The 4th grade has just started their fraction unit, so I was curious how that may impact their work with the Cuisenaire rods. I started just like I did in Kindergarten and 3rd grade, with a notice and wonder:
There were two of the ideas that really struck me as things I want to have the students explore further later: First was, “2 of the staircases (the staggered rods in order of size) could make a square.” They had them arranged on their desks like the picture below…but I want them to answer:
- Is that a square?
- If not, could we make it a square?
Then I started to wonder, do we call it a square? Should we say square face? Then what about area…would we say, “What is the area of the rectangle?”? That feels wrong because they keep calling the white rod a cube (which it is). But then asking about volume is not 4th grade. BUT, the tiles we use for area in 3rd grade are also 3-dimensional. <–would love thoughts on any of that in the comments!
One student noticed that the orange rod was the height of the staircase and I thought of area again since it was said right after the comment above. This idea would be really helpful for the students above when they are determining if their figure is a square.
I loved that one group noticed that any of the rods could be a whole and another group wondered if orange was the whole. Great lead into what I was thinking I wanted them to explore!
I asked them to find values for the rods based on their relationships. Of course the very first group I call on had 2 as the whole, which blew a lot of students minds, so I want to revisit that a bit later and ask them to explain how that works.
All of the other groups had orange as either 1 or 10, so I asked them to find the other values if the orange was 5 and 100. They played with that for a bit and then I began to hear a lot of aha’s, so I set them off to find more and they could have gone on forever.
I left them with the prompt, “Tell me about the patterns and relationships you notice.” and for those who looked like they were struggling to answer that question, I added, “If you are struggling with that, tell me how you could find the rest of the values if I gave you one of them and which one would you want?”
I loved how this student chose the orange, white and yellow as the easiest end, beginning, and half. I also like the red x 2 is purple, but we need to talk through that notation a bit.
This was the most common response, seeing the numbers get smaller as the rod got shorter.
This student’s noticing could be an interesting number choice question to pose: Why do you think groups chose numbers for orange that were doubles or halves of the other numbers we already had?
This student disagreed with the student who gave the responses in the first column because he is determined the white is 1/10 because the orange is 1. Would be great to pair them up and have them come to an agreement.
This student is seeing the white value adding to each value above it to get the next. I also love how she writes notes about how neat her handwriting is:)
I would love to have them play around with this first pattern in this entry! What other relationships could they find after they explored this one?
So much fun! Cannot wait to get into other grade levels to see if I can begin to find a progression of ideas with these rods!
MathMinds 
Wow – as in your last post, Kristin, I’m impressed with how far the students got in just one lesson! Brilliant to start with the notice and wonder – and so good that it turns up lots of what you’d want to explore and more! “Each rod could be the whole” – did they think of that themselves?!
As for the “square” question, I think at a certain point it would be good to bring up. For the most part though, we are all used to ignoring certain parameters and focusing on agreed ones. So, when we think about the rods as lengths, we usually ignore that they have width and depth too. So we call the rectangular prism a square because it’s its 2D aspect that we’re focusing on. Perhaps it’s part of a bigger thing in maths that all representations in the world, whether lines, or circles or whatever are “imperfect” (or rather we agree to ignore things).
That the rods have width and depth turns out to be useful though when we’re looking at area and volume with them!
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Thinking again about their inventiveness in the N&W, I wonder if all the journal writing that the students do helps them to be more reflective at the start of the lesson as well as during and at the end? A kind of intentionality is fostered?
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Yes, I think they always know I am going to ask them to write about something so I think that makes it more meta for them at the end. Sometimes, if I want them thinking about a particular thing, I pose the prompt at the beginning like, “While you are working, be thinking about…” sometimes it depends on what they see.
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Hey there Simon!
Yes, they came up with the whole all by themselves. As I walked by, I heard 1 student say that orange is the whole (they just started their fraction unit so they have been talking about that language) and another said, why does it have to be? Couldn’t this one be the whole?
Thanks for the input on the square vs prism conversation. I really hadn’t even thought about it until a student called the white rod a cube! Then my head started spinning.
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I have always loved reading your blog posts, but I have never been more excited to read them now that you’ve discovered the powerful application of using Cuiseanaire rods to explore all sorts of math concepts. I first learned about them a few years ago and was blown away by how many different concepts can be explored using the Cuisenaire rods. I love that they help students see numbers as existing as a group rather than a collection of ones. Before learning about Cuisenaire rods, in order to help my students learn, let’s say, addition of multidigit numbers, I would use base 10 blocks, including the ones, to model that. Let’s say it was 389 + 278. In the ones column there would be 9 ones and then 8 ones. Many of my struggling learners would count out the full 17, then count out 10 and trade for a 10 rod and then count out the 7 remaining ones. Lots of counting. Now, though, I would have an 8 rod and a 9 rod instead. Seeing the 8 and 9 existing as two groups encourages the students to think of strategies like 8 + 8 + 1 or 9 + 9 – 1 or even 9 + 1 + 8 when they line it up beneath a 10 rod. So much more powerful in encouraging strategic thought! I love it!
