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Elementary Education K-6
Mathematics
Assessment of these competencies and skills will use real-world problems
when feasible.
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7 Knowledge of number sense, concepts, and
operations
"I write rhymes with addition and algebra, mental geometry."
-- Ice-T
1. Associate multiple representations of numbers using word names,
standard numerals, and pictorial models for real numbers (whole
numbers, decimals, fractions, and integers).
Term:
Example:
Whole Numbers
Counting Numbers (1, 2, 1000)
Factions
a/b where a is the numerator and b is the
denominator (2/3)
Dividing Fractions;" The number you're dividing by,
Turn upside-down
and multiply "
and integers).
Decimals
fractions written as powers of 10 (25/100 = 0.25)
Integers
+7, -10
Word Names
One, Two, Five-Hundred
Standard Numerals
1,
Pictorial Models
2,
500
Using pictures to represent numbers
Write an example for each real number: 11, 35, 8
Whole number
Decimal
Integer
Word names
Standard
numerals
Pictorial
2. Compare the relative size of integers, fractions, and decimals,
numbers expressed as percents, numbers with exponents, and/or
numbers in scientific notation.
Fractions, Decimals, Percents
Converting from fractions to decimals to percents:
Operation
Explanation
Example
Convert a
decimal to a
percent
Move the decimal point 2 places to the right
and add a percent (%) sign. If you need to,
add a zero on the back to get the second
decimal place.
.123 =
12.3%
Move the decimal point 2 places to the left. If
you need to, put a zero on the front.
5% = .05
Divide the numerator by the denominator,
using your calculator.
1/8 = .125
Convert a
percent to a
decimal
Convert a
fraction to a
decimal
Convert a
percent to a
fraction
First turn the number into a decimal. Then turn
18% = .18
the number into a fraction by putting it over
= 18/100 =
10, 100, 1000, or whatever number is big
9/50
enough to have enough zeroes for each place
after the decimal.
See http://www.marthalakecov.org/~math/fr_dec_pct.html for additional
examples.
Exponents
Exponential notation is shorthand for repeated multiplications:
103, 36, and 18 as an exponent, in factor form and in standard form:
Exponential Factor
Standard
Form
Form
Form
103
10 x 10 x 10
1,000
36
3x3x3x3x3x3
729
18
1x1x1x1x1x1x1x1 1
Rules for numbers with exponents of 0, 1, 2 and 3:
Rule
Example
Any number (except 0) raised to the zero power is equal to 1.
1490 = 1
Any number raised to the first power is always equal to itself.
81 = 8
If a number is raised to the second power, we say it is squared. 32 is three
squared
If a number is raised to the third power, we say it is cubed.
43 is four
cubed
Scientific Notation
Scientific notation is a way of writing very large and very small numbers
more easily.
5,234 = 5.234 x 103
View this quick video on Scientific Notation from United Streaming.
To login, use your B.E.E.P. username and password:
Video:
http://player.discoveryeducation.com/index.cfm?guidAssetId=8E41
5256-A36C-4D02-A45AD06E54030CC7&blnFromSearch=1&productcode=US
3. Apply ratios, proportions, and percents in real-world situations.
Ratio is the relationship of one number to another
to
4:3
Proportion refers to the relationship between two equivalent ratios
Quick video on proportions:
http://viewer.nutshellmath.com/?solution=67-33-71-41-77-31
Write each as a decimal, fraction, percent, exponent, in Scientific Notation
and Ratio: 12, -240
Decimal
Fraction
12.0
-240.0
Percent
Exponent
Scientific Notation
Ratio
Use Virtual Manipulatives to help solve or convert:
http://nlvm.usu.edu/en/nav/topic_t_1.html
4. Represent numbers in a variety of equivalent forms, including whole
numbers, integers, fractions, decimals, percents, scientific notation, and
exponents. Equivalent Forms:
Write the numbers 25, and -1090 in each equivalent form:
Whole number
Integer
Fraction
Decimal
Percent
25
-1090
Scientific Notation
Exponent
Solve.
In her science class, Leanne was doing an experiment that involved weighing
small objects in grams and then converting those weights into ounces. Her
teacher told her that each gram is equal to 0.035 ounces. Which is
equivalent to 0.035?
a. 7
20
b. thiry-five thousandths
c. 35%
d. 35
100
5. Recognize the effects of operations on rational numbers and the
relationships among these operations (i.e., addition, subtraction,
multiplication, and division). Rational numbers
Rational numbers a real number that can be written as
a ratio of two integers (fraction) excluding zero as a denominator
a/b (a/0 is not allowed)
a repeating or terminating decimal
0.33333
an integer -24
Operations:
Addition (sum)
Subtraction (difference)
Multiplication (product)
Division (quotient)
Order of Operations PEMDAS (Please Excuse My Dear Aunt Sally)
Parenthesis
Exponents
Multiplication*
Division*
Addition**
Subtraction**
When multiplying and dividing, solve whichever comes first from left to right.
Example:
5x9/3x4
15 ÷ 5 / 3 x 4
45 / 3 x 4
3 / 12
45/12
When adding and subtracting, solve whichever comes first from left to right.
Example:
5+9/3+4
9–5/3+4
14 / 3 + 4
4/7
14/ 7
Relationships of operations using integers:
Adding Integers:
Adding with Same signs (find the SUM):
(+) + (+) = +
(-) + (-) = -
(+4) + (+5) = (+9)
(-3) + (-6) = (-9)
Adding with Different signs (find the DIFFERENCE):
Find the difference (subtract) and take the sign of the larger number
(+) + (-) =
Find difference, take sign of larger number
(+5) + (-7) = (-2)
(-) + (+) = Find difference, take sign of larger number
(-4) + (+6) = (+2)
Subtracting Integers
To subtract an integer, add it’s OPPOSITE
5-2=3
5 + (-2) = 3 (Change sign to +)
- 3 – 4= -7
-3 + (-4) = -7
Multiplying Integers
When you multiply two integers with the same signs, the result is always
positive.
