Number and Operations · 7 min read
Decimal Place Value
Section Overview
This section covers the Number and Operations skills in the Grade 5 TEKS (5.2 and 5.3). These standards make up STAAR Reporting Category 1 (Numerical Representations and Relationships) and Reporting Category 2 (Computations and Algebraic Relationships). You will read, write, compare, and round decimals to the thousandths; multiply and divide whole numbers; add, subtract, multiply, and divide decimals; add and subtract fractions and mixed numbers; and multiply and divide with fractions using models. STAAR is taken without a calculator, so each topic shows how to estimate first and check your work.
The Place-Value Chart
Every digit in a number has a place, and the place tells the digit's value. The decimal point separates the whole-number places on the left from the decimal places on the right. In Grade 5 you work with decimals all the way to the thousandths place.
Read the places to the right of the decimal point carefully. They end in -ths: tenths, hundredths, thousandths. Each one is a fraction of 1 whole.
The Value of a Digit
The digit is the symbol (0 to 9). The value is how much that digit is worth in its place. Moving one place to the left makes a digit worth 10 times as much. Moving one place to the right makes it worth 1/10 as much.
| Number | Place of the 5 | Value of the 5 |
|---|---|---|
| 52.13 | tens | 50 |
| 25.13 | ones | 5 |
| 2.513 | tenths | 0.5 |
| 2.153 | hundredths | 0.05 |
| 2.135 | thousandths | 0.005 |
Look at 0.5 and 0.05. The 5 in 0.5 is worth 10 times the 5 in 0.05, because tenths are 10 times as large as hundredths. STAAR questions often ask you to compare the values of the same digit in two places this way.
Standard, Word, and Expanded Notation
A decimal can be written in three ways. Standard form uses digits: 6.284. Word form uses words: six and two hundred eighty-four thousandths. Say and for the decimal point. Expanded notation shows the value of each digit as a sum.
| Form | Example |
|---|---|
| Standard form | 6.284 |
| Word form | six and two hundred eighty-four thousandths |
| Expanded (decimals) | 6 + 0.2 + 0.08 + 0.004 |
| Expanded (multiplication) | (6 × 1) + (2 × 0.1) + (8 × 0.01) + (4 × 0.001) |
| Expanded (fractions) | (6 × 1) + (2 × 1/10) + (8 × 1/100) + (4 × 1/1,000) |
- Step 1: Find the place of each digit. 3 is in the ones, 0 is in the tenths, 7 is in the hundredths, and 2 is in the thousandths.
- Step 2: Write each digit times its place value: (3 × 1) + (0 × 0.1) + (7 × 0.01) + (2 × 0.001).
- Step 3: A zero adds nothing, so you may leave it out: (3 × 1) + (7 × 0.01) + (2 × 0.001).
- Answer: 3 + 0.07 + 0.002, which is the same as (3 × 1) + (7 × 1/100) + (2 × 1/1,000).
Comparing Decimals
To compare two decimals, use the symbols > (greater than), < (less than), and = (equal to). The open side of the symbol always faces the greater number.
- Step 1: Line up the decimal points and write 3.45 as 3.450.
- Step 2: Ones: 3 = 3. Tenths: 4 = 4. Hundredths: 0 < 5.
- Step 3: The first different digit is in the hundredths place, and 0 hundredths is less than 5 hundredths.
- Answer: 3.405 < 3.45, so Marco's ribbon is longer.
Ordering Decimals
Ordering means putting three or more numbers from least to greatest or greatest to least. Use the same line-up-and-fill-zeros method, then compare one place at a time.
- Step 1: Rewrite every time with three decimal places: 12.080, 12.800, 12.085, 12.058.
- Step 2: The ones and tens are the same (12). Compare tenths: Ben has 8 tenths, and the others have 0 tenths, so Ben is the slowest.
- Step 3: Compare hundredths for the rest: Dee has 5, Ana has 8, Cole has 8. Dee is the fastest.
- Step 4: Ana and Cole tie in hundredths. Compare thousandths: Ana 0 < Cole 5.
- Answer: Dee (12.058), Ana (12.08), Cole (12.085), Ben (12.8).
| Runner | Time (seconds) | With zeros added |
|---|---|---|
| Dee | 12.058 | 12.058 |
| Ana | 12.08 | 12.080 |
| Cole | 12.085 | 12.085 |
| Ben | 12.8 | 12.800 |
Rounding Decimals
Rounding gives a number that is close to the exact number and easier to use. First find the two benchmark numbers the decimal is between. Then decide which one it is closer to. Look at the digit one place to the right of the rounding place: if it is 5 or more, round up; if it is less than 5, keep the rounding digit the same.
- Step 1: The rounding place is tenths. The tenths digit is 6, so 8.649 is between 8.6 and 8.7.
- Step 2: Look one place to the right, at the hundredths digit: 4.
- Step 3: 4 is less than 5, so the tenths digit stays 6 and the digits after it are dropped.
- Answer: 8.6 pounds.
Rounding to the nearest hundredth works the same way: 2.955 is between 2.95 and 2.96. The thousandths digit is 5, so it rounds up to 2.96.
- Read the question twice: "value of the digit" is not the same as "the digit."
- Check answer choices that look alike, such as 0.07 and 0.7, by naming the place out loud.
- On a number line, find the halfway mark first, then see which side the number is on.
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