A state department of education, as a question on the public-release form of a statewide standardized test, released a question to the public in which students have to recognize equivalent representations of whole numbers and their place values.
I’m going to use the word DIGIT to mean the numerals 0 through 9. This question gives us a number, 5,360. We are asked to determine the value of the DIGIT 6 in this number.
The value of a DIGIT in any number is based upon its location or placement in the number. That location in the number of a DIGIT is known as its “place” and the total amount that the DIGIT is worth in the number is known as its “place value,” (the VALUE it has because of its PLACE in the number).
For example, there are four DIGITS in 5,360. The 0 is said to be in the ones place, because it is furthest to the right in the number. The lowest place in a whole number is the ones place. Next is the tens place, and the 6 is in the tens place in our number. It has a place value of 60, because we multiply the number in the tens place by 10 to determine its value.
In other words, because the DIGIT 6 is in the tens place in our number, we know there are six 10’s in the number. Since 6 × 10 = 60, we know the value of the “6” in our number is 60.
That’s the answer to our question, but let’s continue.
We have a 3 in the hundreds place, which has a value of 3 × 100 = 300, and a 5 in the thousands place, with a value of 5 × 1,000 = 5,000.
We can also write the number in expanded form. This is another way to show what the number really means. The number 5,360 would look like this in expanded form:
Some teachers say it is OK to leave out the “+ 0” since it really has no value in the number. However, putting it in is definitely mathematically correct, so I figure it’s best to make sure all your bases are covered.
In addition, some teachers say it is OK to leave out the “= 5,360” when writing the number in expanded form. Different textbooks say different things about exactly what to include or leave out when writing a number in expanded form, and it is not our place to throw ourselves into the debate.
We would advise you to ask your teacher how he or she prefers to write numbers in expanded form, and as long as it’s still a mathematically correct formula or style, go ahead and use that. It’s much more important to do the problem as your teacher tells you than it is to discuss what exact form constitutes expanded form.
If you would like additional problems to help you master this skill, go to our online card catalog at VoxLearn.org, select one of the math collections, and enter the search terms “place value” or “expanded form.”
Expanded form vs. Expanded notation
What I called “expanded form” should not be confused with what many textbooks and teachers call “expanded notation.” This is where you take each digit in the number and show it being multiplied by the place.
For example, 5,360 in “expanded notation” would be written as follows:
As with expanded form, described in the main post, some teachers say it is OK to leave out any zeroes (0 × 1 in our example) when writing the number, since the resulting sum is still mathematically equivalent.
Also note that some teachers prefer to group each digit in the number with parentheses, as follows:
Technically speaking, the parentheses are not necessary from a mathematical point of view, but using parentheses may help some students see the groupings better. As always, we would recommend doing it the way your teacher says he or she would like it done.
Try a few on your own
(1) What is the value of the “7” in 57,189 ?
(2) What does the “4” mean in 6,407 ?
(3) Look at this number, which is written in standard form.
(4) In the number 158,942, what digit is in the ten-thousands place?
The Answers
(1) The value of the “7” in 57,189 is 7,000. It is in the thousands place, and to determine the value of digit in the thousands place, we multiply that digit by 1,000. 7 × 1,000 = 7,000.
(2) The “4” in 6,407 means 400. It is in the hundreds place, and to determine the value of the digit in the hundreds place in a number, we multiply the digit by 100. 4 × 100 = 400.
(3) In order to write the number 315 in expanded form, we think about what the number really means. The 3 is in the hundreds place, so that means 3 × 100, which = 300. The 1 is in the tens place, so that means 1 × 10, or 10. The 5 is in the ones place, so that means 5 × 1, or 5. We show the adding signs and get:
That is answer C.
(4) We haven’t covered this so far in this post, but it is really not that difficult to extend what we have learned for smaller numbers without as many digits in them to larger numbers with many more digits.
Starting with the digit furthest to the right in a whole number, the places and place values for additional digits in the base-10 system are as follows:
In our number, 158,942, we are asked which digit is in the ten-thousands place. Starting with the digit furthest to the right, we see that the “2” is in the ones place, the “4” in the tens place, the “9” in the hundreds place, the “8” in the thousands place, the “5” in the ten-thousands place, and the “1” in the hundred-thousands place. Our answer, therefore, is “5”.
Its value would be computed as 5 × 10,000 = 50,000.
Illinois Alignment
Grade Level: 4
Illinois Assessment Framework: 6.4.01 Read, write, recognize, and model equivalent representations of whole numbers and their place values up to 1,000,000.
Illinois Learning Standard: 6.A.1a (K-2): Identify whole numbers and compare them using the symbols , or = and the words “less than”, “greater than”, or “equal to”, applying counting, grouping and place value concepts.
Illinois Learning Standard 6.A.2 (3-5): Compare and order whole numbers, fractions and decimals using concrete materials, drawings and mathematical symbols.