Category Archives: Multiplication

True or False Multiplication Equations

Today,  I was able to pop into a 3rd grade classroom and have some fun with a true or false equation routine! This routine has become one of my favorites, not only for the discussion during the activity, but more for the journals after the talk. I haven’t figured out quite how to use them with the students, but it gives me such great insight into their understandings that I would love to think about a way to have students reflect on them in a meaningful way.  I keep asking myself, what conjectures or generalizations could stem from this work?

I started with 4 x 3 = 3 + 3 + 3 + 3 to get students thinking about the meaning of multiplication and how we can solve for a product using repeated addition. I followed 6 x 4 = 8 + 8 + 4 to see how students talked about the 8’s on the right side. They could explain why it was false by either solving both sides or reasoning about the 8’s as two 4’s in some way.

My final problem was the one below, 8 x5 = 2 x 5 + 2 x 5 + 20. I chose this one because I wanted students to see an equation with multiplication on both sides. Up to this point, I structured them to be multiplication on one side and addition on the other.  There was a lot of solving both sides – I think because of the ease of using 5’s – but, as the discussion continued the students made some really interesting connections about why the numbers were changing in a particular way. I really focused on asking them, “Where do you see the 8 and 5 in your response?” to encourage them to think relationally about the two sides.

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I ended the talk with 8 x 6 = and asked the students to go back to their journals and finish that equation to make it true.

Some students knew it was equal to 48 right away and started writing equations that were equal to 48. For this student I probably would ask about the relationship between each of the new equations and 8 x 6.

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There are so many interesting things in the rest of them, that I am not sure what exactly to ask student to look at more deeply.

In all of them, I see…

  • Commutative property
  • Multiplication as groups of a certain number
  • Distributive property
  • Doubling and halving & Tripling and thirding

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The student below shared this one with the class during the whole class discussion:

8 x 6 = 7 x 10 – 3 x 10 + 2 x 4

From her explanation, she could explain how both sides were 48, but when I asked her how it related to 8 x 6, her wheels started spinning. You can see she played all around her paper trying to make connections between the two. That is the type of thinking I want to engage all of the students in, but based on their own personal journal writing – but what is the right prompt? “Where is one side in the other?” or “How are they related?” <—that one feels like it will lead to a lot of “They are both 48” so I need a follow up.

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I actually left the room thinking about how I would explain how they two sides were related – in particular looking for either 8 groups of 6 or 6 groups of 8 on the right side. I found it was easier for me to find six 8’s, but now want to go back and find eight 6’s for fun. I can see how this could be so fun for students as well, but there is a lot of things going on here so I wonder how to structure that activity for them? Would love thoughts/feedback in the comments!

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Number Talks Inspire Wonder

Often when I do a Number Talk, I have a journal prompt in mind that I may want the students to write about after the talk. I use these prompts more when I am doing a Number String around a specific idea or strategy, however today I had a different purpose in mind.

Today I was in a 4th grade class in which I was just posing one problem as a formative assessment to see the strategies students were most comfortable or confident using.

The problem was 14 x 25.

I purposefully chose 25 because I thought it was friendly number for them to do partial products as well as play around with some doubling and halving, if it arose. When collecting answers, I was excited to get a variety: 370,220, 350 and 300. The first student that shared did, what I would consider, the typical mistake when students first begin multiplying 2-digit by 2-digit. She multiplied 10 x 20 and 4 x 5 and added them together to get 220. Half of the class agreed with her, half did not. Next was a partial products in which the student asked me to write the 14 on top of the 25 so I anticipated the standard algorithm but he continued to say 4 x 25=100 and 10×25=250 and added them to get 350.

One student did double the 25 to 50 and halved the 14 to 7 and then skip counted by 50’s to arrive at 350, instead of the 300 she got the first time. I asked them what they thought that looked like in context and talked about baskets of apples. I would say some were getting it, others still confused, but that is ok for now. We moved on..

Here was the rest of the conversation:

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I felt there were a lot more wonderings out there than there was a need for them to write to a specific prompt, so I asked them to journal about things they were wondering about or wanted to try out some more.

I popped in and grabbed a few journals before the end of the day. Most were not finished their thoughts, but they have more time set aside on Wednesday to revisit them since they had to move into other things once I left.

What interesting beginnings to some conjecturing!

Writing in Math: After a Number String

Many people ask me when and how I use journals in math class. At those moments,  I always seem to have so many reasons that it is hard to pinpoint just one to focus on during the conversation. And even when I seem to find a coherent way of explaining when and how I use journals, I often forget the reasons that seem to happen naturally in the classroom. The other day I had one of those moments that I think Joan Countryman, author of Writing to Learn Mathematics, would classify as continuing the conversation.

