What Does Division Really Mean

Fraction division is a messy business.   Now that we have Common Core State Standards, students must be able to model fraction division in addition to performing the task algorithmically.   This is no simple task for many of them, because it forces them to grapple with the question of what it means to divide.   In order to do it successfully, they must really understand that division represents one of two things.   It can be dividing something into a specified group size to find the number of groups. (I have twenty four cookies and I want to make packages of 2 cookies, how many packages can I make?)  It can also be dividing something into a specified number of groups to find the group size.  (I have twenty four cookies and I want to serve 12 kids, how many cookies can they each have?)  They have to be able to read a problem and figure out which of those two types of problems it is and then form groups accordingly.     Making sense of these ideas and constructing a real understanding of fraction division is hard, even for a lot of adults.

As we were wrapping up our work with these ideas this week, I wanted to do a quick formative assessment to see where everyone was on these ideas.   I took two of the ACE  questions (this is the set of problems from which we draw homework assignments)  from the Connected Math textbook that I use.   Instead of using them as part of a homework assignment, I turned them into a Vote With Your Feet activity.

Vote With Your Feet is a Marzano high engagement strategy that incorporates movement into a lesson. In the activity, students are presented with a multiple choice item.   They move to different locations in the room based on their chosen response to the question (e.g., north wall for A, east wall for B, south wall for C, west wall for D).  The activity gives students a chance to get up and still stay focused on the task at hand.   It also is a really quick formative assessment, taking only a minute or two to see what each student thinks and where their misconceptions are (if the question is well-designed).

Here are the two questions that I posed.

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I had students vote on each of the two items without commenting on their choices.    Afterwards, we debriefed both questions by discussing what problems each model might represent.   Kids talked to partners and in table groups.   The class talked about it together.    As we talked, I asked students to give me two division problems for each model.   What is the problem if you are dividing by a specific group size?   What is the problem if you are dividing by a certain number of groups.

These are two of my favorite problems for a formative assessment on fraction division.   This year, I used them as a Vote With Your Feet activity.   Next year, I might take the four options and make them Quiz/Quiz/Trade cards.   I might take them and just use the model and present it as a “here is the answer, what is the question” formative assessment.   In the meantime, I will probably go ahead and make Quiz/Quiz/Trade cards with problems like this to use as a quick review of fraction division from time to time later in the year.

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Constructing an Understanding of Integer Addition

Making sense of integer addition and subtraction is hard.   The algorithms are complicated and hard to remember.   On their face the results sometimes seem counter intuitive.   Sometimes the answer is positive, sometimes it is negative.   Sometimes the answer gets “bigger” when you subtract and sometimes it gets “smaller” when you add.   Because of all this messiness, students have a really hard time knowing whether an answer makes sense if they haven’t had the chance to build some conceptual understanding before jumping to an algorithm.

I start helping my students make sense of the process using a chip model.    They use black chips to represent positive numbers and red chips to represent negative numbers.    They combine the chips, making zero pairs (one red and one black chip) to find the sum.    I start with the chip model because most students think of addition as combining sets of objects.

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After students have mastered the chip model, I move on to a number line model.   The curriculum that I use (Connected Math) does a really nice job introducing the number line model.    It talks about addition as combining sets by providing a context in which two kids each have a number of video games and then talking about the combined number of video games.    It illustrates the problem using a chip model.   Then, it goes on to talk about addition also being representative of a context in which one “adds on”.   It provides a context in which there is a temperature at sunrise and that is “added on” to as the temperature rises over the course of a day.    They model this problem using a number line.

After introducing students to the number line model, I take them out into the hall to walk the number line.   Prior to this, I have had my student aides create a number line for each table group.    Each number line is created using painters tape (for ease of removal when the time comes).   The numbers on each number line range from -10 to 10.    I begin by modeling a couple of integer addition problems on the line.    I walk forward for positive numbers and backward for negative numbers.

After modeling several problems, I have each group complete a set of problems by walking on the number line.    Each group gets a laminated index card with the problems to be completed and a dry erase marker to record their answer.    The first group member walks the number line to solve the first problem.    The other members of the group check his or her work.   The next person in the group walks the number line for the second problem and the others check the work.    Each problem is completed by a new student until everyone in the group has walked the number line.   At that point, the group rotates through again until all the problems  have been completed.

Once students have mastered integer addition on the number line, they use models to construct an algorithm for integer addition.    They begin with a group of four problems.   They solve these problems using a model.    Then, they identify what is the same about the problems and create two more problems to fit the group.   (All of the problems in the group have addends in which the signs match.)    Finally, they come up with an algorithm for the adding problems within the group.   They repeat this process for a second group of problems in which the signs of the addend pairs do not match.    If students have trouble figuring out the algorithm, I remind them that each number has a magnitude (size) and a direction (sign) and suggest that they consider the two parts separately.   This usually helps them get to the algorithm.

