Field Enactment photo

Fraction Kits – beginnings

So far in summer school, we’ve been working on sociomathematical goals: getting students accustomed to routines and procedures for sharing their thinking in pairs, in written work, and in whole-group dicussions.  Mathematically, the activities and lessons we’ve used to develop these goals have been focused on multiplication and the relationship between skip counting, grouping, and array models of multiplication. This has helped us identify students who need extra support (since multiplication isn’t ‘new’ content for 3rd grade) while also being a slightly more approachable area to start number talks with.

Last week in our classroom, we began work on the content standards that gave our 3rd graders the most trouble on the end-of-year assessments administered by the state and by the school: fractions. Specifically, these students need additional support in understanding what a fraction represents, how fractions can be represented on linear models, and how fractions can be different and yet equivalent. To start this work last week, we began by creating Fraction Kits based on the insight and advice of Marilyn Burns, whose book has been an excellent tool for lesson planning and mapping the mathematical terrain we need to help students explore this summer.

Monday and Tuesday we spent time counting collections, and observing how students were grouping objects.

Wednesday we began math class with a mental math number talk, solving and discussing strategies for 6×7. I was glad to see that students were willing to persist even if they didn’t “just know it” from memory – they were willing to think and find a means of figuring out the problem.

After our number talk, students returned to their seats where each received a set of pre-cut strips of construction paper and a large envelope cut from similar construction paper. They wrote their names on the envelopes and tucked them inside their desks.

Next we went through the process of folding our strips. For each strip, we:

-folded (I first modeled how to fold them)
-predicted how many sections there would be before opening to see
-decided as a group how we might then label each section
-waited for each student to fold, unfold, label each section, and label the back of each section with initials.

Once every student had their fraction pieces folded, labeled, and affixed with initials, we passed out scissors and got them cutting.

Why do, We do, You do?

I want to note that ordinarily, I steer away from “I do, we do, you do” type teaching. However, for creating fraction kits, I modeled first, together we decided how we might name/label, and then asked students to do it on their own. The reason I thought modeling and duplication was fitting in this instance is that our goal was for students to create tools that they could then use to explore fraction relationships. If their kits weren’t created correctly (and some weren’t: a couple students mis-folded and needed help to tape or re-cut later), they might develop some sneaky misconceptions about how much bigger 1/4 was than 1/8, etc.

So why not pre-cut the strips?

Giving the students a chance to fold and cut did elicit some students’ initial ideas about what those weird hyroglyphs called fractions might mean: many students were catching on to the pattern that “8 pieces mean each is called 1/8” and so on. Folding their own strips was also a useful opportunity for them to consider what happens when you take a half and fold it in half. For instance, when we folded the strip into 1/4ths, many students predicted that we would unfold the strip to find 6, 8, or 10 segments, rather than 4. Similar guesses of 10, 15, and 20 came out when we folded the strip into 1/8ths by folding it in  half, half again, and half again.

Wrapping up

Because we opened with a number talk, we only had time to get the strips cut, labeled, and put away on Wednesday. But we did close by asking students to see if they saw any “matches” between strips of different size. This got a few initial ideas out on the table (or in our case, the smart board): two 1/4th sized pieces was the same as one 1/2. Similarly, two 1/8 sized pieces was the same size as 1/4.

8+9: four strategies for counting and adding

Yesterday was our first mental math as a 3rd grade summer school class. This mental math number talk was designed to be a ‘warm up’ to the instructional routine; in terms of the standards, this problem would be well suited for late first grade. However, I wanted a problem that would be slightly easier than the 3rd grade standards, so that the students could focus on developing productive norms for number talks.

I started by discussing with the students what a number talk was, what mental math was, and what their jobs would be: to think about the problem and multiple ways of solving it, to talk at a conversation level to their partner and listen to their strategies, and to be ready to share with the class AND listen to their peers’ ideas during whole-group discussion.

We rehearsed these norms and actions several times, first with questions designed only to get them used to holding their ideas privately until asked to share with a neighbor: “Think of as many animals as you can that are green” and then “Think of something you have exactly 3 of in your house.” Students are generally pretty eager to share responses to these kinds of questions, so they are helpful for giving them a chance to practice holding their ideas private until asked to talk to a partner or share with the class.

We then did one more ‘practice round’ with the problem 2+2 to again reinforce, review, and label productive student actions, like showing on their fist with fingers how many strategies they had, holding their ideas privately until asked, then sharing at a low conversation level with a partner once asked.

