First Fraction Lesson of 4th Grade

The first lesson of a new unit always feels like an entire class period of formative assessment to me, which I love! I think finding out what the students know about a topic, especially if it is the first time it is introduced that year, is so interesting.

Since the first lesson of the 4th grade fraction unit starts with fractions of a 4 x 6 array, we wanted to create an introduction lesson that was more reflective of all of the great work they did with fraction strips in 3rd grade to get a better picture of what they know. In 3rd grade they do all of the cutting of the strips, and since we didn’t feel that was necessary to do again, I created a SMARTBoard file so we could build together. [the file is attached at the end of the post if you want to use it].

I posed this slide to introduce the whole:

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Then I asked this sequence of questions as we built them on the board:

  • If I wanted halves, how many pieces would I have? What is the size of each piece?
  • If I wanted thirds, how many pieces would I have? What is the size of each piece?
  • etc….until they were all built.

I wanted to reintroduce the language of “size of the piece” from their 3rd grade experiences. Every once in a while I would pause and ask how much I would have if I had more than 1 of those pieces to see if they could name fractions over a unit. For example, What if I had 3 of those fourths? How much would would I have? 

Next, we put up the following questions with the picture of the fraction strips we built:

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They recorded them in the journals as a group and then we made a poster to add to as the year progresses. They started with fractions they could show on the fractions strips and an interesting conversation about the fact that we couldn’t list any for 1/8 or 1/12 based on the strips, arose. After talking with their groups, they generated a couple. The conversation about the change in the size of the piece when we make equivalents and how many pieces we would have was really awesome (Yeah, 3rd grade team):)

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This was as much as we could fit into one class period, so we asked them to journal about any patterns they noticed or things they were wondering about fractions.

I apologize for the overload of student journals from this point forward, but there were so many great things to think about in planning the unit from here!

These are things that jumped out at me after reading and leaving notes in their journals, I would love to hear any other things that stood out to you:

  • A lot of talk about “doubling” and “halving” when naming equivalent fractions. Will want to address what is exactly doubling, what that means in terms of the fraction strips, and how it is affecting the numerator and denominator.
  • Interesting noticing and wondering about addition. Some wondering how it works and others thinking they know.
  • Love the even and odd talk throughout!
  • Some wondering about multiplication and division of fraction!
  • The range of fractions – how many we can name, how many unit fractions there are.
  • The size of a fraction in different forms – Is the whole the biggest fraction? Is the numerator smaller than the denominator?

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In case you want to try it out:

SMARTBoard file for 4th grade

SMARTBoard file for 5th grade

PDF file of the file.

Cuisenaire Rods: Fun in 4th

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:

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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!

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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.

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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.

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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.

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This was the most common response, seeing the numbers get smaller as the rod got shorter.

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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?

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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.

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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:)

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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?

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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!

 

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!

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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 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!

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:

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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!

 

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.

 

Dot Image with Halves

Yesterday, I was thinking about some new number routines for the 4th and 5th graders who will soon be in their fraction units. I wanted to find something that both grade levels could engage in so the teachers could try them out and have a common talking point when we met to discuss what we learned about student thinking. I tweeted it out and didn’t want to lose all of the great thoughts so I going to compile them here!

This was the image I created:

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Before launching this with the students, I thought the teacher would establish that a full circle was 1 and ask the question we typically use with our dot images, which is

How many dots do you see and how do you see them?

I thought there could be a variety of responses, but I was most anticipating these two responses:

  • I know two halves make 1 and so the first row is 3. There are 4 rows so 4 x 3 =12.
  • I know two halves make 1, so the first column is 2. There are 6 columns so 6 x 2 =12.

I knew questioning would be important to move from those descriptions to what the equation would look like and imagined we would get to some like this:

  • 6 x 1/2 x 4 = 12
  • 4 x 1/2 x 6 = 12

Other interesting things I am now thinking about thanks to others on Twitter:

Why am I stuck on dots? Circles makes so much more sense!

Yes, why not leave it completely open,not tell them that circle is 1 and talk about the unit?

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It is funny that the same image conjures up different student responses, I always love that! I had not anticipate half of an array. I also love the idea of messing around with quarters here!

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Awesome to have them do some scaling of the dots to find how many are there? And then I can see them saying it is half of the array like Michael suggested once it is uncovered!

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I like the idea of moving them around however the cutting and tearing thing never goes well in my room!! I do love changing the value of the whole a lot!

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I debated this part a lot when I was making it! I thought for the 1st one, a whole number would be best and then move to non!

There is the running list so far! Add away in the comments!