I spent this past week in lovely Calais. While we weren't literally unplugged, connectivity issues lent a bit of a "post-apocalyptic" feel to the first half of the week (minus the mayhem - everyone was making the best of it). No Google Drive access, no internet, no network, no printing, no cell phone coverage, etc... Since my essential tools for working as I travel from school to school are my Chromebook, iPhone and a set of manipulatives, this was a challenge. It forced us to go "old school," and highlighted the power of good manipulatives - especially when there's no power!
By the end of the week, all systems (with the exception of cell service) were back up and running, and there is a lot of good stuff to share out...
Focus
We decided that we would use Monday, Wednesday, Thursday and Friday for coaching. In fifth grade, we focused on developing flexibility of thinking about how we define "one whole" and multiplication of fractions at the Applications/Communications level. In sixth grade, we focused on creating models for order of operations. We decided that I would spend Monday and Thursday in the fifth grade class and Wednesday and Friday in the sixth grade class.I found myself with a hole in my schedule that I was able to fill by working with the fourth graders on division.
Our schedule each day was:
- 8:00 - 9:00 Meeting and planning.
- 9:20 - 10:20 5th or 6th grade math
- 10:50 - 11:30 4th grade math (on Wed. and Thurs.)
- 11:30 - 12:30 Debrief and Plan
Monday
Fifth grade...We decided that we would do a simple, open ended formative assessment when they first walked in. They were asked to brainstorm on a sticky note all of the things that "one whole" meant to them.
This was really interesting and unveiled some misconceptions about "one whole." For example many students thought that one whole meant:
After this, students worked on their projects for the CES "Pi Night" celebration. Students at every grade level were working on "pie" related problems and the problems built on one another so they were all connected. Isn't that a great idea?
The discussion does not need to be done the same day as the formative assessment prompt, if that feels too hard to manage. One way to buy yourself some time to quickly look through and re-write a response is to have some quick activity immediately following the formative assessment prompt (like building something with Cuisenaire Rods, independent work on math fact fluency, a game of number war, fraction war, integer war, etc.).
The fifth grade teacher used the discussion around this activity to revisit their concept of "one whole" taking into account their common misconceptions.
Later, they spent more time working on Tool Building using linear fraction models with the Cuisenaire Rods.
Sixth grade... The sixth grade had just finished up work on surface area of solids and volume of rectangular prisms. They had some really great 3-D posters hanging up in the hallway that I meant to photograph. We decided to do order of operations using the Cuisenaire rods. We began by having students work independently on this formative assessment (entry card - see right). Students were given the expression 4 + 2(9) and prompted for a model, context and to simplify the expression. Before this activity, we needed to familiarize students with the meaning of these terms. With a quick glance, we saw that none of the students had a model, several had a compelling context and all but a few simplified it properly.
Fifth grade... On Thursday, Calais was back online. So the sixth grade class was madly finishing up Pi Night projects, while we decided to get daring with the fifth grade class. We started off revisiting the concept of "one whole." Students were prompted for "Models and Definitions" that represented "one whole" and make a list of "Contexts" for "one whole."
To warm up, we did more Tool Building doing linear fraction models with the Cuisenaire Rods. This time we defined the rods using fractions as the defined lengths (for example, "if the purple rod has a length of two thirds, what is one whole?").
We asked if this represented multiplication and they said, "Yes that's five times twelve." When asked how they know it's multiplication, they pointed out that it represents equal groups.
Then we put up this model again and asked them where the equal groups are:
We were back focusing on sixth grade again. We had looked through the homework (collected on Thursday) and decided to use these two examples for a "My Favorite No" discussion activity.
Students thought independently for 30 seconds about the two students' ideas. Then students collaborated on how the two students could help one another. In our discussion (which I facilitated by asking questions and randomly picking students to share their ideas by drawing popsicle sticks), we started out by pointing out what each student did right. Then looked at what each student needed to learn to improve their work.
We created a model for Student A and talked about how that model could show Student B that the multiplication must be done before addition to simplify this expression.
After that, we began a "True or False" partner activity that will be a multi-day project. We displayed statements and quickly (without too much thinking) had students jot down their initial answers in their notebooks. These were the statements: Order of Operations True or False.
Thanks for reading!
Ellen
This was really interesting and unveiled some misconceptions about "one whole." For example many students thought that one whole meant:
One thing that has not been not broken into parts.We want students to see that "one whole" can be defined multiple ways and that fractional contexts depend on those definitions. To get students to become more flexible about thinking about how we define "one whole," we did some Tool Building with the students using this plan: Cuisenaire Linear Fraction Models.
After this, students worked on their projects for the CES "Pi Night" celebration. Students at every grade level were working on "pie" related problems and the problems built on one another so they were all connected. Isn't that a great idea?