During Number Talks or Number Strings it always seems to happen…one student has a way of solving the problem that, as he or she gets midway through the explanation, the rest of the class begins to disengage either because it is a long explanation or they are lost in what is being said mathematically. Journals help me continue that conversation with the student who is sharing. I attempt to clearly capture what is being said, but ask the student to tell me more in their journal because I am so interested to hear all of their thinking.

This particular string was in a 3rd grade class who has been working with multiplication. I wanted to see how they thought about changing one of the factors in a particular way. This was the string:

3 x 4

4 x 4

6 x 4

12 x 4

The majority of students shared strategies that involved either skip counting or using repeated addition of one of the factors. Some used previous problems (which was my goal) to help them with the new one, however there was one student who started using 5’s for the last two problem instead of either of the factors. He had a very clear way of explaining it, but I could tell many students were beginning to get lost in the explanation. I encouraged the students to ask him some clarifying questions, but that conversation began to stretch this number talk a bit too long time-wise. Not to mention, many had stopped listening at this point.

I was so curious to hear more about his strategy because to be honest, I was getting a bit lost in his explanation of 12 x 4 using 5’s. I told him I wanted to hear all of his thinking but we needed to finish up the number talk to get started with class. I asked him to explain to me what he as doing with 12 x 4 in his journal and I would be sure to check it out later! He went right to work and knocked out this beautifully clear explanation, not just for 12 x 4 but EACH of the problems!

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The thing that I appreciated most was the opportunity if gave me to continue this conversation with him. I could feel he wasn’t done explaining his strategy during the talk and this also gave him the chance to think about how he could clearly communicate it to me in his writing. What a powerful thing for a student to be able to do! It was amazing to me he had done all of that decomposition, adjusting, and adding in his head!

So, if you asked me this week for a reason I have students write in math class, it is to continue the conversations that are not quite ready to end during our class time together.

 

Multiplication: Does Order Really Matter?

Some things I am wondering right now about 3rd grade multiplication…

  • When students notice 4 x 3 is the same product as 3 x 4 and say, “The order doesn’t matter,” how do you answer that question?
  • Is there a convention for writing 4 groups of 3 as 4 x 3?
  • Is there a time, like when moving into division or fraction multiplication and division when the order does matter in solving or in thinking about the context?

Answers I have right now for these questions….

  • Right now, since they are just learning multiplication, I ask them what they think and why.
  • I think there is a bit of a convention in my mind because the picture changes. Three baskets with 2 apples in each is different than 2 baskets with 3 apples in each. Also, when reading the CCSS it seems that way.
  • I am still thinking about division but it makes me think that this would be the difference between partitive and quotative division. I also think when students begin 4th fraction multiplication, they are relating it to what they know about whole number operations, so 4 x 1/2 is 4 groups of 1/2. This seems important.

The 3rd grade teachers and I have been having a lot of conversation about these ideas. The students have been doing a lot of dot images and some feel strongly that the two expressions mean the same thing because they can regroup the dots to match both expressions. Others think they are different because the picture changes. All of this seems great, but then students are taking this reasoning to story problems. For example, given a problem such as, There are 5 shelves with 6 pumpkins on each shelf. How many pumpkins are on the shelves? students will represent that as 5×6 or 6×5. Is that a problem for me, not really if they have a way to get the 30, but should it be? I am not sure.

I went into a 3rd grade classroom to try some stuff out. I told them I was going to tell them two stories and wanted them to draw a picture to represent the story (not an art class picture, a math picture) along with a multiplication equation that matched.

1st story: On a grocery store wall there are 5 shelves. There are 6 pumpkins on each shelf.

2nd story: On another wall there are 6 shelves with 5 pumpkins on each shelf.

I asked them if the stories were the same and we, as I anticipated, got into the conversation about 5×6 vs 6×5 and what it meant in terms of the story. They talked about 5 groups of 6, related the switching of factors to addition and then some talked about 6 rows of 5.

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From this work, many interesting things emerged…

  • Some students had different answers for the two problems. They obviously did not see the two expressions as the same because they struggled with 5 groups of 6 as they tried to count by 6’s and forgot a row.
  • One student said they liked the second problem better because she could count by 5’s easier than by 6’s.

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  • Students skip counted by 5’s but added 6’s when finding the 5 groups of 6. IMG_3127.jpg
  • One student noticed the difference between 5 and 6 and could relate that removing one shelf was just adding a pumpkin to each of the other rows.

