When you hear the word “counting” your mind quickly goes to primary age students. You see the kindergarten students counting, the 1st grade students counting, but what about 6th graders?
Our school is trying to target students’ needs by having tw,o 30 minute target times (RTI), to help move students forward. My Tier 3 group needs a lot of work with number sense and they had been asking me to work with fractions, so I decided to do some Choral Counting with them. (The next day I did choral counting with my whole class. I love things that all students learn something from.) I thought it would be a great way for them to work with fractions, while also engaging students in SMP 3.
To be fair, we have done a lot of work this year with patterns: number patterns with the 100s chart, Pascal’s triangle, the multiplication chart, visual patterns and growth, and structural patterns in equations. Actually, we look for patterns everywhere…in the books we are reading, social studies, science, and across disciplines…we are becoming pattern seekers. This was completely evident during a choral counting lesson that I did with fractions. Patterns and relationships are so important to learning.
We started off…” We are going to count by 2/3 starting at 0.” I had planned how I would record the pattern and thought about what relationships the students might notice: patterns in a column, the denominator remains the same, the numerator is changing by 2 (I wanted to start easy, to build success and ease their working memory so that they could really focus on the relationships.) We started counting. (I was nervous that this was going to be too easy for them.) Hmmm…they were getting stuck in a few places and then I was reminded how important counting is. We are working on fractions, but at the same time we are reviewing multiplication facts, skip counting, and number sense in general. We then had 3 rows completed and I stopped. “What do you notice?”
“We are skip counting by 2.”
“Turn to your math partner. Do you agree or disagree with that statement?”
We then discussed this as a class. Many students agreed and then we got into the discussion of the numerator and denominator. Whoa! Misconceptions have surfaced!
Then a student stated, “We are not counting by 2s, we are counting by 2/3.”
“No, we are adding by 2s. See,” and a student points to the numbers in the numerator.
“If we are counting by 2s, then we should be able to add 2 to the previous number to get the next number. That doesn’t work.”
Ahhh…we have an argument going on. Great! So, I ask the students to sit with their partner and prove or disprove that we are adding by 2.
After some drawings and equations the students come to the realization that we were not counting by 2, but we were counting by 2/3. Though, they all agreed that the numerator is changing by 2.
Then a student said, “I noticed that if you look diagonally down toward the right, the numbers are increasing by 14/3 .” I asked “Is that true for every diagonal?” We tested it out? Yes. “But, why does that work I asked?” Again the students turned to their partners and discussed.
“Well, we are counting by 2/3 and so from 2/3 to 16/3 we keep adding 2/3. That adds up to 14/3.
I write what she said… 2/3+ 2/3 + 2/3 +2/3 + 2/3 + 2/3 + 2/3= 14/3
“Actually if you just multiplied the fraction by the number of times that is much faster,” a student shouts out. Ahhh, yes now we were connecting their previous knowledge about addition and multiplication with whole numbers to fractions.
We continue to look for patterns and notice the relationship down the column, moving diagonally from right to left. We talked about adding and multiplying fractions and then also subtracting fractions if we moved backwards.
Then I placed some boxes in some areas below the 3 rows and asked students to determine which numbers would go into those boxes? They used all the patterns we discussed earlier to determine which number fit and critiqued each other’s reasoning and pointed out more efficient ways.
What I love about choral counting is that it surfaces some misconceptions, allows us to discuss operations, vocabulary, and relationships between numbers. It is not threatening. Every child feels successful and we can really work on justifying their thinking and critiquing the reasoning of others.
Lastly, it is so important how I record students thinking. I have been thinking about this a lot and color has become my best friend. The best statement at the end of our counting session as we looked at the finished board.
“We all know what we were doing, but if someone walked in right now, they would have no idea what all those circles and lines were. But, it looks so cool.”
It looks cool to me too because it reminds me of the rich discussion we just had.
Making connections and looking for patterns. It is so much more than fractions and counting. It is learning how to make sense of the concepts we are exploring.