<<Here>> if you would like to take a look in more detail.
I also promised in that post that students WOULD eventually get to the traditional algorithm. You'll also remember that I said that the traditional algorithm was a 4th grade standard so we are still eeking out as much place value understanding as possible as taught through addition.
If you want to skip to the addition strategy jump below to the large letters that say "STRATEGIES START HERE". Otherwise, keep reading to see how we have bridged the gap between these basic skills and larger sums.
Our second graders have been working towards this goal over the past few months. They have done extensive work in the world of place value and also in mental addition strategies. I am condensing here <in a big way> but one aspect of place value work they have studied is grouping and renaming units. 25 is 25 ones but is also 2 tens 5 ones. 345 is 345 ones, 34 tens and 5 ones or 3 hundreds, 4 tens and 5 ones. Etc.
Additionally, they are thinking of addition and subtraction in terms of place value. For example, if you have 46 and want to add 10 more, a student well grounded in place value knows that 46 is also 4 tens and 6 ones. "+10" is really just adding another ten. The result is 5 tens and 6 ones or 56.
|Here is a look at some of the place value work we have been doing.|
You would likely solve the problem by thinking something like "59 + 10 is 69. Two more is 71." Or, 50 + 10 = 60. 9 + 2 = 11. 60 + 11 is 71." Students have learned ways to note this mental strategies and, as they have, their mental math has really grown.
Strategies Start Here
*If you are REALLY in a hurry, scroll down to the bold conversation below*
I walked into a 2nd grade class last week ready to show them a new way to organize 2-digit numbers vertically **read: leading to the standard algorithm*. Students would be using place value disks to show addition. If you aren't familiar with place value disks, they are foam disks about the size of a quarter that say "1" "10" "100" etc. They are a non-proportional model that students who already understand the magnitude of 1s, 10s, and 100s can use to organize their learning when working with place value.
To add 59 + 12, students would build 59 with 5 tens disks and 9 ones disks horizontally. Below this number they would build 12 by showing 1 ten and 2 ones. Students would then look at the ones and determine how many there were:
T: What is another way to say 11.
S: 1 ten 1 ones.
T: How could we regroup our ones to show 1 ten 1 ones.
S: Trade 10 ones in for a 10 disk
Easy as pie.
So we did. We traded in ones for tens. We found the total number of tens and ones and were pleased to find the answer.
This was the first day of this new instruction on NON-mental addition strategies. We weren't doing any writing or algorithm to accompany the number disks. And something funny happened.
T: Can we build 23 + 72?
S: Do I have to? It's 95. I did it in my head.
T: Alright, I'll give you a harder one. How about 34 + 57.
S: I can do that in my head too!
... thank you place value and mental strategies... No worries, they built them all "because when numbers get larger you will be glad you learned to organize your work!"
So the next day, students were tasked with writing numbers that match their number disks. *Read: The traditional algorithm". Their work in place value again supported this work. Here's what a problem might sound like:
T: Let's build 75 + 18.
S: [Build 7 tens and 5 ones horizontally. Below, they build 1 ten and 8 ones.]
T: How may ones are there in the two numbers together?
S: 13 ones.
T: What is another way to say 13?
S: 1 ten 3 ones.
T: Can you regroup your disks to show 1 ten and 3 ones?
S: [Students trade in 10 ones disks for 1 ten disk]
T: On our paper we can show 13 ones as "1 ten and 3 ones". Let's put the 1 ten in the tens column above [or below, on the line if you are doing "New groups below"] the 7 and the 1. We'll put the 3 ones below to show the total.
T: How many tens do we have in all now?
S: 7 tens plus 1 ten is 8 tens. Add in the new 10 and we have 9 tens.
T: What is the total?
By thinking explicitly about teen numbers as tens and ones, students give meaning to the traditional "carry the 1". Fewer students are making errors like carrying the 1 where no new 10 exists, not carrying a ten when there ARE more than 10 ones and other errors that demonstrate a lack of understanding.
Even better? When a student does make an error, you know that there is a breakdown in place value and addition understanding rather than an error in a procedure. When the algorithm is built on place value and understanding the results will be more consistently successful!