When it was finally time to do the ‘real’ problem for the day (8+9), these norms were pretty firmly in place. I had a few kids whisper “ooh i know it!” to themselves, but using a gesture of “quiet” and modeling my own fist to my chest was sufficient reminder for them to be thinking about strategies, holding their ideas private until called on.

After I saw that nearly every student had at least 1 finger raised (probably 45 seconds), I asked them to turn and talk to a neighbor about how they thought about the problem. The students did so well, and I heard many of them trying to explain their thinking in addition to sharing their answers.

After about 30 seconds of partner time, I called their attention back to me (I use “class class” –> “yes yes” call and response for this) and asked students to share answers. I called on just about every student and got 17 for all but 1, who said 14.

I then asked for volunteers to share their strategies.

The predominant strategy students shared was counting on, either from 8 or 9, using fingers. Two students adjusted and compensated – one by solving 9 + 9 and removing 1, another by taking 1 from the 8 and adding it to the 9 to make the problem 10 + 7. The last strategy (bottom left) was contributed by my coteacher.

Overall, this number talk was a great start to our summer school theme of using number talks in regular instruction to work on the OA strand. I’m looking forward to Monday!



The Final Lap: Comparing and ordering decimals in a 5th grade classroom


Amanda is teaching 5th grade this semester at a local elementary school and recently got to enact the Final Lap task (created by Graham Fletcher) with her students.  Note that Amanda changed the finishing times of the NASCAR cars to minutes, and limited the times to numbers in the hundredth place.

The instructional goal for this lesson was for students to compare decimals to hundredths by reasoning about their size and to recognize that comparisons are valid only when the decimals refer to the same whole. In lessons prior to this, students were introduced to decimals to tenths and hundredths and changing fractions into decimals/decimals into fractions. This lesson introduced students to comparing decimals. Another goal of mine was for students to understand how and why decimals are used. Some students were having difficulty understanding when we use decimals and the importance of them. Showing students a NASCAR race where the times of each car are given as decimals proves that decimals are used in everyday life, for various reasons, and assist us with tasks that must be completed. Therefore the instructional goals of this lesson were met because students compared real life race times (which helped them understand the usage of decimals) and ordered them from smallest to largest/first place to last place.

In the beginning of the lesson, the class created a notice and wonder chart about the NASCAR 3-Act Task video.

I anticipated that students would notice the numbers on the cars, which was good because that’s how we spoke about each of the cars (which got tricky when we added in the decimals we were working with. It was a lot of numbers to write and speak about). Also, I anticipated that a student would wonder, “why are we watching this video” and “who won the race”. This question – who won? – was the first one students asked, and the most to get the “me too!” symbol (sign language we use in class), which made it easy to come back to and highlight as our main question.  During the creation of our chart I did not anticipate a notice about the decimals in the right-hand corner of the screen. Hearing this I was surprised because to be honest I did not catch this the first few times I watching the video by myself. When a student mentioned this as his notice I told him, “Speaking about decimals, I want everyone to keep everything they know about decimals in the front of their brain because it will come in handy.” I did not want to give away anything but I wanted the other students to start thinking about decimals and recalling prior knowledge/lessons.

Once we created a class notice and wonder chart, I asked the students what information/tools do we need to solve this problem. I anticipated the responses would be the lineup of cars and the amount of time each car finished the race. Instead, the only two responses I got were “more of the video” and “a picture of the finish line”.



This probed me to say, “Let’s say technology was not working and the announcers had to take notes of the race. What would we want to see and in their notes?”. I still got answers such as, “the car that won” and “description of which car won and how” but I also got the following response, “which cars finished the race” which lead to “how fast each car finished the race”. Although these weren’t the exact responses, they aligned with the information they needed to solve the problem and what I had to share with them.

When I shared the information with the students I planned that some of them would know the answer right away. Predicting this, I told the students that if they knew the answer please be respectful and keep it in their minds, but if they wanted to share then they would have to use their whisper phones to let it out. During the enactment of the lesson there were students who were so excited they knew who won that they had to let it out so they used their whisper phones. This allowed students to not spoil the answer or persuade any of their peers’ thinking. Having students know the answer right away also led me to stating clear directions of, “proving your answer with work and using the decimals to do so” and extension questions that were as followed, “put the cars in order from 1st to 7th place”, “how many seconds did 1st place win by”, and “how many seconds did the 1st place car beat the 7th place car”.

