Numbers and Operations · 11 min read
Place Value and the Base-Ten System
The test does not only ask you to compute. Many questions show a student's work or a classroom situation and ask what the student understands, which misconception the work reveals, or which representation or example would help most. For that reason, each chapter in this section pairs the mathematics with the errors students (and adults) commonly make and with the way a teacher would address them. Read the Common Trap and Exam Tip paragraphs closely; they reflect the Tasks of Teaching Mathematics angle that runs through the whole test.
This chapter covers outline area I.A, place value. You will review the forms used to write numbers, how multidigit numbers are composed and decomposed, how to find the value of a digit, how exponents describe powers of 10, the ten-times relationship between neighboring places (including decimal places), multiplying and dividing by powers of 10, and rounding.
The Structure of the Base-Ten System
Our number system uses ten digits (0 through 9) and place value. A digit's value depends on its position. Moving one place to the left multiplies the value of a place by 10; moving one place to the right divides it by 10. This single rule explains why the system works for whole numbers and decimals alike.
The decimal point separates the ones place from the tenths place. The places are symmetric around the ones place, not around the decimal point: tens mirror tenths, hundreds mirror hundredths, and thousands mirror thousandths. There is no "oneths" place.
Place-Value Chart
| Place | Value | Power of 10 |
|---|---|---|
| Hundred thousands | 100,000 | 10^5 |
| Ten thousands | 10,000 | 10^4 |
| Thousands | 1,000 | 10^3 |
| Hundreds | 100 | 10^2 |
| Tens | 10 | 10^1 |
| Ones | 1 | 10^0 |
| Tenths | 0.1 = 1/10 | 1/10^1 |
| Hundredths | 0.01 = 1/100 | 1/10^2 |
| Thousandths | 0.001 = 1/1,000 | 1/10^3 |
Memory Aid: Think of the ones place as the mirror. Tens and tenths sit one step away on each side; hundreds and hundredths sit two steps away. The "th" ending always means a place to the right of the decimal point.
Forms of a Number
The test expects you to move fluently among four forms of a number.
Four Forms of the Number 6,048
| Form | How 6,048 Looks |
|---|---|
| Standard | 6,048 |
| Word | six thousand, forty-eight |
| Expanded | 6,000 + 40 + 8 or 6 × 10^3 + 4 × 10^1 + 8 × 10^0 |
| Base-ten (unit) | 6 thousands, 0 hundreds, 4 tens, 8 ones |
Say "and" for the decimal point, then read the decimal part as a whole number followed by the name of the last place: twenty-seven hundredths.
Common Trap: Students often write "four hundred and five" for 405, inserting "and" where there is no decimal point. In formal word form, "and" is reserved for the decimal point. A related error is writing four thousand twenty as 400020, recording each spoken part separately. A teacher addresses this by building the number with base-ten blocks or a place-value chart so students see that 4,020 needs only four places, and the zeros hold empty places.
Composing and Decomposing Multidigit Numbers
A number can be named in many ways using different units. This flexibility underlies regrouping in addition and subtraction.
3,450 = 3 thousands + 4 hundreds + 5 tens 3,450 = 34 hundreds + 5 tens 3,450 = 345 tens
3,450 = 2 thousands + 14 hundreds + 5 tens
The last line shows a nonstandard decomposition: one thousand has been traded for ten hundreds. This is exactly what happens when a student "borrows" in subtraction. Questions may ask which expression is equal to a given number, and the correct choice is often a nonstandard one.
Worked Example: Which of the following is equal to 5,207: 50 hundreds and 27 tens; 5 thousands, 2 tens, and 7 ones; 4 thousands, 12 hundreds, and 7 ones; or 520 tens and 70 ones?
50 hundreds and 27 tens = 5,000 + 270 = 5,270. This does not work; the 2 and 7 were treated as tens.
5 thousands, 2 tens, and 7 ones = 5,000 + 20 + 7 = 5,027. This does not work; the 2 belongs in the hundreds place.
4 thousands, 12 hundreds, and 7 ones = 4,000 + 1,200 + 7 = 5,207. This works; one thousand was traded for ten hundreds.
Converting each choice back to standard form is the most reliable check.
Result: 4 thousands, 12 hundreds, and 7 ones equals 5,207.
Exam Tip: When a question asks how many tens are in a number, it usually means the total number of tens, not the digit in the tens place. In 3,450 the tens digit is 5, but there are 345 tens in all. Read the wording carefully: "the digit in the tens place" and "the number of tens" are different questions.
The Value of a Digit
The value of a digit equals the digit times the value of its place. In 563,412, the 6 is in the ten thousands place, so its value is 6 × 10,000 = 60,000. In 35.06, the 6 is in the hundredths place, so its value is 6 × 0.01 = 0.06.
Because each place is ten times the place to its right, you can compare the values of the same digit in different places by counting places. If one 6 is five places to the left of another 6, its value is 10^5 = 100,000 times as great.
