Grade 11 Exam Cheat Sheet
CHAPTER 1: Logical Reasoning
Conjecture: unproven conclusion based on evidence. It is a hypothesis. It is not fact. A conjecture has
been proved only when it has been shown to be true for EVERY possible case or example.
Inductive Reasoning: makes general conjectures based on specific patterns and recognizing properties.
Additional evidence can further support a conjecture or prove it is not true. This type of reasoning
solves a simpler problem, observes patterns and draws a logical conclusion. Ex. The sum of two
consecutive off numbers is a multiple of 4.
Evidence: you must provide at least 3 pieces of evidence when asked for evidence.
Counterexample: additional evidence that proves your conjecture false. You only need one
counterexample to prove a conjecture false.
Deductive Reasoning: if two things are equal to the same thing, then they must also be equal to each
other. So, if a = b and b = c, then a = c. Example: Mike is in gr 10. All boys in gr 10 are 16. So Mike is 16.
This type of reasoning uses known facts or assumptions in an argument to draw logical conclusions.
Common Errors in Proofs: errors in proofs make the proof invalid.
False Assumptions/Generalizations: assume a trait/characteristic about a large data group.
Error in Calculations: miscalculate the proof (ie. adding instead of subtracting).
CHAPTER 2: Properties of Angles
Equilateral Triangle: a triangle with 3 equal sides.
Isosceles Triangle: a triangle with 2 equal sides.
Scalene Triangle: a triangle with no equal sides.
Right Triangle: a triangle containing an angle that measures exactly 90°.
Obtuse Triangle: a triangle containing an obtuse angle (an angle greater than 90°).
Acute Triangle: a triangle where all angles are acute (less than 90°).
Pythagorean Theorem: is used to solve for a missing side length of a triangle when you have the
measure of the other 2 sides. (a2 + b2 = c2 where a and b are the measures of the legs and c is the
measure of the hypotenuse.
Supplementary Angle: two angles that add up to 180°
Complimentary Angle: two angles that add up to 90°.
Transversal Line: a line that intersects two or more other lines. It creates corresponding angles
Corresponding Angles: are equal when a transversal intersects two parallel lines. They are not equal is
the lines are not parallel! Angles a and e are corresponding, as well as angles b and f, angles d and h,
and angles c and g.
Interior Angles: angles inside the parallel lines. Ex. Angles c, d, e and f.
Exterior Angles: angles outside the parallel lines. Ex. Angles a, b, g, h.
When a transversal intersects two parallel lines :
-alternate interior angles are equal
- alternate exterior angles are equal
- corresponding angles are equal
- interior angles on the same side of the transversal add up to 180°.
The interior angles of a triangle always add up to 180°.
The measure of an exterior angle of a triangle equal to the
sum of the two interior angles that are not adjacent to the
exterior angle.
Convex Polygon: a polygon where all interior angles are less than 180°.
The exterior angles of a convex polygon sum to 360°.
Concave Polygons: have at least one angle greater than 180°.
CHAPTER 3: Acute Angles
SOH CAH TOA for RIGHT ANGLE triangles.
Sine Ratio: opposite side ÷ hypotenuse
Cosine Ratio: adjacent side ÷ hypotenuse
Tangent Ratio: opposite side ÷ adjacent side
Be sure to use the inverses of the function (sin−1, cos −1 , tan−1 ) when determining the size of an angle.
The longest side is always opposite the largest angle. The smallest side is always opposite the smallest
angle.
Sine Law: is used to solve for angles and sides in triangles that are NOT right angles. You can use the
sine law when you know two sides and a matching angle OR you know two angles and any side. Use
cross multiplication to remove the denominators. Solve for the missing side or angle.
Cosine Law: is used to solve for angles and sides in triangles that are NOT right angles. You can use the
cosine law when you know two sides and the contained angle OR all three sides. When using the cosine
law, substitute the known values first. Solve for the unknown side or rearrange to solve for the
unknown angle.
