Competency
002 (Number Concepts and Operations):
The
educator knows concepts related to numbers, operations and algorithms and the
properties of numbers.
The teacher:
A.
Knows how to create, describe, analyze, compare, and model relationships
between operations, algorithms, and number properties for the four basic
operations involving rational numbers, real numbers, and integers, including
real-world situations.
Four basic operations: addition, subtraction,
multiplication, division
Algorithm: a solution with specific steps you
follow (if you follow those steps, you end up with the correct solution). The solution for a long division problem
(with specific steps you take to reach it) are an example of an algorithm.
Number
properties (commutative properties, associative properties, distributive
property, density property, and identity property)
Commutative Properties:
Commutative Property of Addition -
numbers can be added in any order without changing the result
Example: 5+3+2 = 3+2+5
Commutative Property of
Multiplication - numbers can be multiplied in any order, and the result will
remain the same
Example: 5n
× 2 = 2 × 5n
Memorization
tip: When you commute, you are moving- going
from home to work. The commutative
property involves moving numbers around.
Associative Property of Addition - numbers
can be grouped in a sum in any way (using parenthesis) and still get the same
result
Examples:
(3 + 1) + 2
= 3 + (1 + 2)
(2n + 4n) +
6 = 2n + (4n + 6)
Associative Property of
Multiplication - numbers can be grouped in a product in any way (using
parenthesis) and still get the same result
Examples:
(3 × 5) × 2
= 3 × (5 × 2)
5x3(20y)
= 20y (5x3)
Distributive Property - this will
only happen when an expression includes both addition and multiplication. When you have a term multiplied by terms in
parenthesis, you have to distribute the multiplication over the terms inside
the parenthesis.
Examples:
3n (n +5) =
3n2+15n (you have to multiply
3n×n and 3n×5)
2(2+1) = (2
× 2) + (2 × 1)
Rational numbers - any number that can be expressed as the fraction or
quotient x/z of two integers, and the denominator z cannot be equal to
zero. Since z can be equal to 1, all
integers are rational numbers. When
represented as a decimal, a rational number will either terminate (end) after a
finite number of digits or will begin to repeat the same finite sequence of
numbers.
Irrational numbers - any real number that cannot be
represented as a simple fraction or a ratio of two integers. They will not have repeating decimals or
terminal decimals.
Real numbers - any value that can be represented
on a number line, including both rational and irrational numbers.
Integers - whole numbers; numbers that are
not fractions and do not include decimals; can be positive or negative
B. Shows an understanding of
equivalency between mathematical expressions and among different
representations of rational numbers.
Understanding Equivalency Between Mathematical
Expressions
Some
examples of equivalent mathematical expressions:
3/4 = .75
6(3n2)
= 18n2
(4*2) + 8n +
9n2 + 5n = 8 + 13n + 9n2
Understanding Equivalency of
Different Representations of Rational Numbers
Examples:
3 ÷ 4 = .75
100.75 = 100
¾
1. Mr. Jackson posts these numbers on the board. He then asks different students what type of
numbers these are: integers, rational numbers, irrational numbers, or negative
numbers. He asks students to raise their
hand to indicate which type of number it is.
Out of 24 students, 22 raise their hands to indicate these numbers are
irrational numbers. Based on this
information, Mr. Jackson should:
3.33333
1.34343434
100.75
A. Develop a formal assessment that students take
individually so they are not influenced by seeing other students raise their
hands.
B. Explain to the students why their answer is
incorrect and give them additional homework about integers, rational numbers,
irrational numbers, and negative numbers.
C. Since almost all students answered correctly,
commend them and move on to the next lesson.
D.
Find a new method to re-teach rational and irrational numbers, give students
additional opportunities to learn and practice, and then ask individual
students to come to the board to explain why certain numbers written on the
board are integers, rational numbers, irrational numbers, and negative numbers.
Explanation: All of those numbers are rational
numbers, so the majority of the class answered the question incorrectly. D is correct because he needs to re-teach
this concept and give students more practice, and having students come to the
board to explain their thought process will help him know what students
understand, and will also help students learn from each other. A is incorrect because Mr. Jackson does not
need to do a formal assessment- he already knows students do not
understand. B is incorrect because
simply telling students why their answer was incorrect and giving them homework
is not likely to help students understand integers, rational, irrational, and
real numbers. C is incorrect because
most of the students answered incorrectly.
2. Which of the following math problems demonstrates the
commutative property of addition?
A. 3+1 = 2 + 2
B.
3+1= 1+3
C. (3+1) +2=3+ (1+2)
D. 3x + x = 3x + x + 0
Explanation: B is correct because the commutative
property of addition states that numbers can be added in any order and the
result will be the same. None of the
other options demonstrate that.
