Grade 7 Advanced Mathematics Term 3 Revision Guide

Term 3 Exam Coverage and Structure

• Grade Level and Subject: Grade 7 Advanced Mathematics.

• Academic Year: 2024-2025 (Revision for 2025-2026).

• School Group: Applied Technology Schools (ATS) - مدارس التكنولوجيا التطبيقية.

• Modules Covered: Modules 8 through 11.

• Exam Logistics:

  • Total Questions: 25 questions.

  • Format: Multiple Choice Questions (MCQ).

  • Weighting: 4 marks per MCQ.

  • Total Marks: 100 marks.

  • Duration: 120 minutes.

Angle Relationships: Vertical and Adjacent Angles

• Key Definitions and Identification:

  • Vertical Angles: These are opposite angles formed by the intersection of two lines. They are always congruent (equal in measure).

  • Adjacent Angles: Angles that share a common vertex and a common side but do not overlap.

• Mathematical Application:

  • Vertical angles allow for the creation of equations to find unknown measures because they are equal (e.g., if one vertical angle is $52^\natural$, its opposite is also $52^\natural$).

  • Solving for unknowns often involves setting two expressions equal to each other if they represent vertical angles ($52x + 1 = \text{angle measure}$).

  • When angles form a straight line, they are adjacent and supplementary, summing to $180^\natural$.

• Reference: Questions Q5-8 on Page 401 and Q1-4 on Page 401.

Complementary and Supplementary Angles

• Complementary Angles:

  • Two angles are complementary if the sum of their measures is exactly $90^\natural$.

  • Example equation structure: $x + y = 90^\natural$.

• Supplementary Angles:

  • Two angles are supplementary if the sum of their measures is exactly $180^\natural$.

  • Example equation structure: $x + y = 180^\natural$.

• Advanced Equation Solving:

  • Students must be able to write and solve multi-step equations to find angle measures from diagrams.

  • Sample calculations from the material include expressions like $27x + 69 = 180$ or involving fractions like $\frac{x}{2} + 1 = \text{angle}$.

• Reference: Questions Q1-12 on Page 411.

Scale Drawings and Ratio Reasoning

• Concept: Using ratio reasoning to determine actual lengths and areas based on a given scale drawing.

• Application: Converting between a scale (e.g., $1\text{ unit} : 10\text{ units}$) and a different scale or the actual physical dimensions.

• Reference: Questions Q1-6 on Page 433.

Surface Area of Solids Using Nets

• Surface Area Definition: The total area of all the faces on a three-dimensional object.

• Methods: Using nets to visualize the 2D layout of all faces (Front, Back, Top, Bottom, Left, Right).

• Prism Formulas:

  • Total Surface Area = $2 \times (\text{Front Area}) + 2 \times (\text{Top Area}) + 2 \times (\text{Side Area})$.

  • Example components: $\text{Front/Back} = 46 \times 32$, $\text{Top/Bottom} = 46 \times 352$.

• Pyramid Formulas:

  • Surface Area = Base Area + Area of all triangular lateral faces.

  • Triangle area is calculated as $\frac{1}{2} \times \text{base} \times \text{height}$.

• Reference: Questions Q1-4 on Page 495 and Ex 3, 5-6 on Page 491/495.

Volume of Prisms and Pyramids

• Prism Volume: Calculated by multiplying the area of the base ($B$) by the height ($H$) of the prism.

  • $V = B \times H$

• Pyramid Volume: A pyramid has one-third the volume of a prism with the same base and height.

  • $V = \frac{1}{3} \times B \times H$

• Reference: Learn Example 3 on Page 478/479 and Questions Q3-8 on Page 485.

Surface Area and Volume of Composite Figures

• Decomposition Method: Complex figures are broken down (decomposed) into simpler common solids like rectangular prisms, cubes, or pyramids.

• Calculation:

  • Find the volume or surface area of each individual part.

  • Sum the parts to find the total volume.

  • For surface area, care must be taken to subtract any faces that are joined together and hidden within the interior of the composite shape.

