CCSSMashup – Fractions

I never tire of conversations about the 3rd – 5th grade fraction progression because after each one, I leave with the desire to reread the Standards and Progressions with a new lens.

A few weeks ago, a conversation about 3rd grade fractions sent me back to the Standards with a #pairedtexts type of lens. Unlike the hashtag’s typical MO of pairing contrasting texts, I was looking for standards that connected in a meaningful, but maybe unexpected way. By unexpected, I don’t mean unintentional, I mean the two standards are not necessarily near-grade or in the same strand, so the connection (to me) is not as obvious as one standard building directly toward another.

The conversation focused on this standard:

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With that standard in mind, imagine a 3rd grade student is asked to locate 3/4 on a number line on which only 0 is marked.

I expect a student would mark off the 1/4’s starting at 0 and write 3/4 above the point after the third 1/4 segment. What exactly is the student doing in that process?

Is the student adding?

Is the student counting?

Is the student doing both?

How does adding and counting look or sound the same in this scenario? different?

This is where I find pairing two standards fun and interesting to think about because it demonstrates how important seemingly unrelated ideas work together to build mathematical understandings. It is also really fun to think about how a standard in Kindergarten is so important for work in grades 3-5 and beyond.

In this scenario, I think we instinctively believe students are adding unit fractions when asked to place 3/4 on the number line because the standard is in the fraction strand and therefore we consider all of the work to be solely about fractions. We also sometimes impose our thinking on what students are actually doing in this task. For example, you could imagine the student marking off the fourths, stopping after the third one, writing 3/4 and say the student was adding 1/4+1/4+1/4 to get to the 3/4 because they moved along the number line. If this is the case, then the standard would pair with this 4th grade standard:

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Don’t get me wrong, those standards definitely pair as students move from 3rd to 4th grade, however, since the scenario is about a 3rd grade student, pairing it with a higher grade level standard doesn’t seem to make sense in terms of what students are building on. Right here, it is really interesting to pause and think about how building fractions from unit fractions, locating a fraction on a number line, and adding unit fractions are slightly different things a progression.

When I think about the student locating 3/4 in 3rd grade, I hear counting (with a change in units) and would pair that 3rd grade standard with this Kindergarten counting and cardinality standard:

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However, because the 3rd grade work is on a number line and the arrangement and order does matter, I would have to add this 2nd grade measurement standard into the mix, but take off the sum and differences part:

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So, instead of a #pairedtext, I now think of it more as a #CCSSMashup to create this standard:

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With that mashup in mind, I went back to the progressions documents to look for evidence and examples of this.

In the 3rd Grade NF Progression these parts jumped out at me as being representative of this standard mashup:

The goal is for students to see unit fractions as the basic building blocks of fractions, in the same sense that the number 1 is the basic building block of the whole numbers; just as every whole number is obtained by combining a sufficient number of 1s, every fraction is obtained by combining a sufficient number of unit fractions.

The number line reinforces the analogy between fractions and whole numbers. Just as 5 is the point on the number line reached by marking off 5 times the length of the unit interval from 0, so 5/3 is the point obtained in the same way using a different interval as the basic unit of length, namely the interval from 0 to 1 /3 .

There is also a great “Meaning of Fractions” video on the Illustrative Mathematics site that explains this idea with visuals.

There are so many of these great mashups in the standards, especially in the fraction strand, that I find incredibly helpful in thinking about how students coherently learn mathematics.

I look forward to hearing your favorite #CCSSMashup!

Explicit Planning vs Explicit Teaching

Planning is like…..

How would you finish that sentence?

As a facilitator, I use this sentence starter to open Illustrative Mathematics’ 5 Practices Professional Learning. To be completely honest, when I designed the PD I was a little hesitant of using it because I was nervous it was opening a can of worms within the first 5 minutes of the day.  I am, however, always surprised with all of the beautiful analogies participants share and feel challenged each time to come up with something new and better than the one I used in previous sessions. When I first delivered this PD, I started with analogies like a marathon or really hard workout – something that is exhausting, a lot of work, but ends with something I take pride in. While these analogies were accurate representations of how hard I think lesson planning truly is, I was continually unhappy with where students’ ideas fit into my analogy.

My most recent sentence was this…

Planning is like putting together a puzzle. 

When sharing my reasoning with participants for the first time, I included a lot of beautiful words around mathematical connections but in the middle somewhere I used the phrase “making connections explicit” in relation to the puzzle pieces and saw an immediate reaction from a few people in the room. Of course, I had to pause and ask, “Was it the word explicit?” – answered by many nods in the room.

