1. Place Value

1
1. Place Value

• Powers of 10.
• Can help us represent decimals as fractions 0.2,
  0.45, 0.20, 4.6, etc.

2
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

3

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

4
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.

5
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.

6
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.

7
Adding and Subtracting Decimals

• Same idea as with fractions the denominator
  (place values) must be common.
• So, 3.46 2.09 is really like3 2 ones 4
  0 tenths 6 9 hundredths 5.55

8
Multiplying Decimals

• Easiest to see with the area model.
• 2.1 1.3

9
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?

10
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.

11
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.

12
6. Adding and Subtracting Decimals

• Same idea as with fractions the denominator
  (place values) must be common.
• So, 3.46 2.09 is really like3 2 ones 4
  0 tenths 6 9 hundredths 5.55

13
7. Multiplying Decimals

• Easiest to see with the area model.
• 2.1 1.3

14
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.

15
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.

16
6. Adding and Subtracting Decimals

• Same idea as with fractions the denominator
  (place values) must be common.
• So, 3.46 2.09 is really like3 2 ones 4
  0 tenths 6 9 hundredths 5.55

17
7. Multiplying Decimals

• Easiest to see with the area model.
• 2.1 1.3