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

"Area and Word Problems"


NJCCCS
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: Math Syntax

    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.



Of and And, Very Important Signals

      In math, as in English, AND is a signal that at least two ideas are involved. Very often these ideas also involve addition, but not always.

      In math, as in English, OF is a signal that information about 1 item is going to be stated.

      Here are expressions/equations involving of and/or and.

  • five and seven: 5, 7
  • the product of five and seven: (5)(7)
  • the sum of five and seven: 5 + 7
  • the product of two and a number: 2(x)
  • The area of a rectangle is the product of length and width.
  • The area of a rectangle is the product of length and width.
  • The square of the hypotenuse of a right triangle is equal to the sum of the squares of the two adjacent sides.

      Below is a partial list, with mathematical symbols, of operations/ functions which require information provided through an of prepositional phrase.

  • the opposite of: -( )
  • the absolute value of: | |
  • the reciprocal of: 1/( )
  • half of: ( )/2
  • the square of: ( )2
  • the cube of: ( )3
  • the square root of: ( )

      The preposition "of" also acts to nest or arrange information. See how the meaning of the phrase below are changed with the arrangement of the "of."



Summarize the Problem in Code

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

	   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
	


Complete Example 3.

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
 


This Material Is From
Azzolino, Agnes,
"Don't Loose Your Head over This One"
03-11-28
www.mathnstuff.com/papers/condots/whooo.htm.
"Don't Loose Your Head over This One - Answers"
03-11-28
www.mathnstuff.com/papers/condots/whoooa.htm.
"FINALLY PLEASE READILY EXCUSE MY DEAR AUNT SALLY"
2005,
www.mathnstuff.com/math/spoken/here/2class/80/c80pg00.htm.
"For An Afternoon Tradition"
© 00-7-10
www.mathnstuff.com/papers/condots/exr.htm.
"For An Afternoon Tradition -- Answers"
© 00-7-10
www.mathnstuff.com/papers/condots/exrans.htm.
"The Semi or Pseudo Expression or Equation"
The Languages of the Math Classroom
(ISBN: 1-929-870-01-9 1998, Agnes Azzolino)
www.mathnstuff.com/papers/langu/page7.htm.
"The Square of the Opposite" Vs. "The Opposite of the Square"
© 2008,
www.mathnstuff.com\math\algebra\tt09.htm
www.termtiles.com
"writing one operation equations, expressions, statements"
2001, 2007,
www.mathnstuff.com/math/algebra/awrite1.htm.
 
NJ Core Curriculum Content Standards
UPDATE at http://www.nj.gov/education/assessment/updates/update120309.pdf


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