Intermediate II patterns, relations, algebra, time, coordinates (5-6 new curriculum; close to grade 7 old curriculum) Flashcards
How do you group numbers and operations together in math?
Use parentheses:
e.g.: (2+4)/5
Using order of operations, when do you perform operations in parentheses?
This is always the first step.
However, later in math courses such as grade 9, you will learn how to apply rules in order to avoid doing the operations within brackets. This is only done when the operation required within the brackets is not possible at the moment since not enough information is known about the numbers within the brackets.
e.g. 2(x+3) = 2x + 2(3) = 2x + 6
Notice here how we could not add the x to the 3 since the x is unknown. Usually we would try to do this first since brackets need to be dealt with first in order of operations. But there is a property called the distributive property that allows us to write this expression in a different way that is still truthful and maintains the same information as the original expression. Notice how once you know what x is, you can compute the answer in different ways. This is sometimes helpful to have alternative representations.
So basically, brackets first if you are solving the equation to achieve a numerical answer. If you do not have enough information, you can apply distributive property if you wish to represent the expression in an alternative way while waiting for more information on the unknown.
What is an algebraic expression?
An expression that includes at least one variable, and may have constants as well. They are not required to use an operation such as plus, minus, divided by, and multiplied by, but many of them do.
What is a variable
a specific unknown value represented with a letter
How do you show multiplication of two variables?
Place the two letters side by side with no need for a multiplication sign between them:
e.g.:
xy
sometimes they will look like this:
(x)(y)
x(y)
(x)y
How do you show division of two variables?
Use fraction notation.
E.g.:
x/y
sometimes they will look like this:
(x)/(y)
x/(y)
(x)/y
Also the fraction bar can be horizontal, this is just not shown here since I am typing within software without special features to display fractions with a horizontal bar easily.
What is an algebraic term?
The product of a number and a variable.
The number in this case is called a coefficient.
E.g.:
3x is a term, where 3 is the coefficient and x is the variable
3+x is not a term, but two different terms brought together with the addition sign
What is a constant term?
Usually in an algebraic expression, the number that is not multiplied to a variable is known as a constant term, since it does not vary.
In later math courses, when they ask for the constant term, they expect you to expand and simplify and then provide the only constant that is there (rather than just picking any constant within the expression, you must bring all the constants together and provide that as your answer).
Can you replace a variable with a given number?
Yes, you may sub in a number for a variable and then evaluate the expression if possible.
e.g.:
y = x+4
sub in x = 20
y = 20+4
y = 24
When x is 20, y is 24.
But keep in mind that x is not only 20 and that you can also pick x to be any real number, sub it in, and return a value for y.
It is good practice to sub in that number within brackets to ensure that you are properly dealing with negative numbers.
In later math courses you will keep track of what number you subbed in by using function notation. Given f(x), f(3)=10 means that you subbed in 3 wherever there was an x. 10 is the result of subbing in 3 into the equation and may be known as the y value.
How can you show adding 3 a certain amount of times? Let n be the number of 3s that you have to add together.
3n
Remember that repeated addition is really multiplication. This is useful knowledge to apply here, otherwise you would have needed to write 3+3+3+3+……… and it would be tough to show when to stop adding.
If n = 7 then you could show it as repeated addition like this:
3+3+3+3+3+3+3 = 21 notice that there are 7 threes and that the result is part of the skip count of 3s, in fact 3 times 7 = 21.
But the question does not specify how many 3s to add, so we show this number with the variable n.
What is another way to write x/2?
(1/2)x
What is another way to write 3x/2?
(3/2)x
you must use brackets here otherwise it will look like the x is on the bottom of the fraction; this is important to remember if showing a slanted fraction bar
How can you solve an equation? What does it mean to solve an equation?
Try with this one:
2x+5 = 10
Solving an equation means to isolate the variable on one side of the equation with the rest of the numbers on the other side of the equation, and then you can see what that variable is equal to.
In this equation, we will use the inverse operations in order to move the numbers to the other side of the equation.
2x + 5 = 10
We will start by looking at the constant term. The constant term is not touching the variable, it is by itself. The constant term here is 5 and it is added on the left side of the equation. We must do the inverse operation of adding 5 which is to subtract 5. We will do this on both sides of the equation so that the equals sign is still truthful.
2x + 5 - 5 = 10 - 5
Then we will simplify both sides by combining like terms (so the constants on the left are 5-5 = 0 and the constants on the right are 10 - 5 = 5).
