Category Archives: 3rd Grade

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.


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.


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



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.


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!


Extending an Area Task

I just love when students are so excited to extend an activity! During the Notice and Wonder portion of this lesson, a lot of students wondered why those four letters were the ones given. Was it because they are at the beginning of the alphabet? Is it because they have the same area? What would happen with other letters?

Today, Mrs. Sharp gave them the chance to play around with others letters. She asked them to design their own letter and find its area. It is so interesting to see their choice of letters, the way they chose to decompose the shape and their math work all around it. Many of them made multiple shapes because they just wanted to keep going…that is always so AMAZING to hear!

I also see all of this being SO helpful when they find volume of figures composed of two non-overlapping rectangular prisms in 5th grade!

Here are just a few of the creations they came up with in class today:




Area Task in 3rd Grade

Since the 3rd graders just wrapped up their unit on area, I thought it was the perfect time to do a task that hit on some really important ideas about area, while also encouraging them to move beyond counting squares. I wanted to see how (or if) students broke apart shapes to find the area, how they used addition and/or multiplication to more efficiently count squares, and if they used any other strategies such as subtracting out blank spaces or decomposing and rearranging pieces to find the area.

I chose this task from Illustrative Mathematics.

We started with a notice/wonder activity:



They had so many great wonders that inspired me to think about a follow up activity about other letters, but I will chat about that later.

Since they wondered if the letters all had the same square units and if they were all the same, I used that as the lead into the activity. Even as I was giving directions, however, I saw a majority of them start to count squares by 1. I paused them, told them I was so excited to see they knew counting the squares would get them the area and knew they all could find the correct area that way so we were going to try something different. I asked them to find the area without counting all of the square by 1.

In looking at the work, I saw them as a bit of a progression of thinking. I put them in order here of how I see students moving through these ideas about area.

As I expected, some counted each row and added them. It was great they know area is additive, but I would love to ask this student if there is a way they could have used multiplication to  make it a bit easier.

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Some added in chunks, to which I would love to ask the same question. I was excited to see them cutting the shapes up into rectangles in places that made sense.

Scannable Document 8 on Feb 6, 2017, 6_15_11 PM.jpgFrom there, some used a mix of addition and multiplication. I would love to ask these students how they decided where to make their cuts.

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Some students made some larger cuts and I would love to have them meet with the student above and discuss how they decided on their cuts.

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Some used some of the strategies above but also relied a bit on symmetry.

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Finally, I saw 2 students moving squares from the “bumps” to the empty spaces.

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It was always interesting to me that in 5th grade I would still see students find the area of shape on a grid by counting the individual squares even though I know they had better strategies. I think it is the fact that students jump into doing things without thinking about the things first. This is why I think journal writing is so important. It allows students to be more reflective about their decisions.

I asked them to write about which shape was the easiest to find the area and which was the most difficult. It was interesting to see some focus on the size of the number they were working with while others focused on the shape and how it could be partitioned.

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As a follow-up activity, I am going to ask them to choose a letter where they would use the same strategy they used with C and a letter where they would use the same strategy they used with B.


Fraction Talking Points: 3rd Grade

The 3rd grade is starting fractions this week and I could not be more excited. Fraction work 3-5 is some of my favorite stuff. Last year we tried launching with an Always, Sometimes, Never activity and quickly learned, as we listened to the students, it was not such a great idea. We did not give enough thought about what students were building on from K-2 which resulted in the majority of the cards landing in the “Sometimes” pile without much conversation. And now after hearing Kate Nowak talk about why All, Some, None makes more sense in that activity, it is definitely not something we wanted to relive this year!

We thought starting with a set of Talking Points would open the conversation up a bit more than the A/S/N, so we reworked last year’s statements. I would love any feedback on them as we try to anticipate what we will learn about students’ thinking and the ideas we can revisit as we progress through the unit. I thought it may be interesting to revisit these points after specific lessons that address these ideas.

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We were thinking each statement would elicit conversation around each of the following CCSS: 

Talking Point 1CCSS.MATH.CONTENT.3.NF.A.3.C
Express whole numbers as fractions, and recognize fractions that are equivalent to whole numbers. Examples: Express 3 in the form 3 = 3/1; recognize that 6/1 = 6; locate 4/4 and 1 at the same point of a number line diagram.

Talking Point 2CCSS.MATH.CONTENT.3.NF.A.2
Understand a fraction as a number on the number line; represent fractions on a number line diagram.

Talking Point 3CCSS.MATH.CONTENT.3.NF.A.2.B
Represent a fraction a/b on a number line diagram by marking off a lengths 1/b from 0. Recognize that the resulting interval has size a/b and that its endpoint locates the number a/b on the number line.

Talking Point 4: CCSS.MATH.CONTENT.3.NF.A.1
Understand a fraction 1/b as the quantity formed by 1 part when a whole is partitioned into b equal parts; understand a fraction a/b as the quantity formed by a parts of size 1/b.