Worked Example: In the number 77,777, how does the value of the 7 in the thousands place compare with the value of the 7 in the tens place?
The 7 in the thousands place has value 7 × 1,000 = 7,000.
The 7 in the tens place has value 7 × 10 = 70.
The Ten-Times Relationship and Decimals
The outline specifically names the idea that a digit in one place represents ten times what it represents in the place to its right and one-tenth of what it represents in the place to its left. This holds across the decimal point. One tenth is ten times one hundredth; one is ten times one tenth.
Base-ten blocks make this concrete. If a flat (100 small cubes) is defined as one whole, then a rod is one tenth and a small cube is one hundredth. Ten small cubes make a rod, just as ten hundredths make a tenth.
Common Trap: The "longer is larger" misconception leads students to say 0.45 is greater than 0.5 because 45 is greater than 5. Students are applying whole-number thinking to decimals. A teacher can address this by having students write both numbers to the same place (0.45 and 0.50), shade both on hundredths grids, or place them on a number line. The opposite error, "shorter is larger," appears when a student thinks 0.3 is greater than 0.31 because tenths are bigger than hundredths. Both errors come from comparing digits without attending to place value.
Powers of 10 and Exponents
An exponent tells how many times the base is used as a factor. For powers of 10, the exponent equals the number of zeros after the 1 in standard form. 10^1 = 10 10^3 = 10 × 10 × 10 = 1,000 10^6 = 1,000,000 10^0 = 1
Common Trap: Students sometimes read 10^3 as 10 × 3 = 30. Emphasize that the exponent counts factors, not a multiplier. Writing out 10 × 10 × 10 a few times prevents this error.
Multiplying and Dividing by Powers of 10
Multiplying by 10 makes every digit move one place to the left, so each digit's value becomes ten times as large. Dividing by 10 moves every digit one place to the right. Multiplying or dividing by 10^n moves the digits n places.
Effect of Powers of 10 on 4.6
| Operation | Result | Digits Shift |
|---|---|---|
| 4.6 × 10 | 46 | 1 place left |
| 4.6 × 10^2 | 460 | 2 places left |
| 4.6 ÷ 10 | 0.46 | 1 place right |
| 4.6 ÷ 10^3 | 0.0046 | 3 places right |
Teachers often describe this as "moving the decimal point." That shortcut gives the right answer, but the more accurate description, and the one the test favors in teaching questions, is that the digits shift to new places while the decimal point stays fixed between the ones and the tenths.
Worked Example: Compute 3.47 × 10^3 and 52.8 ÷ 10^2.
10^3 = 1,000, so each digit of 3.47 moves three places to the left.
The 3 moves from ones to thousands, the 4 from tenths to hundreds, and the 7 from hundredths to tens. Fill the ones place with 0: 3,470.
10^2 = 100, so each digit of 52.8 moves two places to the right.
The 5 moves from tens to tenths, the 2 from ones to hundredths, and the 8 from tenths to thousandths: 0.528.
Result: 3.47 × 10^3 = 3,470 and 52.8 ÷ 10^2 = 0.528.
Common Trap: The rule "to multiply by 10, add a zero" works only for whole numbers. A student who applies it to 3.4 × 10 writes 3.40, which equals 3.4. A teacher can respond by asking the student to estimate first (3.4 × 10 should be about 34) and then explaining the digit shift. This is a classic example of an overgeneralized rule.
Rounding
To round a number, find the target place, look at the digit immediately to its right, and round up if that digit is 5 or greater; otherwise keep the target digit. Then replace the digits to the right with zeros (whole-number places) or drop them (decimal places).
Conceptually, rounding asks which of two benchmark numbers the given number is closer to. A number line makes this visible: 4,682 lies between 4,600 and 4,700 and is closer to 4,700.
Common Trap: Students sometimes round in a chain, rounding 4,649 first to 4,650 and then to 4,700. Rounding 4,649 directly to the nearest hundred gives 4,600, because 4,649 is closer to 4,600. Another frequent error is changing the digits to the left of the target place, or leaving nonzero digits to the right (writing 4,682 rounded to the nearest hundred as 4,782 or 4,700.82). A number line showing the two possible benchmarks is the best corrective tool.
Exam Tip: When the test asks which numbers round to a given value, find the range. Numbers that round to 3,500 when rounded to the nearest hundred run from 3,450 up to but not including 3,550. Questions often place one answer choice just outside this range.
Key Takeaways
Each place is ten times the place to its right and one-tenth the place to its left, on both sides of the decimal point.
Know standard, word, expanded, and base-ten (unit) forms, and remember that "and" signals the decimal point.
A number can be decomposed in nonstandard ways (5,207 = 52 hundreds + 7 ones), which is the basis of regrouping.
The value of a digit is the digit times its place value; the same digit n places farther left is worth 10^n times as much.
Multiplying or dividing by 10^n shifts digits n places; "add a zero" fails for decimals.
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