Tuesday
The fifth grade teacher used the students' definitions of "one whole" as a "My Favorite No" activity. For those unfamiliar with this activity, here is a video describing how one teacher uses it:The fifth grade teacher used the discussion around this activity to revisit their concept of "one whole" taking into account their common misconceptions.
Later, they spent more time working on Tool Building using linear fraction models with the Cuisenaire Rods.
Wednesday
Then we launched into an abbreviated version of this lesson plan: Order of Operations Instructional Plan. We got through a couple of examples that involved both multiplication and addition, so I gave them one to try for homework: 2 + 9(7). We asked them to come up with a model and then simplify it.
In retrospect, we would have asked students to model a straight addition example and a straight multiplication example as part of our formative assessment, because that might have revealed that we could have moved more quickly through the first few models. As it turned out, the students probably did not need as many examples of the simpler models.
Fourth grade... Later that day, I worked with some fourth grade students who are just beginning long division. We worked on language building using the Cuisenaire rods - What is the dividend? What is the divisor? What is the quotient? Tying the vocabulary to the concrete model and the various symbolic models.
Thursday
Fifth grade... On Thursday, Calais was back online. So the sixth grade class was madly finishing up Pi Night projects, while we decided to get daring with the fifth grade class. We started off revisiting the concept of "one whole." Students were prompted for "Models and Definitions" that represented "one whole" and make a list of "Contexts" for "one whole."
We could see that students had much more to share this time. However, many of the contexts were essentially the same. This was what we were trying to chip away at. We want students to become flexible about how they think about "one whole" so that they develop flexibility around how they think about operations on fractions. The students are at this tricky place where they understand area problems involving fractions, but we need them to move beyond it. In order to do so they need the flexibility to move between area models and linear models.
This was the same thing we were working on at Doty back in January (Fractions Galore! blog post) (and the seventh graders at U-32 were working on last week). And wow, is it hard work!
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| This is a running list that the fifth grade teacher keeps posted. |
After that we took a big breath and launched "The Whole Shebang" a partner activity involving contexts for multiplication of fractions. Going into this activity, students had identified the contexts for fractions listed on the left.
Through this activity, we were trying to get them to move beyond these contexts in a major way.
We took the six contexts contained in this file: Rational Contexts, and cut them into six separate strips of paper (one for each context). We gave each pair of students one set of contexts. To frame the activity, we told the students:
Every context involves multiplying fractions, you need to figure out how to make sense of and create a model for the context described.
Students were asked to:
- Divide the contexts equally between them (with six contexts, this would work for groups of 2 or 3).
- Spend 5 minutes independently reading through "their" contexts, trying to make sense, writing notes and questions, not solving - just working to understand the situation. (During this time the teacher and I circulated to clarify and define any terms that were unfamiliar.)
- Work collaboratively to create:
- A model for the situation
- A symbolic representation and solution
- A define "one whole" in the context of the situation.
This work was fascinating to listen to. The teacher and I felt like we learned a lot about how the students were thinking about contexts for fractions.
Here's an example:
There were two contexts that were "food area" contexts. One involved a pan of brownies. The other involved a pizza. Students used a rectangular area model to represent the brownie pan and a circular area model to represent the pizza. They could explain how the brownie pan represented multiplication of fractions, but not the pizza. One girl who was struggling was asked if it was possible to represent the "whole" pizza as a rectangle. Once she switched to a rectangle she was good to go, but still couldn't explain how the rectangular pizza and circular pizza models related.
Context Theory... Fascinating stuff! For some research on project-based learning that I'm doing at UVM, I've been reading Jo Boaler's work on context theory (linked here). The basic idea is that:
Students interact with the context of a task in many different and unexpected ways... Students are constructing their own meaning in different situations and it is wrong to assume their general familiarity with or general understanding of the context... A consideration of the individual nature of students' learning should precede decisions about the nature and variety of contexts to be used...
(An) activity may have a specific context, the development of the activity must enable students to follow routes which are their own...
Students will transfer from one task to another, even when the external cues are different, when they have developed an understanding of the underlying processes which link the problem requirements and their significance in relation to each other (Boaler, 1998, p. 16-17).
This suggests that engaging students in using whatever models they can to make sense of a variety of contexts that might at first seem dissimilar but all involve multiplicative reasoning with fractions and having them reflect on how those models relate could help those students to transfer that understanding to any context. At least, that's the hope!
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Fourth grade... Later, I spent more time in the fourth grade class. We started with a quick entry card using our last example from Wednesday (18 divided by 3 - looking to see if they could identify the dividend, divisor and quotient and represent the problem as an area model). We saw that some students were still missing the language.
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Fourth grade... Later, I spent more time in the fourth grade class. We started with a quick entry card using our last example from Wednesday (18 divided by 3 - looking to see if they could identify the dividend, divisor and quotient and represent the problem as an area model). We saw that some students were still missing the language.