+
x
x + = + (Same signs = Positive)
- = + (Same signs = Positive)
When you multiply two integers with different signs, the result is always
negative.
x + = - (Different Signs = Negative)
Dividing Integers
When you divide two integers with the same sign, the result is always
positive.
+
/ + = + (Same signs = Positive)
2=5
/
- = + (Same signs = Positive)
-10 / -2 = 5
When you divide two integers with different signs, the result is always
negative.
/ + = - (Different Signs = Negative)
-10 / 2 = -5
10 / -2 = -5
6. Select the appropriate operation(s) to solve problems involving ratios,
proportions, and percents and the addition, subtraction, multiplication, and
division of rational numbers.
Ratios:
A ratio is a comparison of two numbers.
Example:
Jeannine has a bag with 3 videocassettes, 4 marbles, 7 books, and 1 orange.
1) What is the ratio of books to marbles?
Expressed as a fraction, with the numerator equal to the first quantity and
the denominator equal to the second, the answer would be 7/4.
Two other ways of writing the ratio are 7 to 4, and 7:4.
2) What is the ratio of videocassettes to the total number of items in the
bag?
There are 3 videocassettes, and 3 + 4 + 7 + 1 = 15 items total.
The answer can be expressed as 3/15, 3 to 15, or 3:15.
Proportions:
A proportion is an equation with a ratio on each side. It is a statement that
two ratios are equal (3/4 = 6/8 is an example of a proportion).
When one of the four numbers in a proportion is unknown, cross products
may be used to find the unknown number. This is called solving the
proportion. Variables are frequently used in place of the unknown number.
Example:
Solve for n: 1/2 = n/4.
Using cross products we see that 2 × n = 1 × 4 =4, so 2 × n = 4. Dividing
both sides by 2, n = 4 ÷ 2 so that n = 2.
7. Use estimation in problem-solving situations. Estimation:
Estimation is determining an approximate or rough calculation based on
rounding.
Example:
Problem:
You ate at a restaurant and the bill you will pay is $28.90. You want to leave
a tip of 15% of the bill. What is the equivalent amount of the 15% tip that
you want to leave?
Solution:
1) You can round off $28.90 to $30 since it is very close to that amount.
2) Using 10 as our reference number, divide $30/10 = $3. You will retain the
$3.
3) Divide $3/2 = $1.50.
4) Then add $3 + $1.50 = $4.50, which is the estimated tip you will pay for
a $28.90 bill.
5) If you use a calculator to compute the answer for the 15% tip from a
$28.90 bill, you will get $4.33, which is close to our estimate of $4.50.
8. Apply number theory concepts (e.g., primes, composites, multiples,
factors, number sequences, number properties, and rules of divisibility).
Prime
A number that has exactly two factors, 1 and itself
Composite
A number with more than 2 factors
Multiples
A number added to itself a number of times
Factors
number
A whole number that divides exactly into another
Number sequences
term
Numbers expressed in a pattern in the nth
Number properties
Commutative, Associative, Distributive,
Rules of divisibility
A number is divisible by two if it is even.
A number is divisible by three if the sum of the digits adds up to a
multiple of three
A number is divisible by four if it is even and can be divided by two twice.
A number is divisible by five if it ends in a five or a zero
A number is divisible by six if it is divisible by both two and three.
A number is divisible by nine if the sum of the digits adds up to a
multiple of nine. This rule is similar to the divisibility rule for three.
A number is divisible by ten if it ends in a zero. This rule is similar to the
divisibility rule for five
Write two examples for each category:
Prim
Composit
Multiple
Factor
Number
Number
Rules of
e
e
s
s
Sequenc
e
Propertie
s
Divisibilit
y
9. Apply the order of operations. (Please Make Deliveries After Sunset)
(Patrick Makes Delicious Apple Smoothies)
Order of Operations
The precedence set for solving math equations.
In what order should the operations be performed in the expression below?
Solve.
6 x (3 + 2) ÷ 3 - 1
a. +, x,Π,b. +, x,-,Π
c. +,-, x,Π
d. x,+,Π,-
What is the value of the expression 2 + 5 X 32 ?
