CHAPTER 9 Meanings for Operations

ADDITIVE STRUCTURES  JOIN  Involves a direct or implied ACTION in which a set is increased by a particular amount. Set Model Measurement Model

ADDITIVE STRUCTURES  JOIN  There are 3 quantities involved: 1) Initial amount 2) Change amount 3) Resulting amount  EXAMPLES: 1) Caroline has 4 jelly beans. Melissa gives her 5 more. How many jelly beans does Caroline have all together? 2) Caroline has 4 jelly beans. Melissa gave her some more. Now Caroline has 9 jelly beans. How many jelly beans did Melissa give her? 3) Caroline has some jelly beans. Melissa gives her 5 more. Now Caroline has 9 jelly beans. How many jelly beans did Caroline originally have? Initial + Change = Result

ADDITIVE STRUCTURES  SEPARATE  Similar to Join problems in many respects. There is an action that takes place over time, but in this case the initial quantity is decreased rather than increased. Set Model Measurement Model

ADDITIVE STRUCTURES  SEPARATE  There are 3 quantities involved: 1) Initial amount 2) Change amount 3) Resulting amount  EXAMPLES: 1) David has 10 marbles. He gives 6 to Alexa. How many marbles does David have left? 2) David has 10 marbles. He gives some to Alexa. Now, David only has 4 marbles left. How many marbles did David give Alexa? 3) David has some marbles. He gives 6 to Alexa. Now, David only has 4 marbles left. How many marbles did David have to start with? Initial - Change = Result

ADDITIVE STRUCTURES  COMPARE  Comparison of 2 quantities. Set Model Measurement Model

ADDITIVE STRUCTURES  COMPARE  There are 3 quantities involved: 1) Larger amount 2) Smaller amount 3) Difference amount  EXAMPLES: 1) Katy has 7 apples and Martin has 3 apples. How many more apples does Katy have than Martin? 2) Martin has 3 apples. Katy has 4 more apples than Martin. How many apples does Katy have? 3) Katy has 7 apples. She has 4 more apples than Martin. How many apples does Martin have? Larger – Smaller = Difference

ADDITIVE STRUCTURES (CONT.)  PART-PART-WHOLE  Involves a static relationship between 2 sets. There is no direct or implied action. Set Model Measurement Model

ADDITIVE STRUCTURES (CONT.)  PART-PART-WHOLE  There are 3 quantities involved: 1) Part 1 2) Part 2 3) Whole Amount  EXAMPLES: 1) Ashley has 5 pennies and 6 nickels. How many coins does she have all together? 2) Ashley has 11 coins, made up of pennies and nickels. 5 of her coins are pennies. How many are nickels? Part + Part = Whole Whole – Part = Part

COMMUTATIVE & ASSOCIATIVE PROPERTIES OF ADDITION  Read about the commutative, associative, and zero properties of addition on page 177.  We talked a little bit about the importance of these properties already.  For example, how does the commutative property for addition relate to say counting on from first versus counting on from larger?  For example, how does the associative property relate to derived number facts for addition?

SEMANTIC AND COMPUTATIONAL FORMS OF EQUATIONS  A semantic equation is one for which the numbers of a problem are listed in the order that follows the meaning of the word problem.  A computational equation is one for which the unknown is isolated.  For example, ? + 5 = 12 might be the semantic form for a join problem, while 12 – 5 = ? is an equivalent computational form.  Students need to see that there are several ways to represent a situation as an equation.

SYMBOLISM  Contrary to your experience, children do not need to understand the symbols +, -, and = to be able to learn about addition and subtraction.  By first grade, the usage and understanding of these conventions becomes esential.  Help your students begin to understand that symbols are a way to record what they did as they share their thinking.  Make sure you say subtract or minus for the symbol“-”. Resist the temptation to say “take away”. The reason is that some problems such as compare problems don’t involve a “take away” action yet are still subtraction.  Make sure you reflect on the usage of the equal sign. Most students think it’s all about the “answer is coming up” rather than it being read as “the same as”.  For example, =  For example, 7 = 2+5.  For example, ? + 5 = 12.

CGI VIDEOS  Classify the story problem using the various additive problem structures.  Identify the child’s strategy.  Write a semantic and/or computational equation for the problem.