Category Archives: Making Claims

Planning K-5, literally

Tomorrow I have the opportunity to teach a Kindergarten, 2nd and 5th grade class! It is so exciting and interesting to be thinking across all of the grade levels in one day of lesson planning! The most interesting part for me, in thinking through this, is the connections across all of the grades. There is so much potential for conjecture and claim-making supported by their development of proofs.

Background: The 5th and 2nd grade teachers are out at a state math teacher leader meeting so I am teaching instead of the substitute. The kindergarten teacher and I will be teaching it together. I have met with each teacher to chat about where they are within their units and what they have been seeing students do within the current work. I invited teachers both at those grade levels and at other grade levels to pop in if they have the time. I thought it would be great having more people to reflect with after the lessons!

5th Grade: They have just started working with finding a fraction of a fraction using bar models. The initial work is unit fraction of a unit fraction and then moves to non-unit. (My post on that from a couple of years ago on this work, I wish I had done that better, so here is a chance to try something new;) Leigh, the teacher, says they have been really successful in partitioning the bars and arriving at the correct answer. I am thinking about starting with a number routine of either a choral count or a number talk string like 1/2 of 12 = __ of 24… As far as the lesson, I could continue work with this and have students look at noticings after and explore them deeper.They have done these noticings with whole number times a fraction or mixed number, so this could be a revisiting of similarities or differences. OR I could do this cornbread task as a formative assessment as the next piece they will move into is an area model. It may be really helpful for Leigh to see how they are thinking about this before they jump into the work. This is my least planned because I keep bouncing all around with ideas.

2nd Grade: They have been working with even and odd numbers and counting by groups of 2’s, 5’s, and 10’s.  All of this work is within contexts of break a group of students into equal teams or everyone having a partner. Tara, the classroom teacher, said the students are really great at determining whether a number is odd or even, however when asked how many would be on each team, a lot of students struggle. They are great if they know a related double fact, however if they don’t they resort to “passing out” by tallies or drawing the picture and physically dividing the number of things in half. For example, if they do not know 11+11 is 22, then finding the number of on each team become passing out 22 things into two groups to find 11. While they are successful in this, Tara and I were wondering why they do not say 10+10=20 and 1+1=2 so 11+11=22. They are able to add 11 and 11 but unable to decompose it as fluently.

In thinking about this, I am inclined to want to connect that addition to halving. I am thinking a counting collection would be fabulous for this. Give students a collection of things to count. Share how we counted them because I am positive they will not count them by 1’s given a large set. We can share as a whole group, record ways in which we counted and determined if our number was even or odd. Then, put the collection back together, switch with a partnering team and then split the collection into two groups. The share would be, “Could you make two equal groups?” “Was your number even or odd? How did you know?” Record strategies. Ask for noticings/wonderings about how they counted and how they divided into two groups.

Kindergarten: The students in this class have been doing a lot of work with ten frames, dot images, counting jars, etc and having students counting and adding to compose a number. They have just begun working on decomposition of number so I immediately thought about the mice activity in Thinking Mathematically. Linda, the teacher, and I planned to do this activity with the students. In preparation, we read NCTM TCM’s article by Zachary M. Champagne, Robert Schoen, and Claire M. Riddell, Variations in Both Addends Unknown Problems. We are going to use 6 bunnies and see how students show all of the ways the bunnies can be inside and outside in a pen. Instead of just giving a context, I was imagining that the students may need a visual of the rabbit pen so I created this image to launch with a quick notice and wonder:

Screen Shot 2015-12-06 at 9.51.28 AM.png

We will then let the students work on finding the different ways in partners and then come back for a whole group share and record the ideas on the board. We are really looking to see two things….1-how they organize their information and 2- the strategies they use. The students will do a notice/wonder about the recorded information. If there is time it would be great to see if students, when given a different number, would apply any of the strategies and/or organizational tools shared.

Going for a run to think through this a bit more! Would love any thoughts/suggestions, as always!



