My Factor Struggle

It’s been far too long since I last blogged…so now I have an epically long one to write. Sorry! Feel free to skim ūüôā

I have been looking forward to teaching the 4th grade multiplication and division unit all year. Finally, over February break I was able to analyze pre assessment data and make some plans for the unit. I was stoked to begin the unit by working with factors. When I taught this unit last year (for the first time) I hadn’t realized how important the factor standard was. I taught them the vocabulary and we used it, but it wasn’t until we began working with multi digit multiplication and division problems that I realized the importance and benefits of being able to manipulate numbers by using their factors.

So off to the blogosphere and Twitterverse I went. I got a ton of great ideas from Simon Gregg. If you don’t follow him or know his blog, you should check it out (http://followinglearning.blogspot.fr). I was going to start with cuisenare rods.¬†I had never thought to use them to explore equal groups that make a larger number. We definitely didn’t use them last year to find the factors of a number. So we started by exploring. It was super fun! Kids love using the cuisenare rods to explore. They are pretty and fit together in satisfying ways. My students immediately assigned values to the rods, which was fine and actually necessary for our investigation of factors. I asked them to start with the 10 rod and see how many different¬†single color walls they could make that would equal 10. Then we tried 9 and then 7. We discussed what was different about the walls of those three numbers and students were starting to recognize that some numbers have lots of combinations and other numbers don’t really have any at all. I told them they were finding factors and asked them to think about how they might define a factor. We continued our investigation for another day and they’re definitions changed gradually and eventually they were able to explain what a factor was in their own words. Cool! I thought we were in a great place, and ready to move on!

Next up, we were going to explore the Sieve of Eratosthenes.¬†This was an activity I did on paper last year, but this year I was inspired to try a life size version thanks again to Simon. It was fun, and we made a great design on the floor, but it took forever! And it wasn’t permanent. So while we had some good discussion during the activity, not everyone was involved and then we had to pick it up and couldn’t go back to look at the patterns again another day. The goal of doing the Sieve is to find the prime numbers. We had been able to do that, so overall I thought it was a success.¬†IMG_20170302_110349

Now it was time for the activity that I had looked forward to the most! I wanted to create a factor forest with my class. And we would be able to put it up on the bulletin board and parents would be so impressed at parent teacher conference time, go me! If you have not seen the factor forests created by Simon’s students you need to check them out (http://followinglearning.blogspot.fr/2016/02/art-for-maths-sake.html) they are gorgeous!

I started by showing them these inspirational pieces of artwork. We talked about the math that they represented. My kids thought they were the coolest, I totally had them hooked! We made ¬†a plan for our forest. They wanted it to be nighttime, so we would use black paper. They wanted the primes to be stars and the number one would be an owl. They were itching to get started, so I passed out the paper, students picked numbers and away they went. There was about 1 minute of excitement and then groans of annoyance and then the whole class seemed to revolt and give up. It was crazy! I was running around helping students, and they were creating these tiny little trees that could barely be seen and they were mad and didn’t want to do anymore. We stopped and went to lunch. I looked through their artwork, bewildered by the lack of effort in a project that they were so excited about 30 minutes ago. ¬†I complained¬†processed with some of my coworkers and that’s when it hit me. ¬†It wasn’t the art that was frustrating them, it was the math.¬†So I took a moment to reflect on my whole carefully planned week.

I quickly realized that I had planned a week of activities, NOT¬†a week of learning. I had carefully thought out and taught what students would need to know in order to DO each activity. I hadn’t thought about a progression of understanding. And I certainly hadn’t left anytime to be responsive to what the kids needed. Wow, not the teacher I strive to be.

I spoke to the students about their frustration with the factor forest and told them that it was my fault. I hadn’t spent enough time on the math, and so the representation was confusing. We needed to back up. There was a great convo happening on Twitter about the Sieve and I decided to go back to that. ¬†I wanted to stay with the 100s chart representation for awhile. It’s familiar and the students enjoyed looking for patterns. I showed them several different representations of the 100s chart and we did some great noticing and wondering. First we looked at this piece of artwork by Paul Ashwell ()

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It was a great notice and wonder. Students found patterns and then connected the patterns to numbers. They saw the representations of different numbers and began talking about figuring out the number by multiplying the symbols. This is some of their thinking.

