Chapter 9 To accompany Helping Children Learn Math Cdn Ed, Reys et al. ©2010 John Wiley & Sons Canada Ltd.

Guiding Questions 1.What prerequisites are important prior to engaging students in formal work on the four basic operations? 2.What general sequence of activities helps children develop meaning for the operations? 3.What three distinct types of situations lead to subtraction? What four types of structures lead to multiplication? 4.How should thinking strategies for the basic facts be taught? 5.Describe the key thinking strategies for learning basic facts for addition, subtraction, multiplication, and division.

Building a Foundation Knowledge of the basic number facts for each operation provides a foundation for all later work with computation. – Children must develop broad concepts for the operations developed through multiple representations using various physical models.

Helping Children Develop Number Sense and Computational Fluency The instructional goal is that children know how to add, subtract, multiply, and divide More importantly, children should know when to apply each operation in a problem- solving situation Children should also be able to quickly recall basic facts when needed

Development of Number Sense and Computational Fluency Students in grades 1 and 2 will… 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. Students in grades 3-6 will… 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’s Focal Points (2006).

Facility with Counting

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.

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. As with other mathematical content, a variety of problem contexts or situations should be used to familiarize students with the four basic operations.

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.

Developing Meanings for the Operations Concrete—Modelling with materials: Use a variety of problem settings and manipulative materials to act out and model the operation. Semi-concrete—Representing with pictures: Provide representations of objects in pictures, diagrams, and drawings to move a step away from the concrete toward symbolization. Abstract—Representing with symbols: Use symbols to illustrate the operation.

Models for Representing the 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 Comparison Combination Array or area Division Measurement or repeated subtraction Partition or sharing

Addition “How many in all?”

Subtraction: Separation Problems or ‘Take Away’ Peggy had 7 balloons. She gave 4 to other children. How many did she have left?

Subtraction: Comparison

Subtraction: Part-Whole Problems Peggy had 7 balloons. Four of them were red and the rest were blue. How many were blue?

Models for Multiplication

Multiplication: Comparison Problems Hilary spent $35 on Christmas gifts for her family. Geoff spent 3 times as much. How much did Geoff spend?”

Multiplication: Combinations Problems Consider the number of different sundaes possible with four different ice cream flavours and two toppings, if each sundae can have exactly one ice cream flavour and one topping.

Measurement Division: Repeated Subtraction Jenny had 12 candies. She gave 3 to each person. How many people got candies?

Division: Partitioning (or Sharing) Hoda had 15 shells. If she wanted to share them equally among 5 friends, how many should she give to each? One for you, one for you, one for you, etc., and then a second shell to each person, and so on…

Mathematical Properties

Mastering the Basic Facts Ensure that the underlying numerical strategies are developed before you expect mastery Teach children how to retrieve the basic facts using efficient strategies These skills can be developed by teachers when they: – Start where the children are – Build a understanding of the basic facts – Focus on teaching how to remember the facts

Becoming Skillful Children should attempt to memorize facts only after understanding is attained. Children should participate in drills with the intent to develop fluency. Remembering should be emphasized: This is not a time for explanations. Practice opportunities for basic facts should be short (5-10 minutes) and should be given almost every day. Children should try to memorize only a few facts in a given lesson and should constantly review previously learned facts.

Becoming Skillful (cont.) 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. Practice activities should be varied, interesting, challenging, and presented with enthusiasm.

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

Thinking Strategies for the Basic Facts Fact Families-for each basic addition fact, there is a related subtraction fact.

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

Thinking Strategies for the Basic Facts Multiplication-100 facts involving two one-digit factors and their product – Commutativity – 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 the missing factor in the multiplication problem – Repeated Subtraction

An Activity for Application The following slide outlines an activity that can be used to teach the basic addition facts. After you finish playing this game with your partner, discuss your strategies and how you would use or adapt this game for use in your own classroom.

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?

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