# Rhombus vs Diamond

Every year in 5th grade, when we begin classifying quadrilaterals, students will continually call a rhombus a diamond. It never fails. While doing a Which One Doesn’t Belong in 3rd grade yesterday, the same thing happened, so Christopher’s tweet came at the most perfect time! (On Desmos here: https://t.co/rZQhu2SGnR)

Of course I had to pop into the same classroom today and try it out! The lower right was so obviously a diamond to me that I was curious to see if students saw the same thing and if it changed their reasoning about the rhombus as a diamond.

Here are pictures of the SMARTboard after our talk:

After great discussions around number of sides, rotations, decomposition and orientation, they finally got to the naming piece. Honestly, I was surprised names didn’t come up as one of the first things. It started with a student saying the square didn’t belong because it is the only one that doesn’t look like a diamond. The next student said the lower left was the only one “that didn’t have a name.” When I asked him to explain further, he named the square, rhombus, and diamond. Because I knew at the end of our talk I wanted to ask about the diamond vs rhombus, I wrote the names on the shapes. Another classmate added on and said the lower left “may not have a name but it is kite-shaped and looks like it got stuck in a tree sideways.” I asked the class what they thought about the names we had on the board and it was a unanimous agreement on all of them. Funny how quickly they abandoned their idea from yesterday, so I reminded them….they were not getting off the hook that easy;)

“Yesterday you were calling this rhombus a diamond, what changed your mind?”

Students explained that the lower right actually looks like a real diamond and the rhombus doesn’t now that they see them together.

“Can we call both of them a diamond?” I asked. I saw a few thinking that may be a great idea. I had them turn and talk to a neighbor while I listened to them.

We came back and they seemed to agree we couldn’t call them both a diamond because of the number of sides. They were really confident in making the rule that the quadrilateral one had to be a rhombus and the pentagon was the diamond. I pointed to the kite and asked about that one, since it has four sides. “Could we call this a rhombus?” They said no because the sides weren’t equal, so not a rhombus. And because it didn’t have five sides, not a diamond either.

Thank you Christopher! All of these years of trying to settle that rhombus vs diamond debate settled right here with great conversation all around!

Next up, this one from Christopher…

# Fraction & Decimal Number Lines

Recently, our 3rd, 4th, and 5th grade teachers had the opportunity to chat math for 2 hours during a Learning Lab held on a professional development day. It was the first time we had done a vertical lab and it felt like perfect timing as 3rd and 4th grade would soon be starting their fraction unit and 5th would be entering their decimal unit. Prior to the meeting, we read the NCTM article, “Identify Fractions and Decimals on a Number Line” by Meghan Shaughnessy, so we started the meeting discussing ideas in the article. We then jumped into playing around with clothesline number lines and double number lines, discussing what they could look like at each grade level based on where students are in the fractional thinking.

I have co-taught number line lessons in both 5th grade and Kindergarten this year, but both were a bit different in not only number, but organization. In 5th grade we used one clothesline with the whole class, while in Kindergarten we used tape on the floor and students worked in small groups. Leigh, 5th grade teacher, was interested in trying the small group number lines on the floor. As we planned the lesson, the one thing we thought would be difficult about having small groups is getting around to each group to hear their conversations, especially when we were planning cards purposefully to address misconceptions and misunderstandings. However, knowing we would have the two of us circulating, as well as two 3rd grade teachers who wanted to see the lesson (yeah!), we knew we had plenty of eyes and ears around the room to hear the math conversations.

During the lesson, in groups of three, students placed 24 cards on their number line. There were two sets of cards, so after placing all of their cards, each group visited a number line with a different set of cards to discuss. Instead of boring you with all of the number choices we made, here are a few of the choices in cards and the reason(s) we chose them:

1/3 and .3: Students often think these two are equivalent so before the decimal unit we were curious to see how they were thinking around that idea and how they used what they knew about fractions or percents to reason about it.

0.3, 0.33, 0.333, 1/3: The 1/3 and .333 were there to think about equivalency, while the others were there to think about what is the same in each and how much more each decimal has to make it larger. Which you can see caused some confusion here:

2.01 and 2.08: We were curious about the distance they put between these two cards.

