The explanations may certainly be chock full of numbers yet without having the least connection to the way the numbers work. A good example is the standard method for finding 10% of a number: just move the decimal place one digit to the left. The method is nothing but a trick, and our children learn to mistake performance of tricks for understanding of math.
Students need help in constructing mathematical explanations. In an activity I call “The Chocolate Factory,” students pack chocolates in boxes, then in cases, while keeping a tally. At the end of the activity, students will be able to trade and regroup in order to add or subtract.
I usually use beans instead of chocolate because it is less messy and less tempting. I explain that the students are working for Hershey Chocolate Company packing chocolates as chocolate pieces roll down the conveyor belt a la a famous “I Love Lucy” episode. The number of chocolates in each group is simulated by drawing a card from a shuffled deck with no picture cards. A specially made set of number cards with spots but no numerals would be better. Students pack the pieces into boxes of ten pieces each, then pack the boxes into cases of ten boxes each, keeping a running tally in a table on the blackboard.
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Each pair of students shares a set-up: 100 beans, a container capable of holding ten beans to represent boxes, and a larger container to hold ten “boxes.” The teacher explains that the rule of the game is that a “box” can only hold ten beans. Once a box is filled, they begin filling another box, and so on until they have ten boxes. Ten boxes are then packed into a case.
The teacher shuffles the cards and holds the deck face down. The teacher uses any suitable method to select a student to pick a card. The student takes a card from the deck (a five-spot in the example) and shows it to the class. Each pair counts out five beans and puts them in a “box.” The teacher records the five as tally marks in the “leftovers” column. Another student picks a card (an eight spot). The students count out eight beans and the teacher records the tally in the “leftover” column The students use the beans to fill a box, pointing out that they have one full box and three leftovers. The teacher records the result with one tally mark in the “box” column, and three tally marks in the “leftover” column.
It is important to give students experience with “Cases, Boxes and Leftovers” before renaming these columns “100’s, 10’s and 1’s.” Another advantage to using the column names, “cases, boxes, leftovers” is that the activity can be recycled later for teaching any base. I have found it is more helpful to rename the “ones” place “leftovers”. Then it is easy to explain that there are leftovers when the amount is insufficient to fill a box. Thus, there will never be 10 leftovers, because 10 will fill a box, thereby adding 1 to the tally in the “Boxes” column. Converting the final tally in the table to numerals yields 563. Students readily understand that as they accumulate 10 boxes, they transfer those boxes as 1 case and put a tally mark in the “Case” column. It is often at this very point the light bulbs go on, and students see the why carrying works for the first time.
Then we repeat the activity, but the cards now simulate consumed chocolate (yum). A student draws perhaps an eight-spot to stand for eating eight pieces. Students will naturally want to open a box to accomplish this. As they take a box, they erase a tally mark and dump the 10 chocolates (beans) with the leftovers, and record ten more tally marks for a total of 13 tally marks in the leftover column. They continue subtracting in this way. This activity is very similar to most trading activities, but seems to be more effective at building the concept of place value because we avoid giving the columns numerical names at the outset.
With older students we simultaneously keep a record of this computation in the standard algorithm. Again, students often understand regrouping for the first time. We expand and repeat the activity with other groupings which I have carefully planned in advance. I tell students that they have done such a good job that now they work for a more expensive chocolate company, perhaps Ghirardelli, where chocolates are packed in boxes of 5 pieces, and cases of 5 boxes. I give the students 158 chocolates, knowing full well they will again end up with 5 cases, 6 boxes, and 3 leftovers, the same tally as for the Hershey exercise.
In the ensuing class discussion, we talk about why the first 563 (10 to a box) has more chocolate pieces than the second 563 (5 to a box). Students discover that neatly lining up their addition and subtraction columns is not merely for neatness sake, but because the columns have real meaning. Students find they can work just as readily in other bases as long as they remember the basis (pun intended) of the groupings.
If working in base ten, I prefer to name the columns from right to left “leftovers, 10, 100, 1000,” etc. As the Chocolate Factory activity illustrates, the “ones” are “ones” only because there are not enough of them to make a “ten”. They are the ungrouped leftovers, whether in base ten or any other base. In fact, students get very comfortable with working in a variety of bases and discover that for any base (b), the column names will be (from right to left) “(leftovers), (b), (b x b), (b x b x b), and so on. For example, they would name the base 7 columns “(leftovers), (7), (7 x 7), (7 x 7 x 7), and so on.
I like using the parenthesis early on so students become familiar with the parenthesis holding a number just as cupped hands hold an apple, and that the number have different appearances but still be equivalent. In practice I often go beyond leftovers, boxes, and cases, and extend the activity to crates, trucks and warehouses. Just like the the song from School House Rock says, “Don't you worry 'bout the big numbers, they're just bigger, that's all.”
Later the columns can be renamed with exponents, 10^2, 10^1, 10^0 or 7^2, 7^1, 7^0. Then it is a small step to b2, b1, b0, then x2, x1, x0. Students often practice writing in expanded notation without ever grasping real significance of what they are doing. In algebra, many polynomial expressions are really bases in disguise. For example, the base 10 tally and the base 7 tally were both 563. Algebraically both would be expressed as 5x^2 + 6x + 3, where x is the base. The algebraic expression is nothing more than expanded notation. If x is 10, then the expanded notation is (5 x 10^2) + (6 x 10) + (3 x 1). If x = 7, then the expanded notation is (5 x 7^2) + (6 x 7) + (3 x 1). Rewriting numbers as polynomial expressions often makes calculations in different bases much easier, and The Chocolate Factory activity enhances such algebraic understanding.
There is a vendor, Digi-Block, who sells a manipulative that would be ideal for the base 10 chocolate factory. The set has pieces, boxes and cases. Each box holds exactly 10 pieces, and each case holds exactly 10 boxes. Its advantage is that students are prevented from overpacking or underpacking. I have usually had to rely on materials I have scrounged: beans, little cough syrup cups for boxes, and little containers to hold 10 cough syrup cups. If you are looking for basic base ten blocks, Nasco probably has the most complete assortment anywhere.