CHAPTER 4: Oblique Angles
All angles come in pairs. The first set of the pair will occur from 0° – to 90°. The second set of the pair
will occur from 90° to 180°.
SinƟ = sin (180 – Ɵ) when Ɵ = 0 - 90°
CosƟ = -cos(180 – Ɵ) when Ɵ = 0 - 90°
TanƟ = -tan (180 – Ɵ) when Ɵ = 0 - 90°
Sine Law for Obtuse Triangles: you need to watch for an Ambiguous case when you use the sine law to
solve for a missing angle (you don’t need to worry about the ambiguous case if you are solving for a
missing side). The angle measure that your calculator produces will always be acute. You need to check
if an obtuse angle is possible too.
- once you have calculated the acute angle, subtract that angle from 180°. This tells you how \
large the obtuse angle would be.
- Check if the obtuse angle makes sense. Will it still form a triangle or is the angle too large? If
the angle is too large, then the acute answer is the only possibility. If the obtuse angle would
work, then there are two answers for the angle you are trying to calculate.
Cosine Law for Obtuse Triangles: the cosine law works the same way for obtuse triangles. You do not
need to work about an ambiguous case.
CHAPTER 5: Statistics
Mean: average of a group of data. Add all the data points and divide by the number of points.
Median: the value directly in the middle of the data points. List the data points from least to greatest.
Count from the edge to find the middle value. If there is no middle value, add the 2 middle values and
divide by 2.
Mode: the data point that occurs the most.
Outlier: data point(s) that are separated from the rest of the data.
Range: the largest data point subtract the smallest data point.
Frequency Table: is a t-table that lists intervals on one side and frequency of
those intervals. You need to count the number of data points on each interval to
determine the frequency. Data points that land on the interval edge are counted
in the lower interval.
Histogram: is a graph that displays the data of a frequency table. It consists
of bars with no spaces in between. The x-axis is made of intervals and the
y-axis is made of the frequency that occurs on each interval. Be sure to label
your axes!
Frequency Polygon: a line graph drawn from the points on the histogram.
place dots in the middle of the top of each bar. Connect the dots with a
straight line.
Standard Deviation: (σ) measures how spread out the data is around the mean. A low standard
deviation means the data is close to the mean. A high standard deviation means the data is scattered
further away from the mean.
Determine the mean of the data.
- Subtract the mean from each data point and square each answer.
- Add all the squared answers together.
- Divide by the number of data points.
- Take the square root of your answer.
Normal Distribution: a set of data is normally distributed if the
data has a symmetric bell curve as a frequency polygon. The mean
median and mode will be almost equal and fall close to the middle
of the bell curve. The percent of the data within 1, 2 and 3 standard
deviations is shown. To determine if data is normally distributed:
- Calculate the mean of the data
- Calculate the standard deviation of the data
-
Count the number of data points within one standard deviation of the mean. Convert this
to a percent.
If the percent is close to 68%, then the data is normally distributed. If the percent is not
close to 68%, then the data is not normally distributed.
Z-Scores: indicates the number of standard deviations a data point is above or below the mean. Zscores can be used to compare different data sets that do not have the same mean and/or standard
deviation. A positive z-score indicates a data point above the mean. A negative z-score indicates a data
point below the mean. Z-scores can be matched to a z-score table. The z-score table tells you what
percent of data is BELOW that data point.
- Determine the z-score usingthe formula.
- Match the calculated z-score to the table. It should give you a decimal number.
- Multiply the decimal number by 100 to determine the percent.
Confidence Interval: a range in which a particular data point is expected to lie. A confidence interval has
the form:
x ± margin of error (as a percent)
where x is the results of the survey.
Confidence Level: determines how accurate the survey is (usually seen a __ out of_ _ times). Divide the
numbers and multiply by 100 to determine the confidence level.
