Teacher Information: Number Line Bars

This virtual manipulative uses adjustable colorable bars to illustrate arithmetic operations on the number line.

The basic idea is to combine colored bars along the number line to represent particular number operations. To add, say 4 + 7, we can start with bars of length 4 and length 7 (either set the desired lengths in the New Bar window, or drag the arrow-end of any bar to change length), place one to start at the 0-point on the number line, and place the other so that the combined bars reach from 0 to 11, and read the sum. Alternatively, we could just place a 7-bar starting at the 4-point, or place a 4-bar starting at the 7-point, ending up in either case at the 11-point of the number line.

Subtraction is accomplished similarly. Depending on the experience of the student, for the difference 8 - 5, you can use two positive bars and see what the difference is, or you can illustrate that 8 - 5 = 8 + (-5), using a positive bar (pointing to the right) and a negative bar (drag the arrow-end to the left until it changes color and has the desired length). In the first case, place an 8-bar first and use a 5-bar above it, either starting at the 0-point or at the 8-point. It is easy to see then that there is a difference of length 3-and you may want to actually add a 3-bar, placing it so that the 5-bar and the 3-bar together add up to match the 8-bar, illustrating the verbalization of the subtraction process as "8 - 5 is (or names) the number that must be added to 5 to equal 8." Using an 8-bar and a (- 5)-bar, starting at 0, we move with the first bar to 8 and then with the second back to 3.

Multiplication of whole numbers as repeated addition is easily represented here, and the student can easily see that three 7-bars end up at the same point as seven 3-bars, for the product 7*3 = 21. Using the zoom and slider can make it easier to see some products. The same process works just as well with one negative factor (7)(-3) = (3)(-7) = -21.

Likewise, division, either even or with a remainder, say 8/3, is most easily seen as answering the question, "How many copies of 3 are there in 8?" This is obviously equivalent to repeated subtraction for integers, but the idea of seeing how many copies of the divisor can be placed alongside the dividend carries over to the division of fractions while repeated subtraction makes no sense in the context of fractions. With an 8-bar and 3-bars, it is easy to see that the 8-bar can contain two 3-bars, with room left over for a 2-bar, so that the quotient is 2, with a remainder of 2.