Math Reasoning Stages


Tracy Zager (@TracyZager) asked for some thoughts/pushback on these stages of math reasoning imagined as a flow, so here are my thoughts based on my experience in the classroom…

Pattern Sniffing –  After students see a pattern I find they continue, using that pattern, for a while before thinking about a generalization. So, maybe “Extending Pattern Using the Pattern” comes after this in my mind?

Wondering – When they wonder, they definitely look at relationships, but I am not sure they wonder if it will always be true at this point? Now that I just wrote that, I am thinking maybe “Extending Pattern Using the Pattern” comes after this one?

Articulating – “Can I communicate what I am seeing happening in a precise way?” I don’t know if they are thinking too much about it at this point but more seeing it happening? Could

I don’t know where this fits necessarily, if it is embedded in one of them, or if it really fits at all:), but there is a point where mathematically students prove a generalization works with certain number and not others because “the numbers have to work that way” (structure) without the conceptual proof of why that is. For example, “Even dimensions of a rectangle will give you an even area.” Students can make the statement that it has to work every time because when you multiply even numbers it is always an even product….true, but isn’t there proof to that. So, it is like a string of proof by depth?

Investigating and Explaining Why – I feel the relationships and patterns question to themselves comes back up here too.

I love thinking about this process for students and the teacher implications between each step. What questions and/or feedback do we give as students go through this that isn’t too helpful or leading, but not too vague that leave them in one spot spinning wheels? Paging @MPershan…

*Chatting with Tracy after I wrote this, she was focused on mathematicians, not students. I find some holds true in both cases to different sophistications. 

Hope that helps a bit Tracy! Hopping on a plane but as soon as I have wifi I will add a couple more question I have to think about around this!


Last Day of Math Class :(

Today was the last official day I had my students for math. It was a bit sad for me and it was nice to hear some of them say it was “kinda sad” for them too. In moving into a K-5 Math Specialist position next year, I know it will not be the same experience watching a group of students grow over the course of the school year.  It will be great in different ways, but I am really appreciating all of the amazing work my students have done this year.

So…what to do on the last math day after they just had field day yesterday followed by our PBS bowling field trip tomorrow? It is a tough planning!

I first had them look through their two math journals, one from the first half of the year and one from the second half. As their last writing piece, I asked them to write things they noticed in their work over the course of the year after looking through their journal. I only had time to grab one journal today because the end of the year craziness is kicking in, but I plan on following up with a more detailed post later. This one was so powerful and truly gave me goosebumps….

IMG_0955_2After they finished that, I asked them to revisit some of the claims they had written over the course of the year and see if they still thought they were true and could be proven or were not true and needed to be revised. This student had written a claim that when you are multiplying fractions, you could multiply the numerator and denominator to get your answer. As he was proving it just worked for multiplication, he stumbled upon the realization that it worked for division as well. He then worked through a few more division problems and it was such an amazing explanation!

He revised his claim…

IMG_0956_2IMG_0957_2I promise to follow up with some really amazing work they did on the last day when summer is here and there is a chance to breathe 🙂


The Meaning of Subtraction

After a Number Talk a couple of days ago, I blogged about my students’ thoughts around a subtraction problem. Instead of being a talk about subtraction strategies, as I anticipated, it ended up more of a talk about the meanings of subtraction.

After class, I was curious where these meanings of subtraction arise in our curriculum and found this in the 3rd grade Investigations’ Teacher Notes:

gr3Now being in 5th grade, I began asking myself a bunch of questions…If these subtraction meanings arise in 3rd grade, do we ever have students explicitly investigate them? Once they have an efficient strategy to subtract, do we just move on? Do we think that the meanings of subtraction really do not matter if they can read a context and solve for the solution?

After reflecting on my own practice, I honestly think it is a combination of all of these things. I am completely guilty of being satisfied if students are able to understand how to solve a problem, with or without a context, and explain their reasoning. I actually feel quite great about student understandings in that moment, however, I have really seen the importance of having students make deeper connections, create conjectures and justify claims/generalizations. It truly pushes them to think about a deeper meaning of operations and demonstrates the depth of their understanding in developing proof of their thinking.

The day after the Number Talk, I had the class split into two groups and each focused on writing a context that would lend itself to being solved using one of the two strategies for 400-274.