For a long time, the word explicit in relation to teaching held a negative, cringe-worthy connotation for me as well. If ever asked to paint a picture of what explicit teaching looks like in the math classroom, I would describe scenarios in which a teacher is either at the board telling students how to solve a problem or showing a struggling student how to solve a problem because they are stuck or “taking the long way there.” To me, being explicit meant telling students a way to do something in math class – typically in the form of a procedure.

Through teaching a problem-based curriculum [Investigations], designing and implementing math routines such as number talks, and reading Principles to Actions5 Practices and Intentional Talk , I realized that I was guilty of making mathematical ideas explicit every day in my classroom, but not in the way that made me cringe.

I was explicitly planning, not explicitly teaching.

To me, those two phrases indicate a big difference in how I think about structuring a lesson. I have found when teaching a problem-based curriculum, it is easy for ideas to be left hanging and important connections missed, forcing me to explicitly teach an idea to ensure students “get it” before they leave the class period without any understanding of the mathematical goal for the day. Many days, I would find myself frustrated because students would completely miss the point of the lesson, however now I realize this was because I was expecting them to read my mind of what I wanted them to take away from the problem. On the flip side of that coin, however, not teaching a problem-based curriculum and explicitly teaching students how to do the math in each lesson is not an option (and is a topic that could be its own blogpost).

This is exactly why I find the 5 Practices framework invaluable in planning. The framework forces me to continuously think about the mathematical goal, choose an activity that supports that goal, plan questions for students toward the goal, and sequence student work in a way that creates a productive, purposeful discussion toward an explicit mathematical idea. I have learned so much using this framework over and over again in planning for my 5th grade class, collaborating with other teachers and coaching teachers across different grade levels.

Explicit planning is how I would describe the new, open education resource (OER) by Illustrative Mathematics. As a part of the writing team, I explicitly planned warm-ups such as number talks and notice and wonder activities to elicit specific mathematical ideas that play a purposeful role in the coherent plan of the lesson and unit. But not only are the warm-ups explicitly planned, but each lesson and unit tells a mathematical story in which students arrive at a specific mathematical landing point. While they may not all arrive at that landing in the same way, the problems and discussions are structured to ensure students do not leave the work of the day without any idea of what they were working toward.

While I would love to think my blog posts paint a clear picture of explicit planning, I am not that naive. So, what does explicit planning look like in a 5 Practices Framework?

This lesson from Grade 7, Unit 2, Lesson 2 from Illustrative Mathematics’ Middle School Curriculum is one of many in the curriculum. (All images are screenshots from the online curriculum that is linked at the bottom of the post)

Practice 0: Choosing a Mathematical Goal and Appropriate Task

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Lesson Learning Goals

With the goals in mind, the lesson begins with a notice and wonder warm-up that engages students in thinking about tables, followed by two activities that build on those ideas and support the mathematical goals. While both activities demonstrate explicit planning, I am focusing on one for the sake of space.

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Task Statement

 

Practice 1 and 2: Anticipating & Monitoring

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Activity Narrative

 

Practice 3 & 4: Selecting & Sequencing

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Activity Synthesis

 

Practice 5: Connecting

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Activity Synthesis

 

So..Planning is like putting together a puzzle. It is hard, takes time, and is sometimes difficult to figure out where to start. We know all of the pieces connect in the end, but making a plan for all of those pieces to connect takes an understanding of the final picture – the goal. There will be missteps along the way and some parts will take longer than others, but we know it is important to carefully connect each piece to another as one missing piece will leave unconnected ideas and the final picture unfinished. As you work alone, the way the pieces connect to form the final picture may not always be obvious, but as others help us see the pieces in different ways during the process, connections become explicitly clear and the final picture is something in which you can take a lot of pride.

The ‘others’ in my teaching journey have helped me see a difference between explicit teaching and explicit planning. Through explicit planning I have seen the importance in understanding the mathematical goal in a way that enables me to structure activities and lessons that enable students to make important mathematical connections through their own work and discussions. It is so exciting to see IM’s curriculum be a model for how I think about explicit planning in such a coherent, purposeful progression.

Link to Illustrative Mathematics 6-8 Math Curriculum.

Link to the 7th Grade lesson featured in this post.

Choral Count: Trusting Patterns

In the next round of 1st grade videos for my Teaching Channel Math Routines series, I am so excited to release a video of a Choral Count. When I taught 5th grade, I used Choral Counts a lot with decimals and fractions, so using them in the earlier grades was such an interesting parallel to that work.