I have worked as a Math Coach K-5 the last few years and have used them in every classroom practically all year long. I don’t want to mention all the things I use them for because I’m so excited to hear what you come up with!!
In terms of the depth to them, I agree with Simon that there are times when we do disregard the extra dimension. I was using square inch tiles with my students today working on area and even those have a little height to them but I didn’t worry about that.
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Thanks so much Ann! I love the idea of operations with these rods since they were doing so many equal rods! I wonder about bring in the book Equal Shmequal with the rods?? Please keep all of the suggestions coming because I am in love with these things!
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Here are some of the ways I’ve used them with my K-5 kiddos:
Kindergarten –
explored the commutative property by having them build two layers of rods – red and blue is the same as blue and red, etc…when kids are ready they say the numbers
Math facts – We interview students with Math Running Records to determine which strategy they are working on (for addition – progression of plus 1 and 0, plus 2 or 3, make 10, add 10, dbls, dbls+1, dbls+2, decompose to make a 10 with any 9, and decompose to make a 10 when adding with a 7 or 8). The rods are one of the tools we use to teach the strategies to the students (along with rekenreks and 10 frames). Here is a link to my site where I have videos of myself showing how we can teach each addition strategy..need get going on sub and mult). https://goo.gl/5IlvoM
Grade 1 and 2
Double digit addition and subtraction. Since the orange rods are the same size as base 10 rods, we use them interchangeably. For the ones digit, though, we use the rods rather than all ones cubes. When subtracting, we don’t exchange a 10 rod for 10 ones…we just slide over the 10 rod to join the Cuisenaire rod…we line them up to make a total length and then put the amount we are subtracting beneath it to see what the difference is. So awesome to use the 10 as a bridge and add or subtract in chunks rather than counting up or down using fingers.
Grade 3-5
Multiplication – perfect for the area model. We begin by exploring commutative property – you can put a rectangle of 3 -4rods on top of a rectangle of 4 3rods and see that they area is the same. Then, it helps me teach the multiplication math facts by strategy (mult 0, 1, 5, 10, 2, 4, 8, 3, 6, 9, 7). Just today I was showing them with the rods how x8 is x4 doubled by having 8 of the rods and physically just moving half a bit so we could see the 8 groups as 2 set of 4 groups. If the students didn’t know the x4 math fact, I then moved two of the rods to start with dble, then dbled for the 4 and then doubled that for the x8. So cool! For those that were ready, we did two digits by one digit and discussed how we could double the 4’s again or we could break apart by place value position. Love developing that flexibility. I always emphasize that multiplication always results in a rectangle. Here is a video of me showing this from math fact to multidigit: https://youtu.be/pshSjM4SQZU.
Division – The area model changed my brain in how I view division now. We begin by grabbing the rods that are being divided…this could even be in the hundreds using 100s and 10s from Base 10 blocks. Our job is to make the largest rectangle possible with a vertical alignment measurement of the divisor. Once we make the rectangle using the entire amount, the horizontal measurement is the quotient. Students will be able to follow this procedure all the up to high school when dividing polynomials…I had a teacher put on the board x-squared + 5x + 4 divided by x+1. I remember thinking to myself that I was done because I hadn’t done that since high school which was 30 years ago. But then, my brain said grab the rods…xsquared is the 100 flat, 5x would be 5 of the 10 rods, and then the 4 is 4 ones cubes. I then thought that I needed to make a rectangle coming down x+ 1 which is a 10rod and one cube. Within seconds I had a rectangle an the top measurement was x + 4. Not only did I get the answer, but I totally understood why it was x+4. That was the moment I was totally hooked on the area model. How awesome that we can teach students a strategy in grade 3 with basic facts and the very same model can be used through their entire math journey when multiplying and dividing.
Grade 5
Multiplying and Dividing using decimals. The 100 flat is the whole, 10 rods are tenths, and Cuiseanire rods are hundredths using the same area model to explore partial products and quotients.
Those are some of my ideas…can’t wait to see what you come up with!
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Oh my gosh this is an amazing list!! I cannot wait to sit with this and plan!! THANK YOU!!!
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Here’s the video I made of dividing using the rods using the area model…
🙂 Ann Elise
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A couple more things:
for K-2 kiddos, using them to decompose numbers is huge! They can start by building a 5 house. 5 rod on top, then 4 and 1, then 3 and 2, then 2 and 3 and we notice that they are just turned around, 1 and 4 and 5 rod again. We know we found out all the ways 5 can be decomposed. They can write the equations that match and notice the patterns.
K-5 Exploring all the word problem types for addition and subtraction…we’ve been using numbers just within 10 at my grade 3-5 school to focus on the structure of the word problem and not be bogged down with computation. PLUS, the cuisesenaire rods are the exact model! If I had 6 cookies, then you gave me some and now I have 10, I would take a 10 rod and put it on top, then beneath it I would put the six rod. The rod that fits the empty space is the amount that I was given. Here is a blog post I wrote about this initiative. Been so powerful!
https://aerecord.wordpress.com/2016/11/28/why-i-love-tape-diagrams/
Enjoy!
🙂 Ann Elise
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