IMG_0508_2After about 5 minutes, I had each group share their context and they did just what I was hoping. The group with the top strategy had a few contexts that all involved having something and then giving something away or losing something. The bottom strategy had a context involving having $400 and leaving the store with $126 and asked to find how much they spent. The second was much more difficult because they kept arguing (in a great way) that it was an adding up problem, not subtraction.

I had each group read their context aloud twice. The first time I could tell everyone was listening to see if it could be solved using subtraction so for the second time, I gave them a listening focus. I asked them to listen for how the two contexts were different, was something different happening in each? After reading them aloud once more, I had them journal what they thought, were they the same type of subtraction problem or different? (They referred to the problems by the student’s name whose strategy it matched).

I loved this student’s wording of the difference as “things happen”



IMG_0530_2There was an overwhelming “what is left” and “what the middle piece is” theme among all of the responses as the main difference between the two contexts. Knowing that removal is their primary way in which to think about subtraction, we chatted more about this missing piece and they agreed that they think about that context more as adding up, which makes complete sense to me. It was so nice to hear them talking about the way a context can influence how you use subtraction or addition and how it really was doing the same thing.

A lone student also brought out the constant difference meaning of subtraction during the Number Talk, however he was losing the class in his explanation that day. I didn’t want to lose this idea, so while the others worked on the contexts, I asked him if he minded elaborating more on his idea and creating a visual proof for the class to better explain his reasoning. I got this…

IMG_0521_2He did such a great job of showing two different representations, but I was secretly hoping for a number line with a “shift” in the numbers to really show constant difference. He instead showed removal with same difference. I adore the top piece and just as I was thinking of how we could make that more visual for the class today, Simon (of course) responded with a brilliant visual…

At the beginning of class today, I asked the student who wrote the response above, what he thought about this visual and he said, “Well, that is just like what I was saying.” I asked him if he could work on a claim for the way he is thinking about subtraction while I asked the rest of the class to see if they could think of a claim that this representation would support. This was such an interesting reversal of the usual process I use with student claims, but I was excited to try it out!

I got many ideas in terms of the bars such as these…

IMG_0550IMG_0552I then showed them Simon’s second idea…

…and asked them to think of these more as subtraction problems and see what they could come up with in terms of noticings and/or generalizations. I got some awesome responses! IMG_0550 - Version 2

IMG_0548Then we shared the original student’s claim he worked on to see if it matched their thinking…

IMG_0553I love that he was not only thinking about what was happening in the problem, but also why he would want to use this in order to make a problem easier to solve.

These lessons were a beautiful way to work forward and backward in making claims. Thank you Simon for being so amazing, as always, it was not only great learning, but great fun!

My students never fail to leave me with something to think about. One student said he thought of “partial differences” and here is how he explained it (definitely not what I thought when he said the term)

IMG_0547My next task is thinking of questions to ask him about this….


Conjecture or Claim?

I have been having wonderful conversations on Twitter recently with Kassia (@kassiaowedekind), Simon (Simon_Gregg), Mike (@MikeFlynn55), Elham (@ekazemi) around the topic of students making claims, more specifically differentiating between claims and conjectures. I have to admit, I have really just formed my own idea of how I differentiate between the two, so it was nice to hear others’ perspectives around this. I consider a conjecture a noticing they think to be true, more on a case by case basis. A claim, to me, becomes more generalized and then followed with a proof. (I have also had great convo with Malke (@mathinyourfeet) around these proofs w/geometry).

The conversation last night started with Kassia…(Look Kassia, I finally learned to embed tweets:)

  Mike gave us a nice perspective of claims based on his work with Virginia Bastable….

My students have now started to say, “I have a claim to make” when they notice something happening over and over again. In those moments, I don’t really think about “what” they are calling it because I am just so excited to hear them talking about the patterns and regularities they are seeing. But is what they are saying a conjecture or claim? Does it make it to the claim wall to be revisited and proven? This year being my first work in really having students think about making “claims” beyond just noticings, I have made a “Claim Wall.” Students see things happening in certain cases and I ask them if they can write a statement for “any time we…” to see if they can make it more general. I like Simon’s idea to expand on my wall…

We all agreed that the proof piece is the difficult piece of going from being a conjecture or unproven claim to a substantiated, generalized claim. I find my students prove over and over again that it “works here and here and here…” but have trouble with the why. It is hard to do, even as adults putting it into words is difficult.