Starting with the Notice and Wonder chart, I learned that it is a great activity for every single one of my students. Since this activity is “low floor, high ceiling,” students of all skill level can participate. For example, my student that rarely speaks in class due to language disconnect shared a notice about the color of the cars. Also, a student that is usually disengaged during the math block was the student that shared our wonder question, “Who won the race?”. I learned that if you give students the chance to show their knowledge from the very beginning of the lesson they will be more inclined to participate throughout the whole lesson/unit.

Also, I learned that my students are taught to think of math as a question and answer process. From the time that I stated the main question, most of the students were obsessed with getting the right answer. They asked, “what does this video have to do with the math were doing today?” and “Am I right?”. Even during independent work time, multiple students raised their hands just to ask, “Did I get the right answer?”. When I wouldn’t tell them or show them the final answer right away they got frustrated and questioned their work even though it was right. These reactions show me that my students are not regularly learning to value the process and instead focus on the correct answer or algorithm that will get them the “A” or “100 %”.


While sharing strategies, I learned that the students are most comfortable with thinking about whole numbers. Almost all of the students used strategies that sorted the times according to whole number then made a data table from smallest to largest or data table with the smallest whole numbers on top then the largest on the bottom and then wrote 1st, 2nd, 3rd,… next to the times. Also, while sharing their strategies, they called numbers to the left of the decimal whole numbers but they didn’t seem to have a name for the numbers for tenths and hundredths. Some students used these terms without probing, but most students caught on to tenths and hundredths after I went over the terms by showing them the following layout, “whole numbers. tenths hundredths”.

If I were to teach this lesson again, I would do a mini lesson prior to the 3 Act Task. This mini lesson would be on place values including decimals. I would have the students write any decimal they like (keeping it to no more than 3 numbers), write that decimal exactly how they would say it aloud, and then turn and share their number with a partner. This activity would allow for the students to have a productive struggle of what to call each place value. After a class discussion, I would show the students the actual decimal place value system. This would look like “whole numbers . tenths hundredths” or “hundreds tens ones . tenths hundredths”. Students could refer to this place value chart during the 3 Act Task and future decimal lessons. I would include this mini lesson prior to the 3 Act Task because when speaking about the times of each car students would say “26 point 23” or “the first/second number after the decimal”. It wasn’t only until I stopped the lesson to draw and explain the decimal place value system that they started using the language “tenths” and “hundredths”. Some students still fell back to the “26 point 23” or “first/second number after the decimal” but this tells me that they don’t fully understand what the tenths and hundredths represent. Therefore, with these specific students I would have to go deeper into exploring the decimal place value system. If I would have done the mini lesson I think students would understand tenths and hundredths which would have made the task easier to understand, contextualize, and more fluid.

Another change I would make is how I asked the students to show their work. I wish I would have asked the students at first to order them from 1st to 7th place instead of making that an extension problem. I would do this because it would force students to interact with the decimals from the beginning and not just use mental math to figure out who won. Also it would give me more time to focus on strategies because I wouldn’t have to go up to each student who finished and ask them the first extension question.

Based on how the lesson what I would conduct a number talk and have students compare 25.3 and 25.29. This question was my exit ticket, but I ran out of time to hand it out. I would choose this specific problem because I want to see which students understand the place values of tenths and hundredths, and whether they can compare the number of digits with the value of each digit. Also, it would encourage them to recall the names of the place values and use them to describe their thinking.



The Apple Task: Enactment artifacts from a 5th grade classroom

On 3/3/17, I was able to work with an outstanding group of 5th graders at a local elementary school on one of my favorite 3-act tasks by Graham Fletcher – The Apple.

The Apple gets students thinking about division as measurement, and if left to solve the problem in a way that makes sense to them, they tend to generate a range of strategies that can be really useful stuff for making connections in a pre-reveal discussion about their strategies. In particular, this task is useful for making connections between:

  • repeated addition (adding one block at a time until we ‘arrive’ at the apple’s weight)
  • repeated subtraction (taking off a block’s-worth of weight from the apple until we ‘arrive’ at zero)
  • multiplication (often in the form of guess and check after students grow weary of repeated addition), and
  • fraction division using the common denominator method (get a common denominator and divide the numerators).

We started by playing the video for students and asking them what they noticed and wondered. As they shared, we recorded their noticings and wonderings on chart paper:

As a class, we decided on the starred question as the one to solve for the day. Once we’d come to consensus (using words and gestures) for what it would mean to balance the scale,  we asked students to estimate the number of blocks it would take. We recorded their estimates.