After creating the algorithm, students test it on rational numbers (mixed numbers) and verify its efficacy using a number line model.    Finally, they test rational number addition for commutativity.

After students have created an algorithm, I use the foldable shown below to summarize the lesson.

I follow up with this Always, Sometimes, Never exit card.

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What should the answer be? What does it mean to multiply fractions? An exit card or maybe not….

“Does this make sense?”  “What do you think the answer should be?”    “Is this reasonable?”   If I had one of those buttons that you press to answer a question or my own version of a magic 8 ball, these would be my answers.   They are the answers I give when a student asks me if his or her answer is right.   They are (some of the) questions I ask when we discuss the solution to a problem in class    These are some of the comments I put on student work.   I think knowing when one has gone astray in mathematics is important and knowable.   I also think that my students have to be taught that making sense of a problem matters.

As our work with fraction/mixed number multiplication draws to a close, I know which students can correctly model the operation, which students can correctly solve a problem using an algorithm, and which students can solve a problem with an algorithm and then fake a model to fit the answer.   However, I wanted to know whether my students are really thinking about their answers and whether they make sense.    I want to know if they even have a sense of what makes sense when they step outside the realm of whole number multiplication.    Today, I gave them this exit ticket.

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I allocated more time for an exit ticket than I normally would because I didn’t administer this like I would normally administer an exit ticket.   Maybe that means it isn’t actually an exit ticket.   I don’t know.    Anyway, I posed the first question and had students respond to it and then had a discussion with them before moving on the second question.   I wanted to do this for several reasons.   First, this is the first time my 6th graders have done an Always, Sometimes, Never this year so I wanted the chance to debrief what it means to justify something that is always true, something that is sometimes true, and something that is never true.   I wanted to be sure that they realized giving an example that something is true is not sufficient to justify that something is always true.   I also wanted to take time to expose thinking about what it means to multiply.   What happens when both factors are whole numbers?   Are all whole numbers created equal in this scenario?    I wanted to begin to draw out the misconception that some students have that multiplication always means that things get bigger when you multiply.

After the debrief, I posed the second question.    With the second and third questions, I wanted my students to consider whether all fractions are created equal.   Most students initially only considered cases with proper fractions, but reconsidered their thinking when someone raised the question of “what about improper fractions?”.

I don’t know that these are the perfect questions, but I am happy with the conversation that they produced.   My students thought a little bit deeper, asked a few more questions, and looked at things from a slightly different perspective because they considered them.   That is a good thing.

Different Exit Cards For Different Purposes

What exactly do I want to know?    That question has been rattling around in my head a lot since reading Mark’s post on exit cards.   In his post, he talks about different kinds of exit cards – those intending to engage a student’s metacognition, those examining procedural knowledge, those targeting concepts, and those intended to clarify misconceptions.   While I try to incorporate all of those things into my lessons and my assessments, I really  haven’t been as intentional in thinking about the kind of thinking my exit cards elicit as I might be.

As I have been teaching my students about the GCF and LCM, I have tried to incorporate a wider array of exit card types.

Procedural Knowledge

Following lessons on how to find the GCF and LCM, I gave my students two numbers and asked them to find both the GCF and LCM.   This was intended to assess whether they could correctly find the GCF and LCM   I think knowing where students are with regard to a skill is important.   Knowing a skill isn’t enough, though.

Conceptual Understanding

My primary curriculum is Connected Math.   They have several great investigations in which students explore problems utilizing the GCF and LCM.   They pose real questions and students solve the problems.   The questions explore how the GCF and LCM are used in real contexts but don’t specifically direct them to them as solution paths.    Students determine when two ferris wheels of different sizes will align, when two different species of cicadas will simultaneously emerge, how many equal-sized snack packs can be made from a given set of  apples and packages of trail mix, and how many students a set of juice boxes and crackers will feed.

Following this exploration  I asked students what it was about a problem that would indicate that finding the GCF would be a good solution path and what it was about a problem that would indicate that finding the LCM would be a good solution path.   (This is my version of one of the CMP reflection questions at the end of the investigation.)   I wanted to know if they had grasped the fundamental understanding that the LCM is used to solve problems of alignment and that the GCF is used to solve problems of grouping.