32
C.227
47
d. 441
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K
1st
2nd
3rd
4th
5th
add
after*
before
between
(M)
add*
addition (M)
after*
before*
between* (M)
addition* (M)
after*
count
backward
count forward
addend*(M)*
*
array*(M)
denominator*
difference(M)
addend*(M)*
*
array*
calculate*
cardinal
billion
calculate*
cardinal
number*(M)
composite
equal*
even
numbers*
fourth(s)*
group*
(M)
half
(halves)*
hundred(s
)*
count (on)
difference (M)
double (plus
one)
equal*
estimate* (M)
even
numbers*
fact family
fourth(s)*
count (on)*
difference* (M)
double (plus
one)*
equal*
estimate* (M)
even
numbers*
fact family*
fourth(s)*
digit*
estimation*(M
)**
expanded
form*(M)
factor*(M)**
fraction*(M)*
*
minuend*
multiplicand*
number(M)
composite
number*(M)*
*
consecutive*
decimal
number*(M)*
*
denominator
*
number*(M)
**
consecutive*
decimal
number*(M)
**
decimal
point
digit*
dividend(M)
less than*
group * (M)
fraction
multiplication(
digit*
divisible* **
(M)
more than
(not)
equal
number*
(M)
number
line*(M)**
half (halves)*
hundred(s)*
less than* (M)
more than*
near (M)
(not) equal*
number* (M)
number line*
greater than
(M)
greatest
group* (M)
half (halves)*
hundred(s)*
less than* (M)
near* (M)
M)
multiples*(M)
**
multiplier*
number line**
numerator*
place value*
**
divisible* **
divisor**
estimation(M
)**
expanded
notation*(M)
extraneous
information*
divisor* **
equivalent
extraneous
information*
**
equivalent*
greatest
common
odd
number*
one(s)*
opposite
order
pattern*
(M) **
odd number*
one(s)*
opposite*
order*
pattern*(M)**
number* (M)
number line*
(M) **
odd number*
one(s)*
ordinal number
product*(M)*
*
regroup*
relative
size*(M)**
rounding*(M)
*
equivalent*
factor(M)**
fraction(M)**
inverse
operation**
factor (GCF)
inverse
operation*
**
least
common
(M) **
small(est)
subtract
rule
skip-counting*
small(est)*
(M)
pattern* (M)
**
solution*
subtrahend*
sum(M)**
million*
denominator
minuend*
(LCD)
multiples(M)* least
ten(s)*
zero* (M)
subtract*
subtract(ion)*
(M)
sum
ten(s)*
zero* (M)
regroup
rule*
set (M)
skip-counting*
small(est)*
subtract(ion)*
(M)
whole
numbers*(M)
*
multiplicand*
multiplier*
natural
numbers*
numerator*
operation*
common
multiple(LCM
)
million*
natural
numbers*
operation*
sum*
ten(s)*
whole number
(M) **
zero* (M)
**
ordinal
number*(M)
percent*(M)*
*
perfect
number*
place value*
**
prime
**
ordinal
number*(M)
percent*(M)
**
perfect
number*
prime
factorization
(M)
number*(M)* prime
*
product*(M)*
*
quotient*(M)
**
regroup*
relative
size*(M)**
number*(M)
**
quotient*(M)
**
reduce
remainder*(
M)
similarity*(M
remainder*(
M)
rounding*(M)
similarity*(M
)**
simplify*
)**
simplify*
solution*
subtrahend*
value*
whole
numbers*(M)
**
8 Knowledge of measurement
"We only think when confronted with a problem."
-- John Dewey
1. Apply given measurement formulas for perimeter= 2xL+2xW ,
circumference =pi(3.14) x diameter, area= length x width, volume = length
x width x height or displacement, and surface area = the sum of the areas of
all sides in problem situations. Perimeter
The distance around a
shape.
Circumference The distance around a circle.
Area
The size a surface takes up, measured in square units.
Volume
cubic
Amount of space occupied by a 3D object, measured in
units.
Surface Area
squared
Total area of a surface of a 3D object, measured in
units.
Perimeter and Area:
Find the area and perimeter of each rectangle.
7 ft wide, 37 ft long
Perimeter =
Area
=
10 ft wide, 15 ft long
Perimeter =
Area
=
Sachi is building a brick patio and needs to determine its total area. The
dimensions of the patio are shown in the diagram below.
Circumference:
The radius of a circle is 2 inches. What is the circumference?
= 3.14
The diameter of a circle is 3 centimeters. What is the circumference?
Volume:
Find the volume of the rectangular prism to the nearest tenth.
Length
Width
Height
8.8 feet
6 feet
11.5 feet
F. 210.0 feet3
G. 380.3 feet3
H. 596.2 feet3
I. 607.2 feet3
Surface Area:
a = 27 cm
b = 32 cm
c = 27 cm
2. Evaluate how a change in length, width, height, or radius affects
perimeter, circumference, area, surface area, or volume.
A company introducing a new cereal wants to use one of the two boxes
shown. What is the volume, in cubic inches, of the box that will hold the
greatest amount of cereal when full?
3. Within a given system, solve real-world problems involving measurement,
with both direct and indirect measures, and make conversions to a larger or
smaller unit (metric and customary). Solve.
Tina was making punch for a party. She mixed 7 cups of apple juice, 4 cups
of pineapple juice, 8 cups of ginger ale, and 1 cup of lemon juice. How many
gallons of punch did she make?
4. Solve real-world problems involving estimates and exact measurements.
At Smith Elementary School, the number of students in a kindergarten class
ranges between 15 and 20. The school has 5 kindergarten classes.
ESTIMATE the number of kindergarten students who attend Smith
Elementary School.
a. 25
c. 90
b. 75
d. 100
Exact Measurement:
Miguel drives 310 miles on a tank of gas. When he refuels his car, it takes
12.4 gallons. How many miles per gallon did Miguel average on his last tank
of gas?
a. 25 miles per gallon
b. 25.8 miles per gallon
c. 297.6 miles per gallon
d. 3,844 miles per gallon
Tina was making punch for a party. She mixed 7 cups of apple juice, 4 cups
of pineapple juice, 8 cups of ginger ale, and 1 cup of lemon juice. How many
gallons of punch did she make?
5. Select appropriate units to solve problems.
Measurement using customary-to-metric Chart:
View the online Reference Sheet
http://www.thefarm.org/charities/i4at/lib2/metric.htm
Ethan lives at one end of Park Avenue. Brian lives at the other end of the
avenue. It is 5.8 kilometers from one end of Park Avenue to the other. If
Ethan walks 2.79 kilometers toward Brian's house, how many meters does
he have to walk to get there?
