Download Verbal Expressions - Amherst Middle School

Verbal Expressions
An expression is a mathematical phrase (not a complete thought; no verb; no = sign).
An equation is a mathematical sentence (complete thought; has verb; has = sign).
Addition
Verbal Terms
Subtraction
Multiplication
*added to
sum
plus
*more than
add
addend
increased by
*greater than1
total
together
Division
left
times
divided by
difference
multiply
divided into
take away
product
divide
*take from
factpr
each
*less than
multiplied by
quotiend
subtract
groups of
dividend
minus
scale factor
split/group
decreased by
to the power of
divisor
*decreased from
subtracted by
*subtracted from
deduct by
*deduct from
reduced by
*reduced from
1
greater than is most often used as an inequality symbol
*Flip/flop – from, than to
Subtraction – flip flop is important because commutative property does not apply
A number plus 9 would be represented by n+9 not 9+n
9 more than a number = n+9 not 9+n
Ex) 3n+7 can be verbalized multiple ways:
 3 times a number plus 7
 7 greater than 3 times a number
 3 multiplied by a number added to 7
 The sum of a number times 3 plus 7
Ex) 6-3y=-15
6 minus 3 times a number is -15
6 less three times a number is -15
3 times a number less than 6 equals -15
4-1
Expressions, Equations, Inequalities
Expressions - a mathematical phrase that can be simplified or evaluated –by order of operations
2 types of expression:

Numerical expression – no variables, just numbers. Ex) 5+33+(4- -62)3
 Variable expression – has variables (at least one variable) also known as an Algebraic
expression. In this class the only time a variable expression would be simplified would be if
we were given the value of the variable. Ex) m2+3t(m-6)3
Equations - a mathematical sentence that shows equality; made up of two equal expressions.
Ex) n+3=7 (n+3 is one expression and seven is the other expression)
Inequalities – a mathematical sentence where one expression is either greater than, less than,
greater than or equal to, or less than or equal to (>, <, >, <)
4-2
Substitutions for Algebraic (Variable) Expressions
“Simplify or evaluate” – order of operations problem (PEMDAS)
Ex) simplify:
b2+-c if b=7, c= 4
c3
numerator
b + c (the opposite of c)
72+4
49+4
53
53
64
2 -
Ex) simplify:
2a2+b2+(a-b)2 if a=3, b=-5
2(3)2+-52+(3- -5)2
2(3)2+-52+(3+5)2
2(3)2+-52+82
2(9)+25+64
18+25+64
43+64
107
Do not be satisfied with “close,” go for “exact.”
4-3
denominator
c3
- 3
4
16x-4
64
Patterns and Sequences
Arithmetic Sequences (addition (+) only) – each term is found by adding the same
number to the previous term. They have a common difference between each term
(the number you add to each term).
Ex) 8, 11, 14, 17…*rule is add 3 or n+3
Ex) 20, 15, 10, 5…* rule is add -5. Common mistake – looks like subtraction between
terms so it is not recognized as an arithmetic sequence when it actually is because it
is adding a negative number.
Ex) 5, 6, 8, 11, 15 … this is not an arithmetic sequence because the same number
isn’t being added to the previous number. It is a sequence adding 1, 2, 3, 4…
Common mistake - After finding the common difference, students may not identify
the term requested by a question. For example: 1, 2, 3, 4… what is the 10 th term in
the sequence? Students may just locate the next term in the sequence.
Geometric Sequences Multiplication (x) only – each term is found by multiplying the
previous number by the same number of times. They have a common ratio between
each term. It changes each term by the same number of times.
Ex) 1, 2, 4, 8, 16, 32… *rule is multiplying by 2 or 2n
Ex) 40, 20, 10, 5, 2.5… * rule is multiplying by
1
2
or 1 n
2
Ex) 4, -8, 16, -32… *rule is multiplying by -2 or -2n This will most likely be what we see
on the SOL.
4-4
Algebraic Equations
One step algebraic equations
Goals:
 Get the variable by itself – to undo what is being done to the variable, do the
inverse operation.
 Keep the equation balanced – do operations to both sides
*Do not change the sign or number, just the operation.
Ex) 6n= -36
6n 36
= (could also be written 6n÷6 =-36÷6)
6
-
6
n= 6
Check your work:
6(-6)=-36
36=-36
*For organizational purposes, put the variable on the left.
Ex) 16= p-3
p-3=16
p-3+3=16+3
p=19
Check your work:
16=19-3
16=16
*You want to have the variable first in the equation.
Ex) 12=-3-n.
First, get the variable on the left. Next, to get the variable first, change
3-n=12
the subtraction problem to an addition problem. Now because of the
commutative property we can change the order of the addends to put
3+( n)=12
the variable first. Undo what is being done to the variable and do to
n+( 3)=12
n+(-3)-(-3)=12-(-3) both sides.
n=15 (“the opposite of n”=15)
n=-15
Check your work:
12=-3-(-15)
12=12
4-5
Algebraic Equations
Two step algebraic equations
Goals are the same as one step equations.
Ex) 3n-5=-10
3n-5+5=-10+5
3n=-5
If you have multiplication attached to a variable, undo (get rid
of) the addition/subtraction first (unless the multiplication
applies to both the variable and the other term)
3n - 5
=
3
3
-5
n=
or -1.67
3
Check your work:
-5
)-5=-10
3
- 15
-5=-10
3
3(
-
5-5=-10
5+-5=-10
10=-10
With division attached to addition or subtraction, it depends if the division only applies to the
variable or the variable and the other term. If the division only applies to the variable, undo the
addition/subtraction first. If the division applies to both the variable and the other term, undo
the division first.
