1. Place Value Powers of 10. Can help us represent decimals as fractions: 0.2, 0.45, 0.20, 4.6, etc.

Decimals Most decimal numbers are rational numbers: but some are not. A decimal is a rational number if it can be written as a fraction. So, those are decimals that either terminate (end) or repeat. Repeating decimals: 7.6666…; 0.727272… Terminating decimals: 4.8; 9.00001; 0.75

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

When decimals are 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. Write them on a number line. Line up the place values.

Rounding 3.784: round this to the nearest hundredth. Well, 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 Easiest to see with the area model. 2.1 • 1.3 1 + 1 + .1 1 + .3

3. When decimals are 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-how could we use them here?

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

5. Rounding 3.784: round this to the nearest hundredth. Well, 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

6. 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

7. Multiplying Decimals Easiest to see with the area model. 2.1 • 1.3 1 + 1 + .1 1 + .3

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

5. Rounding 3.784: round this to the nearest hundredth. Well, 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

6. 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

7. Multiplying Decimals Easiest to see with the area model. 2.1 • 1.3 1 + 1 + .1 1 + .3