# Category Archives: Math

I am sure we have all seen it at one time or another – those math questions that make us cringe, furrow our brow, or just plain confuse us because we can’t figure out what is even being asked. Sadly, these questions are in math programs more often than they should be and even though they may completely suck, they do give us, as educators, the opportunity to have conversations about ways we could adapt them to better learn what students truly know. These conversations happen all of the time on Twitter and I really appreciate talking through why the questions are so bad because it pushes me to have a more critical lens of the questions I ask students. Through all of these conversations, I try to lead my thinking with three questions:

• What is the purpose of the question?
• What does the question tell students about the math?
• What would I learn about student thinking if they answered correctly? Incorrectly?

Andrew posted this question from a math program the other day on Twitter….

I tried to answer my three questions…

• What is the purpose of the question? I am not sure. Are they defining “name” as an expression? Are they defining “name” as the word? What is considered a correct answer here?
• What does the question tell students about the math? Math is about trying to interpret what a question is asking and/or trick me because “name” could mean many things and depending on what it means, some of these answers look right.
• What would I learn about student thinking if they answered correctly? Incorrectly? Correctly? I am not sure I even know what that is because I don’t know what “name” means in this case. Is it a particular way the program has defined it?

On Twitter, this is the conversation that ensued, including this picture from, what I assume to be, the same math program:

When a program gives problems like this, we not only miss out on learning what students know because they get lost trying to navigate the wording, but we also miss out on all of the great things we may not learn about their thinking. For example, even if they got the problem correct, what else might they know that we never heard?

The great thing is, when problems like this are in our math program, we don’t have to give them to students as is. We have control of the problems we put in front of students and can adapt them in ways that can be SO much better. These adaptations can open up what we learn about student thinking and change the way students view mathematics.

For example, if I want to know what students know about 12, I would just ask them. I would have them write in their journal for a few minutes individually so I had a picture of what each student knew and then would share as a class to give them the opportunity to ask one another questions.

After I saw those the problem posted on Twitter, I emailed the 2nd grade teachers in my building and asked them to give their students the following prompt:

Tell me everything you know about 12.

Ms. Thompson’s Class

Mrs. Leach’s Class

Mrs. Levin’s Class

Look at all of the things we miss out on when we give worksheets from math programs like the one Andrew posted. I do believe having a program helps with coherence, but also believe it is up to us to use good professional judgement when we give worksheets like that to students. While it doesn’t help us learn much about their thinking it also sends a sad message of what learning mathematics is.

I encourage and appreciate conversations around problems like the one Andrew posted. I think, wonder, and reflect a lot about these problems. To me, adapting them is fun…I mean who doesn’t want to make learning experiences better for students?

Looking for more like this? I did this similar lesson with a Kindergarten teacher a few years ago. Every time I learn so much and they are so excited to share what they know!

# 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.

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:

# A Coordinate System

This standard in 5th grade always seemed like so much of a “telling lesson” for me.

I never thought it was really addressed in the spirit of this standard in our curriculum, so I was typically like, “Here is what we call a coordinate grid. These are axes, x and y. We name the points like this…” and so on. It is not my usual approach so it always felt blah for me, for lack of a better word. I told them, they practiced plotting some points, and we played a little bit of Battleship (which was really fun).

Last week, I was planning with Leigh, a 5th grade teacher, and we spent a lot of time just talking about what we appreciate about the grid and how we can develop a sense of need in the students for it. Since they are in the middle of their 2D Geometry unit, we thought this could be the perfect place to plot points that connect to form polygons and look at patterns in the ordered pairs.

The questions we wanted to students to reason about through our intro lesson was:

• Why a coordinate grid?
• Why name a point with an ordered pair?
• What structure do we see?

So, we created this Desmos activity. This was our thinking on the slides and the pausing points we have planned for discussion:

Slide 1: It is really hard to describe a location without guides or landmarks.

Slide 2: Note how difficult it is. Pause and show class the results.

Slide 3: It gets easier. Still need some measurement tool. Notice the intersection of axes.

Slide 4: Note it is a bit easier this time. Pause and show class results.

Slide 5: Much easier because of the grid. Still need a starting point. See it is the distance from axes.

Slide 6: Now it is much easier. Pause show class results. Would love to show all three choices side-by-side (don’t know if this is possible in Desmos).

~Pause~ Ask, “What names of things on the grid would make it easier to talk about the point’s location?” Give students vocabulary and ask them to revisit Slide 6 to describe the location to a partner.

Slide 7: Practice writing some ordered pairs.

Slide 8: Practice writing some ordered pairs.

Slide 9: Start to see some structure in the four ordered pairs of a rectangle.

We are ending with this exit ticket (with grid paper if they choose to use it):

While we are not sure this is the best way to intro the grid, we thought it would generate some interesting conversation. Since we are teaching it tomorrow, there isn’t much time for feedback for change, but we would love your thoughts.

# 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.

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:

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

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

# 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:

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:

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):)

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?

In case you want to try it out:

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:

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!

# 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.

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.
• 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 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:

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.

• 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:

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!

# Which One Doesn’t Belong? Place Value

Since the 3rd grade team begins the year with an addition and subtraction unit in Investigations the teachers and I were having a conversation about how students understand place value. While I don’t see teachers using the HTO (hundreds/tens/ones) chart in their classrooms, students still seem to talk about numbers in that sense. For example, when given a 3-digit number such as 148, students are quick to say the number has 4 tens instead of thinking about the tens that are in the 100. I think a lot of this is because of how we as teachers say these things in our classrooms. I know I am guilty of quickly saying something like, “Oh, you looked at the 4 tens and subtracted…” when doing computation number talks, which could lead students to solely see the value of a number by what digit is sitting in a particular place.

We thought it would be interesting to get a vibe of how this new group of 2nd graders talked about numbers since their first unit deals with place in terms of stickers.  A sheet of stickers is 100, a strip of stickers is 10 and then there are the single stickers equal to 1.

I designed a Which One Doesn’t Belong? activity  with four numbers:  45, 148, 76, 40

I posted the numbers, asked students to share which number they thought didn’t belong, and asked them to work in groups to come up with a reason that each could not belong. Below is the final recording of their ideas:

I loved the random equation for 148 that emerged and the unsureness of what numbers they would hit if they counted by 3’s or 4’s. One student was sure she would say 45 when she counted by 3’s and was sure she would not say 76 or 40, but unsure about the 148. I wrote those at the bottom for them to check out later.

Since the teacher said she was good on time, I kept going. I pulled the 148 and asked how many tens were in that number. I was not surprised to see the majority say 4, but I did have 3 or 4 students say 14. As you can see below a student did mention the HTO chart, with tallies, interesting.

As students shared, I thought about something Marilyn Burns tweeted a week or so ago…

So, I asked the students to do their first math journal of the school year (YEAH!):