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Math for Grade 7
1 Number Sense and Operations
1-1 Integers
1-1 1 Understanding positive and negative numbers
1-1 2 Comparing and ordering integers
1-1 3 Absolute value
1-1 4 Adding and subtracting integers
1-1 5 Multiplying and dividing integers
1-2 Rational Numbers
1-2 1 Understanding fractions, decimals, and mixed numbers
1-2 2 Comparing and ordering rational numbers
1-2 3 Converting between fractions, decimals, and percents
1-2 4 Adding and subtracting fractions and mixed numbers
1-2 5 Multiplying and dividing fractions and mixed numbers
1-3 Exponents and Roots
1-3 1 Understanding exponents
1-3 2 Laws of exponents
1-3 3 Square roots and cube roots
2 Algebra
2-1 Expressions and Equations
2-1 1 Writing algebraic expressions
2-1 2 Evaluating algebraic expressions
2-1 3 Solving one-step and two-step equations
2-1 4 Solving inequalities
2-2 Patterns and Functions
2-2 1 Identifying and extending patterns
2-2 2 Representing patterns with tables, graphs, and equations
2-2 3 Understanding functions and function notation
3 Geometry
3-1 Shapes and Angles
3-1 1 Classifying polygons
3-1 2 Measuring and drawing angles
3-1 3 Understanding complementary and supplementary angles
3-2 Transformations
3-2 1 Understanding translations, reflections, and rotations
3-2 2 Identifying congruent and similar figures
3-3 Perimeter, Area, and Volume
3-3 1 Calculating perimeter and area of polygons
3-3 2 Understanding and calculating the area of circles
3-3 3 Calculating the volume of rectangular prisms
4 Data and Probability
4-1 Data Representation
4-1 1 Collecting and organizing data
4-1 2 Creating and interpreting bar graphs, line graphs, and pie charts
4-1 3 Understanding mean, median, mode, and range
4-2 Probability
4-2 1 Understanding probability as a ratio
4-2 2 Calculating simple probabilities
4-2 3 Understanding experimental versus theoretical probability
5 Problem Solving and Critical Thinking
5-1 Strategies for Problem Solving
5-1 1 Using logical reasoning and critical thinking
5-1 2 Applying the problem-solving process
5-1 3 Using estimation and approximation
5-2 Real-World Applications
5-2 1 Applying mathematical concepts to real-world scenarios
5-2 2 Understanding the relevance of mathematics in daily life
4-2 Probability Explained

4-2 Probability Explained

Key Concepts

1. **Probability**: The likelihood or chance of an event occurring.

2. **Sample Space**: The set of all possible outcomes of an experiment.

3. **Event**: A specific outcome or set of outcomes of an experiment.

4. **Probability Formula**: The formula to calculate the probability of an event is \( P(E) = \frac{\text{Number of favorable outcomes}}{\text{Total number of possible outcomes}} \).

Detailed Explanation

Probability

Probability measures how likely an event is to occur. It is expressed as a number between 0 and 1, where 0 means the event will not occur, and 1 means the event is certain to occur. For example, the probability of flipping a coin and getting heads is 0.5.

Sample Space

The sample space is the set of all possible outcomes of an experiment. For example, when rolling a six-sided die, the sample space is {1, 2, 3, 4, 5, 6}.

Event

An event is a specific outcome or set of outcomes of an experiment. For example, in rolling a die, the event "rolling an even number" includes the outcomes {2, 4, 6}.

Probability Formula

The probability of an event \( E \) occurring is calculated using the formula:

\[ P(E) = \frac{\text{Number of favorable outcomes}}{\text{Total number of possible outcomes}} \]

For example, the probability of rolling a 3 on a six-sided die is \( \frac{1}{6} \).

Examples

Example 1: Calculate the probability of drawing a red card from a standard deck of 52 cards.

Solution: There are 26 red cards in a deck of 52 cards.

\[ P(\text{Red card}) = \frac{26}{52} = \frac{1}{2} = 0.5 \]

Explanation: The probability of drawing a red card is 0.5 or 50%.

Example 2: Calculate the probability of rolling a sum of 7 with two six-sided dice.

Solution: The sample space for two dice is 36 possible outcomes. The favorable outcomes for a sum of 7 are (1,6), (2,5), (3,4), (4,3), (5,2), and (6,1).

\[ P(\text{Sum of 7}) = \frac{6}{36} = \frac{1}{6} \approx 0.167 \]

Explanation: The probability of rolling a sum of 7 is approximately 0.167 or 16.7%.

Analogies

Think of probability as the chance of finding a specific toy in a box of mixed toys. The sample space is all the toys in the box, and the event is finding the specific toy you want. The probability formula helps you determine how likely it is to find that toy.

Practical Application

Understanding probability is crucial in various fields such as statistics, finance, and risk management. It helps in making informed decisions, predicting outcomes, and analyzing uncertainties.

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