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Adding and Subtracting Number to 100

Strand: Number
Outcomes: 8, 9

Step 1: Identify Outcomes to Address

Guiding Questions

• What do I want my students to learn?
• What can my students currently understand and do?
• What do I want my students to understand and be able to do, based on the Big Ideas and specific outcomes in the program of studies?

See Sequence of Outcomes from the Program of Studies

Big Ideas

• Addition and subtraction are related, with subtraction being the inverse of addition.
• The order of the numbers does not matter when you add, but does when you subtract.
• Traditional algorithms are often not the most efficient methods of computing. They are also not naturally invented by students. If the traditional algorithms are not taught to students early on, students will invent or adopt personal strategies that vary with the numbers and the situation. What is important is that the methods used are understood by the user.
• Personal strategies depend on taking apart and combining numbers in a variety of ways and recognizing relationships between numbers.
• Models, be they manipulatives or diagrams, can help a person recognize the operation involved and make sense of a problem.
• Examining any problems for the parts and the whole helps students make sense of the problem and identify the operation required. (See Categories of Addition and Subtraction Problems Based on Structure )

Principles and Standards for School Mathematics states that computational fluency is a balance between conceptual understanding and computational proficiency (NCTM 2000, p. 35). Conceptual understanding requires flexibility in thinking about the structure of numbers (base-ten system), the relationship among numbers and the connections between addition and subtraction. The ability to generate equivalent representations of the same number provides a foundation for using personal strategies to add and subtract, recognizing that for some problems either operation may be used. Computational proficiency includes both efficiency and accuracy. Personal strategies must be compared and evaluated so students adopt methods that are efficient, as well as accurate.

Addition and subtraction problems include four main types:

• join problems involving an initial amount, a change amount (the amount being added or joined) and the resulting amount
• separate problems also involve change as in join problems, but the whole is the result in the join problems, whereas the whole is the initial amount in the separate problems
• part–part–whole problems consider two static quantities either separately or combined
• comparison problems determine how much two numbers differ in size.

Adapted from John A. Van de Walle, LouAnn H. Lovin, Teaching Student-Centered Mathematics: Grades K–3, 1e (pp. 66, 67, 68, 69). Published by Allyn and Bacon, Boston, MA. Copyright © 2006 by Pearson Education. Reprinted by permission of the publisher.

What is crucial is that students are familiar with the relationship between addition and subtraction and all the possible forms these operations take in problems.

By using a variety of problems, students will construct their own meaning for the inverse relationship between addition and subtraction and for the following properties:

• commutative property of addition (numbers can be added in any order), which does not function for subtraction
• associative property of addition (grouping a set of numbers in different ways does not affect the sum) and
• the identity element for addition and subtraction, that is adding zero to or subtracting zero from a number will result in the original or start number.

Students in Grade 2 generally do not know the names of these properties, but certainly learn to recognize and describe them. Grade 2 teachers who have students using the traditional algorithm will note that too many times their students subtract upside down or backwards, as if the commutative property of addition could be equally applied to subtraction. For example, if the equation to be solved is:

53 –29

or

53 – 29 =

students, upon noting that one cannot subtract 9 from 3, without even being aware they are inverting the numbers, may subtract 3 from 9. This creates a difference that is untrue for the equation given. This situation does not occur when students use invented personal strategies. When using the traditional algorithm, the manipulatives will prevent students from making this type of error. Some students will need to be made aware that they unconsciously do this when solving problems without manipulatives, so they can guard against this error.

The use of manipulatives or models helps students understand the structure of the story problem and also connects the meaning of the problem to the number sentence (Van de Walle 2001, p. 108). To develop their understanding of the meaning of operations, students connect the story problem to the manipulatives, connect it to the number sentence and then use personal strategies to solve the problem.

Reflect upon the student who adds

28 +47 615

and prints this incorrect answer: 615 (as all too frequently happens with the traditional algorithm). Errors of this magnitude do not happen when students use personal strategies. Seldom will students use a personal strategy they do not understand. The number sense that students are developing in Grade 2 is critical to their ability to progress to estimating the answer, necessary in Grade 3. Students' understanding of addition and subtraction is enhanced as they develop their own methods and share them with one another, explaining why their strategies work and are efficient (NCTM 2000, p. 220).