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At this point I talked to my lovely math coach Sarah Caban (https://mathontheedge.wordpress.com) and she suggested that I try some number talks using Jo Boaler’s circle number representations from Week 1 of Inspirational Math (https://www.youcubed.org/week-of-inspirational-math/).Screen Shot 2017-03-10 at 10.20.42 AM

I showed a few “numbers” at a time, not the whole chart. These image talks were a hit!

Students could “see” the multiplication of factors within the image. This is when they started to make some real connections. One student even exclaimed, “Hey! These are like those trees!” Yay! They were finally getting it.

We are finally in place where students are seeing that they can manipulate their numerical representations based on what they “see” for groups. I think this will be immensely helpful when we get to multi digit multiplication and division. Next week I’m going to attempt some number strings and I am curious to see if they will use primes and factors as strategies for solving. Wouldn’t that be cool!?

So That’s What the Back of a Geoboard is for…

My fourth grade students are deep in the study of geometry. Our unit is about complete and I am just reviewing major concepts and misconceptions with some of the kiddos. As I was attempting to reteach/reinforce/deepen a select group of students’ understanding of angles as turns, I was reminded of how effective using the back of the geoboard was for this.

When I originally taught this lesson, I was searching for a manipulative that would help students see how angles open, or turn as they become bigger. I was coming up short. A quick tweet tweet to #MTBOS and Kristin Gray (@MindsMinds) suggested using the clock to show this. That got me thinking… I liked the idea of using a circle and mobile hands, but I didn’t want to attach the clock face numbers to it. I was worried about some of my most striving students getting confused by attaching other numbers that were not angle measurements. As I was brainstorming how I could use a brad and attach layers of laminated strips of paper together at a vertex so that students could open them and close them, creating bigger and smaller angles, it dawned on me…the back of the geoboard! That weird circular arrangement of pegs sure does look a lot like a clock.

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Now, if you are like me, you have never ventured to the back side of the geoboard. I had noticed the circular arrangement of pegs before, but had never found a use for it. We’ve made lots of angles using the square side, however. And the geoboard is always an engaging tool for students. So how could I use it to help students think about angles as turns? What could I ask students to help them make some discoveries on their own?

When I was planning this lesson I was at home, and without a geoboard of my own to play with. So I just jotted down some questions to start a conversation with students and decided I would just go from it. I was going to wing it (*gasp*)! I was able to find a great resource to use to help me demo on our Promethean board what students were doing on their individual geoboards  (http://www.mathlearningcenter.org/web-apps/geoboard/).

I should let you know that, up until this point in the unit, I had been really annoyed by how disjointed this unit felt from “regular” math. There was very little work with operations or properties and I was feeling like this unit was just going to set my kids back. I wanted to get to the good stuff! Fractions! Multiplication and division! Geometry was just slowing us down.

Lesson time. I gathered students, and passed out geoboards. I asked them to flip it over and take a look at that circular side. “What do you notice?”

“It’s a circle”

“It looks like a clock”

I asked, “Could we use this side to make angles?”

“Umm, can we have some rubber bands so we can try?” I passed out bands. At this point they started experimenting. They very quickly started making times. Someone made 3:00, another made 6:00. They were excited by this discovery and kept sharing out the different times they were able to create. I stopped them and asked, “Are you creating angles?” Here there was a little “aha!” moment as they recognized that 3:00 is a 90 degree angle, and 6:00 shows a 180 degree straight angle. They even recognized that 9:00 would be a 270 degree angle. At this point I asked them to place one rubber band so that it was “pointing to the 12” and then put another one on top of it. Then I instructed them to take the top rubber band and turn it to show 90 degrees, then turn it more to be 180 degrees and turn it again to be 270 degrees, then turn it again to be back to 12:00. At this students exclaimed, “Hey! 360 degrees and 0 are at the same place!” Wow, I hadn’t anticipated this, but it felt big. “Ok, explain that to a neighbor.” The class erupted in talk and as I listened in, I realized that the idea of 360 degrees being a whole circle and a whole turn around one point was starting to sink in. Many kids were realizing that in a circle, the starting and finishing point are the same. Very cool.

I wanted to keep discovering! “If 3:00 shows 90 degrees, could we figure out the measure of each peg? What is the measure of 1:00 and 2:00?”