What I love most about these conversations is the fact that the next day it continues, but this time with the kids. Simon tweets this morning about a claim that two of his students made while folding paper…

Which coincidentally would help my students tremendously to think about when proving their claim from Friday’s number talk…

The coolest part about this claim was that it stemmed from a multiplication of fraction number talk, yet they proof show division. I loved that. Also loved the explanation that accompanied their statement. I did ask them if this was true for taking half of any fraction because they seemed to be just dealing in unit fractions at this point. So is this a conjecture or a claim? I am not sure. How generalized would make it a claim? Could it be “When taking any unit fraction of another fraction…”

Would love any thoughts, conjectures or claims on this…:)

To be continued…



Geometry Is Worth The Extra Time…

As I am sure many teachers can attest, there is a constant struggle each year between covering content and the precious amount of time we have to engage the students in learning. Prior to the past two years in the classroom, this guilt always seemed to creep up most during our geometry units. I used to feel that once the students could find area, perimeter, and volume, we would move back into our fraction and decimal work because that always took SO much time to develop a deep, foundational understanding. While geometric representations such as an area model support the fraction and decimal work, it is still not the 2D or 3D unit work.  Right or wrong, I felt I had to prioritize to make use of the little time I had for the best of my students. Over the past two years, however, my geometry units have been taking longer and longer because I have started to see things evolve in my geometry units that has me  wanting to kick myself and go back in time to give my past students a different learning experience. From the connections to number and operations to the development of proofs and generalizations have been eye-opening.

As all of these math connections were going through my head, I see this tweet from Malke (@mathinyourfeet)…

mrahhh, it felt like validation in some weird way.

After this Twitter conversation, I started to dig back into my students work to find examples that makes these connections visible.

After doing a dot image as our Number Talk one day, I asked students to see if they saw any connections between the image and our volume work that day. This work shows how students see the commutative property in both, multiplication as groups (like layers in volume) and most importantly puts a visual to how multiplication and its properties “look” in both 2D and 3D.

IMG_7754IMG_7755IMG_7763IMG_7765IMG_7764Then volume led into some great generalizations about how multiplication “works” through looking at patterns, which is extremely important in mathematics in and of itself.  In keeping constant volume (product), students realized they could double one dimension (factor) and half the other. In doubling the volume (product), the students realized they double one dimension (factor) and leave the others the same. T

IMG_7795_2IMG_7797This volume discoveries later let to this claim on our claim wall:

IMG_8148The students extended this area and volume work to fractions/decimals that showed that fractions/decimals act as numbers in operations as well, supporting the structure of our number system.

IMG_7632IMG_7980While we classified polygons, I saw my students develop proofs for angle measures and our always, sometimes, never experience was invaluable. This work in connecting reasonings through visuals of the polygons explicitly supports the Mathematical Practices of using models, perseverance, and repeated reasoning.

IMG_8280_2IMG_8283_2IMG_8423Then our work with perimeter and area solidified the importance of students creating a visual in building number is so important. In a problem with equal perimeter and different area (moving into greatest area), students created a beautiful visual for the commutative property as well as supported students in seeing the closer two numbers (with the same sum), the greater the product.



Developing Claims – Rectangles With Equal Perimeters

Yesterday’s math lesson launched based on this student activity book page in Investigations…


Students read the introduction and I first asked them, “If we were to build these in Minecraft, was feet an appropriate unit of measure?”  Some thought that feet seemed too small for a garden and instead wanted to use yards, that was, until one student schooled us all on Minecraft. Come to find out, Minecraft uses the metric system with each block representing one cubic meter. We then changed the unit to meters and were on our way.

