I have a vivid memory of learning multiplication in third grade. My teacher, Mrs. Bowman, drew three circles on the chalkboard. Then she put five milk bottles in each one. She said this is 3 x 5 which is 15 milk bottles. That's really all I remember about learning about the concept of multiplication, but for some reason, it stuck! I also remember using my Pee-Chee folder to look up the multiplication tables for dividing in fourth grade. I think that is how I memorized them. But times have changed.
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With today's Common Core State Standards, multiplication is more than just memorizing the tables. Students have to understand the relationship between addition and multiplication, multiplication and division. Students have to understand multiplication as equal groups that can be modeled with objects or arrays. Students need to understand how to use the Properties of Multiplication. And yes, students need to fluently multiply within 100 which you can infer as memorizing the multiplication tables to 100.
In this blog post, I show you how I teach the concept of multiplication to my third graders. In my district, we have the Go Math curriculum but are encouraged to use it only as a resource (meaning, we don't have to follow it completely...just use it as a guide). With that said, I do use some of the pages in the consumable math book as well as, the pages from the practice book as homework. I also supplement with resources I created and other resources I have found on Teachers Pay Teachers. I don't want to completely abandon the Go Math book because my students will face similar questions on the SBAC in May.
First, let me give you some background on how I approach math. My district has provided quite a bit of professional development in the area of math since the Common Core was adopted in California. Albeit, opposing types of professional development: one being Lesson Study and the other Direct Instruction. And now this year, we are receiving professional development from Math Solutions, which is a company started by Marilyn Burns. Essentially, the pendulum is swinging not just back and forth but all over the place.
Let's start with Direct Instruction model. About four years ago, my district required all teachers to go through training for Direct Instruction. It uses a step by step approach to teaching students procedures for math. I did like some aspects of the approach but felt it was limited in that the teacher was spoon feeding the students the math and it was very limited on problem-solving. We had even had direct instruction coaches that had to observe us!
But then a couple of years ago, my district included me in another professional development for math. We conducted what is knows as Lesson Study, based on the Japanese model of Lesson Study. From that professional development, I started teaching math differently. If you're not familiar with the Japanese model of teaching, it does not use a gradual release of responsibility model that is widely used in the United States, rather they approach teaching this way: You try it, you all try it, then we try it. It is actually the opposite of how we teach! The Japanese model forces the student to problem solve on their own first. Then, students collaborate with a partner to compare solutions. Finally, the last part is very powerful because as a whole group, the teacher is responsible for guiding the thinking towards the solution by highlighting the different student solutions to the same problem. As you can see, it is very different approach when compared to direct instruction.
What did this lesson study approach look like in my classroom? I would start the lesson with the students on the rug and present to them a problem, in this case, a multiplication word problem. Then, the students returned to their desk with a bag of foam tiles (my go-to manipulative). They had to use the tiles to solve the problem. They would also have a small whiteboard (which I have stored in large baggies with a dry erase marker and a piece of felt for an eraser) or a piece of paper to do any calculations or record their answer. As the students worked, I walked around looking for students who had solved the problem in different ways. I would send those students up to the whiteboard to copy what they had done. I chose between 4 - 6 students.
Then each volunteer explained how they solved the problem. Now that this I am receiving district-sponsored professional development from Math Solutions (a company started by Math Guru Marilyn Burns), I've incorporated "Math Talk" prompts. Math Solutions focuses on teaching students math conceptually with a focus on the Common Core Math Practices. Here are some of the prompts we learned to use:
- Who can repeat what __________ just explained?
- Who can add to that?
- Who has a different way?
There are more prompts that a teacher can use depending on the situation. My district bought us a large poster with the prompts which I have hanging on my math board.
It is not a question of which approach to use. It's a question of what do my students need? This year, I have a class that really struggles with math and new math concepts. Also, just like many teachers around the nation, I am under time constraints because my district uses a pacing guide and expects certain math units completed by a certain date! So I decided to use a direct instruction approach for the beginning lessons of multiplication. Eventually, I would move to the Japanese model once my students had some multiplication strategies to use.
Now that you see my approach to teaching math, here's how I started teaching the concept of multiplication. I first warmed up the group with skip counting. Most students by third grade can skip count by 2s, 3s, 5s and 10s. I found this wonderful FREEBIE on Teachers Pay Teachers which my students used to learn to count by 4s, 6s, 7s, etc. I also use these FREEBIE posters I also found on TpT.
I wanted to first teach the students the concept of equal groups and why it is is a key cornerstone of multiplication. Bear in mind, I have a class in which at least a third of the students struggle with new math concepts so they benefit more from a direct instruction approach. I started the lesson with a Multiplication Concepts of Groups and Arrays PowerPoint that I created. It is actually 2 lessons (Lesson 1 is Equal Groups and Lesson 2 is Arrays).
