Operations Dr. Hailey Developing Meaning and Use of Story Problems

BIG IDEAS Addition can be thought of as physically or conceptually placing two or more quantities together Subtraction can be thought of as taking an amount away from a given quantity, comparing two quantities, or finding a missing part given the whole and the other part

BIG IDEAS Multiplication in grades PK-2 involves counting groups of equal size and determining how many in all Division in grades PK-2 can be thought of as sharing equal amounts among a given number of groups or as repeatedly measuring out the same amount from a given total

BIG IDEAS The operations are related to one another. Addition names the wholes in terms of its parts, and subtraction names a missing part. Multiplication can be thought of as repeated addition. Division names a missing factor in terms of the known factor and the product. Division can be thought of as repeated subtractions. Models can be used to solve contextual problems for all operations

Teaching Operations through Contextual Problems Help children construct rich understanding of the operations Think about children using words, manipulatives, pictures, and numbers to explain how they went about solving a problem **Problem-solving, reasoning and proof, communication, representation, and connections

Teaching Operations through Contextual Problems Important! Explain what they did Why it makes sense Represent and discuss so others can understand top_640x360_ccv2/ab/streaming/myeducationl ab/imap/0525_iPad.mp4 top_640x360_ccv2/ab/streaming/myeducationl ab/imap/0525_iPad.mp4

Addition and Subtraction Understand simple patterns Understand addition as adding to Understand subtraction as taking from Understand addition as putting together and adding to Understand subtraction as taking apart and taking from Kindergarten Pre-Kindergarten

Addition and Subtraction 1 st grade Represent and solve problems Add and Subtract- 20 Addition and subtraction equations Properties of operations Relationships between adding and subtracting 2 nd grade Represent and solve problems Add and Subtract-100 Add and Subtract within 20 fluently using mental strategies Work with equal groups of objects to gain foundations for multiplication

Addition and Subtraction Problem Structures Join/Add to=quantities are physically put together Separate/Take from=part of a quantity is physically taken away

Addition and Subtraction Problem Structures Part-part-whole=Two different things put into group What is known? Opportunities! Examples: Whole unknown—Sam has 4 pennies and 8 nickels. How many coins does he have? One part unknown—Sam has 12 coins. Eight of his coins are pennies and the rest are nickels. How many nickels does Sam have? Both parts unknown—Sam has 12 coins. Some are pennies and some are nickels. How many of each could he have?

Addition and Subtraction Problem Structures Comparison =comparing two quantities. The unknown is the difference between the two amounts Difference unknown—Sam has 12 pennies and Gina has 8 pennies. How many fewer pennies does Gina have than Sam? Bigger unknown—Gina has 8 pennies. Sam has 4 more pennies than Gina. How many pennies does Sam have? Smaller unknown—Sam has 12 pennies. Sam has 4 more than Gina. How many does Gina have?

Typical development First children will do the problem in the order that it occurs—“semantic” Gina has some pennies. Sam gave her 4 more. Now Gina has 12 pennies. ____+4=12 As adults we really think of this problem as 12-4=8 and we can teach our students to do the same

Strategies for Solving Developmental sequence *Direct Modeling *Counting Strategies *Derived Facts Examples of each of these strategies to determine answer to Maggie had 7 bracelets. She bought 8 more bracelets. How many bracelets does Maggie have now?

Strategies for Solving Find the counters on your table. Use them to model each of the following word problems: 1.Todd had 7 cars. He got 8 more cars as a gift. How many cars does Todd have now? 2.Todd had 7 cars. He got some more as gift. He now has 15 cars. How many were in his gift? 3. Todd had some cars. He got 8 more cars as a gift. Now he has 15 cars. How many cars did Todd start with? Which do you think would be more difficult? Why?

Problem Difficulty Easiest-unknown result 7+8=? More difficult-unknown change 7+?=15 Most difficult-unknown start ?+8=15 We need to give children many opportunities for each type to develop proficiency

Problem Difficulty Part-whole-part problems can be difficult because Some children have difficulty putting two smaller categories into on large category-Then one thing belongs in two places!! 3 oranges+4 apples =how many pieces of fruit?

Problem Difficulty Comparison language may be unfamiliar to children Fewer Less than More Bigger Greater

Problem Difficulty The numbers you choose: Look at CCSS to determine the number range your grade should be working Can the numbers be represented with numerals--tens and ones? Draw a picture. Can the numbers easily be decomposed into tens and ones? Can the numbers easily use 5s and 10s as starting points? Can the numbers be noted on a number line?

Problem Difficulty A school of 28 fish was swimming together. Another school of fish decided to join them. The new larger school of fish had 54 fish. How many fish were in the second school of fish? Decompose into tens and ones. Use 5s and 10s as starting points Use a number line to jump Use a hundreds chart to find

Introducing Symbolism + “and” - “subtract” or “minus” You can explain it as taking away but don’t substitute the phrase “take away” for the – because not all subtraction situations are take away = “equals” or “is the same as” (not the end of the problem)

The Commutative Property Commute means to travel or to move around Addition--the numbers being added can move around in position and the sum will remain the same =80 and 50+30= = = =12

The Commutative Property Fold a piece of paper in half Now fold it the other way in thirds. Unfold. You should have 6 “boxes” Put each of these series of numbers in one box each *4+7+6 * *3+7+7 * *5+9+9 * Show your adding strategies and discuss.

Zero Property Sometimes children think that a + sign means the answer is going to be bigger OR That – sign makes numbers smaller. The best way to dispel these myths is to have lots of practice adding and subtracting 0

Laying the Foundation for Multiplication and Division Young children can multiply and divide in context long before they use symbols and are formally introduced to multiplication and division.

Laying the Foundation for Multiplication and Division Multiplication-counting equal groups Division-fairly sharing or measuring out

Children’s Strategies Many of the same tools and strategies used for addition--Number line, hundreds chart, manipulatives, BUT an important tool is the array Groupings in multiple

Arrays nothing

Literature and Music Connection One Hundred Hungry Ants by Elinor J. Pinczes “The Ants Go Marching One-by-One”-Traditional =fvwp