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  • One student showed how he used what he knew about one to switch the factors to make it easier to solve. IMG_3150.jpg

 

But they keep asking Which one is right? and I tell them I don’t have an answer for them. I just keep asking them:

Is the answer the same?

Is the picture the same when you hear the story? 

After chatting with Michael Pershan yesterday, I am still in a weird place with my thinking on this and I think he and I are in semi-agreement on a few things (correct me if I am wrong Michael) …Yes, I think “groups of” is important to the context of a story. I want students to know they can find the answer to these types of problems by multiplying. I want students to be able to abstract the expression and change the order of the factors if they know it will make it easier to solve BUT what I cannot come to a clear decision on is…

If we should encourage (or want) students to represent a problem in a way that matches the context AND if the answer is yes, then is that way: a groups of b is a x b?

What Is It About These Questions?

Today, I gave the 4th graders four questions to get a glimpse into how they think about multiplication and division before starting their multiplication and division unit. Michael Pershan had given the array question to his 4th graders last week and shared the work with me. As we chatted about next steps with his students, I became curious if the students think about multiplication differently depending on the type or setup of the problem.

Here were the questions:

After sorting 35 student responses I found the following:

  • 17 students got the area question wrong but the two multiplication problems on the back correct. Not only correct, but with great strategies based on place value.
  • 8 students got all of the problems correct, however the area was found in many ways, some not so efficient with lots of addition.
  • 10 got more than two of them incorrect. Some were small calculation errors on the back.

So, what makes almost half of the students not get the area?

Here is the perfect example of what I saw on the majority of those 17 papers:

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Then I did a Number String with them to hear how they shared their mental strategies. I wanted to get more insight into some of their thinking because a few students had used the algorithm on the back two problems.

They did great. They used the 10 and 20 to help them solve the problems and talked about adding and removing groups of one of the factors. I was surprised on the final problem of 7 x 18 that no one used the 7 x 20 but instead broke the 18 apart to find partial products.

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This makes me think there is something about that rectangle that makes them not use the 10s to help them decompose for partial products. I would love others thoughts and ideas!

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After reading the comments about area and perimeter, I wanted to throw another typical example of what I saw to see what others think of this (when I asked her she could easily explain partial products on the second and third problem)

 

3rd Grade Dot Image

Since the 3rd grade classes are about to begin their multiplication unit, the teachers and I wanted to hear how they talk about equal groups to get a sense of where they are in their thinking. What better way to do that than a dot image? I chose the first image because of the 3’s and “look” of 5’s and the second image because of the 2’s,3’s, and 6’s, all of which students can count by easily.

Image 1 went relatively the same in both classrooms and much like I anticipated. There were two things that stood out to me as a bit different between the class responses:

  • There were more incorrect answers shared in the 1st class than the 2nd class.
  • In the second class, multiplication came out during the discussion. The “4 groups of 7 and 4 x7 = 28” in the 1st class came out after both images were finished and one student said she knew some multiplication already. She asked to go back to the first image and gave me that.
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1st Class

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2nd Class

After the first image, I anticipated Image 2 would go much the same, however it was quite different.

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1st Class

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2nd Class

After the 1st image, I was really surprised at the difference in responses and I have to say it even felt really different. My assumption at this point is that in the 2nd Class one of the early responses was multiplication.

I am left wondering:

  • Does that early multiplication response shut down other students who don’t know anything about multiplication yet? While I asked her to explain what she meant when she said 4 x 12, I wonder if that intimidated others?
  • How could I have handled that response differently so others felt OK using skip counting or addition to count the dots?
  • Can we anticipate that type of reaction from other students when someone starts the discussion with something that may be beyond where the majority of the class is in their thinking?
  • Was this even the issue at all? Did the 2nd class just see so many more dots and wanted to avoid adding and skip counting?

The 2nd Class ended with a journal entry after a student remarked, “If we know it is 8 groups of 6, then it is also 6 groups of 8.”

I asked if 8 groups of 6 is the same as 6 groups of 8 and the class was split on their response, so they set off to their journals.

The yes’s went with multiplication expressions representing the same product and commutative property:

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I loved this no because the picture changes:

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I am not sure about this argument but I would love to talk to the student a bit more about the bottom part!

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After that talk, I am excited to see what these guys do when they actually start their multiplication unit!

4th Grade Multiple Towers – Pt 2

Due to a schedule change today because of an assembly, Malorie and I did not get a chance to sit and plan together before teaching the lesson. Luckily, she is always so prepared and had read my thoughts on the blog I did yesterday about it so we went into the lesson with a common vision and then talked our way through the shortened class period with the students.

