Thursday, January 26, 2017

Fraction Number Sense and Math Discourse Part 1

fraction number lines
It seems like I have picked a few topics lately that just don't fit into one blog post!  I wanted to share two different "warm ups" I did with fraction number lines--and two different types of "math talk" that resulted from them.  Give these two lessons a try and see what you think!  I went kind of "photo journalism" style because I thought the pictures helped tell the story!  Watch not just for the important fraction understanding--but the immersion in the Standards for Mathematical Process as well!

First of all...if you have followed me for long, you know I love number lines--especially number lines that think "outside the box"--even number lines can be rote and low level--so we want to watch for that!

The other day I gave my students this one...it had the 0 and the 2 marked--and asked student to identify what number they felt that "dot" was showing.  My first step is always to ask them to THINK before they even pick up their pencil.  While they think, I remind them to consider what they know and can tell from just looking at it.  I really think slowing them down before they start writing can lead to deeper understanding and reduce careless errors.  Plus--for those students who ARE slower processors...not having to watch 20 other students get to work feverishly while they sit is SO refreshing and validating for them.  After some think time, they were off!
teaching fractions
 Some asked if they could use rulers...I simply said, "If you think it will help..."
Standards for Mathematical Practice
 I noticed that some students seemed to be putting fractions on their number lines WITHOUT adding the whole numbers first.  I asked, "Are you sure 1/4 goes there?  How do you know?"  Their answers told me a ton about their level of understanding.  Some had already visualized where the "1" went--others were simply putting it "where it looked right".
number lines and number sense
 As with all my number line work, I want them to be able to explain their reasoning both in writing and orally.  As students were working to finish, I had students begin to write their ideas down so they were ready to buddy up.
critiquing reasoning
 When we were all in a good place (in other words, essentially finished), I put students in pairs and trios and gave them the following direction:

You must all come to consensus about what fraction you will assign your dot.  When you are all in agreement (and this took SOME debate in some groups!), you will mark it on the white copy and start explaining your thinking.  When you present, I will call on whoever I want so make sure everyone is accountable for the information and explanation.

(or something to that effect)
teaching fractions
 I circulated and listened and coached and looked for misconceptions and mistakes.  I also looked to see the variety of strategies shown so I could have a variety of ideas to share under the projector.
math talk
For this particular problem, we had some debate.  About half the groups believed the dot to be at 3/4 while the other groups had a variety of answers.  One by one, they presented their solution and defended their logic.  Along the way, there were a few "a-ha's"...and a few stubborn souls who stood their ground despite very good arguments from others!
math discourse
 At one point, we put two different solutions up and had students try to determine which one they felt was "more right"...and which explanations seemed most plausible.
fraction number line
 We also showcased different strategies that seemed to help some groups.  This group used different colors for each step along the way--and other groups agreed that this seemed to make their explanation easier to follow.
standards for mathematical practice
 Finally, we ended up grabbing a spare number line and doing some folding to find those midway points and really "prove" what the 3/4 teams were proposing.  We had a great discussion...and SO much learning happened--and the students did it ALL!  Consider trying the strategy:

1.  Think time
2.  Independent time
3.  Partner/consensus time
4.  Whole group sharing and debating time

Watch how engaged your students will be...and let me know how it goes!
fraction lesson ideas
Looking for the fraction number line resource pictured?
I also have two other number line resources and a bundle of all three...see what you think!  I use them ALL year long.
Want to pin this post for later?  Here you go! And watch for several more fraction posts coming soon!

Looking for an entire fraction UNIT to get students thinking and talking?  Check this one out! 
(the number lines are not a part of this resource)

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