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Students¬†again got to work, excited to figure this out. It didn’t take long for students to reason that if you make the 1:00, 2:00 and 3:00, you have created a right angle and then split it into three parts. Then they reasoned that each part was equal and so must be 30 degrees each. On our geoboards at school, there is a peg in each corner of the circular side of the geoboard, outside of the circle of pegs. Students created an angle here and decided that it showed a 45 degree angle because it was splitting the right angle in half. ¬†I hadn’t even noticed that peg before! And I had not even wanted to venture into the realm of decomposing angles into smaller angles, but that’s where they were headed.

Because I wanted to emphasize the turn of an angle, we put two bands on 12:00 again and “opened” the angle by 30 degrees all the way around to 360. We counted by 30 degrees as we went. I passed out some scrap paper and asked students to make the 1:00, 2:00 and 3:00 angles again. I wanted them to write a number sentence that would represent their thinking. I got a lot of variation here:

30 + 30 + 30 = 90

60 + 30 = 90

3 x 30 = 90

The multiplication representation stuck with me, remember, I had been itching to get to multiplication talk and there it was. It appeared all on its own ūüôā

We created other angles and split them and wrote equations to match their thinking. It was fun. Finally, I decided to let them create their own angles and split them however they wanted. One student did this and I replicated it on the board so that we could talk about it: screen-shot-2016-12-07-at-12-29-51-pm

Whoa! Do you see it? There are four 60 degree angles here. This student saw that there were two groups of two 60 degree angles. I definitely didn’t see that coming, but I sure was thrilled to see it! There it was, the connection to multiplication and the properties of operations that I had been missing all along. It took a student to show it to me. Wow, did that get us talking!

I was so pleased by the discoveries that students made during this investigation. And I was even more pleased that inquiry in math was again validated. I love setting my students free to explore, but sometimes “winging it” doesn’t make me feel like the hyper prepared, forward looking teacher that I am suppose to be. This lesson was a great reminder that I need to always allow room for discovery in math. I have another inquiry lesson prepared for next week. I can’t wait to see what they discover about types of triangles and their angle measures.

They’re Called Studs!

Today in fourth grade we did a number talk using a picture from http://ntimages.weebly.com. We’ve used pictures to do number talks before, but not often. I was excited to see what students would have to say about the Lego image below. I was hoping to get more insight on how they are using the properties of operations and if parentheses would come up again.

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I decided to start by asking, “What do you notice?” I was expecting comments about colors, or number of bricks. I figured I would get some talk about arrays or groups. But this is not what I got. At first, they all approached this talk with a geometry perspective. They saw parallel and perpendicular lines, 90 degree and 180 degree angles. We are in the midst of a geometry unit now, it was awesome that they noticed some of the things we have been talking about, but it was totally unexpected! I wanted to talk about multiplication so I kept pushing them to notice more.

Next, I asked students “What do you wonder?” They’re were lots of great questions, including “I wonder why you are showing us this…” haha! We eventually landed on the question about how many little pegs were in the entire image. This is when I learned that those little pegs are actually called studs!¬†Apparently, fourth grade boys are Lego experts. So I asked, “If we are trying to figure out how many studs¬†are in this picture, show me a representation of what you see.” I passed out white boards. Below are some of their representations.

We talked about how we visualized the studs as a whole group. I tried to record some of their ideas on the Promethean board. I still struggle with recording their thinking…something to keep working on.

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As you can see from their personal boards, there were several ways of “seeing” the studs. I decided to focus in on the kids who first figured out the number of bricks (3×4) and then multiplied that by 4 and the kids who saw an array of studs (6×8). The first students said “3×4 is 12. There are 12 Legos. There are 4 studs on each Lego, so I just need to do 12×4. It’s 48.” Another student had thought the same way, but wrote 3x4x4. So I asked, are those equivalent? There was a resounding yes. I wasn’t convinced, “But where is 12 in 3x4x4?” “The 3×4, duh Ms. Miner!” Ok, moving on.

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We talked about the 6×8. Some students saw it as (6×4) + (6×4); 24 + 24 = 48. Same number of studs, sweet. Side note: some of the kids, without ANY direct instruction, are starting to “recompose” 6×4 + 6×4 into 6×8. Super cool, you can see it on one of the white boards. Kids were rocking the talk, so I decided to keep pushing. “Is 6×8 = 3x4x4? Can you prove it without calculating?”

This is where my mind was blown! They thought to themselves, then turned and talked and then shared out. And guess what they did!? They factored one of the 4s into 2×2 and saw that they could then do 3×2= 6 and 4×2=8 and so 6×8! I didn’t know if we would get there, and it was one of the first shares, and they all were agreeing with their hands and repeating to one another. It was super cool!

