Photo Credit Jim Bentley
I’m working on getting chapter 6 finals graded today and this weekend. Initial results look very good! 🙂
Your kiddos are doing some SERIOUS thinking, and I want to share with you the maturity and growth I’m seeing in EACH child is showing SOOOOO clearly in class. We are thinkers…not just doers.
That said, I wanted to share with you an image.
When we start multiplying fractions, we want to understand what it means. If we take 2/9 of 27, we want to take a part of 27 and not the whole amount.
On the far left, you see 9 groups. The blue counters represent nine groups of one. If we equally share 27 amongst the nine groups, we see three counters in each group. I circled the groups to show they were together and arranged horizontally.
If we take 2 of the 9 groups, we see that 2/9 of 27 is 6.
The middle example shows the traditional algorithm of 2/9 x 27. By decomposing 27 into 9 x3, we’re not only reinforcing multiplication facts, we’re also showing that an equivalent form of one, 9/9 in this example, can be removed from the fraction leaving us with 2×3/1 which is 6.
The final example on the far right shows a method that might be familiar to many of you. It’s called “cross cancelling.”
We looked at this method today and tried to mathematically explain what it means to “cancel” or “cross out” a number and replace it with a 1 or a 3. Most students were not sure what was happening and for good reasoning.
While this method can yield the right answer, it does not explain why. It does not give a mathematical justification when looking at the syntax.
During a three year professional development program focusing on math, I learned that teaching “cross cancelling” is not the best practice to get students multiplying fractions. Syntax-wise, it’s confusing to look at. It does nothing to explain where the 27 or the 9 “went to.” They’re both just “gone.”
PLEASE, if you are helping your child at home with math, emphasize thinking and explaining and justifying each step over producing the right answer. You will help your child so much more in the long run if they understand what they do and why they do it.
We have emphasized “decomposing” numbers all year. 27 can be “decomposed into two factors: 9 and 3, and students know factors are numbers that are multiplied to produce a product. PLEASE emphasize with your child how to use the proper academic language and to show their work carefully and neatly.
Why the emphasis on syntax and deep understanding over just producing an answer? I’ll share with you a TED Talk from a high school math teacher to help explain.
Please, if you have questions or comments about any of the ideas in this post send me an email or give me a call or text. I’d be more than happy to discuss why we use multiple methods and emphasize concepts and process or memorizing steps to yield the right answer.