I gave them 10 minutes, either with a partner or individually, to build as many different rectangular garden designs as they could with 30 meters of fencing. As I walked around, it was interesting to see how students were designing their gardens. Some started building random dimensions and adding/subtracting fence links to eventually hit 30 while others had thought of one rectangle to start and adjusted from there by subtracting/adding from sides. It was fun to see the ones that discovered a pattern in their building and sped through the rest of the rectangles. Some beautiful patterns emerged….

IMG_8854Photo Jan 14, 1 23 24 PM

Photo Jan 14, 1 22 35 PMAfter 10 minutes, I asked them fill in the table on their activity page and chat with their table to see if anyone had any dimensions they didn’t have. As a table, they came up with some noticings based on their table, Minecraft builds, and process of building.

IMG_8862 IMG_8863 IMG_8865 IMG_8866 IMG_8867 IMG_8868 IMG_8869 IMG_8870 IMG_8871 IMG_8872 IMG_8873The class ended after this group work. Talking with another colleague about the lesson at the end of the day, we chatted about what claims could come out of this lesson. It was so interesting to me to think about not only the geometric claims, in terms of rectangles and dimensions, but also the numeric claims that can evolve from these conversations as well. And to think that these number claims are grounded in a visual connection is pretty awesome to me!

I was so I started today by pairing up every two tables to share with one another. One table read their noticings while the job of the other table was to ask clarifying questions. That gave me time to circulate, listen, and choose the claims for our class share out. After the table shares, we reconvened as a whole class and chatted about a few of the claims as a group.

IMG_8853I asked them to think about similarities and if we could combine some of them to form one claim based on our work.

Class ended with a journal entry in which they worked independently to begin to form a claim or the beginnings of one. Here is where I will start tomorrow…

IMG_8855 IMG_8856 IMG_8857 IMG_8858 IMG_8859 How fun!

As a follow up on this post, this is the assessment on their work with perimeter and area. During the class period, only one student started working on finding the square that would give them the largest area with a perimeter of 30. After this idea came out in our class discussion, using fractional dimensions was something that others were thinking about during the assessment.  Here is a student’s (different than the initial student) work that I thought was pretty fantastic:

Photo Jan 16, 11 51 34 AM





Intuition in Learning Math

Yesterday, Malke tweeted this…

i3…and it led to such an interesting conversation that I honestly can say, I had never really given much thought. The conversation is here if you would like to read it now, or you may want to save it until after you read my rambling thoughts. 🙂

At first read of the tweet, my initial thought was how hard it was for me to make a distinction between intuition and making sense of problems. What makes them different? The amount of time it is given? The context of the situation? The math experience behind it? My questions could go on and on. I needed time to process these thoughts and let them sit with me for a bit. I tried reading some of the links to works about intuition in learning, but until I could figure out how I was thinking about intuition and put it in the context of my classroom experiences, the readings were not making much sense to me.

Luckily, I had a lot of car-riding time yesterday to think about this and jot some thoughts down. Disclaimer: these ideas are quite scattered, do not form a cohesive string of thoughts, and you will probably be left with more questions than answers by the end, however it is really fun to think about all of the ways “Intuition” takes shape in learning.

I thought it would be good for me to start with a definition and work from there. First, I tried Webster and got this one:

i2eh. I didn’t like the “without any proof or evidence” piece of this because I feel that our intuitions do come with proof or evidence, they are maybe not explored or articulated yet, however I think they are there. Then I found this one that I felt I could work from a bit better:

iThis definition by no means encapsulates how I envision “intuition”, however it had some really interesting points that led me to more questions….

– Does our intuition mean we have an “understanding”?

– Where does our “conscience reasoning” come from?

– Is our intuition always what is “likely“?

– Where do I see these hunches, inklings, notions in my students’ learning of math?

– Where do I see these same things in myself as a teacher?

Up to this point, I was gaining more questions than answers, so I began thinking about these questions in terms of my students and myself as a teacher. I am a person always in need of examples, so I needed to form some “example cases” to start to clarify these thoughts.

In this first example, from an Illustrative Task, the student was asked to determine if the answer to the problem could be solved using multiplication. Although the student came to the correct answer of 2/40, his intuition is telling him it still doesn’t make sense. Is this intuition based on previous experiences? Are all of our intuitions based on past experiences or non-experiences? Has he never seen an area less than 1 so it can’t make sense? The array has to be whole numbers? For me, the use of miles intuitively feels weird when I read it. Even as an adult, I hear miles and think of distance, bars, and do not like it so much with area.