1 Kilometer = 1,000 meters
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K
1st
2nd
3rd
4th
5th
big(gest)
cent(s)
clock (M)
day * (M)
dime*
equal (parts)*
estimate*
first
fourth*
hour (hand)*
balance
big(gest)*
cent(s)*
clock* (M)
coin (M)
cup (M)
day* (M)
dime*
dollar*
equal (parts)*
balance*
calendar (M)
coin* (M)
cup* (M)
date
day* (M)
dime*
dollar*
dollar sign
equal (parts)*
acute
angle*(M)
**
angle*(M)
base*(M)*
*
closed
figure* **
cone*
congruent
area*(M)**
balance*
capacity*(
M)
Celsius*
centimeter
*(M)
compare
clockwise*
countercloc
balance*
capacity*(
M)
Celsius*
cubic
units(M)
direct
measure*
Fahrenheit
*
(M)
estimate*
estimate*
*(M)**
kwise*
mass*(M)*
longer
longest*
measure
minute(hand)*(
M)
money* (M)
month*
nickel*
first*
foot (M)
fourth*
gram
half-dollar*
half-hour*
hour (hand)*
(M)
foot* (M)
fourth*
gram*
half-dollar*
half-hour*
half-past
hour (hand)*
(M)
cube*(M)
cylinder*(
M)
face*(M)*
*
hexagon*
obtuse
angle*(M)
customary
units*
**
degrees*
direct
measure*
**
distance*
*
quantity*
scale*(M)*
*
square
unit*
units of
measure
penny*
second (hand)
(M)
shorter
size* (M)
small(est)
inch (M)
liter
longer*
longest*
measure*
minute(hand)*
inch* (M)
length (M)**
liter*
longest*
minute(hand)*(
M)
**
octagon*
open
figure*
pentagon*
point* **
Fahrenheit
*
gram*(M)
height*
length*
liter*
volume*(M
)**
week* (M)
year * (M)
(M)
money* (M)
month*
money* (M)
month*
nickel*
reflection(
flip)*(M)*
*
mass*(M)*
*
meter*(M)
nickel*
penny*
pint
pound (M)
quart
quarter
second(hand)*
penny*
pint*
pound* (M)
quart*
quarter*
size* (M)
temperature*
rhombus*
(M)
right
angle*(M)
**
side*
sphere*(M
metric
units*(M)*
*
minute*
nonstandard
units(M)**
(M)
shorter*
size* (M)
small (est)*
temperature
(M)
thermometer
time* (M)
week* (M)
year * (M)
(M)
thermometer*
time* (M)
week* (M)
whole
year* (M)
)
straight
angle* **
symmetry
* **
translation
(slide)*
vertex
(vertices)*
ounce*
perimeter*
(M)**
quantity*
scale*(M)*
*
second*
square
unit*
volume*(M
)**
weight* **
width*(M)
9 Knowledge of geometry and spatial sense
"Geometry is just plane fun."
1. Identify angles or pairs of angles as adjacent, complementary,
supplementary, vertical, corresponding, alternate interior, alternate exterior,
obtuse(obese), acute(a-cute), or right. Identify angles or pairs of angles as
adjacent, complementary, supplementary, vertical, corresponding, alternate
interior, alternate exterior, obtuse, acute, or right.
Adjacent
Two angles are Adjacent if they have a common side and a common vertex
(corner point).
Complementary Angles
Two angles are called complementary angles if the sum of their degree
measurements equals 90 degrees. One of the complementary angles is said
to be the complement of the other.
Supplementary Angles
Two angles are called supplementary angles if the sum of their degree
measurements equals 180 degrees. One of the supplementary angles is said
to be the supplement of the other.
Vertical Angles
For any two lines that meet, such as in the diagram below, angle AEB and
angle DEC are called vertical angles. Vertical angles have the same degree
measurement.
Corresponding Angles
For any pair of parallel lines that are both intersected by a third line.
Corresponding angles have the same degree measurement.
Alternate Interior Angles
For any pair of parallel lines 1 and 2, that are both intersected by a third
line, such as line 3 in the diagram below, angle A and angle D are called
alternate interior angles. Alternate interior angles have the same degree
measurement.
Alternate Exterior Angles
For any pair of parallel lines 1 and 2, that are both intersected by a third
line, such as line 3 in the diagram below, angle A and angle D are called
alternate exterior angles. Alternate exterior angles have the same degree
measurement.
Obtuse Angles
An obtuse angle is an angle measuring between 90 and 180 degrees.
Acute Angles
An acute angle is an angle measuring between 0 and 90 degrees.
Right Angles
A right angle is an angle measuring 90 degrees. Two lines or line segments
that meet at a right angle are perpendicular.
Interior Angles
An Interior Angle is an angle inside a shape.
2. Identify lines and planes as perpendicular, intersecting, or parallel.
Perpendicular lines
Lines are perpendicular to each other if they form congruent adjacent angles
(a T-shape).
Intersecting lines
Two or more lines that meet at a point are called intersecting lines. That
point would be on each of these lines. Lines l and m intersect at Q.
Parallel lines
Lines are parallel if they are always the same distance apart (called
"equidistant"), and will never meet. Always the same distance apart and
never touching.
The red line is parallel to the blue line:
Solve:
The map below shows a portion of downtown Washington, D.C. with named
streets. Which of the following streets appear to be parallel to each other?
3. Apply geometric properties and relationships, such as the
Pythagorean Theorem, in solving problems.
"...The squaw on the hippopotamus is equal to the sum of the
squaws on the other two hides!"
(ie. hypotenuse² = base² + height²)
This mnemonic is the punchline of a very old joke, but its serves its
purpose extremely well! If you've never heard the joke before, there
were 3 Indians who met to trade wives, and the first Indian told his
wife to sit on a blanket made from the skin of a deer, while the
second Indian placed his wife on a hide made of buffalo... (etc)!
Pythagorean Theorem
The area of the square built upon the hypotenuse of a right triangle is equal
to the sum of the areas of the squares upon the remaining sides."
Figure 1
According to the Pythagorean Theorem, the sum of the areas of the two red
squares, squares A and B, is equal to the area of the blue square, square C.
Thus, the Pythagorean Theorem stated algebraically is:
Given the lengths of two sides of a right triangle, use Pythagorean Theorem
to find the length of the third side.
1. a = 3, b = 4, find c
2. a = 3, c = 4, find b
4. Identify the basic characteristics of, and relationships pertaining
to, regular and irregular geometric shapes in two and three
dimensions.
Regular and irregular geometric shapes in two and three
dimensions:
Two-dimensional
A two-dimensional figure, also called a plane or planar figure, is a set of line
segments or sides and curve segments or arcs, all lying in a single plane.
The sides and arcs are called the edges of the figure. The edges are onedimensional, but they lie in the plane, which is two-dimensional.
