Question: What is the purpose of landmark numbers in Investigations? Why do children need to use them? What are ways they are taught, developed, and discussed in the Investigations program?
Elementary Mathematics Coordinator
Milwaukee Public Schools
Note: This response draws mainly from the About the Math in Grade 2: Putting Together and Taking Apart, and the Grades 3, 4 & 5 Landmarks units.
Answer: An important part of students' mathematical work in the elementary grades is building an understanding of the base ten number system. Part of becoming familiar with that system is learning about the relationship of numbers to important landmarks -- numbers that are familiar landing places, that make for simple calculations, and to which other numbers can be related, such as 10, 100, 1000, and their multiples and factors. Familiarity with these landmark numbers is a cornerstone of good number sense -- what one unit describes as involving "a deep understanding of numbers, their characteristics, their place in the number system, and their relationships to one another." (Grade 4, Money, Miles, and Large Numbers, page 9.)
"When solving real problems, people with well-developed number sense draw on their knowledge of these important landmarks. For example, think about how you would solve this problem, in your head, before you continue reading:
If there are about 25 students in a class and 17 classes in our school, about how many students are there altogether?
Many people would use their knowledge that there are four 25's in every 100 to help them solve this problem mentally. Rather than multiplying 17 by 25, they will think something like this: "There are four 25's in 100; there are eight in 200, 12 in 300, 16 in 400 -- and one more 25 makes 425." (Grade 4, Landmarks in the Thousands, Page I-18.)
Efficient strategies to solve computation problems often depend on an understanding of how to use landmark numbers and in order best to find the final answer. Consider a few more examples.
A student with solid number sense and a developing sense of the usefulness of landmark numbers is unlikely to make errors many teachers say they typically see. For example, teachers often describe the following kind of addition error:
First off, many students who are developing strategies based on their own growing sense of numbers, operations, and landmarks would find this answer unreasonable. A first glance at the problem would have led them to an estimate of "less than 200" or "110 and 40, close to 150" or even "about 200." An answer in the 500's would, and should, not make sense.
In fact, these students would likely solve the problem in a different way: 40 is 4 tens so I counted up by 10's: 122, 132, 142, 152. Another child might think of it as 110+40, and then add on the 2. (Russell, 2000.) These students see 40 as 4 tens, are comfortable adding ten (or multiples of 10) to any number, and de- and re-compose numbers in ways that are useful.
To take another example, consider 629 + 72. One student might know that 629 plus 70 is 699 (because 20 and 70 is 90), then add on the remaining 2 to get 701. Another student might say, "Take one from 72, and give it the 629 to make it 630. 630 plus 70 is 700, plus one more (leftover from the 72) is 701."
In subtraction, changing the numbers to landmark numbers is often much easier than subtracting place by place, as in the traditional US algorithm (i.e. "borrowing"). "Think for a minute about this problem: 'If you are 48 years old and I am 62, how much older am I than you?' When you think about finding the difference between 48 and 62, you immediately bring your number sense into play. You recognize very quickly a great deal about these numbers. You might use any of a number of ideas -- 62 is 2 more than 60, 48 is 2 less than 50, 48+10 is 58, 62-10 is 52, and 10 is the difference between 50 and 60.
A lot of the information we use to solve this kind of problem has to do with the relationship between a quantity in the problem and a nearby landmark in the number system. For example, you could solve this problem by thinking that 48 to 50 is 2, 50 to 60 is 10, 60 to 62 is 2 more, so the difference is 10 + 2 + 2, or 14. You used multiples of ten -- 50 and 60 -- as critical landmarks or anchors to which the numbers 28 and 62 are connected." (Grade 4, Money Miles and Large Numbers, page 9.)
To take another example, many children who have learned the traditional rote procedure for solving subtraction problems would solve 1002-997 using the "borrowing" method. "We want children to reason from their knowledge of the place of 1000 as an important landmark in the number system. When students envision where 1000 is placed in the number system, they can easily" (Russell, 1996) solve this problem mentally. They can count up from 997, or back from 1002. Or, they see that 997 is 3 away from 1000, and 1002 is 2 away from 1000, so the distance between these two numbers is 5.
Similarly, "a student who has developed knowledge about 20 and its relationship to 100, who has experience counting by 20's, and knows what the pattern of the multiples of 20 is like, would never make this common error:
Using a written subtraction algorithm -- whether faulty or correct -- is not a sensible approach to solving this problem. Rather, by inspecting the numbers and using knowledge of important landmarks in the number system, students should ... be able to solve this problem mentally...:
'380 to 400 is 20, then 20, 40, two more 20's gets to 440, so that's three 20's. The answer is 60.'" (Grade 3, Landmarks in the Hundreds, Page I-17.)
Consider a "harder," multiplication problem, such as 83 x 25. Rather than using an algorithm to multiply, Investigations students might solve this problem mentally, using reasoning such as this: "I know there are four 25's in 100, so forty 25's (or ten times as many) would be 1000. Eighty 25's would be 2000, and three more 25's makes 2075." (Grade 5, Mathematical Thinking at Grade 5, Page I-18.)
This kind of thinking and reasoning, grounded in the knowledge of landmark numbers, can help students avoid pitfalls. For instance, consider this third grader's solution to the problem 57X4.
He "multiplied the 7 and 4, getting 28, put down the 8 and "carried" the 2, then added the 2 to the 5 before multiplying by 4, giving him an answer of 288." This child knew, when asked, that " two 50's would be 100, so [50x4 is] 200" and that he still needed to solve 7 x 4. "He finished solving the problem, adding the two subproducts in his head. For this student, 57 x 4 should have been an easy mental problem." (Russell, 2000.)
Consider the landmarks these 5th grade students are using to solve 159/13.
