson Sample ages 1-4. 

As I led him, he finally got it! Then we went over the process of his learning again. I wondered if this was embarrassing him, but he was so happy to explain his understanding, I realized that our classroom environment is a safe place to take risks and he felt comfortable.

After he sat down, I posed the question: Does it make more sense to use arrays to group or groups like "Circle and Stars?" They weren't sure, so I rolled the big geodesic die. Up popped 13. So, I wrote 13 groups of... and rolled again. The number 3 popped up. I asked them how I could figure out the product. One of my students offered up the answer of 39 and I started to draw 39 stars.

I acted as if I was confused. I stopped, turned and said " Wait a minute...I have to know the product in order to group! Does this serve as a tool for finding products or not? " 

I got a resounding "NO!" 

I asked, " Then what purpose does it serve?"  I didnt' get answer. So I suggested that maybe arrays would have been the best tool. I quickly drew the array. One student said " That was fast!"

I turned and said, "Yes, and we use arrays as our tool for finding products and unknown factors. The grouping  is simply a model of being able to explain that 39 would be 13 times more than 3. (This supports the standard.)

I brought my student back up to the board to recap the whole process:.Going over the learning process. I think it is good for other students to see how another student processes learning and grows from someone coaching. I am grateful that he felt comfortable and just kept trying to get it. I love the comment from my student about groups being "clusters". That turned up again later in the lesson. It's a good word to describe this and will transfer into graphing.



Grouping 24

10 minutes

Student sample of grouping in notebookI asked students to practice making groups once more. I asked them to draw 24 stars in their notebooks and show the drawing of 6 x 4. I roved around the room and found several good samples of work. They had mastered it. Both of these students grasp and can explain what they did very well. The CCSS promote ownership of learning. Explaining is a way of proving their ownership. This student explains grouping fairly simply. I like how she talks about discovering one of her groups was missing a star.  This Student explains his grouping easily and meticulously. 

I felt the lesson was done at this point and it was time to write about our understanding.

The Writing Assignment: Combined math and writing lesson

30 minutes

CCSS writing standards demand that students write and explain their thinking, procedures and use facts to back up their opinions. This assignment satisfies two standards as well as solidifies the difference between using an array and grouping. It is my hope through this assignment that students become aware that there is more than one way of thinking about meanings of expressions and the placement of numbers. On the last page of the Circle and Stars Lesson, the writing prompt is listed. I am looking for their ability to answer the question and compare the two multiplication ideas. It is also my hope that through writing about the two, they can use critical thinking to understand the idea that array models are easier to identify factor pairs of a product versus using a grouping strategy.

* This transitioned into my writing class very smoothly. I assigned time to plan, and time to write and let them work. After 20 minutes, I stopped them. I asked them to get up and mill to music with their writing in their hands. I wanted them to look at each other's work and then link arms. Eventually their arms were all linked and we formed a circle. Students shared their work aloud. It was a great moment for me to assess their understanding. To integrate writing skills and explanations in math is to deepen their understanding. I used this opportunity to teach them to peer edit. After each student, I talked about saying "wishes and stars." I would say " I wish you would..." for my suggestions to improve, and then say " I really like how you..." to praise good work. After about three students, I asked students to say the "wishes and stars."

They went back to their desks when we were finished and completed their editing process and turned them in. These will be collected into their portfolios for later use.