Talking Point 5CCSS.MATH.CONTENT.3.NF.A.3.D
Compare two fractions with the same numerator or the same denominator by reasoning about their size. Recognize that comparisons are valid only when the two fractions refer to the same whole. Record the results of comparisons with the symbols >, =, or <, and justify the conclusions, e.g., by using a visual fraction model.

Talking Point 6CCSS.MATH.CONTENT.3.NF.A.3.C Express whole numbers as fractions, and recognize fractions that are equivalent to whole numbers. Examples: Express 3 in the form 3 = 3/1; recognize that 6/1 = 6; locate 4/4 and 1 at the same point of a number line diagram.

After the activity, we have a couple of ideas for the journal prompt:

  • Which talking point did your whole group agree with and why?
  • Which talking point did your whole group disagree with and why?
  • Which talking point were you most unsure about and why?
  • Which talking point do you know you are right about and why?
  • Could any of the talking points be true and false?

Would love your feedback! Wording was really hard and I am really still struggling with #4.

If you want to read more about Talking Points for different areas, you can check out these posts:

Rhombus? Diamond? Square? Rectangle?

It happens every year, in what seems like every grade level…students continually call a rhombus a diamond. Last year, when we heard 3rd graders saying just this, Christopher helped the 3rd grade teachers and me put the students’ thinking to the test with a Which One Doesn’t Belong he created.



This year, at the beginning of the geometry unit, we heard the diamond-naming again along with some conversation about a rectangle having to have 2 long sides and 2 short sides. What better way to draw out these ideas for students to talk more about them than another Which One Doesn’t Belong? We changed the kite to a rectangle this time, hoping we could hear how they talked about it’s properties a bit more.

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Overwhelmingly, the class agreed D did not belong because it had “5 sides and 5 corners” and eventually got around to calling it a diamond, which in their words was “not a real shape.”

While we knew a lot of things could arise, our purpose was diamond versus rhombus conversation, so of course the students had other plans and went straight to the square versus rhombus.We wouldn’t expect anything different!:) For every statement someone had about why the square or rhombus did not belong, there was a counter-statement (hence the question marks in the thought bubbles).


Jenn, the teacher, and I were really surprised at how much orientation of A and B mattered to the name they gave the square and rhombus but did not matter for the rectangle. That was just a rectangle, although one student did wonder if a square was also a rectangle (he heard that from his older sister). The students had so many interesting thoughts that we actually had to start a page with things they were wondering to revisit later! That distributive property one blew me away a bit!:)


We then sent them back to journal because we wanted to hear how they were categorizing a square and rhombus. It ended up being really interesting just seeing them try to explain why they were different and change their mind because they just started turning their journals around!

Some stuck with them being different..



Some thought they were different, but one could become the other…



Some were wavering but the square was obviously the “right way.”


Some argued they were the same…

So much great stuff for them to talk about from here! I left wondering where to go from here? In thinking about the math, is it an orientation of shapes conversation? or Is it a properties conversation? In thinking about the activity structure, would you pair them up and have them continue the conversation? Would you throw the rectangle into this conversation? Would you have some playing with some pattern blocks to manipulate? Would you pull out the geoboards? I am still thinking on this and cannot wait to meet and plan with the 3rd grade team!

However, before I left school today, I went back to the 3rd grade standards to read them more closely:Screen Shot 2017-01-05 at 7.07.05 PM.png

and read the Geometry Learning Progressions, only to find this in 1st grade:

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Would love to hear any thoughts and ideas in the comments!



My Beginnings With Cuisenaire Rods

I have never been more intrigued with using Cuisenaire rods in the classroom until I started reading Simon’s blog! I admit, I have read and watched his work from afar…not knowing really where to start with them and was afraid to just jump into another teacher’s classroom and say, “Hey let me try out something!” when I really didn’t know what that something may be. However, after Kassia reached out to Simon on Twitter asking how to get started with Cuisenaire rods and Simon wrote a great blog response, I was inspired to just jump right in!

I am a bit of an over-planner, so not having a really focused goal for a math lesson makes me a bit anxious. I am fairly certain I could anticipate what 4th and 5th graders would notice and wonder about the Cuisenaire rods because of my experience in that grade band, however I wanted to see what the younger students would do, so I ventured into a Kindergarten and 3rd grade classroom with a really loose plan.

Kindergarten (45 minutes)

Warm-up: Let’s notice and wonder!

  1. Dump out the bags of Cuisenaire rods in the middle of each table of 4 students.
  2. Tell them not to touch them for the first round.
  3. Ask what they notice and wonder and collect responses.

Things they noticed:

  • White ones looked like ice cubes.
  • Orange ones are rectangles.
  • End of blue one is a diamond (another student said rhombus)
  • Different colors (green, white, orange..)
  • They can build things (which is why we did no touching the first round:)
  • Orange is the longest.
  • They are different sizes.
  • We can sort them by colors.
  • We can sort them by size.

Things they wondered:

  • What do they feel like?
  • What can we make with them?

Activity 1: Let’s Sort!

  1. Tell the students to sort them by size or color. (they quickly realized it was the same thing)
  2. Discuss their sort/organization and check out how other tables sorted.

