Academic Study Guide: Pearson Mathematics Makes Sense 8

Unit 1: Square Roots and the Pythagorean Theorem

• Square Numbers and Area Models:

  • Verbatim Definition: To square a number, we multiply it by itself. For example, the square of $4$ is $4 \times 4 = 16$. We write this as $4^2$.

  • Perfect Square: A number that is a square of a whole number. $16$ is a perfect square.

  • Geometric Modeling: A square number can be modelled by drawing a square whose area is equal to that number.

  • Side Length and Area: For a square with side length $s$, the area $A = s^2$. Conversely, $s = \sqrt{A}$.

  • Property of Factors: A number is a square number if and only if it has an odd number of factors. For instance, the factors of $16$ are $1, 2, 4, 8, 16$ ($5$ factors; odd). Factors of $136$ are $1, 2, 4, 8, 17, 34, 68, 136$ ($8$ factors; even), meaning $136$ is not a square number.

• Estimating Square Roots:

  • To estimate the square root of a non-perfect square, such as $\sqrt{20}$, identify the closest perfect squares. 16 < 20 < 25. Therefore, \sqrt{16} < \sqrt{20} < \sqrt{25}, which means 4 < \sqrt{20} < 5.

  • Refinement through guessing: $4.4 \times 4.4 = 19.36$ (close to $20$).

• The Pythagorean Theorem:

  • Discovery: In a right triangle, the area of the square on the hypotenuse is equal to the sum of the areas of the squares on the legs.

  • Legs: The two shorter sides that form the right angle.

  • Hypotenuse: The longest side, opposite the right angle.

  • Algebraic Formula: $h^2 = a^2 + b^2$, where $h$ is the hypotenuse and $a$ and $b$ are the legs.

  • Pythagorean Triple: A set of three whole numbers that satisfy the theorem, such as $3-4-5$ ($3^2 + 4^2 = 5^2$) or $7-24-25$.

  • Triangle Classification:

    • In a triangle with sides a \le b < c:

    • If $a^2 + b^2 = c^2$, it is a right triangle.

    • If a^2 + b^2 > c^2, it is an acute triangle.

    • If a^2 + b^2 < c^2, it is an obtuse triangle.

Unit 2: Integers

• Multiplying and Dividing Integers:

  • Sign Rules:

    • The product or quotient of two integers with the same sign is positive: $(+6) \times (+4) = +24$; $(-18) \div (-3) = +6$.

    • The product or quotient of two integers with different signs is negative: $(-8) \times (+5) = -40$; $(+56) \div (-7) = -8$.

  • Multiple Factors: The sign of a product depends on the number of negative factors. An even number of negative factors results in a positive product. An odd number of negative factors results in a negative product.

  • Zero Property: Multiplying any integer by $0$ results in $0$. $(n) \times 0 = 0$.

  • Multiplicative Identity: $(n) \times (+1) = n$.

• Order of Operations with Integers:

  • Steps: (1) Brackets/Grouping Symbols, (2) Multiply and Divide from left to right, (3) Add and Subtract from left to right.

  • Fraction bars act as grouping symbols: Evaluate the numerator and denominator separately before dividing.

Unit 3: Operations with Fractions

• Multiplication:

  • To multiply two fractions, multiply the numerators and multiply the denominators: $\frac{a}{b} \times \frac{c}{d} = \frac{a \times c}{b \times d}$.

  • Simplify by dividing common factors in the numerator and denominator before multiplying.

  • Mixed numbers must be converted to improper fractions first: $2 \frac{1}{2} \times 1 \frac{1}{3} = \frac{5}{2} \times \frac{4}{3} = \frac{20}{6} = \frac{10}{3} = 3 \frac{1}{3}$.

• Division:

  • Reciprocals: Two numbers whose product is $1$. The reciprocal of $\frac{2}{3}$ is $\frac{3}{2}$.

  • To divide by a fraction, multiply by its reciprocal: $\frac{a}{b} \div \frac{c}{d} = \frac{a}{b} \times \frac{d}{c}$.

