Category Archives: Decimal Multiplication

Encouraging Students To Make Deeper Mathematical Connections.

Because of all of the math talk my students do in class every day, they are very comfortable (and flexible) in sharing multiple strategies and solution paths. They can explain others’ strategies in their own words and agree and disagree with one another beautifully, however when asked to make connections between two representations (numerical or visual), I feel like I get very “surface” connections. I will read things in their journals like “They are the same because they have the same numbers” or “They are different because we double and halved the other numbers instead” or “We both used an area model” Something like this…

IMG_0391_2 IMG_0393_2

After going through their journals the other day with Faith, we were thinking and questioning one another about how we, as teachers, can have students dig deeper into the connections. We obviously would like to them to notice them but if not put in a position to make connections, will they on their own? Is there a way to frame a task or question that would push them to think a little deeper about how and why the two representations are alike/different yet still arrive at the same answer? How do we encourage students to make deeper, more meaningful connections when we know they can, but just may not be sure of how to get there?

Instead of a number talk they other day, I did a math routine I named “Where is _____ in _____?” I was hoping the prompt would have them go beyond just looking “at” the representations and look “into” the meaning of each representation. On the board, I posted some examples of their representations from the day before in which they had done a surface job of connecting. I had them work independently for a few minutes and then talk as a table before the group share. I was much happier with the conversation and felt like asking them to look “into” the problems really got them thinking about what the representation was showing.

This an example of two area models in which students the day before had simply said, “We both used the area model” without thinking about how they were related. I love the (.3 x .2) in each of the quadrants of the first grid.

IMG_0388_2This student had a different take on how the two area models were alike, which led to such an interesting discussion! She also did some lovely work with showing how the two distributive properties were within one another through factoring.


This student showed the distributive property and double/halving in a wonderful way…

IMG_0396_2This one was the only student who connected the area model of .6 x .4 to the strategy of .6 x .5 – .06. He showed where the .6 x .5 would be in the model and then scratched out where the extra .06 were coming off to arrive at the answer.


Having students make connections in math is so incredibly important and so difficult to do, especially with so many variations in strategies and representations. I would love to hear other ways to encourage these connections!


Creating Contexts for Decimal Operations

Sometimes I have students engaging in math within a context, however at other times, we just explore some beautiful patterns we see as we play around with numbers. I see a value and need for students to experience both. This week was one of those “number weeks” and it was so much fun!

Over the past few weeks, we have been working on decimal multiplication. If you want to see the student experiences prior to this lesson, they are all over my recent blog posts….it is has been decimal overload lately:) After sharing strategies and connecting representations in this lesson, I was curious how students thought about this problem in a context because up to this point, I had not given them one for thinking about a decimal less than one times a decimal less than one.After they wrote their problem, I asked them to tell me what they were thinking about as they were deciding on the context.

I anticipated that many would refer back to what they know about taking a fraction less than 1 of a fraction less than 1, like in this example…


I love how this one said she knew she “had to start with .4” That shows the order of the numbers in the problem create a context for her. It mattered to her, taking .6 of the .4.


This student went with two different contexts and again saying that he started with the .4. This must be something we have chatted about quite a bit about because it showed up multiple times. I loved how this student said he thought about an area model in creation of his problems.


This student was great in listing all of things he was thinking about as he thought about a context..


I had students who attempted to create a “groups of” context. I don’t know if I ever realized how difficult this and how much I, as an adult, need to be able to create a visual in my mind of what is happening in a problem to make sense of it. Here is one example (not the sweetest context but she thought the Mary HAD a little lamb was clever…) She worked a bit yesterday to show what the representation would be, but kept running into problems with cutting into “.6 pieces.”


And then I have these two that had my brain reeling for a bit, for many reasons. First, does the context work with this problem? Secondly, I knew it sounded like it should work, but when I tried to make sense of it, I couldn’t create a visual. Also, as I read them, I thought I knew where it was going and the question I would pose, but it wasn’t the way they saw it ending. I asked them to create an Educreations about their problem so I could check out their thinking around the context.