1 6/9 and 1.6: We wanted to see how students compared fractions and decimals when they can’t easily convert 6/9 into a decimal. Then, if they began thinking 6/9 and 6/10, how would they decide on the larger fraction and then how much distance do they put between them?

The group below practically had them on top of one another:

While this group had a bit of a space between them:

2.8 and 2.80: Tenths and hundredths equivalency. They all seemed to handle this with ease.

.005 and 1/100: Curious to see the placement in relation to the other numbers. This 1/100 is close to 0 but I wonder about it in relation to the .2. Definitely a conversation worth having!

2.8 and 2 7/8: To see how they compared the 7/8 to the 8/10.

After they visited other number lines, they had a chance to meet with that group and discuss card placements they agreed with and placements they did not. Groups then made adjustments accordingly…

Here was a group’s completed number line and my first stab at panoramic on my phone!

The journal entry we left them with was, “Which cards were the most difficult to place on the number line? Why?”  Many were just as we suspected.

The conversation as I walked back over to the other building with the 3rd grade teachers was, what does this look like in 3rd grade? Could we use array images to place on the line instead of the fractions? Could the pictures include over 1 whole? What whole numbers would we use? Do we play with equivalent pictures with different partitioning? Being mindful of the students’ second grade fraction exposure, below, we are planning on trying out something very soon! I am thinking the cards like these on Illustrative, with the pictures but no fraction names at this point.

CCSS.MATH.CONTENT.2.G.A.2
Partition a rectangle into rows and columns of same-size squares and count to find the total number of them.
CCSS.MATH.CONTENT.2.G.A.3
Partition circles and rectangles into two, three, or four equal shares, describe the shares using the words halves, thirds, half of, a third of, etc., and describe the whole as two halves, three thirds, four fourths. Recognize that equal shares of identical wholes need not have the same shape.
…and then could the journal could be, “Can you name any of the fractions on your number line? How do you know?”…or something like that!

Yesterday, in my planning, I was bouncing around between a couple of ideas for the lessons I was teaching today. I decided to go with this Illustrative task that was the basis for the lesson study project I did last year with The Teaching Channel and Illustrative. I though it would be a great formative assessment as students move from thinking about fraction of a fraction on a fraction bar to an area model using a square unit.

I opened with a choral count in which the students counted by 3/4. I played around with ending each row on a whole number (12/4, 24/4, 36/4…etc) or ending on something not as “nice.” I opted for the second, but in hindsight, probably should have gone with the first. I really wanted more conversation around when the whole number occurred and why and possibly the distributive property (4 x 3/4) + (1 x 3/4) = 15/4, but not as obvious as I felt the first option made it, but it didn’t happen. The count looked like this (I didn’t get a chance to take a pic of the board):

I asked students what they noticed and I got a variety of responses. I got many variations of equations such as 3/4 + 6/4 = 9/4 and 12 x 3/4 = 36/4, and with each one, I asked how we could show that within the count.  One student said she could see 20 x 3/4 =60/4 in the count. Just when I thought she was going to explain by saying we counted by 3/4 twenty times, she surprised me in her wording. She said, “I knew the count was a 4 by 5 array, which is 20 numbers and then each one was 3/4, so I got 60/4.” This struck me different than the skip counting or repeated addition of 3/4 that others were doing and reminded me a a bit of this diagram from the learning progressions:

I think it was the way she described the size of each number in the count as 3/4 that drew me to this diagram.

Next I gave them the task to engage in alone for 5 minutes before I let them move into group work. I was really impressed by the way they jumped right in and easily could find 1/3 of 1/4. I was not surprised that many ignored the “square pan” piece and went with fraction bars. Much of the work looked like this, cut vertically:

While there were also quite a few students who quartered it by cutting vertically and horizontally in half and then splitting that quarter into thirds:

This is something for me to think (learn) more about because if we are thinking area model, the dimensions of the piece, to me, looks like 1/2 x 1/6. That seems like it could be problematic to me when the square has dimensions in units of length.

Another interesting thing that always comes up in this work is the difference between dividing by a fraction and dividing it up into fractional parts. I saw those equations sneaking there way in like this…

That will be something to keep in mind in future lessons. What it means to divide by a number versus divide into parts? Is it different when we are thinking about whole numbers versus fractions? Cool convos to be had around that!