Unit 4: Measuring Prisms and Cylinders

• Nets and Polyhedra:

  • Net: A 2-D diagram that can be folded to make a 3-D object.

  • Polyhedron: A 3-D object with polygons as faces.

  • Regular Prism: A prism with regular polygons as bases.

• Surface Area (SA):

  • Right Rectangular Prism: $SA = 2 \times (lw + lh + wh)$.

  • Right Triangular Prism: Sum of the areas of the three rectangular faces and the two congruent triangular bases.

  • Right Cylinder: $SA = 2\pi r^2 + 2\pi rh$, where $r$ is the radius and $h$ is the height.

• Volume (V):

  • General Formula: $V = Base\,Area \times Height$.

  • Rectangular Prism: $V = l \times w \times h$.

  • Triangular Prism: $V = (\frac{1}{2}bh) \times l$.

  • Cylinder: $V = \pi r^2 h$.

  • Capacity Conversion: $1\,cm^3 = 1\,mL$.

Unit 5: Percent, Ratio, and Rate

• Percents:

  • To represent percents less than $1\%$, use a hundredths chart (where one small square is $0.01\%$).

  • To represent percents greater than $100\%$, use multiple hundred charts.

  • Formula for Percent Increase/Decrease: $\text{Percent Change} = \frac{\text{Difference}}{\text{Original Amount}} \times 100$.

  • Consumer Math:

    • Discount: Amount subtracted from the regular price.

    • Sales Tax: Includes GST ($6\%$ as of 2007) and PST/HST ($5\%$ to $14\%$ depending on the province).

• Ratios and Rates:

  • Two-term Ratio: Compares two quantities with the same unit ($a:b$).

  • Three-term Ratio: Compares three quantities ($a:b:c$).

  • Proportion: A statement that two ratios are equal: $a:b = c:d$.

  • Rate: A comparison of two quantities with different units, such as $120\,km$ in $3\,h$.

  • Unit Rate: A rate where the second term is $1$, such as $40\,km/h$.

Unit 6: Linear Equations and Graphing

• Distributive Property:

  • Verbatim Definition: Multiplication distributes over addition/subtraction: $a(b + c) = ab + ac$.

  • Expanding: Using the property to remove brackets, e.g., $-3(x + 5) = -3x - 15$.

• Solving Equations:

  • Use algebra to isolate the variable by performing the same operations on both sides (preservation of equality).

  • Equations with fractions: Multiply by the denominator to clear the fraction.

• Linear Relations:

  • Table of Values: Lists ordered pairs $(x, y)$ satisfying the relation.

  • Graphing: Discrete data points are not joined by lines if intermediate values are meaningless (e.g., number of students).

Unit 7: Data Analysis and Probability

• Choosing Graphs:

  • Circle Graph: Best for showing parts of a whole as percents.

  • Line Graph: Best for showing trends over time.

  • Bar Graph: Best for comparing discrete data categories.

  • Pictograph: Uses symbols to visually represent counts.

  • Misrepresentation: Faulty scales (starting above $0$), unequal bar widths, or perspective (3-D effects) can mislead viewers.

• Probability of Independent Events:

  • Independent Events: The outcome of one event does not affect the outcome of another.

  • Product Rule: If events $A$ and $B$ are independent, $P(A \text{ and } B) = P(A) \times P(B)$.

Unit 8: Geometry

• Visualizing Objects:

  • Views: Drawing the Front, Top, and Side views of block structures. Internal lines indicate changes in depth.

  • Isometric Drawing: 3-D representation on triangular dot paper.

  • Rotation: Horizontal rotation (vertical axis) or Vertical rotation (horizontal axis).

• Transformations and Tessellations:

  • Transformations: Translation (slide), Reflection (flip), and Rotation (turn).

  • Conservation of Area: The area of a shape remains unchanged during a transformation.

  • Tessellate: Congruent shapes cover a plane with no gaps or overlaps.

  • Rule for Tessellation: The sum of the interior angles at any meeting vertex must equal $360^\circ$.