Three-dimensional
A three-dimensional figure, sometimes called a solid figure, is a set of plane
regions and surface regions, all lying in three-dimensional space. These
surface regions are called the faces of the figure. Each of them is twodimensional. The arcs of curves that are the edges of the faces of the figure
are called the edges of the figure. They are one-dimensional. The endpoints
of the edges are called its vertices. They are zero-dimensional.
5. Apply the geometric concepts of symmetry, congruency,
similarity, tessellations, transformations, and scaling.
Symmetry
The "Line of Symmetry" (shown here in white) is the imaginary line where
you could fold the image and have both halves match exactly.
Congruency
Same shape, same size
Similarity
Same shape, different size
Tessellations
Tessellations are the juxtaposition of shapes in a pattern. A pattern made of
identical shapes. The shapes must fit together without any gaps. The
shapes should not overlap.
Activity:
Try drawing each of these patterns by clicking on this site. Choose a starting
polygon shape, then bend the lines until they look like the design you are recreating, then click the "tessellate" button.
http://www.shodor.org/interactivate/activities/Tessellate/
Transformations
Activity: Click on this site to change various geometric shapes using
rotations, reflections, translations or enlargements - and combinations of
these transformations.
http://www.mathsnet.net/transform/index.html
6. Determine and locate ordered pairs in all four quadrants of a
rectangular coordinate system.
Coordinate System:
A system for specifying points using coordinates measured in some specified
way.
Which point on the coordinate plane below best represents the location of
the ordered pair (4, 3)?
Identify each quadrant.
Which statement describes a way in which one could move from point A to
point B?
a. Move left 3 units, then move
c. Move up 2 units, then move
down 2 units.
b. Move up 3 units, then move
right 4 units, then move
down 2 units.
right 3 units.
d. Move right 4 units, then
move up 3 units, then left 1 unit.
http://www.teachertube.com/viewVideo.php?video_id=13972
http://www.teachertube.com/videoList.php?pg=videonew&tags=601
K
1st
2nd
3rd
4th
5th
above* (M)
above* (M)
above* (M)
acute
acute
acute
behind (M)
below * (M)
beside*
between
(M)
circle* (M)
cone*
corner* (M)
behind*
(M)
below* (M)
beside*
between*
(M)
circle* (M)
cone*
below* (M)
beside*
circle* (M)
cone*
corner* (M)
cube
cup*
cylinder*
angle*(M)**
angle*(M)
base*(M)**
closed
figure* **
cone*
congruent*(
M)**
angle*(M)**
angle*(M)**
base*(M)**
circumference
*(M)
closed figure*
**
cone*
angle*(M)**
circumference*(
M)
congruent* **
coordinates*
dashed line*
diagonal
diameter*
curves
cylinder*
in front of
(M)
inside (M)
left* (M)
corner*
(M)
cup
curves*
cylinder*
in front of
hexagon
left* (M)
line of
symmetry*
parallelogram
plane shapes
cube*(M)
cylinder*(M)
face*(M)**
hexagon*
obtuse
angle*(M)**
congruent* **
cube*
cylinder*(M)
dashed line*
diameter*
dotted line*
dotted line*
edge
equilateral
triangle
horizontal*(M)
intersection*(M)
middle
outside (M)
rectangle*
* (M)
inside* (M)
left* (M)
rectangle*
(M)
rectangular
octagon*
open figure*
pentagon*
face*(M)**
hexagon*
horizontal*(M
**
isosceles
triangle
(M)
right* (M)
shape* (M)
slide*
sort*
sphere*
square* (M)
line of
symmetry
middle*
outside*
(M)
rectangle*
(M)
prism*
right*(M)
shape*(M)
side*
slide*
solid figure
sort*
point* **
reflection(fli
p)*(M)**
rhombus*(M
)
right
angle*(M)**
)
intersection*(
M)**
line* **
line segment*
**
obtuse
line* **
line segment*
**
obtuse
angle*(M)**
parallel lines*
(M)**
triangle*
(M)
under * (M)
rectangular
prism
right* (M)
shape* (M)
side*
slide*
sort*
sphere*
square*
(M)
sphere*
square* (M)
trapezoid
triangle* (M)
turn* **
under* (M)
side*
sphere*(M)
straight
angle* **
symmetry*
**
translation(s
lide)*
vertex
(vertices)*
angle*(M)**
octagon*
open figure*
parallel
lines*(M) **
parallelogram
*(M)
pentagon*
perpendicular
* **
parallelogram*(
M)
perpendicular*
**
plane figure* **
polygon*(M)**
quadrilateral*(M
)
radius**
ray**
triangle*
plane figure*
rectangular
(M)
turn* **
under *
(M)
**
point* **
polygon*(M)*
*
quadrilateral*
(M)
rectangular
prism*(M)**
prism*(M)**
right angle*(M)*
rotation
(turn)*(M)**
scale
model(M)**
scalene triangle
semi-circle
reflection(flip)
*(M)**
rhombus*(M)
right
angle*(M)**
rotation(turn)
similar figures*
**
similarity*(M)**
solid figure*(M)
straight angle*
**
*(M)**
side* **
similar figure*
symmetry* **
tessellation(M)
transformation*
**
similarity*(M)
**
solid figure*
sphere*
straight
angle* **
**
translation
(slide)*(M)**
trapezoid*
triangular
pyramid
two-dimensional
symmetry* **
transformatio
n* **
translation(sli
de)*(M)**
trapezoid*
vertex(vertice
s)* **
vertical*(M)
threedimensional
vertex(vertices)
* **
vertical*(M)
10 Knowledge of algebraic thinking
"Think! Think and wonder. Wonder and think. How much water can 55
elephants drink?"
-- Dr. Seuss
1. Extend and generalize patterns or functional relationships.
2. Interpret tables, graphs, equations, and verbal descriptions to explain
real-world situations involving functional relationships.