I did have to ask them about parentheses too. It had come up when one students told me they thought of (3×4)x4 instead as 3x(4×4). So I asked students what the difference was. They chatted about how the parentheses represent the groups. And so, by changing where the parentheses are, you change the way the group looks. In the slide below you can see the yellow shows 3x(4×4). The blue shows (3×4)x4. They found the groups, not me.

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I know 3x4x4 doesn’t “need” parentheses to be solved. However, students were using them and so it offered us a chance to talk about them (again). I think this is where I want them. They are starting to see that parentheses don’t just mean “do this first,” parentheses ¬†also mean something about groups and the arrangement of the group.

I can’t believe how much these kids already know about multiplication. We have not even started the fourth grade multiplication unit! This is ALL carried over from their 3rd grade year. Thank you Mrs. Tweedie and Mrs. Dunphy! I am stoked to start talking multiplication ūüôā

 

When and How Do We Teach Order of Operations??

In our math PDC group we decided to use string of math equations with our students across the district. The goal was to investigate what students know about the properties of operations and how they use them across the grade levels. The number stings were inspired by @jamestanton¬†(I think…)¬†and then modified to be appropriate for different grade levels.

I accidentally gave the original number string to my fourthies and it was very difficult (obviously, Ms. Miner!). But we talked about it for so long! And the kids were excited and asking questions and taking out manipulative to assist in their explanations and thinking. I couldn’t stop there. I decided to pull a small group of kids and present part of the “correct” number string to them.

The equation we worked on was (7×8) + (8×3). This is the record of their thinking, it’s hard to see what they did first, but I will try to talk about that below.

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https://drive.google.com/file/d/0B3Oqp4y6qfZ7TlFrMmd5aTU2aTQ/view?usp=sharing

This is the audio of our math talk, it’s long and I didn’t trim it so there are a few interruptions. I have pulled out some of the things that stood out to me and wrote about them below.

Students immediately wanted to solve the (7×8). They were unsure of this math fact, so they broke it into ¬†(7×4) + (7×4) and got 56. They knew 8×3 = 24. They added 56+24 to get 80. Awesome! I could tell right away that they are comfortable with decomposing into smaller, more friendly math facts. They are still thinking additively. They were very sure of this answer, and they arrived at it with very little scaffolding from me.

** It was cool that they essentially did what I was asking them to do in reverse. They took 7×8 and broke it down into¬†¬†(7×4) + (7×4). I was asking them to take (7×8) + (8×3) and put it back together (10×8). They didn’t see this connection, but I think we can get there!

I decided to push them. I asked them to think about another way to solve (5:00 in clip). One student (Gavin) suggested switching the factors and multiplying 7×3 and 8×8 and then adding. They were surprised to get an answer that was close, but they knew it wasn’t right. And you can hear one little girl (Emerson) at the start tell him right away that he can’t do that because of the parentheses.

So I asked for another way. I prompted them to think about it in terms of “groups of.” I asked if they could see “7 groups of 3” (from Gavin’s thinking above) in the original problem (7×8). Now here they go on a little tangent about seeing groups within groups, which is very cool and I want to explore later. I think they are at the beginning stages of understanding the associative property but are still thinking additively and so it gets a little jumbled.

Gavin notices the 10, but he still wants to do 8×8 and then subtract 10. At this point I started to erase the parentheses and that’s when Emerson chimes in again. But I got stuck, and quickly put them back on! I think this problem does NEED parentheses. You can come up with multiple answers if they aren’t there, right?

At around 8:15 you can hear me start to lead them to the idea of 10×8. I may not have continued on this path, but I wanted to explore the extent of their knowledge of the properties and so I wondered if they would even be able to see the 10×8 if I presented it to them. At 8:48 you can hear the students start to attempt to explain why it works. Emerson thinks I left out one of the 8s, and Molly and Gavin try really hard to explain where that 8 is. I was impressed that Molly was trying to explain that even though there are two 8s written in the original problem, there are more than two 8s… there are seven 8s and then three more 8s. I should have cued into that more than I did at the time. We then got out some cubes to see if that would help with the explanation. They built and aligned them to look like two separate arrays; 7×8 and 3×8.

Eventually at the end of the video you can hear Emerson ask about the parentheses again. This is where I realized how hung up on the parentheses she was. This is when I started to wonder a few things:

First, it was a content question. What is the reason we can add 7 and 3 before doing the multiplication?  And second, how do I get this student to think flexibly about the parentheses that she is so certain about using? Without overgeneralizing that they can ALWAYS be used flexibly?