2x + 0 = 5
But we do not need to write + 0 since that doesn’t mean anything different from the following:
2x = 5
Then we will recognize that 2x means 2 multiplied by x. We will use the inverse operation of multiplying by 2, and chose to divide by 2. We do this on both sides of the equation to ensure that the equals sign remains truthful:
(2x)/2 = (5)/2
So in this case we can rewrite the left side and compute the right side:
(2/2)x = 2.5
But 2/2 is just 1 so we can write:
1x = 2.5
And then we realize that anything multiplied by 1 is just that anything. For example, 1(3) = 3, 1(4) = 4. So we don’t bother to write the 1.
x = 2.5
And then notice that you have isolated x and solved for x. x is equal to 2.5, so we say the solution to this equation is 2.5
How can you check that your solution is correct?
E.g. somebody states that when 2x + 5 = 10, that x =2.5; can you quickly find a way to state this is true or false?
Yes, you can evaluate the left side and the right side separately by subbing in the value for the variable. Then if the left side equals the right side, then that means the value you subbed in is a solution to the equation:
Left side (represented here as LS):
LS = 2x + 5
sub in x = 2.5
LS = 2(2.5) + 5
LS = (5) + 5
LS = 10
Right side
RS = 10
Thus LS = RS
Therefore x = 2.5 is a solution to the equation 2x + 5 = 10
What is the order of operations?
Brackets
Exponents
Multiplication and Division from left to right
Addition and Subtraction from left to right
What are like terms?
algebraic terms with exactly the same variable
x is NOT a like term to x^2 (x squared)
x is a like term to 2x
2x is NOT a like term to 2y
What are the different properties that you have used before that can be applied to algebraic equations?
Commutative property:
a + b = b + a
and
ab = ba
Associative property:
(a + b) + c = a + (b + c)
(ab)c = a(bc)
Distributive property:
a(b+c) = ab + ac
What does it mean to combine like terms?
Show your understanding with this example:
2x+2y+3 +4x + 5y -2
Terms that share the same variable are added together. Remember what you learned with integers here since it will help you if you should ever need to add negative numbers.
= (2x + 4x) + (2y + 5y) + (3-2)
= 6x + 7y + 1
Note that the three resulting terms can be in different orders and still mean the same thing. This is commutative property:
= 7y + 6x + 1
Sometimes like terms are on opposite sides of the equation but you must combine them. What must you remember to do in order to proceed?
Make sure to use inverse operations to move the entire term to the other side (not just the coefficient!) .
coodinate grid
use coordinates to indicate the location of the point where the vertical and horizontal grid lines intersect
coordinates
ordered pairs (so the order in which you present the two numbers matters) of numbers in which the first number indicates the distance from the vertical axis and the second number indicates the distance from the horizontal axis
(x,y) where x is the first one, and y is the second
positional language
includes these terms:
left
right
up
down
location
the position of a shape in space
can be described using a coordinate grid
x-axis
all the points where y=0 are joined together
y-axis
all the points where x=0 are joined together
origin
(0,0) where the x-axis and y-axis intersect
Cartesian plane
is used to describe a location
2D equivalent of a number line
translation
horizontal and vertical movements as a single movement
horizontal component and vertical component make up the translation
reflection
describes movement across a line of reflection such as across the x-axis or y-axis
rotation
describes an amount of movement around a turn centre along a circular path in either a clockwise or counter-clockwise direction
table of values
first column or row lists the input and the second column or row lists the corresponding term
can be used to represent an arithmetic sequence, and when they do, they will form a line (by definition a line is straight) on the corresponding graph
x = position in the arithmetic sequence
y = term in that position
one-to-one correspondance
shows that for every input, you get a unique output compared to all other possible outputs (for example, if you chose another input and looked at that output), but also the reverse, so that every output has a corresponding unique input
function
a correspondence between two changing quantities represented by independent and dependent variables
takes in an input (within its permissible inputs) and gives one output
(each value of the independent variable in a function corresponds to exactly one value of the dependent variable)
can involve quantities that change over time such as height, temperature, or distance travelled
input –> independent variable –> x
output –> dependent variable –> y
(this used to be taught in the grade 10 curriculum, but is now taught in grade 6 but in grade 6 you do not need to know function notation, just be able to talk about the input and output and related information as described above)
