Why is it important to teach conceptual understanding of multiplication?
Challenge: Print the Math Cards mentioned on page 40. Design a simple activity using these cards that would be meaningful for your students. Share a description of the activity in the discussion board.
Comments
I used the math card activity with my students today and it worked out better than I had anticipated. I decided to use them for a number line activity, since my students are familiar with working with a number line. The activity I usually do with them involves placing cards with values that range between -4 and 4, but I decided that for this activity, I would just put up a line and let them decide what the starting and ending values would be. So, I taped a piece of rope to the wall and gave each student a card and instructed them to work together to place the cards on the number line. I gave them plastic clothespins to hang their cards. There are seven sets of cards and I happened to have seven students so it worked really well. However, this activity could be done with less students or 7 groups of students.
I gave most of them an area model or an array to start off with to place on the number line, as they have worked with those representations of multiplication already. They worked together to place the value of the cards in order and decided that since the highest value was 64, they would make 70 the end value of the number line. When I asked them what the starting number was, they decided it would be zero. Then, I asked them what the halfway point between those two numbers would be and they quickly recognized it as 35.
One student decided then, that we should measure how long the rope was so we could mark the 35 right at the halfway point. Our rope was about 88 inches long so we placed a piece of tape with the number 35 exactly at the halfway point (at 44 inches). Students then decided that they needed to reorganize the numbers they had placed on the number line so they more accurately reflected the markers.
At this point I asked students the value of the card they had placed on the line and how they figured out the value. As I already mentioned, they have worked with area models and arrays in the past, so they all gave explanations that reflected their understanding that length x width = the total number.
I then handed out another card to each student, explaining that the card I was giving them matched a card that was already on the number line in value. As each student placed a new card, I asked them how the visual representation of the number was the same? How was it different? And how would you represent what you see on the cards in a number sentence? Students represented their thinking as number sentences that included addition, multiplication, exponents, and parenthesis. They noticed that although the grouping was different on each card, the value was the same. I repeated this process with the last visual model of each set.
Lots of rich discussions occurred as students talked about the different ways that they were thinking about the grouping that they saw on the cards. Finally, I handed out the traditional multiplication flash cards and students matched them up with the visual representations.
Unfortunately, I couldn’t figure out how to post the pictures of what the activity looked like but you can access them here.I also posted a picture on Twitter.
Thanks for sharing your pictures Patricia. What a great activity. I think this would make a nice article for the Math Practitioner :-)
Patricia,
Thank you for sharing this resource. I decided to use the math card activity at our last Math Professional Learning Community last Friday. Instructors sometimes have the notion that students should never be moved up a level in our AE classes until they know all of their multiplication facts in a certain amount of time.
I gave all the instructors one card (the ones with dominoes) and told them to make a number line (I had hung a clothesline and had clothes pins on the table). They right away came to the front of the room just like students would do and started to place their cards. Through a short discussion, they came to the conclusion that maybe they should put up a zero, but where should it go? So by the end, the instructors had talked through the spacing, what should be their half-way point and how they could adapt this lesson to also include negatives. At that point, I gave them some of the other cards and told them to add the cards to the existing clothes line. We then had a great discussion on multiplication facts, area models, and why it might be helpful to keep a clothesline up in a classroom, to use as a resource, to add to it, to refer to specific numbers and how they work.... They also came up with lots of ideas for differentiation. The visual models help with the conceptual understanding of how multiplication works, and it turned out great having our instructors see this as well first hand.
Mary Jo,
Thanks for sharing your experience in using the math cards in a teacher's workshop. Since I did the activity with my students, I've been thinking about how it might play out with teachers, so I appreciated your input. I was thinking that if the workshop had a large number of participants, maybe there could be more than one number line set up so the groups could be smaller?
Today, I was working with number lines again with my students and we were placing fractions and mixed numbers on a number line. I gave them the condition that the number line couldn't start with zero, so they chose to start with 2 and end the line with eight. It was really interesting to see what a game changer that was as they struggled to figure out what the halfway point would be between those two values. I'm thinking that maybe the next time I do that math card activity with my students I might give them that same condition: that they can pick the starting and ending values but they can't start with zero. I wonder how that would go?
Patricia
Thank you for sharing this, Patricia! Putting the cards on the number line and all of the meaningful, mathy dialogue that had to happen in the process is such a rich experience! I tried the clothespin/number line activity for the first time (or the first time in a while) in Melissa Braaten's session at COABE; the "where does the zero go?" conversation can be really tricky!
I play "go fish" with my student with these cards. It is fun and makes us think, re-think, and visualize. Sometimes I try to push by asking a question like "do you have a multiple of seven?" So far those more open questions have not worked as I had hoped--meaning I did not get the card I wanted; we'll keep practicing. :)
Amy,
One thing that I thought was interesting about Melissa's workshop, is that she had very short number lines and sometimes the value she was asking you to place was actually beyond end of the number line. That is, if you were accounting for precise measurements between values you'd have to actually extend the number line to evenly distribute the values. I haven't tried that yet with my students, but I thought it was a great idea!
Patricia
I wasn't able to attend the COABE, so could someone explain Melissa Braaten's session? Thank you:) There is actually a website devoted to "Clothesline Math," and I've found it to be quite helpful. https://clotheslinemath.com/
One activity that I do with my students that is not really with math cards but I use this activity with all of my algebra students is using an individual xy pegboard. Each student has their own pegboard and I give off random points in the cartesian coordinate system. They just place the pegs and I make sure that they are placing them correctly. I am teaching the four coordinates and how they relate to each other. I have the students place pegs so they cross the x and y planes and explain what the x and y intercept are. We also begin the discussion about how far up and down the x and y axis these points are from each other. Getting started on working on the distance formula. The students really understand the concept of graphing so much better by doing this activity.