Then, we worked through the problem "sixty divided by 5" using the area of a rectangle model and partial quotient recording with the Cuisenaire rods and Base 10 blocks.
- We made 60 the most efficient way (six tens).
- We grouped the 6 tens together (in a 10 by 6 rectangle).
- We looked at this rectangle as a multiplication problem and a division problem.
- We identified the dividend as the area, the divisor as the vertical length, and the quotient as the horizontal length.
- We changed the model to represent 60 divided by 5.
We asked,
Then we asked how we could change the model to represent "60 divided by 5" rather than "60 divided by 6". And they came up with this rectangle-with-a-chimney model:
Then a really interesting question came up that the teacher pounced on as an opportunity for a great discussion. The question was:
What is the dividend (the area) and what is the quotient (the divisor)?As students began to try to build it with fives, we asked them if it would be possible to build it using tens. We asked if the rods had to be vertical, or could we hold them horizontally? This got many of them to this model (which is "60 divided by 6"):
Then we asked how we could change the model to represent "60 divided by 5" rather than "60 divided by 6". And they came up with this rectangle-with-a-chimney model:
Then a really interesting question came up that the teacher pounced on as an opportunity for a great discussion. The question was:
Does this model multiplication? Why not?This was a moment that I wish I had recorded. The students decided that it did not represent a multiplication problem because it is not a rectangle. Then the teacher probed:
Why does a rectangle represent multiplication?This question stopped them in their tracks. They struggled to put words to their ideas. So, we put this model up on the overhead:
Then we put up this model again and asked them where the equal groups are:
They saw that the model did not represent equal groups, so we continued. A student shared out her method of "trading in the last ten for two fives" and adding them on to the left of the tens.
Then we tied the model to the partial quotient recording using the standard algorithm notation. As a next step the teacher might have students who need reinforcement do the Color Coded Division notebook entry for this problem:
Or students who need more challenge could try this three-digit example:
Friday
We were back focusing on sixth grade again. We had looked through the homework (collected on Thursday) and decided to use these two examples for a "My Favorite No" discussion activity.We created a model for Student A and talked about how that model could show Student B that the multiplication must be done before addition to simplify this expression.
After that, we began a "True or False" partner activity that will be a multi-day project. We displayed statements and quickly (without too much thinking) had students jot down their initial answers in their notebooks. These were the statements: Order of Operations True or False.
The goal of this work is to give students a geometric understanding of operations and expressions so that they can to see that "order of operations" are guidelines rather than rules. The "rule" that we want them to see is that:
You can only combine terms that have the same dimensions and that we can use the Commutative and Distributive Properties of Operations to manipulate expressions to create equivalent expressions.This builds on their understanding of how we add, subtract, multiply and divide in our base 10 system. In the future, students will use this same concept to factor, distribute and "combine" like terms in algebraic expressions and equations.
After modeling the first two as a class, students can work at their own pace. Rather than giving them all ten at once, I would feed the statements to them one at a time requiring them to check in with their reasoning before moving on. For each "TRUE or FALSE" statement, the students create a model and use the model think about what the choices are and how they could disprove the statement, if possible.
Here's an example of that process:
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| The expression and statement. |
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| Modeling the expression. Click on the tags to see what each piece of the model represents. Ask: What changes can you make to create an equivalent expression? What are your choices? |
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| Using the Commutative Property to create an equivalent expression. Ask: How do you know it's equivalent? What else could you have done? |
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| Combining 4 and 7 for a sum of 11 and substituting that in to create an equivalent expression. Ask: Why can you combine 4 and 7? How do you know it's equivalent? What other choices did you have? |
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| Multiplying 2 by 3 to get the product of 6 in one dimension. Ask: What operation are we performing when we express 2 times 3 in one dimension? How do you know it's equivalent? What other choices did you have? |
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| Combining 6 and 11 to get a sum of 17. Ask: Why can you combine 6 and 11? How do you know it's equivalent? What other choices did you have? |
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| Multiplying 3 by 8 to get a product of 24 in one dimension. Ask: What operation are we performing when we express 3 times 8 in one dimension? How do you know it's equivalent? |
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| Combining 24 and 17 to get the expression simplified in one dimension. Ask: Why can you combine 24 and 17? How do you know it's equivalent? What other choices did you have? |
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| Ta da! The simplified equation! I have proven that the statement is not true. |
So, you don't "have to" multiply first, but can you? The next step would be to ask students to decide if the "guideline" statement would get them into trouble or not. They should use a similar method.
I would encourage students to play with this, find as many different ways to simplify the expression focusing on creating equivalent expressions. Using a camera (video or otherwise) and equations to document each "step" and uploading that media to their Google Drives would be a great way to curate their work and engage them in some creative expression.
Ahem... assuming their internet access stays reliable.
