Concept #6: Placing Decimals on the Number Line Using the Expanded Form
If I can continue the idea that my students need a common vehicle to move them to a better understanding of numbers, decimals will assist in making the connection between location and place value. We can represent each place value as a multiple of a unit fraction. Each successive place is a fraction whose denominator is a power of 10 where decimals “fill in” the number line between the integers.
2Three main stages that allow for a more detailed placement procedure will really help my students. At stage one, consecutive whole numbers are place in both positive and negative directions. During stage two, we have decimals in the tenths place and no smaller place. Each whole can broken into 10 smaller interval, each having a length of 1/10 of the unit. Notice, this graphic representation was already observed above and students will make the connections between this particular unit fraction set-up on the number line and place value in base 10. The third stage deals with the hundredths place where each tenths place is further broken into ten equally space partitions. Each interval length will now be 1/100 of a unit long. Going through each stage of placing this decimal expansion, the more precise the placement gets. Zooming into a more specific place on the number line will allow students to clearly see that there are space limitations, however we can measure even more precisely the closer we zoom in. We can repeatedly break each resulting unit into ten equally sized pieces seemingly forever. The point needs to be made here that we don’t need very many place values to get a location of a decimal on a number line that is accurate enough for all practical purposes.
In order to locate decimal expansions on the number line we can use a breakdown of a fractional expanded form of 4.375. 4.375 = 4+3/10+7/100+5/1000. There are a series of number lines that are needed here to really bring home this point. In Figure 20, students will see that with each decimal place the number line is zoomed in.
The progression of Figure 20 starts at 4.3, and then zooms in to the next decimal place in the next number line. Each successive decimal place is a new number line. Students will have multiple problems starting with the tenths place. They will be asked to place each number. Next, students will receive a double number line for the hundredths place value. They will then mimic the first two number lines in Figure 20. This will continue up through the thousandths place and stop there. It is important for students to see that after a few decimals places, we can be very accurate in our placement.
Figure 20
Note that the scale, i.e., the size of the unit interval, has been multiplied by 10 in passing from each of these lines to the next. Students are expected to transform rational numbers into their decimal expansions. Placement here will help students make connections between the different forms of these numbers and see that they are indeed the same thing. The process of getting from a fraction to a decimal is computationally a good thing to do. Students will practice a series of important math facts throughout this process as well. Long division is a skill where my students will be able to apply simple rules of mathematics.
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