I checked in with the math coach (Sarah Caban) and she tweeted out for help! Thanks #MTBOS for all of the responses. Check out this collaborative googledoc started by @davidwees! https://docs.google.com/document/d/1N9WvU9ypZCfVM3pJXDapHv0Pssw95hgVcRElMmZF8NE/edit?usp=sharing

I am super excited to continue this exploration with my kiddos! I have so many ideas about where to head next, I wish I could jump into our multiplication/division unit now!

 

 

Ten Times Greater

I’m new to the blogosphere…shoved here ever so gently by my brilliant math coach Sarah Caban (mathontheedge.wordpress.com)! I’ve decided she is the only person in the district who could add one more thing to my teaching life and make me feel good about it. She bribed me to join this crazy¬†awesome PD adventure with chocolate and wine (which I have yet to receive) and then, before I knew it, she had me hopping and skipping through the halls of school musing to myself about the wonderful world of Twitter and the blogosphere. This is what makes her so brilliant…she is a very clever and subtle math attack-er.

So, here goes my first attempt at finding my blog voice ūüôā

Testing is done, routines and procedures have been taught, a math community has been built and it’s time to get down to the nitty gritty. I always start a new math unit (or any unit for that matter) with mixed feelings. I am always excited, but almost certainly nervous and anxious as well. There is pre-test data to sort through and planning to be done. ¬†I want to find the perfect entry point, target my small groups just so and plan jaw-dropping, aha-moment inspiring lessons for each day! These expectations and, let’s face it, my control issues, tend to delay the start of any unit, but eventually I convince myself to just jump in.

Enter 4.NBT.1, my entry point into unit 1. It’s a cool standard, but a difficult one for sure. For those unfamiliar, it requires that my fourth grade students understand the relationships between digits in a number, in particular that they recognize that as you move left on the place value chart, the number will become ten times greater and if you move back, it will become ten times less. ¬†Now, all of my students came to me knowing how to multiply a number by ten. “That’s easy Ms. Miner!” They all knew how to expand numbers using their place values and they even had a conceptual understanding of regrouping, though they have not been taught the addition/subtraction algorithms yet (yay!). They have been primed for success.

So off we went! Diving deep into this standard. We counted around the circle by tens, hundreds, thousands, etc. I recorded patterns and they talked about what they noticed. Really, I was blown away when they discovered on their own that the value was growing by ten times. They practiced and I reflected. Some students started adding ten to numbers rather then multiplying by ten. So we compared and contrasted ten times and ten more. We practiced some more, counted some more! We ¬†worked on the place value chart, used the place value disks, busted out the pennies, dimes and dollars. We worked with the language frame “______ is ten times greater than ______.” It was awesome! I felt like they had it. They were excited and using the terms and answering questions and explaining their thinking. Time for a little formative assessment…

 

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A quick exit slip that asked students to consider the value of a collection of 826 coins. What is the value if they are pennies? Dimes? Dollars? Ten dollars? What is happening to the value of the collection? Explain with words, pictures or numbers. “This will be easy,” I thought, “We’ve been using money like this.” I just needed to document their understanding,¬†and then off to the next concept we would go!

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I’m sure you know what’s coming. I reviewed the exit slips and I was stunned. 2 of my 17 got it. TWO! So of course I looked more closely at the exit slips. I asked myself “what do they know?” (Thanks to Sarah). They almost all got the value correct when the collection was of pennies and of dollars. But when it came time think about the value in terms of dimes and ten dollar bills, they almost all got the value incorrect. Okay… “Do I see any patterns in their work?” AND THERE IT WAS! Nearly every student in the class took the value of the pennies and doubled it to get the value of the collection if it were dimes. Then they took the value of the collection of dollars and doubled it to get the value of the collection as ten dollar bills. It was strange, and I don’t know where it came from and I still can’t figure it out. When does “ten times greater” become “two times greater” in a kiddo’s head? What connection are they making that makes this make sense to them? I encountered this same issue last year and when it showed up again I was baffled. I had definitely done a better job teaching “ten times greater” this year but there it was again.

So now I ask you, Blogosphere, for some help! Has anyone else noticed this misunderstanding with their own students? Do you have any suggestions for next steps? And rather than send you an email, Sarah, I wrote a blog post…just like you asked ūüôā