"Cara thought about the problem as, "how many 13's are in 159?" She knew that ten 13's is 130, that she then had 29 remaining in the dividend, and that she could take two 13's out of 29, giving her an answer of 12 with a remainder of 3.
Armand counted by 13's until he reached 52. He then added 52's until he got as close to 159 as possible (52 + 52 + 52 = 156). He knew there were four 13's in each of the 52's, so he had twelve 13's, with a remainder of 3." (Russell, 2000.)
Another student "just knew" that 13 x 13 = 169. He took away 13, got 156, and said, "So 12 13's make 156, and there are 3 leftover. 12 remainder 3."
"Sound mental strategies such as those described above are used by people who are fluent in computation. When students use strategies based on their own good number sense, they are more likely to solve problems accurately and to make sure their solutions make sense. They tend to see quantities as whole quantities, not individual digits (for example, 629 is about 30 bigger than 600, not 620 is a 6 and a 2 and a 9 in a row), so they are more likely to notice if their results are unreasonable." (Grade 4, Landmarks in the Thousands, Page I-18.) Further, the strategies often "move from left to right, working with the biggest numbers first, an approach that often leads to more accurate calculation." (Grade 4, Landmarks in the Thousands, Page I-18.)
"As students learn about 100 and 1000, how to take these numbers apart into their factors, and how to use them to construct other numbers, they gain the knowledge they need to develop their own strategies to solve problems. They learn what happens when you add or subtract multiples of 10 and 100. They develop good estimation strategies and are less likely to make the kinds of errors that result from the use of faulty algorithms." (Grade 4, Landmarks in the Thousands, Page I-18.)
The Investigations curriculum provides many activities that develop knowledge about important landmarks in our system. The following are some specific examples that develop these ideas, throughout the grades.
The focus in grades K and 1 is on counting, primarily by ones. A solid understanding of the counting sequence, the quantities those counting numbers represent, and the patterns in that sequence (as you count by ones you are adding 1 with each new number; the patterns in the decades such as 29, 30, . . . 39, 40) underlies and supports the work with landmarks numbers in the upper elementary grades. Teachers also see the beginnings of the use of landmarks as students play games and solve problems. For example, a kindergartner who is playing Double Compare, or solving a story problem for 3 + 4 might say, "I know that 3 and 3 is 6; 1 more is 7." A first grader adding 8 and 5 might say, "I know 8 and 2 makes 10. I need 3 more so 11, 12, 13."
Whole investigations in grades 1 and 2 focus on counting by groups such as 2's, 5's, and 10's, as students figure out how many hands or fingers are in the room, count a set of squares, or play games involving money. In the grade 2 number units, "many of the games . . . focus on breaking 100 into parts and on multiples of 10 and 5 as important chunks. Using 100 charts, money, and interconnecting cubes, these activities provide models of two central ideas in working with our number system: how any number is related to the nearest multiples of 10 and how any number can be broken apart into multiples of 10 and 1's (or into other parts that are easy to work with, such as 25's). Most of the addition and subtraction procedures students develop will be based on one of these two ideas." (Grade 2, Putting Together and Taking Apart, Page I-20.)
In grade 3 students also use cubes and 100 charts to skip count. They work with 100 and its factors, and solve problems about money, as they think about how to fairly share a dollar among several people. They explore multiples of 100 with money, the 300 chart, and the calculator, solve word problems, and make a 1000 chart and think about how to find numbers on it.
Also, "in order to use these critical numbers effectively, students must become fluent in finding the difference between a number they are using and nearby landmarks. The game Close to 100 [in grades 3 and 4, as well as Close to 1000 and Close to 0 in grades 4 and 5] provides practice in this skill. ("If I have 48, how much more do I need to make 100? If I have made a sum of 87, how far away is that from 100?") As students play this game, [teachers can] encourage them to use what they know about number relationships. If a student is trying to figure out what is needed to add to 48 to get 100, he or she might count by 10's: "58, 68, 78, 88, 98 -- that's five 10's, and 2 more is 52." Another students might use 50 as a landmark: "I know 50 to 100 is 50, and 48 to 50 is 2, so it's 52." (Grade 4, Money Miles and Large Numbers, page 9.)
In grade 4, students continue to practice skip counting, for example, by Counting Around the Room by different numbers, and considering questions such as: we are on 225 and counting by 25's, how many kids have counted so far? What number will we reach if everyone in our class says a number? How did you figure that out? They also continue to work with factors and multiples as they solve problems involving, say, Frieda and Frogurt the Frogs. If Frieda makes 5 jumps of 20 and Frogurt makes 3 jumps of 25, where will they land? Who will be ahead? By how much? How do you know? "It is also important that students develop some sense of the magnitude and relationships of these numbers: How big is 100? 1000? 10,000? How many hundreds are in 10,000? How far is it from 3000 to 7500? If you added 300 to 7800, what would the result be? For this reason, we spend time in [the fourth grade landmarks] unit estimating large quantities, as well as making a 1000 book and a 10,000 chart, so students can begin to visualize these important relationships." (Grade 4, Landmarks in the Thousands, Page I-18.)
The activities in the grade 5 landmarks unit "are designed to help students develop a sense of the size of numbers up to 10,000. The focus is on 1000 and 10,000 and their factors and multiples. . . . Through the unit, students explore relationships among these numbers. They find ways to draw upon their knowledge of landmarks as they develop strategies for solving computation problems." (Grade 5, Mathematical Thinking at Grade 5, Page I-18.)
In all grades, these kinds of activities support students in developing computational fluency -- the ability to solve problems in efficient and accurate ways, while also showing flexibility -- the ability to choose a solution strategy that best suits the numbers or context of a particular problem.
Megan Murray, TERC