Yes, Rick Astly. But the question at the end, compared to the total time Never Gonna Give You Up, threw me a bit, not where I was going with it….


His Explanation:

The second one tried it out, and wasn’t so sure of his question after messing with it. The wording “.6 as small” was making me think. I was trying to make sense of that wording, do we ever say six tenths times as small? Then does his question referring back to the .4 make sense?

IMG_0345His Explanation:

Definitely a lot for me to think about this week too! I have some amazing work with them connecting representations to write up later…they are just such great thinkers!


Why My Tweeps Deserve a Hug…

By nature, I am not big “hugger” when I first meet someone. Which is why I am always so shocked that every time I meet one of my Tweeps in person, I feel the overwhelming need to give them a hug. I always wonder why that happens….I mean, I have just had online conversations with these people, right? Wrong.

Over the past month it has become crystal clear why my tweeps will always get a hug….

1 – They care about my learning. They challenge my thinking, give me wonderful new ideas, and continually make me feel like I have a team of teachers working with me to improve my practice. For example, just the other day, I threw out this tweet:


and what I got in return was this amazing conversation: AND then two fantastic follow up blog posts: and Talk about professional development at its finest! I LOVED it and left with so many ideas to think about!

2 – They care about my students as much as I do. Recently, our district is working through opening Twitter on our network. Wanting to be as informed and prepared as possible to defend the use of Twitter in schools, I reached out on Twitter with this post:


I tweeted it, went to bed, and woke up to over 40 notifications in response. I had teachers sharing how they use it as PD and in the classroom, as well as superintendents and principals (Thank you @gcouros and @joesanfelippo) offering to chat with my superintendent to offer their experiences. These tweeps are from all over the world, they don’t need to care if the students in my class get Twitter, but they do. They took the time. It is so touching to know that the heart of education is alive and well on Twitter with the students in the forefront.

3- They get as excited and passionate about math work as I do! You can read these fun, nerdy conversations all over my Twitter feed, passionate people talking about what we love, math. After trying Talking Points that I have blogged about recently, I could not wait to tweet @cheesemonkeysf because I was so excited about the conversations in my classroom:


My tweeps have become so much more than online conversations, they are my colleagues, mentors, and coaches. They are open and honest and allow me to feel safe being open and honest.

It is personal. It has changed my practice. It is hug-worthy.

I could really keep going on and on about how much I adore my tweeps, but being a school night, it is time for one more Big Bang rerun and bed!

Happy Almost Friday,


Aurasma Math Work with Decimals

Last year, I started playing around with the Augmented Reality app, Aurasma, in math class. For this particular activity, students worked in pairs to create a video of their explanation and trigger image in which to overlay their response. They swapped their creation with other groups, compared solution strategies, and discussed similarities/differences.

If you have the app Aurasma, simply follow our class channel, “Grays Class,” and scan the trigger images below to hear their explanations.


If you do not have the app, download it here: Aurasma and follow the directions above.

Here are some trigger images to scan and hear solutions:

p2 p3 p4 p5 I definitely look forward to doing more of this work with my students this year and possibly embedding this into a newsletter for parents!


Decimal Multiplication

This is a quick lunchtime post, so not much time to reflect or analyze, but wanted to throw it out there..

We did a Decimal number talk today and ended with the problem 5 x 4.6

I had students double half, use partial products, and use friendly numbers. (Incorrect answers were also written in case you are wondering about the 21.6 and 35…we looked for errors also)


Then one student said that she bumped the 5 up to 6 and did 6 x 4 because she knew that faster, however she couldn’t figure out how to adjust her answer based on what she did. We had already had someone explain how they adjusted from 5 x 5, but this was not the same. She knew that she needed to subtract 1 (because we had already established 23 was correct) but where was that 1 coming from?

I sent them back to their groups to talk about it.

One group had this idea…


I asked them if it always worked so they tried some more problems at the bottom that did not. They tried some more….


They found one that worked….hehe..


The class had to leave for lunch, but I will keep you posted what they come up with…