Fractions to Decimals with Base Ten Blocks

Math 131

Spring 04

Place Value and Fractions and Decimals

Recall the numbers used in Egyptian Hieroglyphics. Like our familiar Hindu-Arabic number system, the hieroglyphic system is a base ten system, i.e. the general grouping scheme follows powers of ten. In hieroglyphics, there are different symbols that represent 1, 10, 100, 1000, etc., and the position of the symbols didn’t matter. In our system, which is a is a place value system, we only use ten different symbols (0, 1, 2, 3, etc.), but the order of the symbols is critical.

The numbers 326 and 623 are different. The value of each digit is determined by its place: in the number 326, the 3 represents three hundreds because it is in the hundreds place. In 623 the 3 represents three ones because it is in the ones place. Other cultures have used place value systems that are not base-ten systems; for example, the Mayans used a base twenty system (which had portions that could be seen as base five) and the Babylonians used a base sixty system. It is from the Babylonian base sixty that we have sixty minutes in an hour, sixty seconds in a minute, and three hundred sixty degrees in a circle.

Sometimes it is helpful to draw a chart to show the place values:

103 102 101

1000 100 10 1

Thousands Hundreds Tens Ones

3 2 6

6 2 3

Note that in this table, as we move from right to left, the value of each column is multiplied by ten. As we move from left to right, the value of each column is divided by 10. We can extend this table to the left by multiplying by 10 and to the right by dividing by 10. Thus we can represent decimals as follows:

103 102 101 1/10 1/100 1/1000

1000 100 10 1 .1 .01 .001

Thousands Hundreds Tens Ones Tenths Hundredths Thousandths

3 2 6

6 2 3

Here we have two numbers in the table. The first is 3.26, which is three ones, two tenths, and six hundredths and can be viewed in all of the following ways:

3.26 = 3 + .2 + .06 = 3 + 2/10 + 6/100 = 3 + 26/100 = 3 26/100.

The second number, 62.3, can be viewed in the following ways:

62.3 = 60+2+.3 = 62 + 3/10 = 62 3/10.

Activity: Fractions to Decimals

In this activity we will use base ten blocks to see visually and kinesthetically how to convert fractions to decimals. You can use physical base ten blocks or virtual ones at http://matti.usu.edu/nlvm/nav/category_g_2_t_1.html (choose base blocks decimals). With the virtual blocks, set Decimals at 3 and click on create problem. Ignore the problems at the left as you do the activity. Note that moving a block to another column breaks it up.

For this activity, we will let the large cube represent one. What do the other pieces represent?

Large Cube ___1____ Flat: As a fraction _________ As a decimal _____________

Rod: As a fraction _________ As a decimal _____________

Small Cube: As a fraction _________ As a decimal _____________

Now in this activity we will start with the fraction, 1/2. Recall that a fraction can be viewed as a division problem: 1/2= . For this activity, we will use a sharing model of division. We are going to start with one large cube and share it between two different piles. We cannot physically split the cube, so we first must replace it with ten flats. Now we can deal these flats like cards into two equal piles. Each pile contains 5 flats. Since each flat represents .1, each pile represents .5, and we have shown that 1/2 = .5. Now also notice that if we were to replace each flat with the equivalent number of little cubes, we would need 500, and each little cube represents 1/1000, so 1/2=500/1000 =.500. In the table we will use three decimal places to be consistent.

Now we will move to 1/3, which is , so we need to separate the big cube into three equal piles. To start this, we first replace the big cube with ten flats and "deal" flats into three piles. We can only deal nine of the flats equally into the piles. Now we must change the remaining flat into rods and deal them into the three piles. Continue, changing the extra rod into little cubes. You should be left with 3 flats, 3 rods, and 3 little cubes in each pile and one little cube left over.

Each pile represents .333, but what about the little cube left over? It shows that we are not done, that .333 is only an approximation for 1/3; it is not exactly 1/3. Now we need to decide whether .333 is the best approximation of 1/3 that can be made with flats, rods, and little cubes. If we did deal out the leftover cube, it would only be dealt to one pile, and we'd have two piles that represented .333 and one pile that represented .334. Since more of the piles would represent .333, this value is a better approximation, but we know that .333 is in fact smaller than 1/3, so we are rounding down.

Also notice that if we changed the rods and flats into little cubes, each of our (rounded) piles would contain 333 little cubes. Since each little cube represents 1/1000, our approximation of 1/3 is equivalent to 333/1000. We are looking at these different ways to represent and approximate a number to help you continue to develop your number sense.

Now continue with the fractions 1/4, 1/5, etc. all the way up to 1/12. Fill in the table on the next page.

Fraction # of Little Decimal to Fraction w/ Exact or Approx?

Cubes 3 Places Denom. 1000 Rounded up or down?

1/2 500 .500=.5 500/1000 exact

1/3 333 .333 333/1000 approx -- rounded down

1/4

1/5

1/6

1/7

1/8

1/9

1/10

1/11

1/12

Now if you have time, you can explore fractions with numerator 2. Let's start with 2/3, which is equivalent to . Here we start with 2 cubes and deal them into 3 equal piles. Can you see why this method will produce a pile with twice as many cubes as there were in each pile for 1/3? How does rounding affect your final answer? Try 2/4, 2/5, 2/6, etc.

To Commentary Doc File PDF File Home