There was one student who was not working with the original whole in her work. She was working with the 1/4 in the first part and then the 1/2 in the second part. When I asked her about how she was determining 12ths, she said it was just like her phone, she took the whole thing and just zoomed in (she did the fingers swipes as you would do on your phone) on the part she needed.

There was some really great proportional reasoning going on with the cost of the cornbread pieces. When the pan price changed from \$12 to \$24, \$6, and \$18, student used great reasoning in relation to \$12. In the example above you can see that work as well as here:

I left them with a question about the denominator, why is not ending up in any of the denominators we are using the problem? I only had a chance to snap one pic before I headed to another classroom.

Lots of great stuff to keep in mind and I think comparing the ways in which the students divided the models would be a really interesting conversation. Then after they move to an area work, I wonder if it would be great to bring that back for a comparison.

Also, I forgot to add that the substitute who was in for Leigh today was so incredible about taking notes and talking to the students because he was going to be in charge of teaching this same lesson to the next math class that came in. He said he learned so much and I thought this could be such a powerful way to have substitutes involved in learning more about the way in which math is taught.

~Kristin

# Planning K-5, literally

Tomorrow I have the opportunity to teach a Kindergarten, 2nd and 5th grade class! It is so exciting and interesting to be thinking across all of the grade levels in one day of lesson planning! The most interesting part for me, in thinking through this, is the connections across all of the grades. There is so much potential for conjecture and claim-making supported by their development of proofs.

Background: The 5th and 2nd grade teachers are out at a state math teacher leader meeting so I am teaching instead of the substitute. The kindergarten teacher and I will be teaching it together. I have met with each teacher to chat about where they are within their units and what they have been seeing students do within the current work. I invited teachers both at those grade levels and at other grade levels to pop in if they have the time. I thought it would be great having more people to reflect with after the lessons!

5th Grade: They have just started working with finding a fraction of a fraction using bar models. The initial work is unit fraction of a unit fraction and then moves to non-unit. (My post on that from a couple of years ago on this work, I wish I had done that better, so here is a chance to try something new;) Leigh, the teacher, says they have been really successful in partitioning the bars and arriving at the correct answer. I am thinking about starting with a number routine of either a choral count or a number talk string like 1/2 of 12 = __ of 24… As far as the lesson, I could continue work with this and have students look at noticings after and explore them deeper.They have done these noticings with whole number times a fraction or mixed number, so this could be a revisiting of similarities or differences. OR I could do this cornbread task as a formative assessment as the next piece they will move into is an area model. It may be really helpful for Leigh to see how they are thinking about this before they jump into the work. This is my least planned because I keep bouncing all around with ideas.

2nd Grade: They have been working with even and odd numbers and counting by groups of 2’s, 5’s, and 10’s.  All of this work is within contexts of break a group of students into equal teams or everyone having a partner. Tara, the classroom teacher, said the students are really great at determining whether a number is odd or even, however when asked how many would be on each team, a lot of students struggle. They are great if they know a related double fact, however if they don’t they resort to “passing out” by tallies or drawing the picture and physically dividing the number of things in half. For example, if they do not know 11+11 is 22, then finding the number of on each team become passing out 22 things into two groups to find 11. While they are successful in this, Tara and I were wondering why they do not say 10+10=20 and 1+1=2 so 11+11=22. They are able to add 11 and 11 but unable to decompose it as fluently.

In thinking about this, I am inclined to want to connect that addition to halving. I am thinking a counting collection would be fabulous for this. Give students a collection of things to count. Share how we counted them because I am positive they will not count them by 1’s given a large set. We can share as a whole group, record ways in which we counted and determined if our number was even or odd. Then, put the collection back together, switch with a partnering team and then split the collection into two groups. The share would be, “Could you make two equal groups?” “Was your number even or odd? How did you know?” Record strategies. Ask for noticings/wonderings about how they counted and how they divided into two groups.