CHAPTER 6: Linear Inequalities
Linear Equation: the form that you need to be able to graph is:
y = mx + b
where m is the slope and b is the y-intercept.
Linear Inequality: a line that has had its equal sign replaced with one of the following:
>
<
≥
≤
greater than
less than
greater than or equal to
less than or equal to
You graph linear inequalities using the following steps:
1. Change the inequality to an equal sign. Isolate y.
2. Plot the y-intercept on the y-axis.
3. Determine the next point on the line by using your slope. (top of the fraction is how much
you rise, bottom of the fraction is how much you run forward ALWAYS).
4. Determine if your line should be solid (≤ or ≥) or dashed (< or >).
5. Test the point (0, 0) in the original inequality by replacing x and y with 0’s. If the inequality is
true, shade the side of the graph that includes the point (0, 0). If the inequality is not true
after substituting, shade the side of the graph that does not include (0, 0).
The shaded area is the solution to the inequality. The line is included in the solution if it is solid. It is not
included in the solution if it is dashed.
When there are two inequalities, the region that is shaded by both inequalities is considered the
solution region. The intersection point is included in the solution (and marked as a shaded dot) if both
lines are included in the solution. If one line is dashed, then the solution does not include the
intersection point and the point is drawn hallow.
Word Problems: when setting up a word problem, be sure to write a defining statement that describes
the word problem. You need to consider restrictions on the variables.
Vertex: is the point of intersection between two or more linear equations.
CHAPTER 7: Quadratic Equations
Quadratic: in standard form is
y = ax2 + bx + c
As a graph, it creates a parabola with a single
line of symmetry (axis of symmetry).
The parabola opens up if a is positive.
The parabola opens down if a is negative.
The y-intercept (where the graph crosses the y-axis) is represented by c. At the y-intercept, x is always
0.
Axis of Symmetry: the vertical line that passes through the vertex around which a parabola is
symmetrical. It is written as x = the x-value of the vertex.
Vertex: the minimum coordinate of a parabola that opens up or the maximum coordinate of a parabola
that opens down. It is a coordinate so it is written as (x, y).
If two points have the same y-value, you can determine the x-value of the vertex by adding the x-value
of both points and dividing by 2.
Domain: represents all the x-values of a parabola. For all parabolas, the domain will be (-∞, ∞).
Range: represents all the y-values of the parabola. If the parabola opens up, the range will be
[y-value of vertex, ∞). If the parabola opens down, the range will be (-∞, y-value of vertex].
X-intercept: where the parabola crosses the x-axis. The x-intercepts are also called roots, zeroes or
solutions. At the x-intercept, y is always 0.
Factoring: reverses the multiplication of polynomials.
1. GCF: greatest common factor. Write the greatest common factor in front of the brackets and
divide every term by the gcf.
2. DOS: difference of squares. Take the square root of both terms. Draw 2 sets of brackets.
Write the first square root in the front of the brackets. Write the second square root in the back
of the brackets. One bracket is addition, one is subtraction.
3. Trinomial: the method used depends on the value of a (# in front of x2)
Cheater Method: use when a = 1. Determine 2 numbers that multiply to ac and add to
b. Write x in the front of both brackets and the numbers in the back.
Box Method: when a ≠ 1. Determine 2 numbers that multiply to ac and add to b. Draw
a box and write your numbers in the box. Determine the GCF of each row and column.
Church Method: when a ≠ 1. Determine 2 numbers that multiply to ac and add to b.
Write the x2 coefficient in the front of both brackets with x. Write your 2 numbers in
the back of both brackets. Divide both brackets by the a-value. Factor a GCF out of one
of the brackets to cancel the denominator.
Backward FOIL: when a ≠ 1. Determine 2 numbers that multiply to ac and add to b.
Split the b-value into two terms using these numbers. Determine the GCF of the first
two terms and the GCF of the second two terms. THE BRACKETS SHOULD MATCH. Use
the brackets as a GCF.
