Decimals and place value

Decimals as rational numbers Some decimal numbers are rational numbers: but some are not. A decimal is a rational number if it can be written as a fraction with integer numerator and denominator. Those are decimals that either terminate (end) or have a repeating block of digits. Repeating decimals: 7.6666…; 0.727272… Terminating decimals: 4.8; 9.00001; 0.75

Irrational numbers A number that is not rational is called irrational. A decimal like 3.5655655565555655556… is not rational because although there is a pattern, it does not repeat. It is an irrational number. Compare this to 3.556556556556556556… It is rational because 556 repeats. It is a rational number.

Comparing Decimals When are decimals equal? 3.56 = 3.56000000 But, 3.056 ≠ 3.560. To see why, examine the place values. 3.056 = 3 + 0 • .1 + 5 • .01 + 6 • .001 3.560 = 3 + 5 • .1 + 6 • .01 + 0 • .001 Think of units, rods, flats, and cubes.

Ways to compare decimals Write them as fractions and compare the fractions as we did in the last section. Use base-10 blocks. Use a number line. Line up the place values.

Exploration 5.16 Use the base 10 blocks to represent decimal numbers and justify your answers. Work on this together and turn in on Wednesday.

Homework for Wednesday Read pp. 308-323 in the textbook Exploration 5.16

Rounding 3.784: round this to the nearest hundredth. 3.784 is between 3.78 and 3.79. On the number line, which one is 3.784 closer to? 3.785 is half way in between. 3.78 3.785 3.79

Adding and Subtracting Decimals Same idea as with fractions: the denominator (place values) must be common. So, 3.46 + 2.09 is really like 3 + 2 ones + 4 + 0 tenths + 6 + 9 hundredths = 5.55

Multiplying Decimals 2.1 • 1.3 As with whole numbers and fractions, multiplication of decimals is best illustrated with the area model. 2.1 • 1.3 1 + 1 + .1 1 + .3

Dividing decimals Standard algorithm—why do we do what we do?

Exploration 5.18 Work on this exploration in class and finish for homework. Part 1: 1-4 Part 2: 1, 2