I’m excited about this fraction lesson on two levels. (1) I taught it to a group of K-8 teachers in our graduate program in an online session. (2) We had a situation occur that confirmed the validity of the activity.
Prior to coming to class, students were given 2 equal sized fractions – 1 divided into 3 equal parts with 2 shaded and 1 divided into 4 equal parts with 1 shaded. (Actually, they were so lightly shaded the students were instructed to re-shade the rectangles. This comes in play later.) They were instructed to have at least 6 of each rectangle cut out and ready to use for class.
We met in an online room and the instructions were to take the rectangle with fourths and fold it so that each strip (or fourth) was divided into equal sized pieces. I didn’t tell them how many pieces or how to fold it. While the students were doing this, I drew rectangles on the white board and told the students to represent their new rectangle on the white board and write the fraction for the new rectangle. Below are the results.
I then asked them to look at the 4 representations and make an observation or statement about what they see. Student A said, “All of our new fractions are equal to the ¼ that we originally had.” I asked her how she knew. She went on to describe how the rectangles were folded differently but each of the fractions were equal to 2/8. I asked for other comments and Student K stated that “if you were to cut off the yellow portions of each of the rectangles and placed them on top of each other, they would all be the same size. That’s how we can say they’re all equal to ¼.” Student S noticed that “every time the numerator goes down one like from 4 to 3, the denominator goes down by 4 from 16 to 12. And then from 3 to 2, the denominator goes 12 to 8.” (I wish I had explored this statement further with a question of “Why do you think this happens?” But I was focused on my goal of this lesson and completely let that nugget pass me by. Maybe I’ll bring this comment back up in another class and we’ll investigate why this happens then.)
At this point I had a goal in mind and they were nowhere near that goal, so I decided to have them try this again. I asked them to get another ¼ rectangle and follow the same instructions as before but to make sure they folded it differently from the first time. I asked similar questions and received similar comments. This was not happening as I anticipated so I decided it was time to move on.
Next I asked them to get the rectangle divided into thirds with 2/3 shaded. I again asked them to fold the rectangle so that each third is divided into equal sized pieces. While they were doing this I set up the whiteboard with 4 rectangles.
This time I asked them to look at their new fractions, the representation of the new fractions and the original fraction of 2/3 and to see if they could relate these 3 to each other. Student N said, “Starting with 2/3, I divided the 3rds into 4 more equal pieces within each third. By adding those pieces it’s almost like I multiplied it by 4 more and 2 x 4 is 8 and 3 x 4 is 12.” Student A stated, “Like Student N, I took my thirds and divided them into 3 equal pieces. So like hers was 3 x 4 is 12. Mine was 3 x 3 was 9. It’s like everyone’s.” Student S said, “They were talking about division I saw mine as 2 times as many. Each third has twice as many so I divided each third into two. So I guess I divided too but I thought of it as multiplication.” YES!! This discussion was just what I had been looking for!
I then asked them to look through the K-5 fraction standard(s) and determine which ones we had been working with up to this point. I wanted to see if they ‘saw’ what I was hoping they would see.
This is what they came up with
3NF.3 – Explain equivalence of fractions in special cases…
4.NF.1 – Explain why a fraction a/b is equivalent to a fraction (n x a)/(n x b) by using visual fraction models, with attention to how the number and size of the parts differ even though the two fractions themselves are the same size. Use this principle to recognize and generate equivalent fractions.
Student S explained 4.NF.1 by saying, “We kind of did this with our last activity of 2/3 times a certain amount. We divided our thirds into more, we multiplied the number of parts in each third. So we did something with that kind of.”
I said that I thought I understood what Student S was saying. I then asked if someone could explain in their own words the same standard.
Student A said, “I can see how students can totally GET multiplying the numerator and denominator by the same thing; they could actually see it holding true. It’s not just something that we say, but they can see that relationship working…I think that this activity would explain why we say ‘whatever you do to the numerator you do to the denominator’ because it actually HAS to be that way when you’re folding the rectangles. You changed each third into 4 smaller pieces. You were doing it to each third. I see students get confused and just multiply either the numerator or denominator by 4 but this would help them see that the entire rectangle’s pieces are being changed.”
Bingo! This was what I had hoped would come from this activity! We then moved on to the second goal of the lesson.
They were instructed to take both rectangles – where 1 fourth was shaded and where 2 thirds were shaded – and fold both sheets in such a way that both rectangles were divided into the same number of equal pieces. This took a little longer for some. I went ahead and drew rectangles on the board while they worked.
