Operations: Meanings and Basic Facts CHAPTER 9 Tina Rye Sloan To accompany Helping Children Learn Math9e, Reys et al. ©2009 John Wiley & Sons

Focus Questions What prerequisites are important prior to engaging students in formal work on the four basic operations? What general sequence of activities helps children develop meaning for the operations? What three distinct types of situations lead to subtraction? What four types of structures lead to multiplication? How should thinking strategies for the basic facts be taught? Describe the key thinking strategies for learning basic facts for addition, for subtraction, for multiplication, and for division. Reys/ Lindquist/ Lamdin/ Smith, Helping Children Learn Math, 9 th Edition, © 2009

Understand meanings of operations and how they relate to one another Grades Pre-K-2 Understand various meanings of addition and subtraction of whole numbers and the relationship between the two operations. Understand the effects of adding and subtracting whole numbers. Understand situations that entail multiplication and division, such as equal groupings of objects and sharing equally. NCTM (2000). Principles and Standards for School Mathematics. Reys/ Lindquist/ Lamdin/ Smith, Helping Children Learn Math, 9 th Edition, © 2009

Understand meanings of operations and how they relate to one another Grades 3-5 Understand various meanings of multiplication and division. Understand the effects of multiplying and dividing whole numbers. Identify and use relationships between operations, such as division as the inverse of multiplication, to solve problems. Understand and use properties of operations, such as the distributivity of multiplication over addition. Describe classes of numbers according to characteristics such as the nature of their factors. NCTM (2000). Principles and Standards for School Mathematics. Reys/ Lindquist/ Lamdin/ Smith, Helping Children Learn Math, 9 th Edition, © 2009

Compute fluently and make reasonable estimates Grades Pre-K-2 Develop and use strategies for whole number computations with a focus on addition and subtraction. Develop fluency with basic number combinations for addition and subtraction. Grades 3-5 Develop fluency with basic number combinations for multiplication and division and use these combinations to mentally compute related problems, such as 30 x 50. NCTM (2000). Principles and Standards for School Mathematics. Reys/ Lindquist/ Lamdin/ Smith, Helping Children Learn Math, 9 th Edition, © 2009

Helping Children Develop Number Sense and Computational Fluency The instructional goal is that children not only know how to add, subtract, multiply, and divide, but, more important, know when to apply each operation in a problem-solving situation. Children also should be able to recall the basic facts quickly when needed. Reys/ Lindquist/ Lamdin/ Smith, Helping Children Learn Math, 9 th Edition, © 2009

Facility with Counting Any problem with whole numbers can be solved by counting, provided there is sufficient time. Because it is not always efficient to solve problems by counting, children need to be able to use other procedures to cope with more difficult computations. Reys/ Lindquist/ Lamdin/ Smith, Helping Children Learn Math, 9 th Edition, © 2009

Experience with a Variety of Concrete Situations Children need to have many experiences in problem situations and in working with physical objects to develop understanding about mathematical operations. Reys/ Lindquist/ Lamdin/ Smith, Helping Children Learn Math, 9 th Edition, © 2009

Familiarity with Many Problem Contexts Problem situations are used in mathematics instruction for developing conceptual understanding, teaching higher-level thinking and problem-solving skills, and applying a variety of mathematical ideas. Reys/ Lindquist/ Lamdin/ Smith, Helping Children Learn Math, 9 th Edition, © 2009

Experience in Talking and Writing about Mathematical Ideas Children need to talk and write about mathematics; putting experiences into words helps with making meaning. Both manipulative materials and problems can be vehicles for communicating about mathematics. Reys/ Lindquist/ Lamdin/ Smith, Helping Children Learn Math, 9 th Edition, © 2009

Developing Meanings for the Operations 1.Concrete: Modeling with materials: Use a variety of problem settings and manipulative materials to act out and model the operation. 2.Semi-concrete: Representing with pictures: Provide representations of objects in pictures, diagrams, and drawings to move a step away from the concrete toward symbolization. 3.Abstract: Representing with symbols: Use symbols to illustrate the operation. Reys/ Lindquist/ Lamdin/ Smith, Helping Children Learn Math, 9 th Edition, © 2009

Meanings for Operations Addition Finding how many in all Subtraction Separation or take away Comparison or finding the difference Part-whole Multiplication Equal groups of objects or repeated addition Multiplicative Comparisons Array or area Division Measurement or repeated subtraction Partition or sharing Reys/ Lindquist/ Lamdin/ Smith, Helping Children Learn Math, 9 th Edition, © 2009

Addition “How many in all?” =5 Reys/ Lindquist/ Lamdin/ Smith, Helping Children Learn Math, 9 th Edition, © 2009

Subtraction: Separation Problems or ‘Take Away’ Peggy had 7 balloons. She gave 4 to other children. How many did she have left? Reys/ Lindquist/ Lamdin/ Smith, Helping Children Learn Math, 9 th Edition, © 2009

Subtraction: Comparison Peggy had 7 blue counters. Richard had 4 red counters. How many more counters did Peggy have than Richard? Reys/ Lindquist/ Lamdin/ Smith, Helping Children Learn Math, 9 th Edition, © 2009

