Effective Collaboration/Communication, Effective Instruction, and Rigor as We Prepare for the HSPA/NJASK - Grades 6-8, 1/2012

• Examine the NJCCCS
   
• Review Order of Operations, Math Syntax
   
• Review Of and And, Very Important Signals  
   
• Summarize the Problem in Code
   

Standard 4.5 – Mathematical Processes Grades 6, 7, 8

COMMUNICATION

___ use reading, writing, discussion, listening and questioning

                to organize and clarify their mathematical thinking

___ communicate their mathematical thinking coherently and clearly

                to peers, teachers, and others, both orally and in writing

___ analyze and evaluate the mathematical thinking and strategies of others

___ use the language of mathematics to express mathematical ideas precisely

REASONING

___ recognize that mathematical facts, procedures, and claims must be justified

___ use reasoning to support their mathematical conclusions and problem solutions

___ select and use various types of reasoning and methods of proof

___ rely on reasoning, rather than answer keys, teacher, or peers,

                to check the correctness of their problem solutions

___ make and investigate mathematical conjectures

___ evaluate examples of mathematical reasoning and determine whether they are valid

REPRESENTATIONS

___ Create and use representations

                to organize, record, and communicate mathematical ideas

                through concrete, pictorial, symbolic and graphic representations

___ select, apply, and translate among mathematical representations to solve problems

___ use representations to model and interpret physical, social, and mathematical phenomena

    Order of Operations is literally the order in which operations are to be performed. The rules are understood by all who write and speak mathematics.

THE ORDER OF OPERATIONS (Topmost First)
• Functions
• Parentheses or other Marks of Inclusion (Innermost First)
• Roots or Exponents
• Multiplication or Division (Leftmost First)
• Addition or Subtraction (Leftmost First)

    Restated it's: Functions, Parentheses first, next Roots or Exponents, then Multiply or Divide (left to righ) then Add or Subtract (left to right): Finally Please, Readily Excuse, My Dear Aunt Sally.

    Restated in a more complete form math syntax is "FIRE My Dear Aunt Sally!" In formal language, top priority first, Functions, Parentheses (and other marks of inclusing, innermost first), next Roots or Exponents (easier first), then Multiplication or Division (leftmost first), then Addition or Subtraction (leftmost first).

    Here's a picture or gif file. Here's a 26-page tutorial on Order of Operations. Here's are two connect the dot puzzles: 1, and 2 and their answers 1 answers and 2 answers.

    The semi/pseudo code looks like this and is "half symbols, half words."

		     50 - 3(notebooks) - 2(CDS)

		 (first) + 3(second) + 4(third)	= 131
	

    Use the semi/pseudo code for the following reasons:

• To present a unified and complete representation of the problem.
• To facilitate, through this intermediate step, the transition from words to mathematical symbolic notation.
• To foreshadow computation on graphing or symbol manipulating calculators.
• Because it helps "decode" a word problem!

    Here are the rules:

• Use words (including nicknames) and arithmetic to express an idea.
• Perhaps encircle the words with parentheses for clarity.
• Use no variable numbers. Use only constant numbers.
• Use order of operations.

Examples 1 and 2

	   1. Compute the change from a fifty dollar bill for 
	the purchase of 3 notebooks at $2.90 each and 2 CDs 
	at $14.50 each.

		     50 - 3(notebooks) - 2(CDS)
		     50 - 3(2.90) - 2(14.50) 
		     $32.60 is the change

	   2. Three consecutive integers are involved.  The 
	sum of the first, triple the middle, and four times 
	the largest is 131.  Find the integers.

		 (first) + 3(second) + 4(third)	= 131
		 (x)     + 3(x + 1)  + 4(x + 2)	= 131
				      8x + 11	= 131
				      8x		= 120
				      x		= 15
			the numbers are 15, 16, 17
	

Write the code and the equation and compute the area.

Compute the area.

    Answer

Area Problems:  

2, 2 answer, 3, 3 answer

Integer Problems:  

1, 1 answer, 2, 2 answer 3, 3 answer

Money and Percent Problems:

1, 1 answer, 2, 2 answer, 3, 3 answer, 4, 4 answer,

5, 5 answer, 6, 6 answer, 7, 7 answer

Other Kinds of Expression Problems:  

1, 1 answer, 2, 2 answer, 3, 3 answer