3. Select a representation of an algebraic expression, equation, or inequality
that applies to a real-world situation.
http://subjectareatestprep.pdshrd.wikispaces.net/file/view/ALGEBRAIC+THINKING.doc
http://www.teachertube.com/viewVideo.php?video_id=140481
K
1st
2nd
3rd
4th
5th
alike
different
match
alike*
different*
match*
alike*
different*
match*
expressio
n(M)
equation
equation*(M)**
expression* **
inequality*(M)**
equation*(M)**
equivalent
forms*(M)**
input
object
output
similar
size
sort*
input*
object*
output
similar*
size*
sort*
input*
object*
output
similar*
size*
sort*
inequality
(M)
pattern(M
)
rule
solution*
order of
operations*(M)*
*
pattern*(M)**
relationship*(M)
**
rule* **
solution*
symbol*(M)
variable*(M)**
evaluate**
expression* **
inequality*(M)*
*
order of
operations*(M)*
*
pattern*(M)**
relationship*(M)
*
*
rule* **
symbol*(M)
variable*(M)**
11 Knowledge of data analysis and probability
"Five out of four people have trouble with fractions."
-- Steven Wright
1. Apply the concepts of range and central tendency; meAn(average),
median(of the road), and mode(as in most).
The mean is just the average of the numbers.
Add up all the numbers, then divide by how many numbers there are.
Example 1:
What is the Mean of these numbers? 6, 11, 7
Add the numbers: 6 + 11 + 7 = 24
Divide by how many numbers: 24 ÷ 3 = 8 (Mean = 8)
For each data set, find the mean, the median, the mode, and range.
1. Maddie scored 7,9,5,3,15 and 15 pionts in 6 basketball games. She wants
to show that she is a valuable player.
2. Mr. and Mrs. Rodriquez collected donations of $50, $125, $10, $210, $50,
$24, and $175 for charity. They wanted to show that they are good
fundraisers.
3. Andy's phone calls lasted 20 min, 7min, 9min, 12min, and 7min. He
wants to show how long a tpyical phone call is.
4. A golf team had scores of 383, 392, and 401 strokes in a tournament. The
lowest score wins. The team wants to show that it played well.Range
2. Determine probabilities of dependent or independent events.
Probability:
Probability is the likelihood of something happening. For instance, when you
roll a pair of dice, you might ask how likely you are to roll a seven. In math,
"something happening" is called an "event."
Solve:
A cell phone company manufactures 1000 cell phones each day. Of those
1000, about 30 are returned each day needing adjustments or replacement
parts. If you purchase a cell phone from this company, what is the
probability that your cell phone will be defective?
3
1000
3
100
3
10
7
10
3. Determine odds for and odds against a given situation.
Solve:
Each customer at Blakely’s Hardware gets one chance to spin the arrow on
the spinner shown below. The customer then receives the discount displayed
on the section where the spinner arrow lands.
If the sections on the spinner are congruent, what is the probability that the
spinner arrow will land on a section for a 5% discount?
F.
1
4
G.
1
2
H.
3
10
3
8
4. Apply fundamental counting principles such as combinations to
solve probability problems.
Probability =
Number of ways and Event can occur
Total number of possible Events
Example:
The probability of drawing a red marble at random is:
number of red marbles
4
--------------------------- = --total marbles in jar
10
Since 4/10 reduces to 2/5, the probability of drawing a red marble where all
outcomes are equally likely is 2/5. Expressed as a decimal, 4/10 = .4; as a
percent, 4/10 = 40/100 = 40%.
Solve:
A game show has a large spinner that players spin during their turn. What is
the likelihood of the pointer of the spinner landing on a slot worth more than
$500?
a. 3/16
b. 1/4
c. 3/13
d. 4/13
5. Interpret information from tables, charts, line graphs, bar graphs,
circle graphs, box and whisker graphs, and stem and leaf plots.
Box and Whisker Graph
A box and whisker graph is used to display a set of data so that you can
easily see where most of the numbers are.
Example:
Solve:
Draw a box-and-whisker plot for the following data set:
4.3, 5.1, 3.9, 4.5, 4.4, 4.9, 5.0, 4.7, 4.1, 4.6, 4.4, 4.3, 4.8, 4.4,
4.2, 4.5, 4.4
Stem-and-Leaf Plot
A stem-and-leaf plot, in statistics, is a device for presenting quantitative
data in a graphical format.
Solve:
The numbers of points scored by some football teams in the first game of
the season are shown in the stem-and-leaf plot below. What is the mode
score?
a. 4
c. 20
b. 14
d. 35
6. Make accurate predictions and draw conclusions from data.
Mrs. Robertson, a teacher, saw the chart below in her local newspaper. She
came to the conclusion that the number of failing students decreased from
1985 to 2005.
Which is a true statement?
a. The graph is not misleading. The number of failing students decreased.
b. The graph is misleading because the number of years represented by
each bar is not the same. This makes it look like the number of failing
students has decreased.
c. The graph is misleading because it is important to include students
making
Ds in a survey about failing students and this graph does not include
students making Ds.
d. The graph is not misleading because as a teacher at Oakwood School,
Mrs. Roberson knows many students whose grades have improved in the
past several years.
2. Determine probabilities of dependent or independent events.
3. Determine odds for and odds against a given situation.
4. Apply fundamental counting principles such as combinations to solve
probability problems.
