thinking with and/or knowing numbers is valuable. I also don’t know of many instances in Maths where trying to emphasize one particular strategy w.r.t. I don’t know an answer to how HS teachers use Number Talks.
– Are there mathematical concepts that present themselves later in Middle School or High School in which known and derived facts would be useful? – Using known or derived fact and compensation are invaluable for students when working with both whole numbers, fractions and decimals. I tell myself all of the time, but in the moment I always forget. – I would much prefer if I remembered to use the word “sum” instead of “answer”…. – Subtraction will be an interesting one to try out next. – Students can be flexible with fractions if you push them to be. – Do they really “think” about the fractions when the denominators are the same? Can they reason if that answer makes sense if they are just finding equivalents and adding. – Students LOVE having the same denominator when combining fractions. What I learned (and questions I still have) from this little experiment: There were many more students who used problems we had previously done. Huge variety on this one and I thoroughly enjoyed it! From 6/8 + 5/8 = 11/8 to decomposing to combine 3/4 and 2/8 to get the whole and then 3 more 1/8s = 1 3/8. I waited….finally a student said, “It is just a 1/4 more than the previous problem so it is 1 1/2″ and another said each 3/4 is 1/4 more than a 1/2 so if you know 1/2 + 1/2 = 1 then you add 1/2 because 1/4 + 1/4 = 1/2.” I had to record that reasoning for the class bc it was hard for many to visualize. I noticed that when the denominators are same, they don’t really “think” about the fractions too much. Got some grumbles on this one, because it was “too easy” – 6/4…Duh! The class shook their hands in agreement and they were ready to move on to something harder. Some used 2/4 + 3/4 to get 5/4 while others decomposed the 3/4 to 1/2 + 1/4, added 1/2 + 1/2=1 and added the 1/4 to get 1 1/4. It was to the effect of,”I know a 1/4 is half of 1/2 so the answer would be a 1/4 less than 1.” Some said they “just knew it because they could picture it in their head” I asked if anyone used what they knew about the first problem to help them with the second problem? Hands went right up and I got an answer that I wish I was recording. Majority reasoned that 1/2 was the same as 2/4 and added that to 1/4 to get 3/4. Thumbs went up and they laughed with a lot of “this is too easy”s going around.
I thought I would try a Number Talk the following day to see…. Don’t get me wrong, I loves his use of equivalency and I am a fan of improper fractions, however I started wondering to myself if it would have been more efficient (or show that he actually thought about the fractions themselves) if he used a fact he may have known such as 3/4 + 3/4=1 1/2 to then add an 1/8 on to get 1 5/8? Or used 3/4 + 1 = 1 3/4 and then took away an 1/8? Is that the flexibility I want them using with fractions like I do with whole numbers? He used 6/8 as an equivalent of 3/4, added that to 7/8 and ended with an answer of 13/8. For example, a student was adding 3/4 + 7/8. To build efficiency, I don’t want them to the treat the final problem in the progression as a “brand new” problem in order to reason about an answer.Īlong these lines of thinking, as I observed students working the other day, I realized that students weren’t using this same use of known/derived facts when working with fractions.
This progression leads them to use a known or derived fact (18 x 20) in order to solve 18 x 19. I don’t want them treating every problem as if they have to “start from scratch” adding all or adding on. In the younger grades, I would like to see students using the double known fact of 7+7=14 to know 7+8=15. I feel like the problems I have been using are crafted to use the answers from previous problems to reason about the ending problem. Through doing Number Talks with students K-5, I started to realize that one thing I look for students to use in our whole number computation discussions is using known or derived facts to come to a solution.
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