Resources and Next Steps...
Fourth Grade... Student can work on solidifying their understanding of how the area of a rectangle model relates to division. They could start using an exploration of some of the divisibility rules as a "bigger idea" behind the practice (looking for these patterns might help to hook some who "get it").
Color Coded Division - Two Digit
Color Coded Division -Three Digit
Fifth Grade... In fifth grade, students will continue to work on the "The Whole Shebang." Tool Building on linear models will continue. It would be great to work in some linear contexts here, too. For example:
Another thing that the fifth grade teacher let me share with you is her "badging system" (pictured on the left).
Students are given license to "skip" steps when they can communicate their reasoning. Please note that there is an expiration date! I love this and have been looking into creating digital badges for our WCSU Non-Negotiable skills.
Sixth grade... Students will continue on on the TRUE or FALSE activity.
A good open ended formative assessment would be to have student reflect on what "=" means. This follows along the lines of defining "one whole" as we did with the fifth graders this week. It should be an ongoing topic for discussion. They should be building on their definition and its applications. We want students to bring in properties of operations and concepts involving "equivalence" rather than simply representing a relationship between a "problem" and its "answer."
As Tool Building, the teacher will alternate presenting them with concrete models and asking them to express them symbolically with presenting them with symbolic representations and asking them for the concrete model. Then they will simplify by creating equivalent expressions.
For example:
Eventually, we want the students to realize that the "rules" are guidelines and will develop a conjecture that although these guidelines are not incorrect, a better "rule" is that:
All grades... Here is Mahesh's write up of the 12 some odd math games you can make out of War. Ultra versatile for those of us with students focusing on fluency with different skills.
Color Coded Division - Two Digit
Color Coded Division -Three Digit
With your finger measure the length of the blue rod. That represents a length of one and one half.
What context could this represent? Could the blue rod represent one and one half meters of yarn? What if I used one-third of this length of yarn to make a macaroni necklace? What rod would represent what I used for the necklace? (Light green)
What rod represents one whole? (Dark green) Prove it. What does "one whole" represent in this context? Is it one whole meter or the whole length of yarn?
What is the length of the light green rod? How is the light green rod related to the blue rod? How do I see this relationship in the model? How can I express this relationship symbolically?Eventually, these students could work on the "Linear Contexts for Operation on Fractions" that we modeled in the"Fractions Galore!" post from Doty. At some point (or maybe as a break from the context insanity) the fifth graders will tackle multiplying mixed numbers using the area of a rectangle paper model.
- Cuisenaire Linear Fraction Models
- Rational Contexts
- Contexts for Operations on Whole Numbers
- Linear Contexts for Operations on Whole Numbers
- Area Contexts for Operations on Fractions
- Linear Contexts for Operations on Fractions
- Video on Multiplying Mixed Numbers
- Fractions Galore! Blog Post on Connecting Linear and Area Models for Fractions
Another thing that the fifth grade teacher let me share with you is her "badging system" (pictured on the left). Students are given license to "skip" steps when they can communicate their reasoning. Please note that there is an expiration date! I love this and have been looking into creating digital badges for our WCSU Non-Negotiable skills.
Sixth grade... Students will continue on on the TRUE or FALSE activity.
A good open ended formative assessment would be to have student reflect on what "=" means. This follows along the lines of defining "one whole" as we did with the fifth graders this week. It should be an ongoing topic for discussion. They should be building on their definition and its applications. We want students to bring in properties of operations and concepts involving "equivalence" rather than simply representing a relationship between a "problem" and its "answer."
As Tool Building, the teacher will alternate presenting them with concrete models and asking them to express them symbolically with presenting them with symbolic representations and asking them for the concrete model. Then they will simplify by creating equivalent expressions.
For example:
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| Ask: What expression does this represent? What are my choices for how to simplify it? What would an equivalent expression be? |
You can only combine terms that have the same dimensions and we can use the Commutative and Distributive Properties of Operations to manipulate expressions to create equivalent expressions.From there, I would revisit the expressions that they created to represent the surface area of a rectangular solid, thinking about how "order of operations" applies. Then, the sixth grade will need to tackle contexts involving for multiple operations (using the same context theory we've been using with the fifth grade students). They can move on from there to creating equivalent variable expressions using the commutative and distributive property and representing those relationships in tables.
- Order of Operations Instructional Plan
- Order of Operations True or False
- What does "=" mean? Formative Assessment Think Prompt
- A video that demonstrates using Cuisenaires to introduce variables (I apologize, but the footage is not great).
All grades... Here is Mahesh's write up of the 12 some odd math games you can make out of War. Ultra versatile for those of us with students focusing on fluency with different skills.
Feedback
As always, send me your feedback, comments, questions and/or any resources you'd like to share out. edorsey@u32.org.Thanks for reading!
Ellen
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