Each time that I posed this question (I teach four sections of this class), I left plenty of time for students to answer it and also for us to debrief the question.   I wanted the individual information, but I also wanted to de-privatize student thinking.   After about five minutes of thinking and writing, we discussed the question.   It was eye-opening for me.   Every student had been able to work through the investigations pretty successfully, but only about a third of them were able to articulate how they knew what approach to take.   This was surprising because of the discussions that we had during each investigation.   When we explored the question of ferris wheel alignment, I knew which students had jumped to the LCM and intentionally did not have them talk about their thinking until the end.   Instead, I drew out students who had struggled a little, who had to draw a picture or a model to figure out what was happening.   Only after we talked through their thinking, did I call upon students who had seen finding the LCM as an appropriate strategy.   I used a similar strategy for the other investigations.   I expected that the act of modeling the situation would help the students who struggled a little bit with the problem to see that they were looking for alignment.   Yet, when the time came, it was difficult for them to articulate it.    I’m not sure what this says about their depth of understanding vs their ability to translate their understanding into words.   I am certain, however, that posing the question and asking them to respond to it pushed them to think more deeply and that the ensuing discussion was more important for them than it was for the ones who were able to readily answer the question.

Clarifying Misconceptions

Having taught these concepts for several years now, it has become clear that students confuse the GCF and the LCM fairly frequently.    They don’t seem to deeply grasp the fact that the GCF is a factor and must be less than or equal to the numbers and that the LCM is a multiple that must be greater than or equal to the numbers.   They don’t always make sense of the problems they face and don’t have those alarm bells screaming that something doesn’t make sense when they try to find/use the wrong one.   Over the last few years, I have tried to make this thinking a lot more explicit as we discuss the concepts in class and on the quizzes/tests that I give.    Last week, I posed this question on a quiz:

  • Choose all that apply. When finding the GCF of two numbers,
    1. the GCF is always less than the two numbers
    2. the GCF may be less than the two numbers
    3. the GCF may be equal to the two numbers
    4. the GCF is never less than the two numbers
    5. the GCF is always greater than the two numbers
    6. the GCF is never equal to the two numbers

Despite my efforts to draw this out in our previous discussions, the response was only fair.   I decided I needed to keep working with this with my students so I incorporated the same sort of question into an exit card following a lesson dealing with fraction subtraction.   This is the exit card.

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LCD Always Sometimes Never Exit Card Select All That Apply

My TakeAways

As I have explored using different kinds of exit cards, I have learned a lot about what my students know and what they don’t know.   I think looking deeper gave me a better understanding of my students’ thinking and gave my students a better  understanding of the math.

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Pondering the GCF & LCM

Some ideas are a little bit like raindrops that have splashed onto my puppy in a summer rainstorm.    They land lightly on her but are quickly shed as she shakes her body in a happy dance.   They touch her for only a moment and are gone.   Some ideas, though are more like a piece of sticky gum on the sidewalk on a hot summer day.    They seem to be permanently attached.    For some reason, that is how Mark’s post on exit cards has been for me these last few weeks.   I keep pondering what exactly I should want to know about my students.

This question has been dancing around in my head as I have been planning the first unit for my sixth graders.   This particular unit incorporates the ideas of the Greatest Common Factor and the Least Common Multiple (among other things).   For some reason, a lot of students seem to lump the two ideas into a single muddy mess.   I think part of the problem is the language – the words are so similar and yet so different (greatest/least, factor/multiple).    Even parsing out just the factor/multiple piece is tricky.   So many students confuse the two words.    They have trouble making them their own.  I also think part of the problem is that students have some measure of difficulty determining just what a Greatest Common Factor or a Least Common Multiple is.   I don’t mean the question of how you find them, I mean the question of exactly what are they and why should anyone care.

As we explore this larger question in the unit, we will do so in a problem context using Connected Math.

  • We’ll start with a scenario in which two siblings are going to ride two different Ferris Wheels, one larger and one smaller, at a carnival.   Given the amount of time for a single revolution for each Ferris Wheel, students will consider how long it will take for the two siblings to both be at the bottom of the revolution simultaneously and how many rotations each wheel will have made.    I usually see a fairly wide range of strategies as students tackle the question.    Some students will start making a list of the times for each sibling to return to the bottom and then find the common time in each list.   Some students will use this strategy but will organize the information in a table (which makes it so much easier to make sense of the information).   Some students will really struggle to make sense of it and we’ll need to talk about what kind of strategies they might use to help make sense of the situation.   Eventually, they try something like drawing a picture or modeling the scenario using some different colored cubes.    A few will immediately see that they need the LCM.
  • Next, we’ll move onto an exploration of the life of two different cicada species, one of which emerges every 13 years and one of which emerges every 17 years.   Students will explore the question of how often both species emerge simultaneously.    Generally, the thinking will be similar to that in the Ferris Wheel problem but it will come together a little more readily because of the discussion that ensued after the Ferris Wheel problem.
  • Our next exploration will consider a scenario in which a girl is planning snacks for an upcoming hike.    She has a specified number of apples and a specified number of bags of trail mix and needs to determine how many snack bags she can make if each snack has the same amount of food and there is no food left over.    Initially, some students want to apply that same hammer – they want to find the LCM because that worked on the last two problems so it must be the way to go on this one as well.   Then, they get an answer that doesn’t make sense – how can you have more snacks than you do apples?   I like when this happens because it makes them step back and try to actually make sense of the problem – to realize that sense-making is what math is really all about.   Some students will reach for some blocks, using one color for apples and another for trail mix.   After they play around with it for a few minutes, they usually realize that they are dividing into groups and move forward without actually modeling the whole problem.   Some students will do the same kind of modeling just sketching or tallying and reach the same end result.   A few students will just seem to see it and will find the GCF.    I always try to bring out the different strategies as we debrief the solution and ask a lot of questions about why they chose that strategy and what that strategy showed them.   We follow that up with a scenario in which the canary ate some of the trail mix and the numbers suddenly get messier.   The idea remains the same, though.
  • Finally, we consider a scenario in which someone has donated some cans of juice and some crackers to the school for snacks on a field trip.    Students need to find out how many kids can be served by the donation if each kid is going to get the same amount.   This problem seems to come together pretty quickly since it is the second GCF problem and follows a pretty solid debrief of the first problem.

