October 4 Educ 325 Developing Whole Number Operations and Reading Fluency

Developing Operation Sense Recognizing real-world settings for each operation Developing an awareness of models and properties of each operation Recognizing relationships among the operations Understanding the effects of an operation

Curriculum Focal Point Develop strategies for adding and subtracting whole numbers on the basis of earlier work with small numbers (digits 0-9) Use a variety of models, including discrete objects, length-based models (e. g., lengths of connecting cubes), and number lines, to model “part-whole,” “adding to,” “taking away from,” and “comparing” situations to develop an understanding of the meanings of addition and subtraction and strategies to solve such arithmetic problems

Introduce Operations with Word Problems Learning about operations should be based on meaning and understanding This begins with an exploration of real world settings or story problems Children have already constructed meaning for mathematical operations before they enter school through real-life, informal experiences such as sharing cookies or combining collections of cars or Barbies

Real Life Word Problems Allow students to see more personal relevance which enables them to more easily analyze the problem and its component parts Routine, opening activities like the calendar can provide problem contexts Children’s literature, classroom events, and sharing materials all provide meaningful, engaging problem contexts

Model for introducing word problems Real World Setting or Problem MODELS Concrete Pictorial Mental Images Language Symbols Written text

Understanding Addition and Subtraction – Types of Problems (pp ) Join Problems – items are added or joined to a set the starting amount, the change amount, and the resulting amount Separate Problems – items are removed from a set the starting amount, the change amount, and the resulting amount

More problem types Part-Part-Whole problems – there is no action but rather a relationship between a set and its two subsets Involves three quantities, the whole and two parts Compare Problems – there is no action, instead they involve comparisons between two different sets Three quantities are two wholes and a difference

Using models to solve addition and subtraction problems Direct Modeling using real world materials or concrete manipulatives Using Measurement Models (p. 136) - lengths are used to represent quantities in the problem

AFTER MUCH MODELING – Writing Number Sentences p. 137 Concrete model Semiabstract Model Symbolic

Understanding Multiplication and Division Fundamentally different from addition and subtraction due to the different types of quantities represented Problem 1: Peter has 2 cookies Amy gives him 3 more. How many cookies does Peter have now? Problem 2: Peter has 2 bags with 3 cookies in each bag. How many cookies does Peter have?

How are problems similar and different? Both have numbers 2 and 3 but what do the numbers “2” and “3” represent in each problem? Multiplication is more complex because there are more factors to pay attention to Multiplication also requires a different type of counting

Making transition from Adding to Multiplying Encourage children to use the phrase “groups of” to indicate creating a number of EQUAL GROUPS Help children understand the meaning of each quantity 2 x 3 is two groups of 3 objects 3 x 2 is three groups of 2 objects

Types of Multiplication Problems Equal groups problem – based on making a certain number of equal-sized groups The three numbers in the problem represent the number of groups, the size of groups, and the total number of objects Area and Array problems – finding the area of a rectangular region or finding the total number of objects in a rectangular display Area can be found by counting OR by multiplying the length by the width

Multiplicative Comparison Problem – involves comparing two quantities multiplicatively Describe how many times as much one quantity is compared to another Combination problems – involves different combinations that can be made from sets of objects Most difficult type of multiplication or division problem to model

Introducing children to division Clarity and familiarity of language used in division problems is very important Children experience division throughout their everyday life but they might not recognize it when they are sharing food or toys with others Use familiar language such as “shared by” or “equal groups” to make sense in children before ever introducing term “divided by”

Division with Remainders Children experience real life situations with remainders in sharing cookies and having some “left over” – engage in discussion about what to do with the cookie that is left over This can lead to discussions about what to do with “leftovers” in division Sometimes leftovers will just be set aside and sometimes it may be appropriate to make unequal groups (put 5 children in one car and 4 children in the other)

Types of Division Problems Equal Groups Problems – splitting a larger group into several smaller groups The three numbers represent the number of groups, the size of the groups, and the total number of objects PARTITIVE division problems – total number of objects is partitioned into a specific number of groups – also known as fair sharing MEASUREMENT division problems – total number of objects is measured out into groups of a certain size – also known as repeated addition

Area and Area problems- involve finding one of the dimensions of a rectangular region or rectangular array when the total number of objects in the arrangement is given Multiplicative Comparison problems –how many times as much one quantity is compared to another quantity, is known, as is one of the quantities being compared Eric ate twice as many cookies as Patrick did. If Eric ate twice 18 cookies, how many cookies did Patrick eat?

Combination problems – involve different combinations that can be made from sets of objects, such as the number of outfits that can be made from 2 shirts and 3 pairs of pants. In division, the total number of combinations is known, as the number of one of the objects being combined How many pairs of pants are needed to make 6 different outfits by using 2 shirts?

Modeling to Solve Multiplication and Division problems Pp Key Language Terms pp. 146