Bb3AJhpIQAA1EexIn class each day we do Estimation 180. In this example, I specifically look at Day 23. I never really thought much about this, but I think it is intuitive of the student to look at the size of the item being packaged in another when thinking about capacity and volume. The student doesn’t look at the color of the paper or types of lines on it, but instead intuitively looks at the size of each part. It is something that happens so fast, that perhaps these are the quick, small moments in my classroom where students are acting intuitively.

i5On Day 36 in the example below, a student’s reasoning is that it is “usually a weird number” and on Day 37 says “My family never gets blow pops so I guessed.” Both of these seem to stem from experience/non-experience.  The student has noticed that in previous days the packaging numbers have not been “friendly numbers” so the students is intuitively thinking it is a non-friendly number. If he/she was given this problem on the first day of the estimation questions, would their intuition have led them in a different direction? Day 37, shows a glimpse into, what I call, non-experience. This student intuitively goes to guessing because of never having them at home before, however is still only 20 off. This then leads me to question, that if a student doesn’t intuitively think of a reasonable estimate, do they then move into a more concrete strategy? To be within 20, I feel like the student used counting in the picture to some extent.

i4In this second grader’s work, I see such an interesting intuition in the second part of this question…

BEG7yU8CcAAsz00She seems to think that she cannot possibly know all of the possibilities because she is not older. She associates getting older as getting closer to knowing “all” of something. Little does she know that as you get older, you find you know less of “all” than you originally thought 🙂 Ha!

Here is where I get muddled between intuition based on experience and making sense based on prior knowledge. I asked this question to my students last year and here is one example of a student who went beyond yes or no and started to give a proof. I would love to hear others thoughts on intuition here….


This example below makes me think about how conjectures are made by students. Do conjectures stem from intuition and proofs that we can’t fully explain? The student said that when comparing two fractions, with the same numerator, she can subtract numerator from the denominator and the smaller difference is the larger fraction. Is this false intuition in dealing with the numerator and denominator as whole numbers? Thinking you can just pull them apart and operate with them as wholes?BVcOikSIgAA5AT1

This one may not be an example of intuition, but it is how I think about my own intuitions in learning….I try to explain them, prove them, revise them, edit them….. I would love every student paper to look like this…


In writing so far, I am really thinking that a student’s intuitions in learning math come from a “conscience reasoning” based on  prior experiences and exposure. I could truly rack my brain over this for a while, but before I do, I wanted to think about myself as a teacher also. How much do we do as teachers that is intuitive? How does that intuition change as we evolve as educators?

There are many things I do during the course of the school day that just feel like routine or habit. The hard part is deciphering where it is not just habit or routine, but instead intuition.

When writing my lesson plans, I would say I use my intuition often in the respect of anticipation.  From the minute I read the lesson, I have intuition on how I feel the lesson will “go over” with the students. I have a gut feeling if they will be interested in it, which students will be able to easily enter into the problem and which will struggle, and what strategies will emerge. All of these anticipations are based on my experiences with the students. So are these anticipations, intuitions?

As the lesson is happening, I think so much of my questioning is based on my intuition. I could not possibly have a list of questions to ask students during the course of every lesson, I have to rely on my intuition. As a student is explaining something, I am thinking to myself, “I think it would be interesting to ask _______.”  This is something that has definitely evolved based on my experience, however because of the “newness” of every day and every class, I have to rely on intuition of similar case scenarios. Something like, “I asked this question the other day and it got me nowhere, how can I ask it differently to push student thinking?” This inner dialogue during a lesson happens in an instant which makes me believe it is intuitive.

If all of this is true, then I would say that when I first started teaching, my intuitions were not as fine-tuned as they are now. Does that makes sense, can you fine-tune intuition?  Is there a point beyond thinking something is a good idea/bad idea or makes sense/doesn’t make sense that is still intuition but a more detailed, specific intuition? Intuitively, I think there is:)

A lot to think about still….Thanks to Malke, Tracy, Simon, Bridget, Kassia for a great (to be continued) conversation!