Kindergarten: The students in this class have been doing a lot of work with ten frames, dot images, counting jars, etc and having students counting and adding to compose a number. They have just begun working on decomposition of number so I immediately thought about the mice activity in Thinking Mathematically. Linda, the teacher, and I planned to do this activity with the students. In preparation, we read NCTM TCM’s article by Zachary M. Champagne, Robert Schoen, and Claire M. Riddell, Variations in Both Addends Unknown Problems. We are going to use 6 bunnies and see how students show all of the ways the bunnies can be inside and outside in a pen. Instead of just giving a context, I was imagining that the students may need a visual of the rabbit pen so I created this image to launch with a quick notice and wonder:

We will then let the students work on finding the different ways in partners and then come back for a whole group share and record the ideas on the board. We are really looking to see two things….1-how they organize their information and 2- the strategies they use. The students will do a notice/wonder about the recorded information. If there is time it would be great to see if students, when given a different number, would apply any of the strategies and/or organizational tools shared.

Going for a run to think through this a bit more! Would love any thoughts/suggestions, as always!

~Kristin

# How Planning Mistakes Can Lead To Great Student Thinking….

The other day I did this fraction clothesline activity with a 5th grade class and today I had the chance to do it again with another 5th grade teacher, Leigh. It is always so nice to get to have a do-over after having time to reflect and think more about what the students thought about both during and after the activity.

I really thought the conversation was great during the clothesline activity, but it took too long the first time. We noticed that some students began to disengage. To try and improve upon that, Leigh and I decided to give only one card to every pair of students instead of each student having one. However, due to us wanting to keep a few important cards we wanted to hear them talk about, some pairs had two.

I also did not like my placement of 0 being at the very end of the left (when looking at it) end of the string. I moved it out some and talked about the set of numbers that falls on either side of the 0. I felt much better about that this time around!

In the planning of the first clothesline activity, we took fractions from the work the students had been doing with percents and decided on putting 100% in there, completely thinking it would be at 4/4. As the student placed it, however, I started realizing that I never thought about the difference of 100% in terms of the area representations the students had been using versus 100% when talking about distance on a number line. But now, having time to reflect on the card, I thought it would make a great journal entry!

As we neared the end of the card placements, I handed the 100% card to a student and told her it was going to probably cause a lot of discussion but just put it where she thought it went. She said she got it, walked up there and placed it on top of the 2 (the highest number on the line). There were some agree signals going on and some other hands that shot right up to disagree. We talked about it a bit and then we asked them to journal their ending thoughts so we could move on with the rest of the lesson about different sized wholes.

Some thought that 100% was at 4/4 on the number line because it equals 1….

Some thought it was at 4/4, but because of the conversation became a bit unclear…

Some thought it goes on the 2 because it is the biggest number on the number line…

Some related it to different contexts with different wholes…

And one student said it can be anywhere with beautiful adjustments as it moves….

What a great day revisiting my planning mistake!

-Kristin

Today, I had the chance to plan and teach with a 5th grade teacher and it was awesome! Last week, this class had just finished a bunch of 100s grid shading in thinking about fraction/percent equivalencies, so we picked up planning the lesson in Investigations with the fraction/percent equivalent strips. Instead of the 10-minute math activity, we thought it would be really interesting to do the clothesline number line to kick off the class period.

We chose fractions (and one percent I will talk about later) based on the fractions the students had been working with on the grids. We chose fractions based on different comparison strategies that could arise such as:

• Partitioning sections of the line
• Distance to benchmarks
• Equivalent Fractions
• Common Denominator
• Greater than, Less than or equal to a whole or 1/2

We settled upon the following cards:

1/4, 3/4, 4/4, 1/3, 4/3, 5/10, 2/5, 100%, 3/8, 1 5/8, 1 7/8, 4/5, 11/6, 1 6/10, 1/10, 9/8, 12/8, 2

Then, we moved into the fraction cards. We gave each pair of student two cards. In hindsight, for times sake, I would probably only do one card per pair. I gave them one minute to talk about everything they knew about the fractions they had and then we started. I asked for volunteers who thought their card would help us get started and called on a boy with the 1 7/8 card. He went up and stood all of the way to the right and said he couldn’t put his on. I asked why and he said that since the cards were all fractions the line could only go to 1 so his is more than one and can’t go on here. I asked if anyone in the class had a card that may help us out and a student with the 2 card raised her hand. She placed her card all of the way to the right, said “maybe it goes to two” and the other student placed it just to the left of it because, “it is only 1/8 from 2.” Awesome!

