Memory Aid | Mathematics — Secondary 1 & 2

To compare numbers, it is better to use only one type of notation. Since the set of |\mathbb{Q}| contains the most elements with values that are easily illustrated, fractional notation |\left( \dfrac{a}{b} \right)| with |b \neq 0| will be used.

In word problems, it is important to fully understand the context in order to choose the best approach. Thus, it is useful to follow these steps:

1. Create the chain of operations by highlighting keywords

2. Solve by following the order of operations

The concept of a percentage is an example of a proportional situation. To properly build the proportion to find the desired quantities, follow these steps:

1. Identify the quantity given and associate a percentage with it.

2. Identify the quantity sought and associate a percentage with it.

3. Construct the proportion according to the following model: ||\displaystyle \frac{\color{red}{\text{Quantity given}}}{\color{blue}{\text{Quantity you’re looking for}}} = \frac{\color{red}{\text{Its percentage}}}{\color{blue}{\text{Its percentage}}}||

4. Solve the proportional situation.

We express a ratio using two superimposed points or a fraction.

|a : b| is the part-to-part ratio.

|\displaystyle \frac{a}{a+b}| is the part-to-whole ratio,

where |a| and  |b| are parts of the same whole and generally prime (simplified ratio)

By definition, parts |a| and |b| of a ratio |a:b| are like units, thus, entering the units associated with each part is unnecessary.

Generally denoted |a / b|, the rate creates a relationship between two unlike quantities or measurements.

We call this a unit rate if |b=1.|

For a situation to be proportional, the associated graph needs to:

• Pass through the origin |(0,0).|

• Be represented by a straight line.

Once the situation meets these criteria, use cross multiplication or the multiplicative coefficient to solve the problem.

For a situation to be inversely proportional the associated graph must have:

• A decreasing curved line.

• A line that does not touch the x-axis and y-axis.

Once the situation meets these criteria, it can be solved according to |x y = k|, where |k| is a constant.

Knowing the names of each of the components in an algebraic expression helps in understanding their specific roles. These components have specific names:

• Unknown: Desired numeric value.

• Variable: Letter used to identify the unknown.

• Coefficient: Multiplicative factor placed in front of the unknown.

• Terms: Parts of an expression or equation that are separated by addition or subtraction.

• Constant term (constant): Term consisting only of a number or where no variable appears.

• Like terms: Terms made up of the same variables that are also assigned the same exponents.

• Algebraic expression: Combination of several terms where the total is not known (no = sign).

• Degree: In a monomial, it corresponds to the sum of the variables’ exponents. In a polynomial, it corresponds to the highest degree among the monomials of which it is composed.

• Algebraic equation: Combination of several terms with a known result (with an = sign).

To simplify an algebraic expression, follow the order of operations:

• Multiplication and division: Apply to the coefficients, regardless of the terms.

• Addition and subtraction: Apply to the coefficients of like terms.

• Evaluating an algebraic expression: Substitute the variables by the values given.

Generally, a problem using algebra can be solved by following these steps:

1. Identify the variables and unknowns.

2. Create the equation according to the context of the situation.

3. Simplify the resulting equation.

4. Solve the equation by isolating the variable.

5. Validate the answer using the initial equation.

Length Units

Area Units

To define various plane figures and find missing measurements, reference is made to particular types of lines and line segments:

• |\color{blue}{\text{Diagonal}\ (\overline{BD})}|: Line segment that connects two vertices which are not adjacent.

• |\color{red}{\text{Median} \ (\overline{DF})}|: Line segment that connects a vertex with the midpoint of its opposite side.

• |\color{green}{\text{Perpendicular bisector}\ (\overline{FH})}|: Line segment that is perpendicular to another segment and which divides the latter into two equal parts.

• |\color{fuchsia}{\text{Bisector}\ (\overline{DE})}|: Line segment that divides an angle into two equal parts.

• |\color{orange}{\text{Height}\ (\overline{DG})}|: Line segment starting from the vertex of a figure or a solid that is perpendicular to its base.

Regular polygons all have the same properties:

• Every side has the same measurement.