I was surprised to see not many sorted them into piles because that is normally how they sort things. I am wondering if the incremental size difference between each rods made them do more of a progression of size than sort into piles? Some groups worked together while others like making their own set with one of each color (and size) and keep making more of those!

Activity 2: Let’s Make an Orange!

Since a lot of students kept mentioning that the orange was the longest, I decided to see if they could build some trains (as Simon calls them) that made an orange.

My time was running out, but it left my mind reeling of where I wanted to go next! My inclination is to ask them if they could assign numbers to some of the rods or if they could build some trains the same length as the different colors? I would love to hear which piece is their favorite piece because a lot of them found the smallest cube really helpful when building the orange.

3rd Grade (60 minutes)

Warm-up: Let’s notice and wonder!

Things they noticed:

  • Groups were the same color and length.
  • Blue and white is the same length as the orange rod.
  • Kind of like adding.
  • White is 1 cm.
  • Go up by one white cube every time.
  • Odd + even numbers
  • 2 yellows + anything will be bigger than 0.
  • 1 white + 1 green = 1 magenta

Things they wondered:

  • Is red 1 inch?
  • How long are the rods altogether? (Prediction of 26 or 27 in wide)
  • Is orange 4 1/2 or 5 inches?
  • Why doesn’t it keep going to bigger than orange?

Activity 1: Let’s build some equivalents!

I found 3rd graders love to stand them up more than Kindergarteners:)

Activity 2: Let’s assign some values!

After they built a bunch, I asked them to assign a value to each color that made sense to them…this was by far my favorite part – probably because it was getting more into my comfort zone!


Again, time was running out, but next steps I am thinking…

  • What patterns and relationships do you see in the table?
  • What columns have something in common? Which ones don’t have anything in common? Why?
  • What if I told you orange was 1? What are the others?
  • What if orange was 2? What happens then?

Thank you so much Simon for all of inspiration and Kassia for the push into the classroom with these! Reflecting, I was much more structured than Simon and Kassia, but I look forward to a bit more play with these as the year goes on! I look forward to so much more play with the Cuisenaire rods and continuing Cuisenaire Around Ahe World!

Number Talk Karaoke

It is always so fun when I have the chance to hang out with my #MTBoS friends in person! This summer Max was in town, so I not only got to have lunch with him but also meet his amazing wife and puppy!  Of course, during lunch, we chatted a lot about the math work we are doing with teachers and some of the routines we are finding really valuable in their classrooms. From these two topics of conversation, Number Talk Karaoke emerged.

We both agreed that while Number Talks are invaluable in a classroom, it can be challenging to teach teachers how to use them in the classrooms. As much as we could model Number Talks during PD and show videos of them in action, it is still not the same as a teacher experiencing it for themselves in their classroom with their students. There is so much to be said for practicing all of the components that are so important during the facilitation with your own students.

That conversation then turned into two questions:  What are these important components? and How do we support teachers in these areas?  We discussed the fact that there are many books on mathematical talk in the classroom to support the work of Number Talk implementation, however the recording of student explanations during a Number Talk is often left to chance. What an important thing to leave to chance when students often write mathematics based on what they see modeled. We brainstormed ways teachers could practice this recording piece together, in a professional development setting, where students were not available.

Enter Number Talk Karaoke.

During Number Talk Karaoke, the facilitator:

  • Plays an audio recording of students during a Number Talk.
  • Asks teachers to record students’ reasoning based solely on what they hear students saying.
  • Pair up teachers to compare their recordings.
  • Ask teacher to discuss important choices they made in their recording during the Number Talk.

Max and I decided to get a recording and try it out for ourselves. So, the next week, I found two of the 3rd grade teachers in my building who were willing to give it a go!

They wanted to try out the recording piece themselves, so they asked me to facilitate the Number Talk. They sat in the back of the room, with their backs to the students and SMARTBoard so they could not see what was happening. All they had in front of them was a paper with the string of problems on it.

Before seeing our recording sheets below, try it out for yourself. In this audio clip of the Number Talk, you will hear two students explain how they solved the first problem, 35+35. The first student explains how he got 70 and the second student explains how he got 80.

Think about:

  • What do you think was really important in your recording?
  • What choices did you have to make?
  • What question(s) would you ask the second student based on what you heard?

The talk went on with three more problems that led to many more recording decisions than the ones made in just those two students, but I imagine you get the point. I have to say, when I was facilitating, I tried to be really clear in my questioning knowing that two others were trying to capture what was being said. That makes me wonder how this activity could be branched out into questioning as well!

Here was my recording on the SMARTBoard with the students:

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Here are the recordings from the two teachers in the back of the room:



We sat and chatted about the choices we made, what to record and how to record certain things. We also began to wonder how much our school/district-based Number Talk PD impacted the way we record in similar ways.

Doesn’t this seem like a lot of fun?!? It can be done in person like mine was, or take the audio and try it with a room of teachers, like Max did! <– I am waiting on his blog for this:) Keep us posted, we would love to hear what people do with this!