We went along with the rest of the cards and so many amazing conversations, agreements and disagreements happened along the way. There are a few things that stand out in my mind as some great reflections on the activity:

1. A student had placed 5/10 halfway between 0 and 1. The next student placed 2/5 just to the left of the 5/10 because, “I know 2 and a half fifths is a half so that means that 2/5 has to be less than 5/10. It is a half of a fifth away.” The NEXT student volunteered and placed 3/8 overlapping just the edge of the 2/5 card on the left. I was expecting percentages to come out, since that was their most recent work with those fractions, however the student said they knew 3/8 was an 1/8 from a half and 2/5 was a 1/10 from a half and an 1/8 and 1/10 are close but an 1/8 is just a little bit further away. Awesome and definitely not what I expected!
2. I wish I had not put the zero so far to the left. Looking back I am wondering if that instills misunderstandings when they begin their work with negative numbers on a number line similar to the original misconception that launched the activity with the 1 7/8.
3. Oh, the 100% card….complete mistake on my part, although it may have been a great mistake to have! In the first class, the student with the 100% card came up and said, “I have 100% and that is 100/100 which is 1” and put it in the appropriate place on the line. Just as she did that, I started thinking how I never really thought about the distinct difference between percent in relation to area (like the grids they had been shading) and 100% when dealing with distance on a number line. No one seemed to notice and since I didn’t know exactly what to ask at that point because I was processing my own thoughts, I waited until another student placed 4/4 on top of it and erased it from my immediate view!
• I stayed for the next class and this time I was prepared for that card and now really looking around to see what students’ reactions were when it was placed. As soon as the student placed it at the 1 location, I heard some side whispers at the tables. I paused and asked what the problem was and they said, “100% is the whole thing.” The next student who volunteered had the 2 card, picked up the 100% card on the way to the right side and put the 2 down and the 100% on top. Lovely and just what I was thinking.

I have never had students reflect on the difference of talking about percentages with distance versus area because I had never thought about it! It definitely feels like an interesting convo to have and a great mistake that I am glad I made!!

I will be back in another 5th grade class tomorrow and will see what happens…it could make for a great journal writing!

-Kristin

# Fraction/Percent Equivalents

It goes without saying that I miss talking 5th grade math with my students each day. But I am so lucky this year to have a new, wonderful teacher in 5th grade who lets me plan and teach some lessons with her! This lesson was one of her first lessons of Unit 4, Name That Portion.

Since in 4th grade the students do a lot of work with comparing fractions, we designed a Number Talk string in which students were comparing two fractions. We wanted to hear how they talked about the fractions. In the string we used a set with common denominators, common numerators, and one unit from a whole. On each problem we were excited to hear talking about the “size of the piece” being the unit and the numerator telling us how many of those pieces we have. Our 4th grade teachers really do a beautiful job with this work. They also used equivalents to have common denominators to compare and a few used percents, since they had done a some grid work with that they day before.

They then worked in pairs to shade 1/8 and 3/8 and we came back to discuss. We noticed as we walked around that the shading was wonderful on their papers, but when asked to write the fraction and percent, most were blank. I remember this lesson from last year during decimals where the same thing happened. So, we asked them what they thought the fraction was as we got these three answers…

12 r4/100

12 1/2/100

12.5/100

They were not overly comfortable with any of them so we asked them to journal which one “felt right” to them and why…

We loved to see what they knew about decimal fraction relations, but we especially liked the “it sounds more fifth grady to use 12.5.”

-Kristin

# #IntentTalk Chapter 1

Since it was a bit too much for me to continually tweet, I decided to do a quick blog at lunch!

Principle 1: Discussions Should Achieve a Mathematical Goal

The first week of school my mathematical goals revolve around discussions about students’ mindset in terms of math, as well as the mathematical practices. I found this year that Talking Points and one of the tasks I found on Fawn’s blog fostered those goals. I blogged about it here. My tables are all arranged in groups and the students know from the very first day that discussions will be a huge part of our work together.