Following this sequence, I expect some of the students will have made sense of the concepts of GCF and LCM, but some of them will not have.    Some of them will still struggle on the next novel scenario to figure out how to solve it.   I am wondering if my question (exit ticket) needs to be something like this.

What kind of problem requires you to find the LCM?   What kind of problem requires you to find the GCF?

I want to know if they have been able to generalize the concept.    Do they realize that LCM problems are really problems of alignment and that GCF problems are really problems of grouping (dividing)?   This is a shift for me.   In years past, my exit card would have been another problem and I would have evaluated their mastery based on their ability to solve that problem.   With this question, I am asking them to think on a different level.   I want them to consider what is the same about LCM problems, what is the same about GCF problems and how are the two different.

I’m not sure if this will be my exit card question, yet.    I have a little more time to ponder.   Am I asking the right question?

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Timber! DI Decimal Division Jenga

When I first started teaching middle school, I guess you could say I was a generalist.  I tended to see my students as a single body.  I thought about what they knew or didn’t know as a group.   I would look at mastery levels on tests and then decide that I needed to do some reteaching on specific concepts based on the performance of the whole group.

Over the last few years, I have shifted my focus to the specific.   Now, I track each student’s performance on each of the standards or skills that I address in my course.  I use an excel spreadsheet with the various standards/skills as the column headings and the kids (grouped by period) as the row headings.   At the start of the year, I give a pre-assessment for the course so that I know where everyone is starting.   After each quiz or test, I update levels of mastery in the spreadsheet based on each student’s demonstrated mastery.   I use this data on a daily basis to ensure that each student is working on things at his or her level.    My goal is to have every single student with full mastery by the end of the year.   I don’t always completely meet this goal, but I come a lot closer to it than I did before I started doing this.

As I said in yesterday’s post, I use the last five to ten minutes of class every day for review and reteaching.   I try to make the review focused and fun.   Since it is a review/reteach, I have already taught the concept conceptually.   Sometimes, the review continues to be conceptual.   Sometimes, it  is just working with a skill.

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One of the activities I use is Decimal Division Jenga.  I like the use of a game to practice.   Everyone knows a game is more fun than a worksheet and Marzano’s research supports this in The Highly Engaged Classroom.

 

I have decimal division problems on Jenga blocks.   Students pull blocks from the Jenga tower and have to do the decimal division problem.   They check their work with a calculator.   If they got the problem correct, they keep the block.   If they made a mistake, they put the block back on top of the tower.   If a player topples the tower, he or she must put all of his or her blocks back and rebuild the tower (the other players keep their blocks).   Their are several free blocks in the tower.   No player is allowed to have more than 3 free blocks.   If he or she does, the free block must be replaced and another block drawn.   The player with the most blocks when I call time is the winner and receives a piece of candy.

Decimal Division Jenga

I have three versions of this game.   The first version has problems in which a decimal is divided by a whole number.   The second version of the game has problems in which a decimal is divided by a decimal.   The third version of the game has problems in which a decimal is divided by a decimal resulting in a solution with zeros in the quotient.   I assign students to the appropriate version of the game based on their pre-assessment results initially.   I circulate between the games and monitor their work.   I work with students individually on their specific errors.   As students demonstrate sufficient levels of mastery over time, I shift them to a more difficult version of the game.   Eventually, everyone is playing only the most difficult level of the game.  You can download the file with the problems for all three levels by clicking on the link below the photo.

I let students know which version of the game they are playing using an index card system. I have an index card for each student with his or her name on it.   I write the name of the game they are playing in colored ink.   The color of the ink matches a piece of paper under the Jenga tower.   They match the color on their card to the color of the paper and go to the correct game pretty seamlessly.

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