I don’t want give away what you will see in the video, but Heidi’s recent post about trusting patterns when multiplying by 10s made me think about two particular student journal entries after the count.

In this count, students counted by 10s starting at 4 as I recorded their count and I stopped them at 154. They discussed patterns they saw in the count and afterwards, I asked them to journal about any new or extended patterns they think may happen if we continued counting. One thing I stress in their journal writing is the fact that it doesn’t have to always be written in words, we can explain mathematics with numbers as well. These two students didn’t include any description of what they noticed, but from their recording, I can assume some things about what they saw.

Through Heidi’s lens of trusting patterns, I watched this student record her extended count. Instead of writing every number in its entirety, this student wrote out all of the tens and then went back to add all of the 4s on. I asked her how she could show the pattern she noticed and she quickly went back to underline all of the tens. She trusted the pattern that the tens would continue going up by 1 and the ones place would stay the same throughout the rest of the count since we were counting by 10s. The pattern she noticed and trusted worked, would continue to work if she kept counting, and is something teachers could build on by counting by different multiple of tens.

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Now, this student below invented another pattern that doesn’t keep the count in tact but brings up an interesting connection to the work above. Instead of looking at the tens going up by 1 and ones place staying the same, this student added 100s going down each column. In his work, the tens and ones stayed the same while the 100’s increased by 1 with each jump down.

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The more I looked at these, I thought about how to support students in later grades who are unsure whether to trust a pattern or even why patterns work in the first place. For example, and not that counting by 7’s would be the way to go, but for Heidi’s problem of 2,000 x 7, I wonder how a choral count could support students who were struggling to explain multiplying by 10s. The student in Heidi’s example obviously has a grasp of the idea, but what about those who don’t?

Let’s see what we notice here:

7       14     21    28     35

42     49     56    63     70

77     84     91    98    105

112   119   126   133   140

Could students pull out where the multiples of 10 would show up if we kept counting?

Could they translate that pattern to equations?

Could they connect this additive thinking to multiplicative thinking?

Could they apply that same understanding to counting by another number but keeping the same structure of 5 in each row that is shown here?

All of these questions are so interesting to me and leave me wondering if we did more of this work in the earlier grades the impact it would have in later grades. Thank you Heidi for sparking such a great convo on Twitter and giving me a different lens for student work in the earlier grades!

 

 

True or False Equations: Kindergarten

Yesterday, the Kindergarten math routine videos I recorded were posted. This set of routines were so incredibly fascinating to me and completely out of my comfort zone – such an incredible learning experience. In addition to the 6 videos that are posted, there are about 12 more in my Google Drive that I had to chose between, each with its own unique student responses that would be so interesting to discuss.

With all of the videos and associated student work, I am just finding some of the work I thought I forgot to collect after the routine. This particular set of work is from the True or False Equation routine.  This is probably my favorite routine in the set because it really pushes me to think about the language, recording, and understandings students have around the meaning of the equal sign.

The final equation, as anticipated, caused a bit of a controversy. Since the class was split on whether 2+3 equaled 1+4, I asked the students to explain their reasoning in their journal.

This response is reflective of the student’s experience with equations. How much do we record, or ask students to record, equations that only have three numbers? I would guess many students only see equations with two addends the majority of the time.

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I liked the “same thing” and “I used my fingers” here. This is the language piece I find so interesting. What does it mean to be the same? In this case it could mean the same amount or looks the same. She could find the amount on her fingers or the two ways of showing the expressions on each hand would look the same in the end. A small, but important distinction I think.

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This student appears to have related this to the first problem in the string, seeing both as the same since 5=5.

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I really like this one because of the arrangement of dots. This student seems to think of exactly the same as the same amount since the dots look different in the way they are drawn. The dots are great because they look the way a student would easily subitize an arrangement.

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This student broke the equation apart and set each side equal to 5 and showed with circles that each side was 5.

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I was inspired by this number talk to dive even further into what it means to the be the same. I brainstormed some thoughts here and then tried an activity about what it means to be the same that ended in this work: (I haven’t gotten around to a blog on this one, but will soon!)

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Today’s Number: Making Connections

The Investigations curriculum and Jessica Shumway’s book, Number Sense Routines contain so many wonderful math routines. Routines designed to give students access to the mathematics and elicit many ways of thinking about the same problem. One of the more open routines, is Today’s Number. In Today’s Number, a number is posed to the class and the teacher can ask students for questions about that number, expressions that equal that number, or anything they know about that number. I love this routine, and while it is more commonly used in the primary grades, I used it often in my 5th grade classroom. While I would capture so much amazing student thinking, I always felt like all of that great thinking was left hanging out there. I could see some students were using what they knew about operations and properties to generate new expressions for the given number, however I wondered how many saw each expression as individual, unconnected ideas.