I used the first part that taught equal groups. The PowerPoint is interactive and animated to demonstrate the concept of equal groups. There are slides in which students interact with each other and talk about equal groups and what each number in a multiplication sentence means (it also introduces the vocabulary of factors and product). There is a place in the PowerPoint in which the students now work at their desk with a manipulative. In my case, I use the bags of foam tiles that came with our Go Math program (at least that's one good thing about it!). Each student gets his/her own bag for some hands-on practice.
I instructed the students to take out 12 tiles. I demonstrated how to put them into equal groups of 6. Then I had them practice putting the same 12 tiles into groups of 4, then 3, and 2. Each student had a dry erase marker to draw the circles on their desk around the groups so they could visually see the groups. From there we practiced using the vocabulary: 3 groups of 4, 6 groups of 2, 4 groups of 3, etc. When I could determine that the majority of the class understood equal groups, we practiced with a printable I created to go along with the PowerPoint. They had to draw a model for a particular multiplication sentence using equal groups. Then they answer questions about their model. I collected the printable so I could quickly assess student understanding of equal groups. The students also worked on similar problems in their consumable math book. The homework also followed up on this concept of equal groups.
The next day I reviewed equal groups again. This time, it was time to directly teach the concept of repeated addition. By the way, there is controversy about whether we should even teach multiplication as repeated addition! But don't panic, I would say yes, go ahead and teach it as repeated addition. You can read the linked article to see why there is a controversy about this!
So I posed a problem for them: Looking at the equal groups on your desk, how could you quickly find the total without skip counting? Eventually, we talked about how we could just add each group together (3 + 3 + 3 + 3 = 12). What is the purpose of this? It's to get kids to see multiplication from many perspectives: what it is and what it isn't. For example, if you had 3 groups of 2 and 1 group of 3, would that be multiplication? Some would say yes and some would say no. No, because the groups aren't equal. Yes, because you can multiply the equal groups and then add the unequal group. The point is, addition is related to multiplication and the students need to know that! At this point, I am not using a Lesson Study approach to math, but rather more direct instruction. Why? Because I need the students to have some strategies for multiplication. As I indicated before, at least a third of my class would just sit there helpless because they just don't have enough conceptual understanding or "tools in their math belt" to get started.
Using a Numberline
You would think that such a great visual that has been used since first grade and maybe even kinder, would work wonderfully to teach students how numbers are multiplied. Nope, using a number line is tricky and confusing to kids. Because I knew the difficulties of using a number line for multiplication, again I went to a direct instruction model. What difficulties would they encounter?
- not starting at zero
- not skip counting correctly
- counting the starting number as part of the skip counting
- confusing "jumps" with "how many to jump"
In my instruction, I gave each student these wonderful and colorful number lines which I laminated. I like the fact they only go up to 30, which is a good number when dealing with multiplication. Here are the steps I taught them. For example, for 3 groups of 4 or 3 x 4:
- Circle the first number. Think of it as jumps.
- Underline the second number. Think how many each jump.
- Put a dot on the zero.
- Jump to the number 4 (as in this example of 3 x 4) and put a dot.
- Then draw an arched line back to the zero. Label it with 1.
- Then count ahead another 4 spaces or lines and put a dot.
- Draw an arched line back to the previous dot, label it with a 2
- Continue the process until you have 3 jumps (as in this example of 3 x 4).
- What is the final number?
By following these steps, a lot of the confusion is cleared up and students use the number line properly from the get go. Even with these steps, I did have a few students use the number line incorrectly. I helped those students individually while the rest worked on similar problems in their math book. I prefer to sometimes use the consumable Go Math book to save on copying paper. I hand pick which problems the students would work on. This particular math program has a very poor design when it comes to independent practice. It expects the students to make conceptual leaps with some of the word problems that are presented. They're not ready yet! So we do not work on those types of problems yet. I just want to make sure the students understand the concept of multiplication and ways to approach it.
We finally arrived at the point in which I can now switch to a Lesson Study approach to teaching math. Now that the students are armed with multiplication strategies, we can leave the direct instruction model to the Japanese model of forcing the students to struggle and solve it on their own.
We start out on the rug in which I present a problem on the whiteboard such as the one below.
Then I quickly review some of our multiplication strategies: equal groups, skip counting, repeated addition, and number lines. With that, I instruct the students to use the foam tiles to solve the problem any way they can. As this was happening, I would search for students who used equal groups and send them to the board to copy their solution below the problem. I looked for other students who also used equal groups but put the groups in a different way to go up to the board to copy their solution. Finally, if any of the students lined up their tiles in an array, I would have that student go up and copy it, too! Once I had several solutions, we came back as a class and I had each of those students explain how they arrived at their solution.
The other students had to listen because they knew I would use the Math Talk prompts (who can repeat that, who can add to that, etc.). For the student who made an array, I had also taken a video of him making the array with my phone.
But first, I had him come up and explained what he did. Would you believe me if I told you that he actually used the term rows! Yes, he did! I said I had to record his thinking on the board so I wrote: 4 tiles in 3 rows. I explained to my class that this student just discovered another way to multiply: arrays! At this point, I explained that arrays are groups but they are formed with equal rows and columns and have a rectangular or square shape. I then showed my class the video of the student making an array which you see below.