Articulating Claims in Math

This summer I was fortunate to hear Virginia Bastable keynote about the work in her book Connecting Arithmetic to Algebra. If you have not read this book, it is a must! It explores the process by which we have students notice regularities, articulate claims, create arguments and representations, and make generalizations.

9780325041919It is something, as elementary school teachers, we need to really be thinking about more in our math classes. Are we creating environments that encourage students to think about the math behind the strategies and make generalizations based on the properties of operations? I have taken this recent reading and made it a priority in my classroom.

I always have students notice and discuss patterns and regularities but I don’t often have them create generalizations for us to revisit as we move through the year. For example if their claim works for whole numbers, shouldn’t I revisit that as we work with fractions? Does your claim still hold true?

As a class routine, I posted this on the board and asked students to fill in the blanks to make it true:

12 x 4 = ___ x ____

Quickly, students wrote down answers, had their hands up, and one student blurted, “This is easy, you don’t even have to solve it!” Typically blurting out answers before others are done thinking drives me a bit crazy, but this time I was thinking…Yes! I asked who else thought the same thing. I had at least one hand up at each table so I asked them to discuss with their table how that is possible. We came back together and each table said they could double/half to fill in the blanks. I took answers on the board and got the expected: 6 x 8, 24 x 2, 48 x 1, 3 x 16 and then I even had a 96 x 1/2 and 36 x 1 1/3! I asked about the 48 x 1…did you get that by double/halving from the original problem? What is happening there? They noticed that it was x 4, ÷ 4, and then the same with 3 x 16. I asked them to take some individual time to see if they thought their strategy would always work and could prove it with a representation. They then talked at their tables and I asked each table to write a claim, something they think is true about this work.

I got some who kept solving problems to prove it works:


I had a couple try out the representation (exclaiming how hard it was to draw what is happening:)


Here are some of their claims:



This is the class list from my second period class. I especially liked that one of them said it only worked with multiplication. How fun to revisit!

IMG_8107Instead of losing these, I started a Claim Wall to post and have students add to and revisit throughout the year. I am trying to think through how to have students comment on them, possibly agree/disagree post its?


If you would like more information about Virginia’s work, there are courses available here: Check it out, great stuff!


Making Mathematical Connections

Each day I start the class with a Number Talk. I thought to continue building our multiplication strategies and make connections to our volume work, I would do Dot Quick Images. This is one of the images that I did yesterday:


In this image I hoped to bring out the commutative and associative properties (not by name, but the idea of what is happening in each) within their solutions as well as the use of the 3 x 3 array to get the number of groups in the picture and 3 x 4 array to get the number of dots in each group.  This would be the moment when I wish I took a picture of the board with their responses, but in the flow of the lesson, forgot. Many said they did 12 x 9 to get 108. I especially loved that some said they said they didn’t know 12 x 9, so did 12 x 10 and took a group of 12 away:) I said that when I read 12 x 9, I think of 12 groups of 9, trying to elicit the commutative property. I had them talk to their neighbor and we agreed this picture looked like 9 groups of 12, but there was a way to make it look like 12 x 9. I wrote both on the board agreed the amount of dots didn’t change, just the way we looked at it did. This went on into another image and we began our first lesson on volume. I blogged about that work here.

So, after chatting with a colleague after the lesson, we thought it would be interesting after the work yesterday, to reflect back to that number talk. Today I put the same image back up and I did a much better job of pulling out the (3 x 4) x 9 through better questioning and because they were solid in the answer, they could reason about it a bit deeper. I also told them to be thinking about our lesson yesterday to see if they could see any connection between the two. I finished the number talk and gave them 2 minutes to reflect in their journal about any connections they saw. Here is what we shared as a class…

IMG_7753 IMG_7754 IMG_7755 IMG_7762 IMG_7763 IMG_7764 IMG_7765 IMG_7766 IMG_7767 IMG_7769

So much to love here…..I loved the idea of layering the dot arrays to make a box. I loved the connection to the properties in each….