5. Interpret information from tables, charts, line graphs, bar graphs, circle
graphs, box and whisker graphs
http://www.worsleyschool.net/science/files/box/plot.html
and stem and leaf plots.
http://www.icoachmath.com/sitemap/problemslink.aspx?Search=stem-andleaf%20plot&grade=0
6. Make accurate predictions and draw conclusions from data.
K
1st
2nd
3rd
4th
5th
graph (M)
table* (M)
**
bar graph*
certain
chart**
bar graph*
certain*
chart**
axis(axes)* **
chance*(M)
combinations(
arrangeme
nt*
axis(axes)*
arrangement
*
break* **
tally
(mark)*
tally
equally likely
graph* (M)
impossible
equally likely*
impossible*
less likely*
M)
coordinates
data*(M)**
**
(squiggle)
break(squig certainty*(M
gle)* **
)
(table)*
less likely
picture graph
table* (M) **
tally (mark)*
tally (table)*
unlikely*
pictograph
picture
graph*
survey
table* (M) **
tally (mark)*
tally (table)*
equally likely
input*
line graph(M)
median*(M)**
mode*(M)**
ordered pair *
**
certainty*(
M)
chance*(M)
circle
graph* **
combinatio
ns
chance*(M)
circle graph*
**
coordinates*
**
data*(M)**
data
unlikely*
output*
pictograph*(M)
**
point* **
range*(
coordinates
* **
data*(M)**
data
collection*(
M)
diagram*(
M)
double-bar
graph*(M)*
collection*(M
)
diagram*(M)
double-bar
graph*(M)**
equally
likely*
event*
function*(M)
**
*
grid(plane)*
equally
likely*
event*
function*(M
)**
grid(plane)(
M)**
labels* **
**
increments
input*
intervals
labels* **
likelihood*
**
line
likelihood*
**
line
graph*(M)*
*
mean*(M)*
graph*(M)**
location
mean*(M)**
ordered
pair* **
organized
*
median*(M
)**
data* **
output*
pie
mode*(M)*
*
ordered
pair* **
organized
data* **
pie
chart*(M)
predict*
probability*(
M)**
randomly
chosen**
ratio**
chart*(M)
point* **
predict*
probability*
(M)**
range*(M)*
*
survey*(M)
tree
diagram*
scale**(M)
stem-andleaf
plot**
survey*(M)
tree
diagram* **
trend line**
unorganized
data**
**
Venn
Venn
diagram*(
M)
x-axis* **
y-axis* **
diagram*(M)
verify(M) xaxis* **
y-axis* **
12 Knowledge of instruction and assessment
"I never got a pass mark in math ... Just imagine --mathematicians now use
my prints to illustrate their books."
-- M.C. Escher
1. Identify alternative instructional strategies. Performance assessments,
dramatic productions, key word alternative yet valid in relation to the
content objectives.
2. Select manipulatives, mathematical and physical models, and other
classroom teaching tools. Counters, marbles, cards, probability cubes, cards,
bears, blocks,
3. Identify ways that calculators, computers, and other technology can be
used in instruction.
http://www.edutopia.org/multiple-intelligences-learning-stylesquiz?gclid=CNfzwIXv66ECFQifnAod6CW3JA
4. Identify a variety of methods of assessing mathematical knowledge,
including analyzing student thinking processes to determine strengths and
weaknesses. AREAS of opportunity!!
Gardners 8 intellegences.
Howard Gardner of Harvard has identified seven distinct intelligences. This
theory has emerged from recent cognitive research and "documents the
extent to which students possess different kinds of minds and therefore
learn, remember, perform, and understand in different ways," according to
Gardner (1991). According to this theory, "we are all able to know the world
through language, logical-mathematical analysis, spatial representation,
musical thinking, the use of the body to solve problems or to make things,
an understanding of other individuals, and an understanding of ourselves.
Where individuals differ is in the strength of these intelligences - the socalled profile of intelligences -and in the ways in which such intelligences are
invoked and combined to carry out different tasks, solve diverse problems,
and progress in various domains."
Gardner says that these differences "challenge an educational system that
assumes that everyone can learn the same materials in the same way and
that a uniform, universal measure suffices to test student learning. Indeed,
as currently constituted, our educational system is heavily biased toward
linguistic modes of instruction and assessment and, to a somewhat lesser
degree, toward logical-quantitative modes as well." Gardner argues that "a
contrasting set of assumptions is more likely to be educationally effective.
Students learn in ways that are identifiably distinctive. The broad spectrum
of students - and perhaps the society as a whole - would be better served if
disciplines could be presented in a numbers of ways and learning could be
assessed through a variety of means." The learning styles are as follows:
Visual-Spatial - think in terms of physical space, as do architects and
sailors. Very aware of their environments. They like to draw, do jigsaw
puzzles, read maps, daydream. They can be taught through drawings, verbal
and physical imagery. Tools include models, graphics, charts, photographs,
drawings, 3-D modeling, video, videoconferencing, television, multimedia,
texts with pictures/charts/graphs.
Bodily-kinesthetic - use the body effectively, like a dancer or a surgeon.
Keen sense of body awareness. They like movement, making things,
touching. They communicate well through body language and be taught
through physical activity, hands-on learning, acting out, role playing. Tools
include equipment and real objects.
Musical - show sensitivity to rhythm and sound. They love music, but they
are also sensitive to sounds in their environments. They may study better
with music in the background. They can be taught by turning lessons into
lyrics, speaking rhythmically, tapping out time. Tools include musical
instruments, music, radio, stereo, CD-ROM, multimedia.
Interpersonal - understanding, interacting with others. These students
learn through interaction. They have many friends, empathy for others,
street smarts. They can be taught through group activities, seminars,
dialogues. Tools include the telephone, audio conferencing, time and
attention from the instructor, video conferencing, writing, computer
conferencing, E-mail.
Intrapersonal - understanding one's own interests, goals. These learners
tend to shy away from others. They're in tune with their inner feelings; they
have wisdom, intuition and motivation, as well as a strong will, confidence
and opinions. They can be taught through independent study and
introspection. Tools include books, creative materials, diaries, privacy and
time. They are the most independent of the learners.
Linguistic - using words effectively. These learners have highly developed
auditory skills and often think in words. They like reading, playing word
games, making up poetry or stories. They can be taught by encouraging
them to say and see words, read books together. Tools include computers,
games, multimedia, books, tape recorders, and lecture.