Factored Form: factored form of quadratics lets you determine the x intercepts.
y = a(x – r)(x – s)
where a is a coefficient, r and s are x-intercepts.
Vertex Form: lets you determine the vertex of the quadratic.
y = a(x – p)2 + q
where a is a coefficient, p is the x-value of the vertex and q is
the y-value of the vertex. So vertex is (p, q). NOTE THE
SIGN CHANGE ON P!!!
To Graph From Factored Form
1. Plot the x-intercepts on the x-axis (remember to change the sign of r and s).
2. Determine the x-value of the vertex using: x-value of vertex = (r + s)/2
3. Determine the y-value of the vertex by substituting the x-value into the given equation and
calculating.
4. Plot the vertex and connect the dots.
To Graph From Vertex Form
1. Plot the vertex (p, q). Remember to change the sign on p!
2. Move one box to the right of the vertex and up/down the a-value.
3. Move one box to the left of the vertex and up/down the a-value. Connect your points.
To Determine Factored Form From a Graph
1. Write the factored formula.
2. Fill in r and s with the x-intercepts.
3. Select any other point on the graph, except the x-intercepts. Substitute the x- and y-values
from the coordinate into the formula. Solve for a.
4. Rewrite the factored formula, filling in a, r and s.
To Determine Vertex Form From a Graph
1. Write the vertex formula.
2. Fill in p and q with the vertex.
3. Select any other point on the graph, except the vertex. Substitute the x- and y-values
from the coordinate into the formula. Solve for a.
4. Rewrite the factored formula, filling in a, p and q.
CHAPTER 8: Proportional Reasoning
Unit Price: determines the price of one unit of a product to allow comparison between different
products. Comparisons can only be made if the unit price is calculated in the same size of unit.
Sometimes you may need to convert units to before you are able to compare them.
You determine unit price by dividing the cost by the number of units.
Unit Price =
𝐏𝐫𝐢𝐜𝐞
𝐓𝐨𝐭𝐚𝐥 𝐍𝐮𝐦𝐛𝐞𝐫 𝐨𝐟 𝐔𝐧𝐢𝐭𝐬
Remember, money is always on the top of the fraction (in the numerator) and time is always on the
bottom of the fraction (in the denominator).
When you are converting between units, create 2 fractions. The first fraction is the conversion ratio that
will be provided. The second fraction is what you're trying to convert. The units must be the same in
the numerators of both fractions. The units must be the same in the denominators of both fractions.
Cross multiply to solve for x.
conversion
ratio
1 kg
1000 g
1 kg
1000 g
=
=
x
500 g
amount to be
converted
x
500 g
You can also use the same process to determine equivalent rates. For example, if a butcher shop sells 4
lb of chicken for $12.57, how much would it cost for 10 lb?
$12.57
4 𝑙𝑏
𝑥
= 10 𝑙𝑏
You can cross multiply to determine the price of 10 lb of chicken.
Scale Factor (k): compares the measurement of a shape in a diagram or model to the measurement of
the actual shape.
- linear scale factor = k
- area scale factor = k2
- volume scale factor = k3
You can calculate scale factor using the following formulas:
Linear
Area
Volume
k=
𝑑𝑖𝑎𝑔𝑟𝑎𝑚
𝑜𝑟𝑖𝑔𝑖𝑛𝑎𝑙
k2 =
𝑑𝑖𝑎𝑔𝑟𝑎𝑚
𝑜𝑟𝑖𝑔𝑖𝑛𝑎𝑙
k3 =
𝑑𝑖𝑎𝑔𝑟𝑎𝑚
𝑜𝑟𝑖𝑔𝑖𝑛𝑎𝑙
When a scale factor is between 0 and 1, the diagram will be smaller. When a scale factor is greater than
1, the diagram will be larger.
To determine the measurements of the diagram or model, use the following formulas:
Linear
Area
Volume
2
diagram = k(original)
diagram = k (original)
diagram = k3(original)