Extending Our Understanding of fractions

…..Inquiry learning is all about giving students the skills, the disposition and the opportunity to investigate – to find out information, making meaning and take action based on what’s discovered….‘ Murdoch, Kath. The Power of Inquiry. pp. 40.

My third graders had been extending their understanding of fractions last week.

What does a fraction mean?

How does fraction work?

Why do we need fractions?

When do we use fraction in real life?

My teacher’s questions have been moved from form to more application concepts (function and connection).  Fraction is not a new concept for them.  Having them reviewed what they’ve learned (prior- knowledge) makes it easier to see what needs to be developed and how they can extend their knowledge.

This is what they shared as I asked them questions.

What does a fraction mean?

It’s a part of a whole piece. It’s a half, a quarter or one-third. It’s when you share something to more that one person.  It’s cutting a shape into equal parts.

Everyone has their own way to explain their understanding of fractions.  They mention the key words – a part of a whole and equal.

How does it work?

I gave them the fraction manipulatives.  I encouraged them to use manipulatives and draw pictures to share their understanding.  They shared the examples on how it works to their peer.  This way let develop their reasoning/explaining skills and let other give feedback.

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As they used the maths manipulatives to show basic fractions which they’ve learned in previous grade level, some of them showed and mentioned: ‘A half is equal to two-fourth. One-third is equal to three-ninth. etc.’

How do you know?

One child explained, ‘Because both fractions show the same size.

I got them continue sharing other equivalent fractions using the manipulatives.

This is the point which I actually wanted them to reach.  Extending their think to equivalent fractions.

What does equivalent fraction mean?

I got them continue developing their understanding of what equivalent fraction means and how to find equivalent fractions.

We watched a video on BrainPop Jr. to help visualize the explanation. Then, we did the quizzes and played the game.

screenshot_2(Click on the picture to watch the video)

  So, how can you find equivalent fractions?  How do you know that one-fourth is equal to two-eighth or three-twelfth?

…….From the picture…… cause we use manipulatives…..

Then, what happens if there is no manipulatives and no pictures? How can you find out?

Silent.

I gave them a hint.  Take a look at the numerator and denominator.  How do they relate?

One child screamed with big eyes, ‘Ahh…you just need to double them.

What do you mean?

Double the numerator and denominator.‘ He repeated.

I invited him to come to the board explain his thinking.

He used the example of one-fourth is equal to two-eighth.  It seemed that everyone agreed with his ‘theory’.

But…one-fourth is also equal to three-twelfth.  How can it be?  I continued questioning their understanding.

This time a girl from the other side of the table came up with the idea that you can multiply both numerator and denominator by three.  She came to the board explain her thinking.

So, it’s time for making conclusion then.  How do find equivalent fractions?

Most of them understood the idea of multiplying both numerator and denominator with the same number.  Two students still needed to use the manipulatives to visualize this concept.  Slowly by the end of the week, they were able to understand the concept and rule.

Thinking, reasoning and questioning their thinking are required in order to construct their knowledge of what equivalent fraction means (extension of how fraction works).

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Students enjoyed sharing their thinking about equivalent fractions. Which scoops of ice cream are equal to the cone?

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As you know that my kids love to play the musical swapping game.  Each of them got a flower with a fraction in the middle of the petal.

They had to swap the flower as the music was played on.  When the music stopped, they had to write one equivalent fraction on the petal.

They chose to listen to Heroes by Måns Zelmerlöw.

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They got to share their understanding of equivalent fractions on Seesaw.

COMPARING FRACTIONS

After the students got the idea of equivalent fractions, they continued to apply their new understanding into comparing fractions.

I wrote two basic fractions on the board.  One-third and one-eighth.  Which one is bigger?

I could see all students were eager to answer.

‘One-third of course.  If you have a pizza and you share with three people, then you will get bigger piece than sharing with eight people.’

Ah…this one is too easy. This means they get the concept of sharing equal parts to others.

Then… one-third and four-sixth.  Which one is bigger?

Silent.  I love the silent part which means they are thinking and questioning.

Using the thumb sign, they share what they think they know. Some said that one-third is bigger.  Some said that four-sixth is bigger. A few said unsure or ‘I don’t know’.

How can we find out?

We were going back using pictures and maths manipulatives.

‘Ahh….four-sixth is bigger,’ one showed her finding using manipulatives.

I gave them more fractions to compare and let them explore. Once they get familiar with the idea of comparing fractions, I asked, ‘Is there other way to compare fractions? What if there is no picture and manipulatives?’

Hint: How does this connect to equivalent fractions?

I took a simple example.  One-half and one-eighth.  One half and six-eighth.

One-half is definitely bigger than one-eighth.

The boy with his round eyes concluded, ‘The bigger the denominator is the smaller piece you get.’

Everyone agrees with his ‘theory’. 🙂

Then, what happens to one-half and six-eighth.  Does the same rule apply?

Yes…No..I don’t know.

I asked them to prove using the manipulatives and they found out that one-half is less than six-eighth.

Then, he came up with another theory, ‘When the numerators are the same like one half and one-fourth, you will ALWAYS get a bigger piece with the smaller denominators.  BUT it is not always like that if both numerators and denominators are not the same.

He stressed out the word ‘always’ ini his explanation.

Some of them understood and some were confused.

Let’s go back and look at the one-half and six-eighth.  By using the maths manipulatives, six-eighth is bigger than one half.

What is one-half equal to?  two-fourth, three-sixth, four-eighth….

We can compare fractions when the denominators are the same.

Slowly I heard a few students saying ‘Ahhh…..’

‘So, a half is equal to four-eighth and four eighth is smaller that six-eight!’ exclaimed a girl.

You got it!

It’s time for snowball game.  I wrote a pair of fractions on the paper.  Each of them got one and they should compare them.  Once they finished showing which fraction is bigger, they crumpled the paper and ready to play.

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They threw the paper and they had to check if the answer was correct or not. They put a smiley face if is it correct and a sad face if it is wrong.

They were encouraged to check on the way of comparing fractions.

At the end of the game, we collected the paper and discussed some of the fractions which had sad faces on the paper.

Another game which was also played to get everyone active in sharing their thinking was ‘putting fractions in order’ game.  I think I just made this name up.

Each of them wrote a fraction on a piece of paper (They got to choose any fraction they want). They were then showed the fraction to each other.  I asked one student to stand up and find someone who has smaller fraction.  This game allows everyone to think and be active.  After 5 rounds, I had four kids in a group and asked to compare their fractions.  They had to stand in a line to show who has the biggest fraction among them.  We did around 4 to 5 rounds and always mixed the kids.

Isn’t learning fraction fun?! 🙂

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