M2N2e: ways to compensate

M2N2. Students will build fluency with multi-digit addition and subtraction.
e) Use basic properties of addition (commutative, associative, and identity) to simplify problems (e.g. 98 + 17 by taking two from 17 and adding it to the 98 to make 100 and replacing the original problem by the sum 100 + 15).
Students, and adults, often use different mental computation strategies. The one that is discussed in this standard is often explained by using the associative property of addition: 98 + 17 = 98 + (2 + 15) = (98 + 2) + 15

However, we can also explain it slightly differently. "98 + 17" means we are putting together 98 and 17. If we pretended 98 were 100, that means we actually have 2 more than we are supposed to. So, if we don't want to change the final answer, we have to make 17 smaller by 2. In other words, 98 + 7 = (98 + 2) + (17 - 2). In general, if we added a number to one of the addends, we have to subtract the same number from the other addend to compensate.

What about subtraction? Let's think about 83 - 18. Subtracting 20 mentally is much easier. But if we subtract 20 instead of 18, we will be taking away 2 more than we are supposed to. So, to compensate for that, we must make the starting number bigger by 2, too. That is, 83 - 18 = (83 + 2) - (18 + 2). Alternately, you might think if we make 83 into 89, then there will be no re-grouping needed. But, in that case, you are starting with 6 more. So, if we want to keep the answer the same, we must take away 6 more than 18 as well. Thus, 83 - 18 = (83 + 6) - (18 + 6). As it turns out, for subtraction, if we add (or subtract) the same number to both the minuend and the subtrahend, the difference stays the same. This idea is sometimes called the equal addition principle of subtraction.

What about multiplication? How do we compensate? Let's think about 35 x 16. If we had 70, it might be easier to multiply mentally. But if we realize that 35 x 16 means 16 groups of 35 [I'm using the Japanese convention of writing the number in a group first]. So, if we make 35 into 70, you are actually putting 2 of those 35's together, and there will be only 8 groups. Or 70 x 8. Thus, we see that 35 x 16 = (35 x 2) x (16 ÷ 2). In general, if we multiply a factor by a number, then we must divide the other factor by the same number to keep the product the same.

For division, let's think about 112 ÷ 14. One way to interpret 112 ÷ 14 is to figure out how many in each group if we split 112 into 14 equal groups. The answer should be the same if we only consider 7 groups with a half as many total. So, 112 ÷ 14 = 56 ÷ 7. In general, if we multiply (or divide) both the dividend and the divisor by the same number, the quotient does not change. In the GPS, this particular idea is actually explicitly mentioned in M4N3(d). I sometime call this relationship the equal multiplication principle of division. Probably the most common place where we see the use of this principle is with problems like 2400 ÷ 400.

When you look at these four ways of making compensations, you notice that there are parallels between addition/multiplication and subtraction/division. With addition and multiplication, we do "opposite" to the two numbers to keep the result the same. However, with subtraction and division, we do the same to both numbers. Although only the division situation is mentioned explicitly in the GPS, looking at these compensation strategies may be useful in helping students develop a deeper understanding of the four arithmetic operations and how they may relate to each other.

M1N3 f - Mastering the basic addition and subtraction

M1N3. Students will add and subtract numbers less than 100, as well as understand and use the inverse relationship between addition and subtraction.
f. Know the single-digit addition facts to 18 and corresponding subtraction facts with understanding and fluency. (Use strategies such as relating to facts already known, applying the commutative property, and grouping facts into families.)
Many of today's elementary school mathematics textbooks discuss a variety of thinking strategies children can use to figure out the basic addition facts. Some textbooks even organize their addition units according to those strategies: add 1/2, doubles, doubles plus/minus 1, make 10, etc.. Young children often "invent" these strategies. In fact, these strategies are the results of children's developing number sense. Enriching children's number sense, for example, composing and decomposing numbers (MKN2b, M1N3c), is a major emphasis in primary grades. Thus, this particular standards has two purposes: helping children master basic facts and helping them further their number sense.