• Reference: Questions Q1-6 on Page 503.

Probability and Likelihood

• Degrees of Likelihood:

  • Impossible: Probability of $0$.

  • Unlikely: Probability less than $0.5$ (closer to $0$).

  • Equally likely to happen as not to happen: Probability of $0.5$ or $50\%$.

  • Likely: Probability greater than $0.5$ (closer to $1$).

  • Certain: Probability of $1$ or $100\%$.

• Theoretic vs. Experimental Probability:

  • Theoretical Probability: Based on mathematical reasoning: $P(E) = \frac{\text{number of favorable outcomes}}{\text{total possible outcomes}}$.

  • Relative Frequency (Experimental): Based on actual trials or historical data: $\frac{\text{number of times event occurred}}{\text{total number of trials}}$.

  • Law of Large Numbers: As the number of trials increases, the relative frequency (experimental probability) tends to get closer to the theoretical probability.

• Complement of an Event: The probability of an event not occurring. $P(\text{not } E) = 1 - P(E)$.

• Reference: Pages 514, 527, 528, 533, 537, and 545/546.

Compound Events and Sample Spaces

• Sample Space Tools:

  • Organized Lists: Writing out all possible outcomes.

  • Tables: Useful for events involving two independent triggers (e.g., rolling two dice).

  • Tree Diagrams: A branching visualization of all possible outcome sequences.

• Outcomes: For a coin toss, outcomes include $H$ (Heads) and $T$ (Tails). For two coins: $HH, HT, TH, TT$.

• Reference: Questions Q1-2 on Page 557 and Ex 3, 3-5 on Page 552/557.

Simulations of Probability

• Design: Creating a model (e.g., using a spinner or coin) to mimic a real-world event.

• Example: If the probability of rain is $40\%$, use a spinner with 5 sections where 2 sections represent "Rain" and 3 represent "No Rain".

  • $P(\text{Rain}) = \frac{2}{5} = 0.4$

  • $P(\text{No Rain}) = \frac{3}{5} = 0.6$

• Reference: Questions Q1, 2, 10 on Page 567/572 and Learn Ex 1 on Page 561-564.

Sampling and Population Inferences

• Sampling Methods:

  • Unbiased Sample: A representative sample where every member of the population has an equal chance of being selected. Inferences from these samples are valid.

  • Biased Sample: A sample that does not represent the population accurately. Inferences made from biased samples are invalid.

• Predicting Populations: Using the proportion found in a random sample to estimate characteristics of a larger population.

• Variation: Understanding how different random samples of the same population can yield varying results.

• Measures of Center and Variation:

  • Center: Mean and Median.

  • Variation: Range and Interquartile Range (IQR).

• Visual Overlap: Comparing two data distributions (box plots or dot plots) to judge how similar or different two populations are.

• Reference: Pages 583, 591, 601, 611, and 615/617.

Term 3 Exam Coverage and Structure

• Grade Level and Subject: Grade 7 Advanced Mathematics

• Academic Year: 2024-2025 (Revision for 2025-2026)

• School Group: Applied Technology Schools (ATS) - مدارس التكنولوجيا التطبيقية

• Modules Covered: Modules 8 through 11

Exam Logistics

• Total Questions: 25 questions

• Format: Multiple Choice Questions (MCQ)

• Weighting: 4 marks per MCQ

• Total Marks: 100 marks

• Duration: 120 minutes

Angle Relationships

Vertical and Adjacent Angles

• Key Definitions:

  • Vertical Angles: Opposite angles formed by the intersection of two lines; they are always congruent.

  • Adjacent Angles: Share a common vertex and side but do not overlap.

• Mathematical Application:

  • Vertical angles create equations to find unknown measures (e.g., if one is $52^<br>atural$, the other is also $52^<br>atural$).

  • For adjacent angles on a straight line, they are supplementary and sum to $180^<br>atural$.

• Reference: Questions Q5-8 on Page 401 and Q1-4 on Page 401

Complementary and Supplementary Angles

• Complementary Angles:

  • Sum of measures is $90^<br>atural$ (e.g., $x + y = 90^<br>atural$).