Ex)
÷ applies to variable and other term
÷ applies just to variable
m5 = 10
3
3 m5 (
)= 10(3)
1
3
m-5=-30
m-5+5=-30+5
m=-25
Check your work:
- 25  5 = 10
3
- 30 = 10
3
-
10=-10
n
-6=3
5
n
-6+6=3+6
5
n
=9
5
5 n
( )=9(5)
1 5
n=45
Check your work:
45
-6=3
5
9-6=3
3=3
4-6
Inequalities
An inequality is a mathematical sentence with unbalanced expressions using either greater than,
less than, greater than or equal to, or less than or equal to signs (>, <, >, <). Always read signs left
to right.
If the big part of the symbol is on the left, it is greater than. >
If the big part of the symbol in on the right, it is less than. <
Goals:
 Get variable by itself
 Keep the inequality “unbalance” by the same amount it was initially.
Ex) q+-6>4
q+-6-(-6)>4-(-6)
q>10
Always check your work. In this example try any number greater than 10.
Check:
12+-6>4
6>4
Graphing inequalities
Whatever number you are graphing, put in the middle of the numberline.
> or < exclude the number when graphing. Use “open dot”
> or < include the number when graphing. Use “closed dot”
The arrow on the graph will match the > or < symbol.(only when variable is on the left)
Ex) x>-4
-
6
-
-
5
-
4
-
3
2
n<5
3
4
5
6
7
p< -1
-
y>7
4
3
5
-
6
-
2
7
1
0
8
1
9
10
Common mistake: students circle the wrong answer. i.e., n<9. Students circle 9 but n is less than
9. Just plug in answers to check.
4-7
Inequalities
Any time you multiply or divide both sides of an inequality by a negative number, switch the
inequality symbol.
3
p
>4
-3
-3 p
( )<4(-3) *multiplying by a negative so the sign is switched
-1 - 3
Ex)
p<-12
Check your work:
-
14
-
13
-
12
-
11 -10
- 15
>4
-3
5>4
4-8
Functions
Relation – any set of ordered pairs. For each “x” member there may be many “y” members.
Ex) {(5,2), (-7,3), (6,-4), (5,9), (1,2)}
{ } symbol means “set containing”
All functions are relations but not all relations are function.
A function is a relation in which there is one and only one “y” value for each “x.” There are no
repeats in the ”x” values. The “x” value is referred to as input (domain) and the “y” value is the
output (range). Functions can be graphed on a coordinate plane because input and output
become ordered pairs. Linear functions are functions that create lines.
Elementary function:
Input
X
Rule
Output
Y
When deciding what number to put in (input) remember that the ordered pair should be small
enough to plot on a coordinate plane (usually -1, 0, 1)
Ex) The rule is 3x+1
y=3x+1
If x =
1
y=3(-1)+1
0
y=3(0)+1
1
y=3(1)+1
y=-3+1
y=-2
y=3(0)+1 y=1
y=3+1
y=4
Ordered
pairs
x
y
1
2
0
1
1
4
Ex) Are these relations also functions?
1) {(6,8), (-3,4), (5,2), (6,3)} – This is not a function because the “x” value of 6 has two “y” values
(8 and 3).
2) {(3,4), (2,-4), (1,0), (-2,4)} – This is a function because there are not repeating “x” values (even
though there are two “y” values that are the same).
4-9
One Step Algebraic Word Problems
Step 1 – make a “let” statement (letting a variable represent the unknown amount in the
question)
Step 2 - do the elementary math. The algebraic method of solving is always the inverse of the
elementary way. (step 4 helps with this if you are having trouble)
Step 3 – create an algebraic equation (inverse of elementary). If you are having trouble with step
3, use step 4 to help (trial and error)
Step 4 - write out a verbal sentence from the algebraic equation
Step 5 – solve the algebraic equation and write a completion statement.
The right side of any one step algebraic equation (when keeping the equation balanced) will be
the elementary problem
Ex) Jim’s 3 fish tanks contain an equal amount of fish. If Jim has a total of 27 fish, how many fish
are in each tank?
Steps:
1) let f= number of fish in each of the 3 tanks
2)
27
=9
3
3) 3f=27
4) Three fish tanks times the number of fish in each tank equals 27 fish
5) 3f=27
3 f 27
This is the same as the elementary problem in step 2.
=
3
3
f=9 Jim has 9 fish in each of his 3 tanks.
Ex) There were 7 apples. Fred ate 2 apples. How many apples were left”
1) Let a = number of apples left
2) 7-2=5
3) a+2=7
4) The number of apples left plus the 2 that were eaten equals 7 apples.
5) a+2=7
a+2-2=7-2
a=5 There are 5 apples left.
4-10
Two Step Algebraic Word Problems
For two step problems, make sure you talk yourself through the elementary math as well as the
algebraic equation.
Circle the numbers you have “given” in the word problem from your elementary math. This will
give you the numbers for your equation. From there you can use trial and error. Combine with
the variable (from the let statement) to make your algebraic equation and check to see that it
makes sense.
Ex) It costs $12 to attend a golf clinic with a local pro. Buckets of balls for practice during the
clinic cost $3 each. How many buckets can you buy at the clinic if you have $30 to spend?
Steps:
1) let b= number of buckets of balls you can buy
2) 30-12=18
18
=6
3
3) 3b+12=30
4) Cost of one bucket (3) times the number of buckets you can buy (n) plus the cost of the
class (12) equals total spent (30)
5) 3b+12-12=30-12
This is the same as the elementary problem in step 2.
3b=18
3b 18
=
3
3
This is the same as the elementary problem in step 2.
b=6 You can buy 6 buckets of balls
4-11