Question: How do students build an understanding of place value in Investigations?
Answer: To understand place value is to understand the structure and sequence of our base ten number system. As students count, interpret the values of written and spoken numbers, decide which number is larger or smaller, and explore relationships among numbers, they are developing a picture of our number system.
At first this picture is a single sequence of counting numbers: 1, 2, 3, 4, 5 ... Young children become familiar with this sequence, and through many experiences with quantities, they gradually come to see that each number in the sequence represents an amount that is one more than the previous counting number. Even before students can count very high themselves, they begin to appreciate that the numbers can go on forever, with each number representing "one more."
In our base ten number system all these ones are organized in a particular way -- in tens and multiples of ten. Our written numerals and most of our spoken numbers reflect this organization. As young students are developing their understanding of the sequence of ones, they also begin to make sense of how our written and spoken numbers are structured. A first grader who is determined to count to 100 sometimes says "47, 48, 49, what's the next one?," and as soon as "50" is supplied, the student takes off again, "51, 52, 53, 54 . . ." until the next multiple of ten is reached. This student has recognized that each decade has a familiar pattern. A second grader exploring different ways to count a large group of objects comes to understand that each number in the counting-by-tens sequence 10, 20, 30... represents a quantity that is 10 more; another student exploring a variety of number patterns notices that each time 10 is added to a two-digit number, the tens digit increases by 1 and the ones digit remains the same. Through experiences such as these, students begin to recognize "place value" -- that where a digit appears in a number determines its value. The "2" in 24 represents 20, the "2" in 247 represents 200.
Understanding place value, however, is much more than knowing how to break numbers into hundreds, tens and ones. Some students who can recite "247 has 2 hundreds, 4 tens, and 7 ones" may have no idea about the relationship of 247 to the number system -- for example, that it is 100 more than 147 and 10 less than 257, that it is almost 250, or that it is about halfway between 200 and 300. In the Investigations curriculum, understanding place value is linked to understanding how a number is composed and to knowing its relationships to many other numbers. For example, understanding the place value of a number like 126 is studied in a larger context that includes exploration of the following questions:
- Where is the number in the number system? (Is it bigger than 10, bigger than 50, bigger than 100?)
- How can the number be pulled apart into additive components that are easy to work with? (For one problem, it might be convenient to pull 126 apart into 100 and 25 and 1; for another problem, it might be convenient to think of it as 120 and 6.)
- What are the factors of the number? (What numbers can you count by to reach the number exactly?)
- What are the multiples of the number? (What happens when the number is doubled or multiplied by 10 or by 100?)
- What is the relationship of the number to important numbers in our number system, such as multiples of 10, 25, or 100?
Throughout work on the number system and the place value of numbers, the emphasis is on making sense of the structure of the number system and on developing a large repertoire of number relationships that students can build on to solve problems. One focus of this work is on what we have come to call landmarks in the number system -- familiar anchor points in the sequence of numbers, such as 10, 25, 100, 1000, .1, and their multiples. Students study these numbers, their factors and multiples, and their relationships to other numbers.
Because operations with numbers such as tens and hundreds make for simple calculations, place value plays a critical role throughout the grades in the development of computation strategies. Consider the following addition and multiplication problems and sample strategies:
364 + 347
- Break the addends into hundreds, tens, and ones, then combine the parts: 300 + 300 = 600, 60 + 40 = 100, 4 + 7 = 11, 600 + 100 + 11 = 711
- Transform the problem into an equivalent problem that uses landmark numbers: 364 + 347 = 361 + 350 = (350 + 11 + 350) = 700 + 11 = 711
24 x 21
- Break one of the numbers into 10's and 1's and multiply the other number by each of these parts: 21 x 24 = (10 x 24) + (10 x 24) + (1 x 24) = 240 + 240 + 24 = 504
- Change one factor to a landmark number, then adjust: 24 x 21 = (25 x 21) - (1 x 21) 20 x 25 = 500, so 21 x 25 = 525, 525 - 21 = 504
All these strategies rely on knowledge of place value: understanding how numbers are composed of tens and multiples of tens, knowing the place of a number in the number sequence, knowing the relationships of numbers in the problem to landmark numbers, understanding how landmark numbers behave when they are added or multiplied.
Place Value at Each Grade in Investigations
- Students begin learning about the number sequence and about relationships among the counting words, the written numbers, and the quantities they represent. Students count by ones as they construct and compare quantities, create sets of a given size, and find how many in sets. Most of their work with quantities involves amounts up to about 20 or 25; they count orally to numbers 25 and higher.
