Finally, we are making progress. Several weeks ago C was not convinced that division and fractions are related. At the beginning of class, she was still not convinced.

The definition of a fraction is: , where a and b can be any integer (b cannot be zero). The symbol for division is .

What is similar between the symbol for division and a fraction?

Our notation gives a hint there is a relationship between fractions and division.

Next, I drew a square and tasked the girls with dividing the squares into four equal parts. I drew the first division.

How many different ways can you divide the square into four equal parts?

I asked, “If you were to give me one piece of one of the squares, how much would you give me?” With a little hesitation, C ventured “1/4.” I replied, “yes, you divided one square by four. One of the divided sections of the square is 1/4. Can you see that there is a connection with fractions and division?” C nodded.

“If you gave me 2 sections, how much of the square are you giving me?” J replied “Two quarters” “Exactly, how much of the square are two quarters?” J said “one half” “Are two quarters the same as one half?” They all nodded.

I handed out four cards with the fractions 1/4, 1/3, 1/5 and 2/3 written. Place these fractions in order from smallest to largest. With lots of conversation and disagreement, they placed them as follows:

Nothing was ordered accurately. “Is 2/3 smaller than 1/3?” C responded, “Yes.” I drew a cookie and divided it in thirds. “I’ll give you one third. Who got the bigger piece?” C responded you did. “Is 2/3 smaller than 1/3?” In unison, the girls replied ‘Ohhhhh”

“Is 1/3 smaller than 1/4?” J said, “Yes because 3 is smaller than 4.”

There you have all the misunderstandings about fractions in one little statement. Fractions throw everything one thought they knew about numbers out the window.

I have two cookies. I am going to give C a third and you a quarter. Who got the bigger piece?

They all said, “C.” “Is 1/4 bigger than 1/3?” Again, “Ohhhh.” “Are your fraction cards in the right order from smallest to largest?” They rearranged the cards in the correct order. They transferred their understanding of the magnitude of 1/3 and 1/4 to 1/5 and 1/4. We were making progress in understanding fractions.

“If we were to put these on the number line, where would you place them?” My question was received with blank stares. I pulled out Guy’s fraction circle from a few weeks ago. We constructed the number line with his circle, one row of fractions at a time. We also referred back to the fraction strips we created. We started with one-half.

Next thirds

Adding quarters

And fifths

Now the number line was beginning to make sense. The children could see how the fraction boxes lined up on the number line. They could tell the magnitude of the fraction with cookies, fraction cards and the number line.

In the past four weeks we have explored three questions: What is a fraction?Which fraction is bigger? Where do the fractions fall on the number line? These are not easy questions and have taken us time to explore. We are not ‘done.’ While I know the children are beginning to understand, these concepts will need to be revisited over and over again. To quote the Carpenters, “We have only just begun”

To make it fun, I gave them one more representation of dividing a square into four equal parts: The Greek Cross and Square.

I gave them each four of the following shapes and asked them to join them together to make 1 square.

We ended well and I am feeling relieved that fractions are starting to make sense. The girls have taught me so much about the misunderstandings about fractions. Each misunderstanding required me to dig deeper. Their misunderstandings took us on great detours and conversations. With deliberate interjections and cycling back and forth between representations, I can see the light bulbs turning on. We are really close to broaching the subject of equivalent fractions. I can hear M already, ‘what?”