Expectations from the Number and Operations Standard

1
Expectations from the Number and Operations
Standard

• Principles and Standards for School Mathematics
• National Council of Teachers of Mathematics 2000

2
Understand numbers, ways of representing numbers,
relationships among numbers, and number systems.

• Grades 3-5
• Describe classes of numbers according to
  characteristics such as the nature of their
  factors.
• Grades 6-8
• Use factors, multiples, prime factorization, and
  relatively prime numbers to solve problems.
• Develop meaning for integers and represent and
  compare quantities with them.

3
Understand meanings of operations and how they
relate to one another.

• Grades 6-8
• Understand and use the inverse relationships of
  addition and subtraction, multiplication and
  division, and squaring and finding square roots
  to simplify computations and solve problems.

4
The Algebra Standard

• Principles and Standards for School Mathematics
• National Council of Teachers of Mathematics
• 2000

5
Instructional programs from pre-kindergarten
through grade 12 should enable all students to

• Understand patterns, relations, and functions.
• Represent and analyze mathematical situations and
  structures using algebraic symbols .

6
Instructional programs from pre-kindergarten
through grade 12 should enable all students to

• Use mathematical models to represent and
  understand quantitative relationships.
• Analyze change in various contexts

7
Three Aspects of Algebraic Thinking

• Equality
• Change
• Generalizations

8
Types of Patterns

• Repeating Patterns-have a core with a repeating
  element
• ab-two repeating elements in core.
• Growing Patterns-have a core with a growing
  element
• YBBYYBBBYYYBBBB

9
Number Patterns

• Circle 2 on your hundred board, then cover or
  mark out every multiple of 2 remaining on the
  board.
• Circle 3 on your hundred board (the next unmarked
  number following 2). Cover or mark out every
  multiple of 3 on the board, noting some have
  already been marked during the first step.

10
Number Patterns (cont)

• What is the next unmarked number at the top of
  your board? Cover or mark out the multiples on
  the board.
• Continue the process until the entire board is
  circled or marked.
• Is there a pattern in the circled numbers? What
  is it?

11
Consider the fractions 1 2 3 4 5 6
7 8 99 9 9 9 9 9 9 9 9

• Think of each fraction as a division problem and
  compute each quotient.
• Do you observe anything of interest?
• Suppose a different denominator had been
  selected, for example 11.

12

• Look at the fractions through 1/11 through 6/11
  repeating the process outlined above.
• Do you observe any patterns?
• Can you predict 7 11 from the information
  gathered?

13
Palindromes

• These are word palindromes
• WOW TOOT NOON
• Are the following number palindromes?
• 6556 1231 44 232 9887
• How would you define palindrome? Circle all
  palindromes on the hundred chart.

14
Palindromes

• Consider 57. Can we transform 57 to form a
  palindrome?
• Start with 57 75 132 (not a palindrome)
• Try 132 231 363 (Palindrome!)
• We call 57 a two-step palindrome.

15
Palindromes

• Choose another number and see how many steps it
  takes to form a palindrome.
• Investigate other numbers on the hundred board.
  Mark one-step, two-step, three step palindromes
  using different colors for each category. Is
  there a pattern?