Principle 2: Students Need to Know What and How to Share

To support this in my classroom at the start of the school year, I have the students agree upon our class norms. They originate after doing a Number Talk together and reflecting on what we expect as a group during our discussions. I reference these norms throughout the course of the school year.

Principle 3: Teachers Need to Orient Students to One Another and the Mathematical Ideas

I find a lot this happens during our Number Talks and in then daily in our journal reflections. This is such a focus on my planning of questioning. Asking things such as, “Can you re-explain their thinking in your own word?” or “Did something ____ say change your mind about that way you were thinking?”

Principle 4: Teachers Must Communicate That All Students Are Sense Maker and That Their Ideas are Valued

I think this principle emerges during our very first round of Talking Points of the year when the students my go around their circle with no commenting from others. It allows students the opportunity to speak their ideas without judgement or comment. Being able to change their response on the second round, lets the group know that as we make sense of problems and listen to others, we change our mind, just as we do when we make errors. The freedom I see in their journal entries also reinforces the idea that I value their thinking and know that there is reasoning behind everything they write and do in my class.

“Talk is an important way to build that sense of community and to help children grapple with important mathematical ideas.”

-Kristin

# Last Day of Math Class :(

Today was the last official day I had my students for math. It was a bit sad for me and it was nice to hear some of them say it was “kinda sad” for them too. In moving into a K-5 Math Specialist position next year, I know it will not be the same experience watching a group of students grow over the course of the school year.  It will be great in different ways, but I am really appreciating all of the amazing work my students have done this year.

So…what to do on the last math day after they just had field day yesterday followed by our PBS bowling field trip tomorrow? It is a tough planning!

I first had them look through their two math journals, one from the first half of the year and one from the second half. As their last writing piece, I asked them to write things they noticed in their work over the course of the year after looking through their journal. I only had time to grab one journal today because the end of the year craziness is kicking in, but I plan on following up with a more detailed post later. This one was so powerful and truly gave me goosebumps….

After they finished that, I asked them to revisit some of the claims they had written over the course of the year and see if they still thought they were true and could be proven or were not true and needed to be revised. This student had written a claim that when you are multiplying fractions, you could multiply the numerator and denominator to get your answer. As he was proving it just worked for multiplication, he stumbled upon the realization that it worked for division as well. He then worked through a few more division problems and it was such an amazing explanation!

He revised his claim…

I promise to follow up with some really amazing work they did on the last day when summer is here and there is a chance to breathe 🙂

-Kristin

# Patterns and Perseverance

Today in math was a test in perseverance. The students were working on the growth pattern of an animal called the Fastwalker. It was fairly easy for them to complete up to the 10th year, graph it and answer the questions regarding the line they graphed. The book did not require them to do any generalizing of a rule, however they had other plans! Here is a completed table of one of my students:

We worked on this up until lunch, discussed the line and how it was different than the constant rate of change graphs we had seen earlier. They kept asking if there was a rule for this one, so I asked them to play around with it and see what they thought. One of the students noticed that if you added all of the terms before with the term number you were trying to find, it gave her the height, so she started adding to see if it worked for the 100th term (nothing like starting small:).

While she was working on adding, another student, who had done a consecutive sums task earlier in the year in RTI w/me, realized there was an easier way to add those numbers, and you can see on the top of the page where he started playing around with pairing up the numbers:

It was so interesting to see the groups working with them and asking questions as they tried different ideas. The two of them finally saw that pairing up the numbers was working and began to generalize based on what they had done with the numbers. It was awesome because they began generalizing based on an even or odd number term because of the pairings and needing to divide the term number by 2. At the bottom of the first paper earlier in this post, you can see she wrote an even and odd rule for the pattern, while this student realized that if should work with even and odd because the decimal didn’t make a difference.

Shew…..they were sooo proud of themselves (and I was so proud of them) at the end of all of this hard work! The student who did the paper above said, “Wow, that one problem took us almost two hours!” And it was SO worth it to see the accomplishment on their faces. THIS is the reason we must make time for students to investigate their own mathematical curiosities and give them the time they need to persevere through these problems!

-Kristin