Clearly, we want children to be able to recall the basic addition facts quickly, and some people may wonder why we need to bother with these different strategies. There are many reasons to include students' invented strategies in primary grades mathematics instruction, but to me the following three are the major reasons. First, these strategies are natural for children. If we take the idea of "starting with where children are," then we should think about how to take advantage of children's natural thinking processes. Another reason is that these invented strategies are the results of and promote further development of children's number and operation senses. I believe that the ability to see numbers and calculations flexibly is a powerful mathematical tool. If that is the case, it seems to make little sense to squash children's natural ability to think and force them to memorize the basic facts first then try to teach these flexible ways of thinking later. Such an approach seems to be rather inefficient. Finally, I believe that a major reason we teach mathematics in elementary schools is to help students become better thinkers. Thus, we should be always encouraging students to think. Quick recall is a goal, but if we want students to continue developing their thinking ability, we must dedicate some time in mathematics classrooms that focuses on children's thinking.

Anyway, although these strategies should be discussed as children naturally "invent" them, there is one particular strategy that should be treated intentionally. That strategy is the make-10 strategy. For example, 9 + 6 can be thought of as (9 + 1) + 5 = 10 + 5 = 15. For subtraction, like 13 - 8, children can think 13 - 8 = (10 - 8) + 3 = 2 + 3 = 5, or 13 - 8 = (13 - 3) - 5 = 10 - 5 = 5. 10 is such an important number in our numeration system. Thus, developing the ability to think with 10 systematically must be a major goal of mathematics teaching. For some of the invented strategies, I don't think it is necessary for all children to be able to use them. However, the make-10 strategy is mathematically so significant that all children should understand and be able to use it effectively. This way of thinking also helps students to go beyond the counting-by-one approach. If we consider older students counting on their fingers a problem, we have to offer them an alternative that can be just as effective and perhaps more efficient. The make-10 strategy is one such strategy.

M3N5 - What are decimal fractions?

M3N5. Students will understand the meaning of decimal fractions and common fractions in simple cases and apply them in problem-solving situations.

When the GPS was first released, some people wondered what the phrase found in this standard, "decimal fractions," meant. If you research the Internet, you will find that "decimal fractions" are fractions with powers of 10 as denominators. This interpretation was emphasized in the 2008 revision of the GPS. Thus, M3N5(b) states, "Understand that a decimal fraction (i.e. 3/10) can be written as a decimal (i.e. 0.3)." The corresponding standards in the original GPS, M3N5(c), stated, Understand a one place decimal fraction represents tenths, i.e., 0.3 = 3/10."

However, I believe this was an unnecessary change which actually made the revised GPS a bit incoherent. It seems clear that the phrase, "decimal fractions," in the original GPS was used to mean "decimal numbers." Although the phrase "decimal fractions" isn't commonly used in the existing literature, when it is used, it typically means decimal numbers - or fractional quantities expressed in decimal format. Clearly, it is important for students to understand the equivalence of 3/10 and 0.3, but separating out fractions with powers of 10 as denominator seems to make a little sense mathematically. Furthermore, there are other statements in the GPS where this interpretation of "decimal fractions" creates some problems.

For example, the first sentence describing Grade 3 Number and Operations states, "Students will use decimal fractions and common fractions to represent parts of a whole." By examining the actual standards, we notice that students are also introduced to decimal numbers in Grade 3, but if we interpret "decimal fractions" as fractions with powers of 10 as denominators, then there is no reference to decimal numbers in the description of the standard. Similarly, the description of Grade 4 Number and Operations states, " Students will further develop their understanding of addition and subtraction of decimal fractions and common fractions with like denominators." However, students are to learn addition and subtraction of decimal numbers, M4N5.

In fact, everywhere except in M3N5, the GPS makes much better sense if we interpret "decimal fractions" to mean "decimal numbers." This is a great example how a simple phrase plays an important role in interpreting the standards. I really wish the state DOE will actually publish a document that will further elaborate what they meant.