Using Video as an Example
What was great about the video, is that he did take some excellent steps to construct his array. First, he put down a tile for each row. Then he completed the first row. From there, he just copied the first row to the other rows by adding tiles. He recounted just to make sure he did have an equal number as the previous row. So we named this method of constructing an array "The G Method" after his first name. Needless to say, he was very proud. I was very proud! This boy is an English Language Learner who finally got the eyeglasses he needed (2 months of school with no glasses)! He is reading below grade level and sometimes struggles with math. But here he shone as a bright star to the entire class.
From there, we continued to practice forming arrays to match a multiplication sentence. Once I knew the students had gotten it, I had the work with a partner on a more complicated multiplication fact such as 8 x 6 which required making a larger array. Working as partners, they could quickly assemble the array and produce a product. Finally, I assigned them independent work from the consumable which you see below.
Properties of Multiplication
I will be continuing this math lesson with the second part of my Multiplication Concepts of Groups and Arrays PowerPoint, which I will use as a review of arrays. So what is left to teach about multiplication? A lot! I will be teaching separate mini-lessons on the Properties of Multiplication. I also have developed a PowerPoint for the Properties of Multiplication (Zero, Identity, Commutative and Associative Properties).
I also have some follow-up activities to reinforce those properties: Properties of Multiplication Practice. These reinforcement activities include:
- Bookmark to use as a study guide
- Flap book with the definitions and examples of the properties
- Mini-book of the properties
- Venn Diagrams to compare the properties
- Commutative Property Match Up Center
- and Multiplication Hero!
The last one has been a favorite of my students. It uses this page-sized X for multiplication. On it, the student writes the definition of each of the properties and an example. Then the X is cut out and taped to the student's chest (one piece of masking tape folded over is enough). To play Multiplication Hero, two students face each other. One student reads a definition or gives an example of one of the properties by reading his partner's X that is taped to his/her chest. The other student has to name the property. If the answer is correct, they cross their arms across their chest and say multiplication hero! They continue to switch partners. This is a fun game to play for a review.
These properties are very important to teach and to learn. Why? Because they make learning the multiplication facts easier! Because of the Zero and Identity Properties of Multiplication, we know that any factor multiplied by zero is zero, while any factor multiplied by 1 is that factor. But the most important property is the Commutative Property.
With the help of the Zero Property, Identity Property, and Commutative Property, learning the multiplication facts become easier. Knowing that 7 x 4 produces the same product as 4 x 7 reduces the number of facts to memorize when you take into account the entire multiplication chart. Watch this YouTube video that explains how the entire multiplication chart can be brought down to 6 facts to memorize.
Memorizing those Facts!
Eventually, all students do have to memorize the multiplication facts. I use various approaches for the students to master the facts. One is to hold them accountable for studying. Learning to memorize is an important life skill. The act of focusing on something for a length of time until it becomes committed to long-term memory requires discipline and good habits. At Back to School Night, I give each parent a folder with my Multiplication Homework Activity Chart. I once again remind the parents during parent conferences that this homework will now begin. I explain how the chart works. Essentially, the student must complete 3 activities (or more) per week to make a tic-tac-toe. The activities are varied with different approaches to learning the facts. I provide copy masters to make flash cards, paper dice, premade recording sheets and links to many multiplication websites. The students do this as weekly homework for 6 months.
Every day I give the students a one minute timed test to see if they have their facts memorized. Any student who passes goes on to the next multiplication table. Once they've reached ten, they have the option of continuing to 12. I also have a Level 2 Homework Activity Chart for those who need to be challenged. Learning the facts is self-paced and the chart provides many different ways to memorize them. This resource comes with everything you need to get your students studying the multiplication facts. There is also a bilingual English-Spanish version as well.
But that isn't usually enough. I also teach my students strategies or tips for each multiplication table. For example, doubling the 2s will give you the four times table. Example: 4 x 6 is like 2 x 6 doubled. 2 x 6 = 12, then double it and you get 24! The same trick works for the 3s and 6s. I have created a Multiplication Tips and Strategies Chart of these tips and strategies to give to each student which they put in their math folder. I also send an additional copy home. I've also turned the chart into Multiplication Tips and Strategies Posters that also hang on my math board for reference. You can try out the Multiplication Tips and Strategies Posters SAMPLER here.
Usually, around the beginning of December when I start teaching division, I notice that some students are stuck on a particular multiplication table and can't pass the timed test. So that is when I bring out the Multiplication Practice Cards for select students. I already have these stored in baggies with a dry erase marker and felt eraser. I give the student the set for a particular multiplication table and they take it home to study. When I started this process last year, it helped jump start those kids that were stuck.
As a grade level, we also reward all the third graders who have memorized their facts to 10 with a popsicle party in March. The goal is automaticity by March.
If you need more ideas or resources for multiplication, consider following my Pinterest Boards below. I just created a NEW BOARD exclusively for the PROPERTIES OF MULTIPLICATION!
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Multiplication and Division Pinterest Board
Properties of Multiplication Pinterest Board