I then asked them how they decided what they were going to do. Student A said, “I wanted them to be equal. I chose 12 because 12 is a multiple of 3 and 12 is a multiple of 4 as well.” Student T stated, “I thought about our previous problems and I had folded both of them into 12ths before so I knew that I could do it again.” Student K pointed out that one might also do this activity by trial and error. To be honest this is exactly how I solved it because I wanted to see what a child might do who didn’t know about common denominators. I first folded the thirds in half horizontally and the fourths in half horizontally. That gave me 6ths and 8ths. I folded the thirds again (just repeating the first fold) and then one more time horizontally. That gave me 12ths. I looked at the fourths that had been folded into 8ths and realized that I couldn’t turn it into 12ths; but if I started over with a new fourths rectangle, I could fold it into thirds horizontally it would also have 12ths.
After this discussion I then asked If we wanted to compare 8/12 and 3/12, which fractional amount is greater and why or are they equal?
Student A stated that they are not equal. “8/12 is greater than 3/12.” I asked her how she knew. “I’m looking at my models and 2/3 is 8/12 and there is more shaded than the ¼ or 3/12 on my other rectangle. But if you look at my drawing you can see that 8/12 is 5/12 more than 3/12.”
Again, we moved on to another problem. I asked, “Can you fold both rectangles in such a way that the shaded amount on each rectangle is divided into the same number of pieces?”
Then something beautiful happened. Notice that some students already have their drawings up while the 4th student is still folding her rectangles. I notice that Student K doesn’t have 2/3 shaded on her first rectangle so I asked a tricky question, “Hey Student K, did you do the actual folds on your rectangles or did you just start drawing it on the board first?” She used a technique she’s seen me use often – answered my question with a question, “did you ask us to fold it so that you have the same number of shaded pieces? They each have 2 pieces shaded; they’re just not of equal size.” I then repeated my question, to which she confessed that she didn’t fold the papers. After looking at the other drawings she realized that she didn’t shade in 2/3 of the first rectangle. So she went to correct it by shading in the 2nd third. I told her to go back and do the problem again with her paper rectangles.
Notice what happened. Again Student K did not use the paper rectangles. Instead she hurriedly divided the 2 eighths into shaded equal pieces because she already had 4 shaded equal pieces on the first rectangle. As a result she didn’t think to continue the new horizontal lines. This confirmed for me the importance of students using the actual paper rectangles and folding them. If you fold a horizontal line in the shaded section, it’s going to go all the way across the sections that are not shaded as well.
Once all 4 students completed their drawings and fractions I asked them to use words to discuss their 2 fractions? Student N said, “My fraction 4/6 was orinigally 2/3 and I went ahaead and divided it more so 4 out of 6 parts are shaded. And in the picture on right, it started as ¼ and I divided that ¼ further into smaller parts of fours. So it pretty much mulitplied itself by 4 going from ¼ to 4/16. So you can see how ¼ times 4 on both the numerator and denominator is 4/16.”
Student A stated, “Both of my numerators are 6 and I can see from the shaded parts that I have a common numerator. We can compare the 6/9 to 6/24. We know 6/9 Is greater but typically we compare with the denominators the same but now we are comparing the numerators.”
I then said let’s assume a student has had a week or two of doing similar activities to what we’ve been doing with folding rectangles, representing them on paper, writing the fractions, discussing them, comparing them. How might one know which fraction, 6/9 or 6/24, would be greater and why, without the rectangles?
Student S said, “In one of them you imagine folding the fractions into much more pieces. So they are all smaller pieces than in the other rectangle where you folded it only into 9 pieces. So the pieces are much bigger than the other pieces. So 6 bigger pieces (6/9) is much bigger than 6 much smaller pieces in 6/24.”
What standards are we addressing now?
3.F.3.d. Compare two fractions with the same numerator or the same denominator by reasoning about their size. Recognize that comparisons are valid only when the two fractions refer to the same whole. Record the results of comparisons with the symbols >, =, <, and justify the conclusions, e.g., by using a visual fraction model.
I was very pleased with how this lesson progressed with developing fractional understanding of equivalence and fraction comparisons. But I was also delighted that we were able to develop such deep understanding in an online setting. We were once timid about doing any of our program online because we felt that it would not be as ‘good’ as face to face classes. This shows that it can be done. My brain does not automatically think in an online format yet, but it’s getting better and I’m no longer afraid to try new things.