• Every angle has the same measurement.

• The sum of the interior angles can be calculated using the formula: |(n-2) \times 180°|, where |n| is the number of sides.

• They are formed by a set of isosceles triangles, except the hexagon that is formed by equilateral triangles.

• They each have a different name depending on the number of sides.

• The apothem is the line segment connecting the center of the polygon with the midpoint of one of its sides.

• The apothem is perpendicular to the side it touches.

To clearly distinguish the properties of different line segments in a circle, use the following terms:

• |\color{orange}{\text{Chord} \ (\overline{CF})}|: Line segment which connects any two points on the circle.

• |\color{red}{\text{Diameter}\ (\overline{DE})}|: Line segment which connects any two points of the circle passing through the centre.

• |\color{green}{\text{Radius} \ (\overline{AO})}|: Generally noted |r| , it is a line segment which connects the centre of the circle to any point on the circle.

• Circumference of a Circle |=| outline of the circle |= 2 \pi r.|

• |\color{fuchsia}{\text{Arc of the circle} \ \overset{\huge\frown}{\small {AB}}}|: Curve joining two points on the circumference of the circle
  ||\displaystyle \frac{\color{fuchsia}{m \overset{\huge\frown}{\small {AB}}}}{\color{fuchsia}{m \ \angle AOB}} = \displaystyle \frac{\text{Circumference}}{360^\circ}||

• Area of a Circle |=| surface covered by a disc |= \pi r^2.|

• Area of ​​a Sector: Portion of the circle which is delimited by two radii. ||\displaystyle \frac{\text{Area of the sector} AOB}{\text{Area of the circle}}= \displaystyle \frac{\color{fuchsia}{m \ \angle AOB}}{360^\circ}||

Since it is a decomposable figure, it is necessary to work with ​​each face’s area. Thus, the area formulas of plane figures will be used.

It is preferable, for a decomposable solid, to work with the area of ​​each face rather than with the total area of ​​each solid used to build it. In other words, the area formulas of plane figures will be used.

Use the following steps to find a missing measurement:

1. Identify the measurements given.

2. Determine the formula to use.

3. Replace the known variables.

4. Isolate the desired variable.

Here are the names of the different types of angles according to their measurements:

• A zero angle: angle that measures |0^\circ.|

• An acute angle: angle that measures between |0^\circ| and |90^\circ.|

• A right angle: Often represented using a black square, it is an angle that measures exactly |90^\circ.|

• An obtuse angle: angle that measures between |90^\circ| and |180^\circ.|

• A straight angle: angle that measures exactly |180^\circ.|

• A reflex angle: angle that measures between |180^\circ| and |360^\circ.|

• A full (or complete) angle: angle that measures |360^\circ.|

Here are some definitions related to pairs of angles:

• Adjacent angles: a pair of angles that share a common vertex and side and which are located on either side of the common angle.

• Complementary angles: two angles where the sum of their measurements is |90^\circ.|

• Supplementary angles: two angles where the sum of their measurements is |180^\circ.|

Two lines intersected by a transversal form pairs of remarkable angles. If the lines are parallel, several congruent angles can be found.

Thus, |d_1 // d_2| and |d_3,| is a secant, as follows:

The following angles are congruent:

• Alternate-Interior angles |(\color{redorange}{m\angle BEG} = \color{fuchsia}{m\angle CBE})|: Angles which are on either side of the transversal, that do not share the same vertex, and are inside the parallel lines.

• Alternate-Exterior angles |(\color{green}{m\angle ABF} = \color{orange}{m\angle DEH})|: Angles which are on either side of the transversal, that do not share the same vertex, and are outside the parallel lines.

• Corresponding angles |(\color{red}{m\angle ABC} = m\angle BED)|: Angles which are on the same side of the transversal and do not share the same vertex. One angle is inside the parallel lines and the other, outside.

• Vertically Opposite angles |(\color{blue}{m\angle FBE} = \color{red}{m\angle ABC})|: Angles which share the same vertex and where the sides of one of the angles are the extension of the sides of the other angle.