Subtraction: Part-Whole Problems Peggy had 7 balloons. Four of them were red and the rest were blue. How many were blue? Reys/ Lindquist/ Lamdin/ Smith, Helping Children Learn Math, 9 th Edition, © 2009

Models for Multiplication 3 x 2 2 x 3 Arrays Equal Groups of Objects Reys/ Lindquist/ Lamdin/ Smith, Helping Children Learn Math, 9 th Edition, © 2009

Multiplication: Combinations Problems Consider the number of different sundaes possible with four different ice cream flavors and two toppings, if each sundae can have exactly one ice cream flavor and one topping: Reys/ Lindquist/ Lamdin/ Smith, Helping Children Learn Math, 9 th Edition, © 2009

Hilary spent $35 on Christmas gifts for her family. Geoff spent 3 times as much. How much did Geoff spend?” Multiplication: Comparison Problems Reys/ Lindquist/ Lamdin/ Smith, Helping Children Learn Math, 9 th Edition, © 2009

Measurement Division: Repeated Subtraction Jenny had 12 candies. She gave 3 to each person. How many people got candies? Reys/ Lindquist/ Lamdin/ Smith, Helping Children Learn Math, 9 th Edition, © 2009

Partitioning (or sharing) Division: Gil had 15 shells. If he wanted to share them equally among 5 friends, how many should he give to each? One for you, one for you, one for you, etc., and then a second shell to each person, and so on… Reys/ Lindquist/ Lamdin/ Smith, Helping Children Learn Math, 9 th Edition, © 2009

Mathematical Properties Reys/ Lindquist/ Lamdin/ Smith, Helping Children Learn Math, 9 th Edition, © 2009

When to Develop Basic Fact Fluency and Efficiency The process of developing fluency with basic facts for immediate, automatic recall can begin as soon as children have a good understanding of the meaning of the operations and the symbols. Children should be able to : ▫ state or write related facts, given one basic fact. ▫ explain how they got an answer, or prove that it is correct. ▫ solve a fact in two or more ways. Reys/ Lindquist/ Lamdin/ Smith, Helping Children Learn Math, 9 th Edition, © 2009

Overview of Basic Fact Instruction Start where the children are. Build understanding of the basic facts. Focus on how to remember facts. Reys/ Lindquist/ Lamdin/ Smith, Helping Children Learn Math, 9 th Edition, © 2009

Thinking Strategies for the Basic Facts Addition-100 facts involving two one-digit addends and their sum. ▫ Commutative Property ▫ Adding One and Zero ▫ Adding Doubles and Near Doubles ▫ Counting On ▫Combinations to 10 ▫ Adding to 10 and Beyond Reys/ Lindquist/ Lamdin/ Smith, Helping Children Learn Math, 9 th Edition, © 2009

Thinking Strategies for the Basic Facts Subtraction-100 facts involving the difference between one addend and the sum for all one-digit addends ▫ Subtracting One and Zero ▫ Doubles ▫ Counting Back ▫ Counting On Reys/ Lindquist/ Lamdin/ Smith, Helping Children Learn Math, 9 th Edition, © 2009

Reys/ Lindquist/ Lamdin/ Smith, Helping Children Learn Math, 9 th Edition, © 2009 Thinking Strategies for the Basic Facts Multiplication-100 facts involving two one-digit factors and their product ▫ Commutative Property ▫ Skip Counting ▫ Repeated Addition ▫ Splitting the Product into Known Parts ▫ Multiplying by One and Zero ▫ Patterns

Thinking Strategies for the Basic Facts Division-90 facts (no division by zero) involving the quotient of one factor and the product for all one-digit factors -Fact Families ▫ Find missing factor in the multiplication problem ▫ Repeated Subtraction Reys/ Lindquist/ Lamdin/ Smith, Helping Children Learn Math, 9 th Edition, © 2009

Principles for Basic Fact Drill Children should attempt to memorize facts only after understanding is attained. Children should participate in drill with the intent to develop fluency. Remembering should be emphasized: This is not a time for explanations. Drill lessons should be short (5-10 minutes) and should be given almost every day. Children should try to memorize only a few facts to a given lesson and should constantly review previously memorized facts. Reys/ Lindquist/ Lamdin/ Smith, Helping Children Learn Math, 9 th Edition, © 2009

Principles of Drill Children should develop confidence in their ability to remember facts fluently and should be praised for good efforts. Records of their progress should be kept. Drill activities should be varied, interesting, challenging, and presented with enthusiasm. Reys/ Lindquist/ Lamdin/ Smith, Helping Children Learn Math, 9 th Edition, © 2009

21 or Bust! Objective: Using a game to develop logical reasoning and to practice addition. Grade Level: 3-4 Instructions: Play this game with a partner: Enter 1,2,3,4, or 5 in your calculator. Give the calculator to your opponent, who adds 1,2,3,4, or 5 to the displayed number. Take turns adding 1,2,3,4, or 5 to the total. The first player to reach 21 wins! If you go over 21, you “bust” or lose! What is a winning strategy? Reys/ Lindquist/ Lamdin/ Smith, Helping Children Learn Math, 9 th Edition, © 2009