We then asked students what information they would need to solve this problem – and why they would need it. Most students suggested fairly quickly that we would need to know the weight of the blocks and the weight of the apple – but I asked several students to argue for why we needed both and not just one. This took some time, but by the end it seemed most students understood why both weights were important, and what significance they had mathematically and contextually.

Then, we gave them the information. (I wrote the info down in the wrong ‘spots’ on the “what do we need to know” chart at first, so I had to rewrite.)

Once we gave them this information, we gave students about 2-3 minutes to begin a strategy on their own piece of paper – enough time to make a plan, but not enough time to follow it all the way through.

After a few minutes passed, we asked them to talk to the other students at their tables and discuss their strategies, and come to a decision about how they would use one or more of the strategies at their tables to solve the problem. They were also tasked with representing one strategy on a group white board. Here’s what those white boards looked like by the end:






Multiplication strategy: what number multiplied by 3/8 gets us 5 and 2/8?


These students started by repeatedly adding 3/8 until they grew weary of it, and decided to start multiplying. They tried 1, 2, 3, 5, 10, 13, and then finally did 13 blocks plus 1 more block.


Repeated addition of 4 blocks at a time.

I LOVED this strategy. This group was led by two students in particular whose really wanted to find a combination of blocks that would add up to an easy ‘split.’ So they settled on 4 blocks, since 4 groups of 3/8 makes 12/8, which would only need 1/2 an ounce to get to the next whole. They knew, then, that two groups of 4 blocks would ‘make up’ the next whole.

Once they started down this path, however, they found themselves in improper fraction land quite quickly, which it turned out was not a land they had adventured in many times before. Out of the 4 students in the group, 3 of them were quite insistent that 4 and 16/8 oz (which they arrived at by repeatedly adding 4 blocks 4 times – so, 16 blocks) was not yet enough weight to equal the apple. They had some difficulty moving past this moment, and ran out of time to come to consensus as a group about what should happen next – though they agreed that it would likely entail adding single blocks to finally get to 5 1/4 oz, since most of them were sure that 4 more would be too many.


Repeated addition, using a choral count representation

This was a strategy we were really hoping would turn up, since the students had just done a choral count by 1/8 in the moments preceding the 3-act task. Interestingly, this strategy did not represent the first strategy all of the students in the group had used; several had used a division strategy, but because they could not figure out how to represent or explain how that strategy worked (which was what we insisted they be able to do with any strategy they put on their board), they resorted to using a repeated addition strategy. The challenging thing in this group’s case was figuring out in all of this representation where the answer was.

In addition to these three group boards, there was one last group who had worked hard to find solution strategies independently, but who had run out of time to get much onto their board before we discussed strategies:

The students had about 20 minutes in their groups to produce the representations above, and then we took about 10 minutes to look across the first three boards above, first looking at the choral count strategy, then the repeated addition of 4 blocks, then finally the multiplication strategy. The questions I asked of each group were –

  • where is the apple represented in this group’s strategy?
  • How did they know where to stop?
  • What were they looking to have happen by [adding 1/8 at a time…adding 4 blocks at a time…multiplying numbers by 3/8]
  • How did they know they had done enough?
  • Based on all of this work, what is the group’s answer to the question?

Though I’m not confident that all of the students came to consensus about whether or why multiplication was more efficient than repeated addition, they all agreed that in this case, anyway, it produced the same result. The group who was adding ‘batches’ of 4 blocks at a time seemed to understand by the end that their method was functioning pretty well, but that it might require more adjustment because of undershooting or overshooting the perfect amount – a problem the choral count group didn’t have, but that the multiplication group did (they found that 13 was close, so they adjusted from there).

Afterward, we gave them exit tickets to see how they would solve a related problem – and to see how they were making sense of the fraction 3/8 as it relates to 1. Here are a few examples of student work :


All in all, this task was a great way to feel out students’ thinking about the relationships between addition, subtraction, multiplication, and division, and whether those relationships behave predictably with fractions. It was also a great way to peek into how students are marshaling procedures like finding a common denominator as steps to solving a problem, rather than as a procedure for it’s own sake. It’s clear that we have more work to do to build common understanding about some of these ideas, but I loved the opportunity we got to ask students to completely fly blind into material they’ve never ventured into before, and see what sense they made of it.

Counting up by 1/4 from 1/4

Note an additional representation on the right side (the open number line) where we explored what the “hop” was from 1/4 to 1 and 3/4.