Logical -Mathematical - reasoning, calculating. Think conceptually,
abstractly and are able to see and explore patterns and relationships. They
like to experiment, solve puzzles, ask cosmic questions. They can be taught
through logic games, investigations, mysteries. They need to learn and form
concepts before they can deal with details.
At first, it may seem impossible to teach to all learning styles. However, as
we move into using a mix of media or multimedia, it becomes easier. As we
understand learning styles, it becomes apparent why multimedia appeals to
learners and why a mix of media is more effective. It satisfies the many
types of learning preferences that one person may embody or that a class
embodies. A review of the literature shows that a variety of decisions must
be made when choosing media that is appropriate to learning style.
Visuals: Visual media help students acquire concrete concepts, such as
object identification, spatial relationship, or motor skills where words alone
are inefficient.
Printed words: There is disagreement about audio's superiority to print for
affective objectives; several models do not recommend verbal sound if it is
not part of the task to be learned.
Sound: A distinction is drawn between verbal sound and non-verbal sound
such as music. Sound media are necessary to present a stimulus for recall or
sound recognition. Audio narration is recommended for poor readers.
Motion: Models force decisions among still, limited movement, and full
movement visuals. Motion is used to depict human performance so that
learners can copy the movement. Several models assert that motion may be
unnecessary and provides decision aid questions based upon objectives.
Visual media which portray motion are best to show psychomotor or
cognitive domain expectations by showing the skill as a model against which
students can measure their performance.
Color: Decisions on color display are required if an object's color is relevant
to what is being learned.
Realia: Realia are tangible, real objects which are not models and are useful
to teach motor and cognitive skills involving unfamiliar objects. Realia are
appropriate for use with individuals or groups and may be situation based.
Realia may be used to present information realistically but it may be equally
important that the presentation corresponds with the way learner's
represent information internally.
Instructional Setting: Design should cover whether the materials are to be
used in a home or instructional setting and consider the size what is to be
learned. Print instruction should be delivered in an individualized mode
which allows the learner to set the learning pace. The ability to provide
corrective feedback for individual learners is important but any medium can
provide corrective feedback by stating the correct answer to allow
comparison of the two answers.
Learner Characteristics: Most models consider learner characteristics as
media may be differentially effective for different learners. Although
research has had limited success in identifying the media most suitable for
types of learners several models are based on this method.
Reading ability: Pictures facilitate learning for poor readers who benefit
more from speaking than from writing because they understand spoken
words; self-directed good readers can control the pace; and print allows
easier review.
Categories of Learning Outcomes: Categories ranged from three to
eleven and most include some or all of Gagne's (1977) learning categories;
intellectual skills, verbal information, motor skills, attitudes, and cognitive
strategies. Several models suggest a procedure which categorizes learning
outcomes, plans instructional events to teach objectives, identifies the type
of stimuli to present events, and media capable of presenting the stimuli.
Events of Instruction: The external events which support internal learning
processes are called events of instruction. The events of instruction are
planned before selecting the media to present it.
Performance: Many models discuss eliciting performance where the
student practices the task which sets the stage for reinforcement. Several
models indicate that the elicited performance should be categorized by type;
overt, covert, motor, verbal, constructed, and select. Media should be
selected which is best able to elicit these responses and the response
frequency. One model advocates a behavioral approach so that media is
chosen to elicit responses for practice. To provide feedback about the
student's response, an interactive medium might be chosen, but any
medium can provide feedback. Learner characteristics such as error
proneness and anxiety should influence media selection.
Testing which traditionally is accomplished through print, may be handled by
electronic media. Media are better able to assess learners' visual skills than
are print media and can be used to assess learner performance in realistic
situations.
from "The Distance Learning Technology Resource Guide," by Carla
Lane
Competency 12.
Knowledge of instruction and assessment
Identify alternative instructional strategies.
Alternative instructional strategies:
http://spacegrant.nmsu.edu/NMSU/fac_dev/grasp_alternative.pdf
Select manipulatives, mathematical and physical models, and other
classroom teaching tools.
Virtual Manipulatives:
http://nlvm.usu.edu/en/nav/topic_t_1.html
Math Manipulatives:
http://math.about.com/gi/o.htm?zi=1/XJ/Ya&zTi=1&sdn=math&cdn=educat
ion&tm=6&f=22&tt=14&bt=0&bts=1&zu=http%3A//mathcentral.uregina.ca/
RR/database/RR.09.98/loewen2.html
3.
Identify ways that calculators, computers, and other
technology can be used in instruction.
An Article on calculators, computers, and other technology used in
instruction.
http://www-users.math.umd.edu/~dac/650/bowespaper.html
4.
Identify a variety of methods of assessing mathematical
knowledge, including analyzing student thinking processes to
determine strengths and weaknesses.
This article gives good information about
“ASSESSING TO SUPPORT MATHEMATICS LEARNING”
http://www.nap.edu/openbook.php?record_id=2235&page=67
Other Online Resources:
Math Reference Sheet:
http://fcat.fldoe.org/pdf/fc8miref.pdf
Virtual Manipulatives:
http://nlvm.usu.edu/en/nav/topic_t_1.html
A Maths Dictionary for Kids (This is a great interactive site for kids and
adults):
http://www.teachers.ash.org.au/jeather/maths/dictionary.html
Fractions, Decimals, Percents:
See http://www.marthalakecov.org/~math/fr_dec_pct.html for additional
examples.
Scientific Notation from United Streaming.
To login, use your B.E.E.P. username and password:
Video:
http://player.discoveryeducation.com/index.cfm?guidAssetId=8E415256A36C-4D02-A45A-D06E54030CC7&blnFromSearch=1&productcode=US
Proportions from United Streaming:
http://viewer.nutshellmath.com/?solution=67-33-71-41-77-31
Gallon Man
http://www.twogetherexpress.com/gallon%20man2.htm