After I read Connecting Arithmetic to Algebra, I had a different ending to Today’s Number, an ending that pushed students to look explicitly at relationships between expressions. I tried it out the other day in a 3rd grade classroom.

I asked students for expressions that equaled Today’s Number, 48. I was getting a lot of addition, subtraction and multiplication expressions with two numbers, so I asked students if they could think of some that involved division or more than two numbers. I ran out of room so I moved to a new page and recorded their ideas.

After their thinking was recorded, I asked the students which expressions they saw a connection between. This is where my recording could improve tremendously, but I drew arrows between the two expressions as students explained the connection.

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In case the mess is hard to see, these are some connecting ideas that arose:

Commutative Property: 3 x 16 and 16 x 3, 6 x 8 and 8 x 6, 12 x 4 and 4 x 12

Fraction and Fraction addition: 48/1 and 24/1 + 24/1 and 24 + 24

Subtracting from 100 and 1000: 100-52 and 1000-952

Multiplication and Repeated Addition: 4 x 12 and 12+12+12+12

Adjusting Addends in similar ways: 38+10 and 18 + 30, 40 + 8 and 48+0

Other ideas that I don’t particularly know how to categorize:

10 x 4 + 8 = 10 x 4 + 4 x 2

58 x 1 – 10 = 58 – 10

The second page got even more interesting:

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“Groups of” and Decomposition: 7 x 4 + 2 + 18 = 14 + 14 + 2 + 10 + 8 . This student saw the two 14s as two groups of 7 and then the 18 decomposed into 10 + 8.

Halving and Halving the Dividend and Divisor: 192÷4 = 96÷2. This student actually used the 192 to get the expression with 96.

Another variation of the one above was 200 ÷ 4 – 2 = 100 ÷ 2 – 2.

Other cool connection: 

96 ÷2 = (48 + 48) ÷ 2; This student saw the 96 in both expressions since they were both dividing by 2.

I think asking students to look for these connections pushes them to think about mathematical relationships so expressions don’t feel like such individual ideas. I can imagine the more this is done routinely with students, the more creative they get with their expressions and connections. I saw a difference in the ways students were using one expression to get another after just pushing them try to think of some with more than 2 numbers and some division.

Number Talk: Which Numbers Are Helpful?

I think Number Talks are such a powerful routine in developing students’ fluency and flexibility with operations, but maybe not for the reason most think. One of the most highlighted purposes of a Number Talk is the ability to elicit multiple strategies for the same problem, however, an even more important goal for me during a Number Talk is for students to think about the numbers they are working with before they begin solving. And then, as they go through their solution path, think about what numbers are helpful in that process and why.

The struggle with trying to dig deeper into that thinking is simply, time. If the opportunity arises, I ask students about their number choices during the Talk but often students just end up re-explaining their entire strategy without really touching on number choices. Not to mention the other 20ish students start losing interest if they take too long. I do think it is a particularly tough question if students are not used to thinking about it and when the thinking happens so quickly in their head, they don’t realize why they made particular choices.

Last week in 2nd grade I did a Number Talk with two problems, one addition and one subtraction. During the addition talk, I noticed students using a lot of great decomposition to make friendly numbers (the term they use to describe 10’s and 100’s).

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During the subtraction problem, I saw the same use of friendly numbers, however in this one I actually got 100 as an answer. My assumption was because the student knew he was using 100 instead of 98, but got stuck there so went with 100 as the answer. I was really impressed to see so many strategies for this problem since subtraction is usually the operation teachers and I talk endlessly about in terms of where students struggle. I find myself blogging on and on about subtraction all of the time!

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When the Number Talk ended, I looked at the board and thought if my goal was to elicit a lot of strategies, then I was done – goal met. However, I chose the numbers in each problem for a particular reason  and wanted students to dig more into their number choices.

This is where I find math journals to be so amazing. They allow me to continue the conversation with students even after the Number Talk is finished.

I went back to the 100, circled it and told the class that I noticed this number came up a lot in both of our problems today. I asked them to think about why and then go back to their journal to write some other problems where 100 would be helpful.

Some used 100 as a number they were trying to get to, like in this example below. I really liked the number line and the equations that both show getting to the 100, but in two different ways.

