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Wednesday, 17 February 2016

Adding Large Numbers

Sometimes parents want to know how to do this 'new math'  Actually, there is nothing new about what we are doing in our class.  We are adding large numbers.  The difference from when most adults learned how to add big numbers, to now, is that we are asking the students to understand what they are doing.  We do not introduce the algorithm until the students can both demonstrate and explain their understanding.
I'm going to walk you through today's lesson.
Here we are going to add two large numbers: 26 and 32

We are using an open number line  so that we can see the addition of one number to another.

I asked the students which number they would like to put onto the number line first and the child chose 26.  We then 'jumped' 32 more.  It is difficult for us to jump that far and know where we are going to land so we jumped in smaller amounts.  We jumped 10 to land at 36, another 10 to land at 46 and then 10 more to land at 56. (NOTE: 10+10+10 is 30)  We felt that we could jump 2 more easily because we know how to skip count even numbers so we landed at 58.  We proved that 26 add 32 is 58.

We then reminded ourselves that in adding, the order of the addends is not important, so we tried it again, starting with 32 and then jumping 26.  Notice that the girl who was helping suggested that  she could jump by groups of 3 to make up the 6 at the end.

In 'teacher talk', the students are able to decompose and compose numbers.  It is a very important part of understanding numbers, which we call number sense.  It is also important that students are able to make those 'jumps' rather than counting on.  It demonstrates that they are visualizing.  Think of it this way:  I started at 14.  Can I visualize what 10 more is?  How about 10 less?  Could I visualize 2 more than 14?  How about 2 less?  If the student is doing this and especially if they can explain or prove that the number they are sharing would be the correct one, then they are confident in their number sense.

It they are looking up and just counting on...their number sense is still as the beginning stages and they need more practise.  We have been practising since September here at school.  Not all students are confident yet.  It may be time for some home practise.

Here is one student explaining their thinking for 38 add 83.  She said "I put 38 on the number line and then I need to jump 10 8 times to add 80."  Can you see how she was able to decompose 83 into 8 tens and 3 ones?
This boy said "I started at 51 and jumped 2 10's and then 3 small jumps and it ended up at 77"

Actually 51 add 23 is NOT 77, but now I can see where his thinking was wrong.  His three small jumps were of 3, not 1.

You might say: "Why bother?  Just teach them the algorithm.  It worked for me!"

The answer is that we want students to be confident with numbers, to understand their patterns, to be able to compose and decompose them because when these students are working with algebra in 5 short years, those are the things they will be asked to do with unknowns.

We want them to have confidence that they know how to do this with our number system so they their concrete thinking can move them into the abstract thinking of algebra.

It's not so far away!  When these students are in the first month of Grade Three, they will be writing a provincial exam known as the SLA.  This kind of thinking and explaining is part of that exam.  It is the way that number sense is revealed.  In the most important way possible, we are working to ensure that students understand what the algorithm is asking them to do.  We don't want them to say "I just do this" without understanding why they do it.

The other thing that is different from mathematics classes of twenty years ago, is that we offer a number of ways or strategies for students to use.  We suggest that they try them but settle on the one that works for them.  This way I call the pull down method, but some people call it branching.  It doesn't matter, but it might be easier for some students to understand.

The two numbers are written horizontally and then the student identifies the tens in both numbers, pulls them down (by drawing lines) and adds them.  They say "20 add 30 is 50", to demonstrate that those digits are in the tens' place.  They do the same with the ones, saying '6 add 2 is 8' and then finally saying that '50 add 8 is 58."

Here's one girl's explanation:
How I pull down - I thought what's 5+2. Then I thought what's 1+3. That's how I did math.
What will I give her as feedback?  I'll write: Remember that it really is "What's 50+20"  Those digits are in the ten's place.  

Why is that important?  Think about what will happen if the digits are '8 and 9 in the ten's place.  What does the student need to recognize then?

This girl wrote "I know because I remember the old pull down and I did that but changed the numbers"
To most of you that won't make sense, but because I have heard her talk about this, I know that she is trying to explain that this method 'old pull down' is something she knows will work every time, no matter which numbers she uses in the equation.

This girl said "I really like the pull down strategy.  It's fast and easy. I learned it in Grade One."

This is a variation of the pull down method and I introduced it this week because I would like to see if the students are ready to have the algorithm introduced to them.

They are again decomposing the big numbers and adding.  
Here's what it might sound like:
20 add 30 is 50 (and they would write the 5 in the ten's place)
6 add 2 is 8 (and they would write the 8 in the one's place)
26 add 32 is 58!

This is the step before introducing writing the two addends vertically.  They need to recognize the place value of the numbers to ensure that they are line up correctly.

LONG explanation but I hope that helps you all understand what your kiddos are doing in math class.

How would you use these strategies for subtraction?

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