• Supplementary Angles:

  • Sum of measures is $180^<br>atural$ (e.g., $x + y = 180^<br>atural$).

• Advanced Equation Solving:

  • Write and solve multi-step equations (e.g., $27x + 69 = 180$).

• Reference: Questions Q1-12 on Page 411

Scale Drawings and Ratio Reasoning

• Concept: Use ratio reasoning to determine actual lengths/areas from scale drawings.

• Application: Convert between scales (e.g., $1 ext{ unit} : 10 ext{ units}$).

• Reference: Questions Q1-6 on Page 433

Surface Area of Solids Using Nets

Definition and Methods

• Surface Area: Total area of all faces of a 3D object.

• Using Nets: Visualize the 2D layout of all faces (Front, Back, Top, Bottom, Left, Right).

Formulas

• Prism: Total Surface Area = $2 imes ( ext{Front Area}) + 2 imes ( ext{Top Area}) + 2 imes ( ext{Side Area})$.

  • Example components: $ext{Front/Back} = 46 imes 32$, $ext{Top/Bottom} = 46 imes 352$.

• Pyramid: Surface Area = Base Area + Area of all triangular lateral faces.

  • Triangle area = $rac{1}{2} imes ext{base} imes ext{height}$.

• Reference: Questions Q1-4 on Page 495 and Ex 3, 5-6 on Page 491/495

Volume of Prisms and Pyramids

• Prism Volume: $V = B imes H$ (base area $B$ and height $H$).

• Pyramid Volume: $V = rac{1}{3} imes B imes H$.

• Reference: Learn Example 3 on Page 478/479 and Questions Q3-8 on Page 485

Surface Area and Volume of Composite Figures

• Decomposition Method: Break complex figures into simpler solids (e.g., prisms, cubes, pyramids).

• Calculation Steps:

  • Find volume/surface area of each part.

  • Sum the parts for total volume; subtract hidden faces for surface area.

• Reference: Questions Q1-6 on Page 503

Probability and Likelihood

Degrees of Likelihood

• Impossible: Probability of $0$.

• Unlikely: Probability < $0.5$.

• Equally likely/not to happen: Probability of $0.5$.

• Likely: Probability > $0.5$.

• Certain: Probability of $1$.

Probability Types

• Theoretical Probability: $P(E) = rac{ ext{favorable outcomes}}{ ext{total possible outcomes}}$.

• Relative Frequency: $rac{ ext{times event occurred}}{ ext{total trials}}$.

• Law of Large Numbers: More trials result in closer approximation to theoretical probability.

• Complement of an Event: $P( ext{not } E) = 1 - P(E)$.

• Reference: Pages 514, 527, 528, 533, 537, and 545/546

Compound Events and Sample Spaces

• Tools for Sample Space:

  • Organized Lists: List all possible outcomes.

  • Tables: Useful for independent events (e.g., rolling dice).

  • Tree Diagrams: Visualize all possible outcomes.

• Examples: Outcomes for two coins: $HH, HT, TH, TT$.

• Reference: Questions Q1-2 on Page 557 and Ex 3, 3-5 on Page 552/557

Simulations of Probability

• Design: Create a model (e.g., spinner, coin) to represent real-world events.

• Example: For a 40 ext{%} chance of rain, use a spinner with 5 sections (2 for Rain, 3 for No Rain).

• Reference: Questions Q1, 2, 10 on Page 567/572 and Learn Ex 1 on Page 561-564

Sampling and Population Inferences

• Sampling Methods:

  • Unbiased Samples: Representative samples valid for inferences.

  • Biased Samples: Unrepresentative samples invalid for inferences.

• Variation: Acknowledge differing results from random samples of the same population.

• Measures of Center/Variation: Mean, Median (Center) and Range, IQR (Variation).

• Visual Overlap: Compare data distributions to judge similarities/differences.

• Reference: Pages 583, 591, 601, 611, and 615/617