I have been assigned a room full of bright six-graders. They are successful; achieving high proficient scores – they are students who are “on the bubble” of scoring at the advanced level. A review of their work reveals the problem: Operations with fractions.

My goal is to help them create a stronger conceptual understanding of fractions, in order to allow them to be more successful in their fraction interactions!

The initial lesson involves some baby steps, in order to formatively assess the understandings about equal sharing, naming shares and rebuilding wholes. Materials will include the students themselves and a fair share box (see attached image- thanks to Jere Confrey).

I tell them they are going to gather in teams and ask them to come up with ways that we can divide the class so that each team has an equal number of students, WITHOUT COUNTING FIRST. *At this point I am not looking for anything but suggestions on grouping.*

I will use thinking questioning to get them started, clarifying q’s to help them express ideas and repetition: students repeating what others have said. The students will then be asked to critique, analyse and decide how best to group.

They will use a method they decided upon to create the teams. *We may need time for trial and error here, as well as how to divide ‘leftovers’ when the class does not divide evenly!*

Once they have successfully created even teams, I will pose questions (I will use the family feud style of questioning, so every student gets a chance to answer, can ask for help, can challenge another group’s answer, etc)

First question: Have we divided the teams evenly? Are they ‘fair’? (There may be discussion about this, especially since students sometimes want ‘smarter’ kids in their group!)

We will have to talk about ‘leftovers’ and how to divide them. (Options: let the students adjust groups until we have even amounts, or give the extra students jobs before the exercise begins so we don’t have to divide any bodies!)

Q: (work on in groups) How would you describe your team in relation to the rest of the student teams in the room? ( this is where I will go around and listen to what the students think about their relationship to the other groups. I want to find out about their level of expertise with the language of groups- I anticipate that I will get anything from ‘one group of five students out of 20’ to ‘1/4 of the whole class’.

I will have the students share their comments and ideas. I will then let the students give each other feedback on the reasonableness, accuracy, and will take the opportunity to guide the vocabulary for the discussion, by restating, and asking students to restate.

*At this point, I expect to have identified any conceptions about fair sharing and language of sharing the students have. *

Now we will fill in the chart:

How many students in each team?

How many teams?

What was the total number of students in the class

So what can we say about each team?

I will have the students tie their comments to the chart, walking through the naming conventions they use. If there is not a connection to fractions in the discussion, I will shape my questions: ex. Do they see another way we could name each group in relation to the whole?

Once we have a fraction on the table, we can start rebuilding. (At this point, I would have them go back to their seats.)

I would *flip* the table at this point, to show the connection to the fractions they usually see. I want them to make the connection from part to whole.

The next step will be to reconstruct our groups: if one group is 1/nth of the whole class, how will we talk about two groups in relation to one? What about the whole class in relation to one group? *(n here represents how many groups a class actually comes up with, but this could also be the goal, to start with two teams, three teams, four teams, and have the students introduced to the mathematical extension of ‘n’!)*

After this discussion, I would reinforce the problem solving with the penguin problem, (how do we share 30 snowballs among three penguins?) using small groups. At this point, I will be checking in with each group to hear what they are saying and to reinforce the ideas of sharing, naming and reconstructing. I will be listening for understanding.

All of this should only take a fraction of the lesson time. I am asking the students to think about fractions in a primary way, to connect the more elaborate work of operating with fractions. The students will be encouraged to use the language of fractions, wholes and reconstructing as they work with grade level problems involving fractions. Students will be encouraged to break down fractions or reconstruct them to assist in solving problems, with drawings, with manipulatives, or using the fair share table. Then they could begin to operate with the fractions, understanding how to combine groups, and in further lessons, what happens when groups are multiplied and divided.

*I see my understanding of how children instinctively do this work informing my ability to identify students who have “lost” or buried the skills and to assist in reconstruction of this knowledge to enhance and strengthen their future conceptual understanding.*

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