Developing Operation Sense

Key Aspects to Developing Operation Sense

  • Recognizing real-world settings for each operation

  • Awareness of models and properties

  • Recognizing relationships among operations

  • Understanding the effects of an operation

    • Having conversations with students can help them understand this

A Model for Introducing Operations with Word Problems

Word Problems

Should come from familiar real-world experiences

  • Everyday classroom opportunities

    • Calendar

    • Morning routine

    • Classroom events

    • Children’s Literature

  • Allow children to make decisions about sharing and distribution of classroom materials.

  • Avoid pressure to memorize basic facts!

    • Meaning develops over time

    • Refrain from using “keyword” strategies

      • because the words can be used in different ways to mean different things

Types of Addition and Subtraction Word Problems

  • Join and Separate: action words

  • Part-Part-Whole and Compare: no action words/action words don’t affect the quantities

Examples of Problem Types

  • Be able to tell what type of problem one is and be able to write a problem when given a type

Join:

  • start quantity, change quantity, end quantity

  • Join: Result Unknown (JRU)

  • Join: Change Unknown (JCU)- not telling what the change is explicitly

  • Join: Start Unknown (JSU)- not telling what you’re beginning with

Separate:

  • Separate: Result Unknown (SRU)

  • Separate: Change Unknown (SCU)- not telling what the change is explicitly

  • Separate: Start Unknown (SSU)- not telling what you’re beginning with

Part-Part Whole:

  • there are no action words

  • whole unknown: the whole (end part) is unknown

  • part unknown:

Compare:

  • Difference Unknown: create a train and see the difference between the two trains

  • Larger Unknown

  • Smaller Unknown

Structure of Addition and Subtraction Problem Types

  • can use an organizational map with students, especially if they’re struggling

Strategies Children Use to Solve Story Problems

  • Direct Modeling: representing quantities with concrete objects

  • Counting Strategies:

    • counting on, counting back, skip counting

  • Derived Facts: to build on or modify facts that are already known

  • Recall: retrieving facts already known to solve the problem quickly

  • Inverse operations: something that will undo what is done— addition and subtraction, multiplication and division.

Transition from Adding to Multiplying

  • use the phrase “groups of” to indicate creating a number or equal groups

  • understanding the meaning of each quantity

    • communicative property: you can change the order of the numbers and get the same answer

      • The associative property states numbers can be regrouped with addition or multiplication, and the answer will not change.

      • The commutative property states the numbers can change positions with addition or multiplication, and the answer will not change.

      • multiplication: factors

      • addition: addends

Types of Multiplication Word Problems

  • Equal Groups Problems: Based on making a certain number of equal–sized groups. The numbers in the problem represent:

    • Number of groups

    • Size of the groups

    • Total number of objects

  • Area and Array Problems:Involve finding the area of a rectangular region or finding the total number of objects in a rectangular array.

  • Multiplicative Comparison Problems: Involve comparing two quantities multiplicative.

    • How many times as much one quantity is compared to another quantity

    • “twice as much, — times more”

  • Combination Problems (Cartesian products): Involve different combinations that can be made from sets of objects.

  • how many different types of sundaes can be made with the 3 different toppings and 3 different flavors?

Types of Multiplication Problems

  • Number of groups: 2 bags of oranges

  • Size of the groups: 3 oranges in each bag

  • Total number of objects: unknown, total size we are looking for

    • always trying to find an unknown number of objects

Types of Division Word Problems

  • Equal Groups Problems: Involve splitting a larger group into several smaller groups. Numbers in the problem represent:

    • Number of groups

    • Size of the groups

    • Total number of objects

  • Partitive Division Problems – the total number of objects is partitioned into a specific number of groups.

  • Measurement Division Problems – the total number of objects is measured out into groups of a certain size.

  • Area and Array Problems: Involve finding one of the dimensions of a rectangular region or of a rectangular array when the total area or total number of objects in the arrangement is given.

  • Multiplicative Comparison Problems: Involve comparing two quantities multiplicative.

  • Combination Problems (Cartesian products): Involve different combinations that can be made from sets of objects.

    • moving where the unknown is

Division Problem Types

  • 6 oranges: total

  • 2 bags: number of groups

  • number of oranges in the bag: unknown

    • in every problem, you’re solving for the size of the group

Avoiding Misconceptions and Dead Ends

Misconceptions and instructional “dead ends” with multiplication and division:

  • “Multiplication makes bigger”

  • “Division makes smaller”

  • Division means only fair shares (or partitive division)

Modeling Multiplication and Division:

Developmental levels in understanding multiplication

Problem: “If 8 plates hold 4 cookies each, how many cookies are on all the plates?”

  • Level 1:A child sets out 8 plates, puts 4 cookies on each plate, and counts the total number.

  • Level 2: A child makes 8 groups of 4 without using separate objects for plates.

  • Level 3: A child makes one group of 4 and recounts it 8 times, keeping track of how many groups have been counted by using fingers or another memory device.

CRA:

  • concrete: using physical objects

  • representation:

  • abstract: thinking in their heads, skip-counting

  • Advanced levels include

    • Counting by fours

    • Counting on when they cannot recall the next multiple (4, 8, …9, 10, 11, 12)

    • Adding fours

    • Using derived facts: using what you know to figure out another fact

      • the video of the boy doing 12×6 and taking 12×5=60 and adding on 12 to figure it out.

For the Test:

  • know the types of problems and their structures

  • be able to create your own problems