We started with choral counting by 3 and then 30. We chose to include 5 multiples in each row in hoping the students would see a pattern in the “friendly numbers” if they struggled to see patterns in other places. After the list was up, they took a few minutes to write  down any patterns they saw in their journals and we shared out. This was the count and then notices Malorie typed as they were sharing:

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I could have stayed with these patterns all day long, but we decided to let students revisit these patterns over the course of this unit. There are some really great ones up there like adding the digits and that number is also a multiple of three and if you add any of the numbers up there, the answer will also be a multiple of 3. They made some predictions about numbers that would show up later in the list and talked about connections between the two lists. Multiplying by 10 and the zero “put on the end” was definitely the most popular noticing of the class period, which was a nice lead into our stacking of the boxes of oranges.

We explained the boxes of oranges held 30 oranges and asked them to estimate how many boxes (represented by the post-its) would be needed to stack up to Mya’s shoulder. Malorie and I quickly tried to figure out if we wanted to ask about number of boxes or number of oranges and decided to give them the option of either. It worked out beautifully because the majority of students told us boxes along with how many oranges it would be as well. As Malorie stacked the post-its, the students counted along. Some were counting post its while I heard others counting oranges. We stopped approximately halfway, took adjustments on their estimates and continued. We finished with 480 oranges, 30 boxes. As we were running out of time, we decided to end by asking if there were any equations they could think of to represent the oranges and boxes.

I was surprised to get division first…480 ÷ 30 = 16 and then 16 x 30 = 480.

This was unfortunately where we had to leave off because the classes were shortened and then next class was waiting at the door. So we met during her planning later that day to regroup and find our starting point for the next day.

Tomorrow we have decided to lead with a talk about the orange tower we built today. This is where we had a big discussion around what we want to really have students think about….the book seems to really focus on the multiplying by a multiple of 10, like what would be the 20th multiple, the 30th multiple, etc but we want to play around with the properties a bit more here.

So, we decided to ask students what equations or expressions they could use based on the tower to arrive at the 330 post it. We are thinking we may get some things like this:

30 x 11

(3 x 10) x 11

(3 x 11) x 10

(30 x 10) + 30

(30 x 10) + (1 x 30)

We want to set them equal to one another and ask how they could prove are the same answer. We are hoping to see the associative and distribute properties come out. We definitely could get some division too and that could make it really interesting!

We decided to go with this because Malorie says that she often sees students “putting one zero” on the end of any number when they multiply by any multiple of 10. For example, when multiplying 40 x 30, they will just get in the habit of putting one zero because they don’t see that 10 x10 happening. This is where being able to think about 40 x 3 x 10 is helpful for students.

Next, they will make their multiple towers based on these numbers we will assign to partners: 15, 16, 32, 28, 24, 48, 18, 36, 35, 70, 45, 14. Since Doubling/halving is coming up, we thought this could bring out some of those ideas. Then we will ask them to come up with expressions for any number they choose in their tower. We had planned for that to be in their journal, but I wonder if that is something they could hang up next to their tower for a notice/wonder walk around the room? Hmmm…have to ask Mal in the morning!

~Kristin

Follow Up…today we did a choral count 6 and then 60 and the pattern-finding was even better then yesterday. It was always amazing to me how students notice new things and then based on how we record. This time Malorie did four multiples in each row so in addition to finding new patterns, they also started comparing it back to what they noticed yesterday during the 3 and 30 count. Even though the 3 and 30 count was not up on the board anymore, they still remembered the patterns and numbers like they were right in front of them because they were so excited about them. So cool to see/hear. 

Next, we looked back at the multiple of tower to 30 on the wall. I asked the students if they could find an equation with the answer of 330 (which was on the tower) using numbers on the tower and gave them time to jot some ideas in their journal. 

As the students started to share, I realized it was a really Badly Worded Question. As they shared I was getting “ways to get 330” but not relating anything to the tower, which we were hoping would happen. We wanted to see the associative and distributive properties emerge, but were getting addition and subtraction equations. 

So, as I got “420-90=330 and 270 + 60=330…” etc, I asked “How would would you show that action on the tower….and then they were on a roll. I didn’t want them to think what they had written was not valued so we asked them to share what they had originally written and we built on from there! There is SO much to work with here so I cannot wait to plan for some exciting algebraic work. 

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I had to plan with another teacher so I left as the students began working on their multiple towers. After they finished their own towers, we were going to ask them write equations for a number on their tower based on moves on their tower. I have to check back in with Malorie in the morning! 

~Kristin