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This student got to 100 in two different ways also. I thought this was such a clear explanation of how he decomposed the numbers to also use 10’s toward the end of their process as well.

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This student used the 100 in so many ways it was awesome! She got to 100, subtracted by 100 and adjusted the answer, and then added up to get to 100.

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While the majority of the students chose to subtract a number in the 90’s, this student did not which I find so incredibly interesting. I would love to talk to him more about his number choices!

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I didn’t give a clear direction on which operation I wanted them to use, so while most students chose subtraction because that was the problem we ended on, this one played around with both, with the same numbers. I would love to ask this student if 100 was helpful in the same or different way for the two problems.

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As I said earlier, this is a really tough thing for students to think about because it is looking deeper into their choices and in this case apply it to a new set of numbers. This group was definitely up for the challenge and while I love all of the work above, these two samples are so amazing in showing the perseverance of this group.

In this one, you can see the student started solving the problem and got stuck so she drew lines around it and went on to subtract 10’s until she ran out of time. I love this so much.

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This student has so much interesting work. It looks as if he started with an addition problem involving 84, started adding, then changed it to subtraction and got stuck.

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This is what I call continuing the conversation. They wrote me notes to let me know Hey, I am not done here yet and I am trying super hard even though there are mistakes here. That is so powerful for our learners. So while there was no “right” answer to my prompt, I got a glimpse into what each student was thinking after the Number Talk which is often hard to do during the whole-group discussion.

If you want to check out how I use journals with other Number Routines, they are in the side panel of all of my videos on Teaching Channel. 

The Equal Sign

True or False?

5 = 5

5 = 4 + 2

2 + 3 = 1 + 4

After reading so much about the meaning of the equal sign in books such as Thinking Mathematically and About Teaching Mathematics , I anticipated students may think each was false for different reasons…..

5 = 5: There is no operation on the left side.

5 = 4 + 2: The sum comes first or 4+2 is not 5.

2 + 3 = 1 + 4: there is an operation on both sides or because 2+3 =5 (and ignore the 4) or because 2+3+1 ≠ 4.

While I anticipated how students may respond, I was so intrigued by the number of students (probably about 75%) that said false for 5=5. They were about split on the second one, but for many reasons – not many of them being that 5 ≠ 6. The final one left many confused, in fact one student said, “Well now you are just trying to confuse people by putting two plus signs.” So cute.

As they explained their reasoning, my mind was reeling….

  • What questions do I ask to get them to:
    • Think about what the symbols mean?
    • Talk about what is the same?
    • Realize the equal sign in the first one is not a plus sign, so there is no answer of 10?
    • See the equal sign to not mean “the answer is next”?
    • What wording do I use for the equal sign?
      • “The same as” felt wrong because the sides do not look the same in both cases….so, is “Is the same amount” a helpful way for them to think about it?

I got back to my room and starting thinking about what learning experiences would be helpful for students in building their understanding of the equal sign? I talked through it with some colleagues at school and reached out to those outside of school, I needed some serious help!

I started playing around with some cubes and realized how interestingly my thinking changed with each one. I didn’t take a pic of those cubes so I recreated them virtually to talk thru my thinking here.

The first set represents 5 = 5. I can see here where “the same as” works for the equal sign because there are 5 and they are all yellow. But what if I put 5 yellows on the left and 5 red on the right? Then they are the same quantity, but do not look the same.

The second set represents 2+3=5 and is definitely the one students are most comfortable seeing and representing as an equation. It looks and feels like composition to me so I can definitely see why student think the equal sign means “makes” or “the total is.” It looks like 2 and 3 more combine to make 5.

Something interesting happened with the green set. I made two sets of 5 and then broke one set to make the right side – felt like decomposition. I can see why it would feel differently to students. I also realized that when I look at them, I look left to right and much of that lends itself to the way I was thinking about what was happening.

The last set I made by taking my 2 sets of 5 connected cubes and breaking each set differently. Again, “the same as” doesn’t work for me here really well either because they don’t look the same.

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Still thinking of next steps because I always like to put context into play with these types of things, but I am finding that very difficult without forcing the way students represent their thinking which I don’t want to do.

Right now, things I am left thinking about before planning forward:

  • What do students attend to when we ask if things are the same?
  • Our language and recording is SO incredibly important.
  • How can these ideas build in K-1 to be helpful in later grades?
  • If I am thinking of moving students from a concrete to more abstract understanding, how does that happen? Is it already a bit abstract in the way the numbers are represented?
  • Do we take enough time with teachers digging into these ideas? [rhetorical]

I look forward to any thoughts! So much learning to do!