Advanced Addition and Subtraction

Grouping, Addition and Subtraction with Beans, Cups, Bowls and Tin Cans

Now that we have completed the basics of beginning addition and subtraction, we continue advancing the skill in this area with the use of a place value board and counters.

Anything can be used as math manipulatives, although these marshmallows proved to be a bit large to fit ten of them in a cup for the tens side. Mini marshmallows worked better..

I begin by having them put a few counters in the ones and a few cups of ten of the same item in the tens column. It really doesn't matter what you start with. Then I have him subtract one at a time from the ones column until he runs out of ones. He now has to dump the contents of the tens cup on the ones side of the board. This can be played in reverse too, adding one at a time. When ten are reached on the ones column, they are scooped up and put in a cup to be placed in the tens column.

We play this plus-one and minus-one game a few times.

We then play the game again, recording on strips of paper how many items (ones) and how many cups (tens) are in each step of the game. We examine the columns of figures for possible patterns. Can you predict the next number in the series? You can also play the minus-one game and record the patterns. Are the patterns the same for the plus-one and the minus-one games?

At some point bowls can be added for a hundreds column. We run up and down the amounts with these as well, noting patterns. Are they the same? They soon find out that one can avoid lengthy counting by noticing the patterns within the numbers. Understanding place value they begin to understand helps them to know the amounts without having to count every bean.

At this point I need to take some time out to make sure he knows enough about place value that he can accurately identify the larger and smaller of two numbers so that when he make up his own addition and subtraction problems, he can be sure to put the larger number and small number in the correct places.

 So now we play a place value game, which also covers less than/more than symbols to indicate who won.

I favor working with beans, cups and bowls on a place value board as a beginning for all basic math. One day we use it with multi-digit addition with carrying. This day I gave him the problem 28 +54 = 

and so he set out 2 cups and 8 beans on the board.

and then he added to them 5 cups and four beans. He quickly saw that there were more beans than available spots in the beans, or ones column and so he set them to the side.

I had him fill up the ones column...

put them into a cup (or trade them for an already filled cup) and then put the remaining two into the beans/ones column.

We then filled in the answer on a chart. He will do this hands-on regrouping for a time and then when he feels he wants to fill out the chart without using the beans, I will show him the more traditional way of carrying. By this time, it will seem obvious or second-nature to carry, or change the ones into tens as needed.

Next we add a tin-can for the thousands column and we roll the die to make up addition problems. For the first row, we roll for times for the four place values ones-thousands. Next we roll three times for the next row. Now we add (or subtract) them.

We begin recording these problems, too.

Grouping, Addition and Subtraction with Chips
The chips to which they will now work with provide the necessary link between concrete representation of beans and cups and the more abstract numbers representing unseen quantities.

We begin with chip trading in order for him to learn that each chip, as we progress to the left, represents ten times the chip to the right.

We begin by his putting seven chips in the green and blue columns and we add one, two or three to the green column, and once he get ten greens he trades them in for a blue chip.

He also trades ten blue chips, when he gets them, for one red chip. We play this game until he can easily trade back and forth when adding and subtracting one to three chips at a time. It is sort of a banking game.

Once he proficient at this, he can begin adding and subtracting, by the roll of four die for the top amount and three for what is added or subtracted.

When he feels comfortable doing so, he can begin to create his own problems in base ten.
He can also learn that he can verify his addition answers by subtracting and verify his subtraction problems by addition.

"Students must decide when, if ever, they wish to abandon aids to computation." -Mathematics; a Way of Thinking, Bob Baratta-Lorton

source: Mathematics; a Way of Thinking, Bob Baratta-Lorton