A clear understanding of place value builds the foundations of decimal system familiarity. However, many children have trouble learning how to shift a multiple of ten from one column to the next. Discovering that the “ten” part of “eleventy-one” actually shifts to represent one unit in the hundreds column represents a big leap in the child’s mathematical understanding.
How to help them make that leap?
To begin, the child needs to learn that each separate digit in a number actually represents a multiple. Manipulatives are particularly effective in helping to teach this lesson. Set out one hundred and ten popsicle sticks. Have the child bundle together ten groups of ten popsicle sticks with elastic bands, emphasising that each bundle of ten popsicle sticks represents ten loose popsicle sticks. You may need to let the child bundle and unbundle the ten popsicle sticks several times before the association is solid. Leave one group of ten popsicle sticks loose.
Once the child understands that a bound group of ten popsicle sticks represents the same thing as ten loose popsicle sticks, it is time to start working with columns.
You can draw the next example on the blackboard for the class as a whole. Have the children each copy your columns onto their own pieces of paper.
Draw two columns: with the right-hand column representing the ones and the left-hand column representing the tens. Write “Ones” over the ones column, and “Tens” over the tens column. Now write a two-digit number in the columns. Emphasise to the child that only one digit can go into each column: the furthest right digit always goes in the furthest right column, and the digit next to it goes in the column next to it. A column can NEVER contain more than one digit.
Explain that the column furthest to the right is the ones column: this digit represents the number of SINGLE popsicle sticks. Then explain that the column on the left is the tens column: this digit represents the number of bundles of TEN popsicle sticks. Bundles only ever go over the tens column, loose popsicle sticks only ever go over the ones column. Have the child count out the number of popsicle sticks and popsicle stick bundles the number represents, and place them above each column. For example, the number 35 would have three popsicle stick bundles above the tens column, and five popsicle sticks above the ones column. Repeat this part of the lesson until the child is always counting out the correct number of bundles above the tens column and single popsicle sticks above the ones column. End this lesson by writing the number ten, or 1-0, into the two columns. Make sure the child understands it is okay not to put any popsicle sticks into the ones column.
Now comes a tricky step but a major one. Take the time to ensure the lesson is absolutely solid.
Take the children through the numbers one through nine. Ask what happens when we reach ten. You will probably have to remind them more than once that only one digit can ever “fit” inside a column. Even though the child does have ten loose popsicle sticks, the two digits of “10” can’t fit into one column. You might also have to remind the child that one bundle of bound popsicle sticks represents the same thing as ten single popsicle sticks. At the same time, you will probably also have to remind the child that only bundles go over the tens column. Once the child understands that all ten sticks are replaced by one bundle which then moves over to the next column, repeat the lesson by counting up to twenty. Once this part is solid, reverse the lesson by counting back down to ten and then to one: here the child has to grasp the idea that a bundle of ten popsicle sticks can be unbound and one taken away, to move back into the ones column as nine loose popsicle sticks. End this lesson when the child solidly grasps the idea that ten loose popsicle sticks in the ones column always become one bundle in the tens column instead; and a bundle from which a popsicle stick has to be taken always becomes a bunch of loose popsicle sticks and moves over to the ones column.
Once the child is absolutely secure in counting up and counting down, replacing loose popsicle sticks with bundles of ten in the tens column and then replacing a bundle of ten with loose popsicle sticks in the ones column, it is time to venture the next step: simple addition and simple subtraction.
To begin, always use numbers for addition where the individual digit totals are less than ten, and numbers for subtraction where both digits of one number are larger than both digits of the other number.
Start by having the child write the digits of the first number in the appropriate columns and represent that number with popsicle sticks and bundles. Then have the child write the number to be added below it: and do the same thing with the popsicle loose sticks and bundles. Then have the child count up the number of bundles (tens) and loose sticks (ones), and write the result below the two numbers: this third number is what you get when you add the first two numbers together. It may be useful to the child to count up to the third number and realise that its popsicle stick breakdown, counting up, is the same as its popsicle stick breakdown when adding the first and second numbers together. Much later on you will want to show that the first number plus the second number is always the same as the second number plus the first number, but this is a concept that can wait for now.
Similarly, for subtraction, have the child write the digits of the first, larger number in the appropriate columns and represent that number with popsicle sticks and bundles. Then have the child write the number to be subtracted below it. This time, the child takes away a number of bundles and loose popsicle sticks representing the second number. Then have the child count up the remaining number of bundles (tens) and loose sticks (ones), and write the result below the two numbers: this third number is what you get when you subtract the second number from the first number. Again, it may be useful to count backwards, popsicle stick by popsicle stick, to show that you get the same number the long way as with the column “shortcut”.
Make sure the child is absolutely secure in these concepts before introducing addition and subtraction which crosses columns (carrying and borrowing).
In adding, the key is that every single time ten or more is reached in one column, the ten loose sticks are replaced by one bundle, which is then moved over to the tens column. Let the child know that this is called “carrying”, because you “carry” a new bundle from one column to the next one over, so that you never end up with more than one digit in a column. A very few children might independently make the conceptual leap into the hundreds column at this point. Let them.
For subtracting, the concept is a little more difficult. How do you take away popsicle sticks when there aren’t enough popsicle sticks in that column to take away from? Here, again, we return to the lesson that one bundle can always temporarily be replaced by ten loose sticks, so long as ten or more loose sticks don’t end up in the column. Treat it at first as a special case: you can always borrow one bundle of sticks from the next column over if you have to, so long as you turn it into loose sticks in the loose stick (ones) column, and so long as the number of loose sticks which remain in the column after you have taken those for the second number away are nine or fewer (ONE digit). For simplicity, you may wish to use an example such as 20 minus 1: which the child will already have encountered from the forward and backward counting earlier. You may run into some interesting experimentation by the child here. Some children wonder what would happen if you borrow more than one bundle. Let them try, and discover for themselves that after you change over all the extra tens back into bundles, you end up with exactly the same number.
For children who enjoy storytelling aspects to learning, stories can be drawn around the different columns to animate what the numbers are doing. For example, one number may be friends with another number and want to visit; or maybe nine numbers is the most that will fit into one room before you need to build a new room next to it.
Above all, have patience! These steps may not come quickly, but once they are learned thoroughly, they will stick firmly: and that thorough learning will provide a solid foundation for the child’s mathematical abilities through the rest of their lives.