Finally, to derive angle measurements, it is sometimes helpful to remember that the sum of a triangle’s interior angles is |180^\circ.| For other polygons, apply the following formula:

• The sum of the interior angles of a polygon |=(n-2)\times 180^\circ| where |n| is the number of the polygon’s sides.

Denoted |t_{(x,y)}|, a translation is an isometry since the measurements of the angles and corresponding sides remain identical.

A translation is generally defined by a translation arrow.

Denoted |r_{(O,\text{degree})}|, a rotation is an isometry because the measurements of the angles and corresponding sides remain the same.

The rotation is defined by an angle of rotation.

Denoted |s_{\text{axis}}|, a reflection (symmetry) is an isometry since the measurements of the angles and corresponding sides remain the same.

A reflection is defined by a line of symmetry.

|h_{(O,k)}| establishes a similarity between the two figures since the corresponding angles are congruent and the corresponding sides are proportional.

​All Cartesian planes have the same characteristics:

• Quadrants: they represent each of the four divisions of the Cartesian plane.

• The x- axis: horizontal axis associated with the independent variable |(x).|

• The y- axis: vertical axis associated with the dependent variable |(y).|

• The origin: meeting point of the two axes with the coordinate |(0,0).|

• Coordinates |(x,y)|: any point on the Cartesian plane has a pair of coordinates based on its value on the x-axis |x| and the y-axis |y.|

• The axes: each of the axes is represented by a graduated straight line.

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To fully understand probability, it is important to differentiate between different types events:

• Impossible: probability is equal to 0 (0%).

• Certain: probability is 1 (100%).

• Probable: probability is between 0 and 1 (between 0% and 100%).

• Elementary: contains a single element.

• Compatible/Incompatible: events where intersection is not empty/intersection is empty.

• Dependent/Independent: when the 2nd draw’s outcome is influenced by the 1st draw or when the 2nd draw’s outcome is not influenced by the 1st draw.

As in many fields, theory and practice often give two different results:

• Experimental probability: Probability that is obtained following an experiment.

• Theoretical probability: Probability that is obtained as a result of the theoretical analysis of the possible outcomes.

||\mathbb{P} = \displaystyle \frac{\text{Number of desired results}}{\text{Total number of possible results}}||

In addition, the number of possible results will be influenced if the draw is made with or without replacement.

Here are two sampling methods that are frequently used:

• Random: Elements are chosen at random, without a precise methodology.

• Systematic: Elements are chosen respecting a precise frequency.

To ensure the credibility of a survey, certain pitfalls should be avoided during the setup, completion, and analysis of the survey data. For example, the following sources of bias are common:

• Sample size: ensure that enough people are interviewed so that the results are representative of the population.

• Formulation of questions: Ensure that the questions do not suggest a position or specific opinion (i.e., avoid leading questions).

In general, we can define the type of variable studied using the following qualifiers:

• Qualitative: When the answer is a word or a phrase.

• Quantitative Discrete: When the answer is a numerical value that is a whole number (belonging to the set of integers |(\mathbb{Z}).|)

• Quantitative Continuous: When the answer given is a numerical value that can have decimal form (belonging to the set of real numbers |(\mathbb{R}).|)

Once the data has been collected, it must be analyzed to draw satisfactory conclusions. To do that, some numerical values can be used:

• Mean |= \displaystyle \frac{\text{Sum of data}}{\text{Total number of data}}|

• Range |= \text{Maximum value} - \text{Minimum value}|

• Minimum |= \text{Smallest value in the distribution}|

• Maximum |= \text{Largest value in the distribution}|

Use the following list to ensure that all elements are included when setting up a bar graph.

1. Build a distribution table.

2. Label the axes and give the title of the graph.

3. Ensure that an appropriate scale (graduation) is used and enough space is available to write the different terms/values of the survey.

4. Associate the size of the bars with the frequency of each of the terms/values.

Use the list below to ensure all elements are included when setting up a pie chart.

1. ​Build a distribution table with the corresponding central angles.

2. Draw each of the sectors using their respective central angles.

3. Add a legend and a title.

4. Make sure the percentages are written in each of the sectors.