Decimals and place value Decimals as rational numbers

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. 565565555655556… is not rational because although there is a pattern, it does not repeat. It is an irrational number. • Compare this to 3. 556556556… 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. 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 • As with whole numbers and fractions, multiplication of decimals is best illustrated with the area model. 1 • 2. 1 • 1. 3 1 +. 3 + 1 +. 1

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