- Students become familiar with numbers up to about 100. They count quantities up to 50 or more, and they count orally and read, write, and sequence numbers to 100 or higher. Students do most of their counting by ones. They begin to develop meaning for counting by small, familiar numbers such as 2's and 4's.
- Students work with addition combinations of 10. They begin to learn about ways that numbers are built from tens and ones as they engage in activities such as organizing sets of objects in tens and ones and exploring patterns of tens and ones on the 100 chart.
- Students become familiar with numbers into the low hundreds. They develop a sense of the size of these numbers as they work with objects, 100 charts, and coins; they break numbers into tens, ones, and other parts that are easy to work with such as 5's or 25's (27 is 20 + 7, or 10 + 10 + 7, or 25 + 2); and they find how far numbers are from the nearest multiple of 10 and from 100 (how far is 72 from 80? 90? 100?). Students find patterns based on repeatedly adding or subtracting tens (53 and 10 more is 63, 10 more is 73, etc.).
- Students work with landmarks up to 100, such as 5, 10, 25, and 100 as they engage in activities that include playing games in which they collect 25, 50, and 100 objects, reasoning about addition combinations of 10, and finding addition combinations of 100 (92 + 8, 37 + 13 + 50, 25 + 75). Students count by ones and by familiar numbers, such as 2's, 4's, 5's, and 10's.
- Students become familiar with numbers into the low thousands. As they work with number charts and other materials, they develop a sense of the size of these numbers, they break these numbers into parts such as 100's, 10's, 1's, 25's, and 50's (253 is 200 + 50 + 3, 5 x 50 + 3, 25 x 10 + 3, 240 + 13), and they find how far these numbers are from 100, 1000, and other landmarks (253 is 53 more than 200, 3 more than 253, 7 less than 260, 47 less than 300).
- Students explore patterns of hundreds, tens, and ones in the numbers up to 1000. They become familiar with patterns based on repeatedly adding or subtracting tens, hundreds, and other landmarks (400 - 10 is 390, 10 less is 380, another 10 less is 370, and so on), and with patterns of factors and multiples of landmarks (there are 5 20's in 100, 10 in 200, 20 in 400, 40 in 800).
- Students reason about counting sequences that involve landmarks. (If we're counting by 25's and we're up to 325, how many numbers have we said? How do you know? What's a number larger than 500 that we'll say? What's a number smaller than 500 that we won't say?)
- Students work with numbers to 10,000 and higher. They develop a sense of their size, they break them into parts involving landmarks (304 is 300 + 4, 3 x 100 + 4, 12 x 25 + 4), and they find how far these numbers are from landmarks (how far is 604 from 650? from 700? from 1000? from 2000?).
- Students explore patterns of thousands, hundreds, tens, and ones in the numbers up to 10,000. They investigate patterns based on repeatedly adding or subtracting landmarks (100 more than 572 is 672, 100 more than that is 772, and so on), and patterns based on factors and multiples of landmarks (4 x 25 is 100, so 8 x 25 is 200, 12 x 25 is 300...40 x 25 is 1000).
- Students reason about counting sequences that involve landmark numbers. (If we're counting by 25's and we're up to 775, how many numbers have we said? How do you know? Will we say 1000? Will we say 2000? How many more numbers until we say 1225?)
- Students extend their understanding of the number system as they work with tenths and hundredths.
- Students become familiar with numbers up to 1,000,000, with much of their work on numbers up to about 10,000. They develop a sense of the size of these numbers, they break these numbers into parts involving landmarks (2637 is 2000 + 600 + 37, 2500 + 137, 100 x 25 + 5 x 25 + 12), and they find distances from these numbers to landmarks (how far apart are 43 and 100? 43 and 1000? 43 and 10,000? 43 and 100,000?).
- Students explore patterns based on repeatedly adding or subtracting the same amount from different landmarks (such as repeatedly subtracting 10's from 100, 1000, and 10,000). They also explore patterns based on factors and multiples of landmarks: 4 x 25 is a factor pair of 100, so factor pairs of 1000 include the factor pairs that are ten times 4 x 25, for example, 10 x 4 x 25 or 40 x 25.
- Students reason about counting sequences involving landmarks that extend to 10,000 or more, that begin at numbers other than 0, and that go backwards (e.g. If we start at 1000 and count back by 75, will we end up on 0?).
- Students deepen their understanding of tenths and hundredths. They explore relationships among numbers such as .01, .1, 1, and 10.
Marlene Kliman, TERC