Using What We Know about Fractions Equivalent to One Half

Fractions just started to get more interesting for my fourth grade class.  The kids started to get excited about the discoveries they were making.  What a complete 180 from the week before!  (Get it?  180 degrees?  One half?)

I started out today's lesson with a "warm up that is also a hint."  I wrote 2 and 9 on the board, and under that I wrote 7 and 8.  I asked which ones were far apart and which ones were close together.  I wanted them to be able to recognize a fraction like 1/20 has numbers that are far apart, so it is smaller than a half, and a number like 3/4 has numbers close together, so it's greater than a half. 

In the past when I asked my class, "What do you notice about the numbers in the fractions," they would struggle, looking for odds and evens and other coincidences that weren't actually moving their thinking in a direction that would help them with this concept.  With this warm up, they actually got the concept halfway through the lesson, and they OWNED it.  They felt like they were discovering this on their own.  Really powerful!  

This was even after I made a joke and asked, "Which numbers are close together, and which are far apart.  Not literally."

1                              2
3  9

They thought it was funny.

The next step, after identifying those fractions that were equal to a half, was to identify fractions that are greater and smaller than half.  I gave them another chance, and allowed students to get manipulatives to test out a few of the fractions on the page.  They did a better job with the materials this time, maybe because their product was an individual one instead of a group one.

So once again, I asked students to memorize two fractions, this time one that was greater than a half and a fraction that was less than a half, write them on a poster and randomly stick them on the poster.  Once again, I was able to rapidly sort them and once again, I asked them to figure out how I was able to do it.  And once again, they made some impressive observations that I was able to commemorate in an anchor chart. 

They each copied their favorite method, and went back to work on their papers with a new found strategy and way to explain how they know what they learned.

As promised, I posted the coloring page they started the day before for you to download for free for a limited time. 

Here are the finished results!

The kids had their quiz on fractions greater than, less than, and equal to a half, in preparation for moving on to, well, the rest of fractions, haha.  Luckily we still get to "limit" the fractions that our fourth graders are exposed to.  For example, 7ths, 9ths, 11ths, and pretty much anything over 12ths are not in the CCSS. That takes a lot of the pressure off of focusing on the algorithm as the only tried and true method of finding an equivalent fraction.

I'm loving the more exploratory methods I'm "allowed" to focus on this year instead.  There will be plenty of time later to multiply the numerator and denominator by the same number.  Because for now, they seemed to deeply understand the concept of half, greater than half, and less than half; they all passed their quiz, woo!  Next week I might post the quiz as a freebie too, but it's also hand written.

More Equivalent Fractions: Discussing Nonexamples

Our last fraction lesson was about creating equivalent fractions.  I wanted the kids to discover for themselves what happened when they broke 1/2 of a colored figure down into smaller, equal sized pieces.

Today I had them vote on which they thought were right and wrong, and although the majority knew, not all of them did.  I separated out the majority of the correct figures and put them on the other poster.  I put only one correct example on this other poster so the kids could compare the non-examples to a correct one.

I wanted to hear students explain why their result was a correct fraction or not.  Again, I used the definition of a fraction of a whole as something that is cut into equal sized pieces.  Meg over at Fourth Grade Studio has been blogging about fractions and put so much emphasis on having kids explain their thinking that it made me realize I need to do more of that. 

For a few of these samples, the creators were able to explain, "No, I made a mistake.  This piece is bigger than that piece."  I was happy to see those kids say so, because they demonstrated their understanding of fractions, even if they aren't successfully folding them.

A few kids were too shy to "admit" to their sample, (I purposely did not ask them to put their names on them since this was an introductory lesson, and I planned on mistakes happening that we could learn from) so I asked for volunteers to talk about what they noticed in those figures instead.

Of the figures shown, (to the right) there were two that were the most obviously wrong to the class.  One was figure w (green, right, middle) which the creator quickly spoke up and said, "Yeah, I forgot to split up the boxes on the colored side."  So she had the concept; she simply needs to work on rechecking her work generally.

The other obviously wrong figure was figure v (bottom left).  The class mentioned that not only were the shapes different sizes, it also did not have half shaded.  The child who drew it stayed silent; he'd correctly created a different figure and this was an "experimental" and rushed attempt, so I knew he also had the concept already as well.

The other figures resulted in rich conversation.  Some of the kids were stumped with figure y.  They "wanted" it to be right, because it was such a clever way of creating 1/2, but when I wrote a number in each box and asked, "Is box 1 the same size as box 2," they conceeded that they were not.  A couple students pointed out that boxes 1 and 4 are equal however, which made me smile because I remembered doing proofs in high school geometry class about the exact same principle!

At that point more kids said, "then figure x is also wrong, because that [box 1] is smaller than that [box 3]!" I labeled those boxes with numbers as well so the rest of the class could see it and got more "ohhhhhh"s!

To finish the lesson, I asked students to draw fractions that are equivalent to 1/2.  There are a lot more who can, but still several who are either still drawing 1/2 using different shapes instead of dividing the boxes further, some who are starting with a "creative" shape that can't be divided into more than 2 equal pieces (shows left) and others who are stuck on where to begin to create a perfect array with a set number of boxes that they have in mind.

I can't wait to try Meg's next lesson on using fraction stations with my class after school vacation gets out.  I've got the manipulatives for 3 of the stations, but I'm planning to substitute one of them with my Trading Fractions product.  I'm excited about keeping most of the class busy, but still completely engaged in something that all ability levels can access, and only managing a small number of kids with the Trading Fractions.

I'm actually almost looking forward to this vacation being over just for this lesson!