For Teachers’ Eyes Only

Throughout the Common Core Georgia Performance Standards for Mathematics in K-5 you read the phrase “based on place value, properties of operations, and/or the relationship between addition and subtraction; as well as multiplication and division”. What does this mean?

Let me provide a context to which we can all relate. Most all of us were taught “THE” properties in one to two days each year we were in school. There was one page in the book that described each property – commutative, associative, distributive, identity – and provided one to two examples of each. Then on the next page in the book we were given numerous examples and then we had to record what property was being presented in each example. Not only was it boring, but rarely did we see a purpose.

So when I taught 6th grade I decided to make it more ‘exciting’. I typed up a list of examples, cut them apart, handed one out to each student then had them walk to the designated part of the room for that property. Yes, students may have enjoyed it a bit more, but they did not really make any connections between what they were ‘matching’ and the other problems they did in mathematics the other 178 days of the school year.

I recently taught a workshop on how Number Talks help students develop understanding of these properties. I stumbled a bit on the first time of presenting this idea, but improved with practice and additional thought.

Doubles/Near Doubles    999+999

I honestly do not know what 999+999 is without thinking about it. But I can determine it rather easily. 999 is really close to 1000 and I automatically know that 1000 + 1000 is 2000. So now I know that 999 + 999 is 1,998. Wait a minute, how did that happen?

999 + 999 = 999 + (1 – 1) + 999 + (1 – 1)  Behind the scenes I added a ‘zero’ to each 999         without changing the value of the problem. This can be done because zero is the Identity Element of Addition. Zero does not change the identity of any number that it is added to.

999 + 999 = (999 + 1) – 1 + (999 + 1) – 1 This is where I said, “I know that 1000 and 1000 is 2000.” But what went on behind the scenes? After I added the 0 to each 999 in the form of (1-1) I then moved my parenthesis to add in a different order, using the associative property of addition.

999 + 999 = 1000 -1 + 1000 – 1 = 1000 + 1000 – (1 + 1) There are two properties that I used to write this equation string. I swapped the first -1 with the second 1000 using what is called the commutative property of addition. Then I should have been left with -1 -1. Where did            – (1 + 1) come from? I could have thought in my head “2000 minus 1 is 1999 and then minus 1 again is 1998.” But I actually thought in my head “2000 minus 2 is 1998”. I was using the distributive property of multiplication over addition. If you can remember back to algebra the negative is to be distributed to each term within the parenthesis – ( 1 + 1 ) = -1 -1.

In this one simple problem I used 4 mathematical properties ‘behind the scenes’. Why do I keep putting it like that? Teachers need to know what is ‘allowed’ and ‘not allowed’ so that they can assess on the spot whether a student is using a mathematically valid strategy. Yet, the students do not necessarily need to know the names of these properties depending on their level of understanding.

So when a student talks through the problem like I did originally, they have in fact used the CCGPS for their grade level that says students should use strategies based on place value, properties of operations, and/or the relationship between addition and subtraction; multiplication and division. I have discovered that the difficulty associated with these particular standards is that teachers may not always recognize unspoken properties. I hope that this series of conversations will be helpful.

I stated that I do not believe that students need to know the names of these properties; however, when a student changes a problem by switching the placement of two numbers the teacher should ask why they did that and does it change the value of the problem, why or why not? After this ‘switching of numbers’ becomes so commonplace in the classroom then the teacher can introduce the official name of commutative property of addition (or multiplication). I know that tacking on ‘of addition’ or ‘of multiplication’ is time consuming but if you consistently use the whole name of the property once it’s introduced at some point one of your students is going to ask, “Is there a commutative property of subtraction?” What a beautiful question! Respond with, “That’s an excellent question. Talk with your neighbor about this question and see if you can determine if there is such a property and be able to explain why and why not?”

I will continue to discuss some of the Number Talk properties in following posts. But in the meantime, you have homework.

Solve the following problems mentally and record your thinking. Then go back and look at your recorded thinking and see if you can determine how you used strategies based on place value, properties of operations, and/or the relationship between addition and subtraction; multiplication and division.

a. 58 + 36     b. 158 + 221    

I’ll assign some subtraction problems next time.

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