Place Value and Algebraic Thinking

Surprisingly, an important foundation to algebraic thinking is the ability to simply read a number correctly. Teachers often blow off reading numbers, but when mathematical expressions are correctly read, the math speaks. It almost screams, “This is how you solve me!” Even before students encounter mathematical expressions,simply reading a number can clarify the difference between mere digits, and the number itself. For example, 392.67 as digits is “three nine two point six seven”, and as a number, “three hundred ninety-two and sixty-seven hundredths.” The worst habit is mixing digits and numbers, or putting "and" in the wrong place. The worst way to read the example is "three hundred and ninety-two point six seven," yet math teachers model the poor reading of numbers everyday. Because the placement of the word “and” marks the boundary between the wholes and the part of a whole, students need to develop facility with the use of “and”.

Another important by-product of the chocolate factory activity is a keener sense of rounding. They learn that rounding a number is a matter of distinguishing place value. After work with cases, boxes, and leftovers (loose ones), children can readily accept the renaming of the place value columns as “10 x 10,” “10,” and “1,” or even, for example, “7 x 7,” “7,” and “1” depending on the base. Then it is an easy precursor to exponents to rename the columns as 10^2, 10^1, 10^0 or 7^2, 7^1, 7^0 (or whatever, depending on the base).

Students often practice writing in expanded notation without ever grasping the 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 this could 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 10^1) + (3 x 10^0). If x = 7, then the expanded notation is (5 x 7^2) + (6 x 7^1) + (3 x 7^0). Rewriting numbers as polynomial expressions often makes calculations in different bases much easier, and the Chocolate Factory activity enhances such algebraic understanding.

The Chocolate Factory: Place Value in Algebraic Thinking

Even good math students may begin studying algebra with deficiencies in their understanding of place value. The following activity offers middle school students concrete, hands-on experience with the concept of place value and an opportunity to express the concept algebraically. In the process, students clarify their understanding of the difference between numeral and number while enhancing their number sense. Hopefully students will be able to replace the blind procedure of borrowing and carrying with genuine understanding.

Although students may easily name the place-value column for any particular digit in a number, they often do not understand the significance of the names and they cannot explain mathematically why regrouping works. Students need help in formulating a mathematical explanation to replace the non-math explanation they usually hold. In an activity called “The Chocolate Factory”, students pack chocolates and then tally the number of boxes and cases.

Practically speaking, I usually use beans instead of chocolate. I tell students they are packaging chocolates as the chocolates come down an assembly line. Each piece of chocolate is a “one” or a unit. Students pack the pieces into boxes of ten pieces each, then pack the boxes into cases of ten boxes each, keeping a tally in a table:

I give students opportunities to gain experience adding and subtracting “Cases,” “Boxes” and Loose Chocolates.” Another advantage to using the column names as shown in the table is that the activity can be recycled to teach any base. I have found it is more helpful to rename the “ones” place “loose chocolates”. Then it is easy to explain that there are loose chocolates when there are not enough chocolates to fill a box. There will never be 10 “loose chocolates”, because 10 will fill a box, thereby adding 1 to the tally in the “Boxes” column.

If the above table represents base 10, converting the above tally to numerals yields 562. 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. Often at this point the light bulbs go on, and students see the concept of carrying for the first time. We continue packing chocolates and "carrying" for a while. Then we "eating" the chocolates, metaphorically speaking. Perhaps starting from the above tally, I ask them to take out 8 pieces of chocolate. They 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 loose chocolates, resulting in a total of 12 tally marks in the “loose chocolates” column. This activity is very similar to other trading activities used to teach place value, but seems to be more effective at building the concept of place value because we avoid giving the columns numerical names at the outset.

Simultaneously, we keep a record of each subtraction in the standard algorithm. Students will often understand regrouping for the first time as they compare the physical packing of the chocolates with the ongoing mathematical representation. Three things are going on at the same time, the packing activity, the data record in terms of a tally and the data record in terms of the standard algorithm. We expand and repeat the activity with other groupings which I have carefully planned in advance. For example, if we repeat with groupings of 7, then 7 chocolates make a box, and 7 boxes make a case. I give the students 289 chocolates, knowing full well they will again end up with 5 cases, 6 boxes, and 2 "loose ones". Sometimes I like to use the word "leftovers" for"loose ones."

Students instantly want to know how they got the same apparent number, 562. In the ensuing class discussion, we talk about why the first 562 (10 to a box) has more chocolate pieces than the second 562 (7 to a box). We discover that the reason we line up columns for addition and subtraction is not merely for neatness sake, but because the grouping determines the mathematical meaning of the columns. Students find they can work just as readily in other bases as long as they remember the basis of the grouping (pun intended). I prefer to name the columns from right to left in base ten as: loose, 10, 100, 1000 etc. If, we are working in, for example, base 7, the column names are "loose", 7, 49, 343, etc. Eventually, we can work in base y with column names y^0, y^1, y^2, y^3, etc. As the Chocolate Factory activity illustrates, the “ones” are “ones” only because there are not enough of them to make the next grouping level. They are the ungrouped loose ones, whether in base ten or any other base.

The Vital Place of Place Value

Perhaps one of the most important foundational concepts in mathematics is place value. As the Massachusetts Department of Education rightly observes, “The subtly powerful invention known as place value enables all (my emphasis) of modern mathematics, science, and engineering. A thorough understanding removes the mystery from computational algorithms, decimals, estimation, scientific notation, and—later—polynomials” (Massachusetts Department of Education (2007). In fact, it is when students first meet polynomials in algebra, that the lack of a proper grounding in place value becomes painfully apparent. Most likely a significant number of the difficulties that students experience with math may be traced to place value.

I reviewed the state standards of various states with regard to place value. I looked for an explicit reference to “regrouping,” the current term for what we used to call “borrowing” and “carrying.” My survey of state standards resulted in a mixed bag. Some states require students to do little more than name the place value of a particular digit. Other states expect students to use various means to model place value. Alaska asks students to not only perform the operations of addition and subtraction, but to explain those operations.

State standards have their utility, but apparently whatever the specific state standard, students are able to follow the regrouping recipe without having any real understanding of why the recipe works. In fact, adults of all ages add and subtract by mindlessly following the recipe. Most adults, and of course, all children could do with a solid grounding in place value.

I have a number of activities I use to make place value explicit. Tomorrow I will tell you about an activity I like to call “The Chocolate Factory.”