Pengi Math (Grade 7)

In this Grade 7 Pengi Math lesson, students learn to define integers and locate them on the number line, including negative numbers and their opposites using the additive inverse property. The lesson covers absolute value as a measure of distance from zero and explains why a greater absolute value means a lesser value among negative numbers. Students also practice comparing and ordering negative integers, building foundational skills for Chapter 1: The Integer System and Operations.

In this Grade 7 lesson from Pengi Math Chapter 1, students learn to add integers using number line models and arrows to visualize the process. They apply rules for adding integers with the same sign and with different signs, then explore the additive inverse property, where a number and its opposite sum to zero.

In this Grade 7 Pengi Math lesson from Chapter 1: The Integer System and Operations, students learn to multiply integers by applying sign rules — same signs yield a positive product, different signs yield a negative product. Students practice evaluating expressions with multiple factors by counting negative factors to determine the sign, and extend their understanding to repeated multiplication using exponents. Special attention is given to distinguishing between expressions like -3² and (-3)² to avoid common errors with negative bases.

In this Grade 7 lesson from Pengi Math Chapter 1, students learn to divide integers by applying sign rules and understanding division as the inverse of multiplication. The lesson also covers special cases such as division by zero and division by negative one.

In this Grade 7 Pengi Math lesson from Chapter 1: The Integer System and Operations, students apply integers to real-world contexts such as temperature changes, elevation, and financial transactions. They calculate net change, final values, and mean rate of change using all four integer operations. Multi-step word problems challenge students to connect integer arithmetic to everyday situations.

In this Grade 7 Pengi Math lesson from Chapter 2, students learn to define rational numbers as ratios of integers and distinguish them from irrational numbers by identifying terminating and repeating decimals. Students practice converting fractions to decimals using long division, with an extension activity covering how to convert repeating decimals back into fractions.

In this Grade 7 Pengi Math lesson from Chapter 2: Rational Numbers and Exponents, students learn to add and subtract rational numbers, including fractions, decimals, and negative values. The lesson covers converting between fraction and decimal forms to solve mixed-form problems and finding the distance between rational numbers on a number line.

In this Grade 7 Pengi Math lesson from Chapter 2: Rational Numbers and Exponents, students learn to multiply and divide fractions and decimals with positive and negative signs, including the correct placement of negative signs in fractions. The lesson also covers simplifying complex fractions and evaluating algebraic expressions that involve rational numbers.

In this Grade 7 Pengi Math lesson from Chapter 2: Rational Numbers and Exponents, students learn to identify perfect squares, evaluate square roots, and estimate square roots of non-perfect squares by locating them between consecutive integers. The lesson also introduces cube roots, including their application to negative numbers, and applies both square and cube roots to solve real-world area and volume problems.

In this Grade 7 lesson from Pengi Math Chapter 5, students learn to define and work with key algebraic vocabulary including variables, terms, coefficients, and constants. They practice translating verbal phrases into algebraic expressions and evaluating those expressions using substitution. The lesson also builds real-world understanding by interpreting the structure of expressions in applied contexts.

In this Grade 7 Pengi Math lesson from Chapter 5: Algebraic Expressions, students learn to factor algebraic expressions by identifying the Greatest Common Factor (GCF) of terms and applying it as the reverse of distribution. The lesson covers factoring linear expressions, handling negative coefficients, and working through special binomial cases. Students also apply factoring to interpret rectangle dimensions in real-world contexts.

In this Grade 7 Pengi Math lesson from Chapter 5: Algebraic Expressions, students learn to identify like terms by matching variables and exponents, then simplify expressions by combining them using the Commutative and Associative properties. The lesson covers simplifying expressions with integer and rational coefficients, distinguishing between equivalent and non-equivalent expressions, and working with multi-variable expressions.

In this Grade 7 Pengi Math lesson from Chapter 5: Algebraic Expressions, students learn to apply the Distributive Law to expand linear expressions, including those with positive and negative numbers, variables, and fractional coefficients. Using area models as a visual tool, students develop a concrete understanding of distribution before simplifying expressions by combining like terms.

In this Grade 7 Pengi Math lesson from Chapter 6, students learn how to represent the solution of a two-step equation on a number line and connect symbolic algebraic solutions to their graphical representations. The lesson also covers how to distinguish between equations with a single solution and inequalities with solution sets, including the interpretation of open and closed points. Students practice verifying their algebraic solutions visually, building a foundation for graphing inequalities.

In this Grade 7 lesson from Pengi Math Chapter 6, students learn to interpret and use inequality symbols (<, >, ≤, ≥), graph inequalities on a number line using open and closed circles, and translate verbal phrases into inequality statements. Students also verify solutions by substitution and solve one-step inequalities using the Addition and Subtraction Properties of Inequality.

In this Grade 7 Pengi Math lesson from Chapter 6, students learn to solve two-step inequalities by applying the Multiplication and Division Properties, including the rule for reversing the inequality sign when multiplying or dividing by a negative number. Students also practice using the Distributive Property to simplify inequalities and graph solution sets on a number line. The lesson extends to real-world problems with inequality constraints, helping students avoid common errors like forgetting to flip the inequality sign.

In this Grade 7 lesson from Pengi Math Chapter 7, students learn to identify and define acute, obtuse, right, and straight angles, then explore the properties of complementary, supplementary, vertical, and adjacent angles. Students apply these angle relationships by writing and solving algebraic equations to find unknown angle values.

In this Grade 7 Pengi Math lesson from Chapter 7: 2D Geometry and Measurement, students learn to classify triangles by their sides and angles, apply the Triangle Inequality Rule to determine whether a triangle can exist, and use the Triangle Angle Sum Theorem to find missing angles. The lesson also covers properties of exterior angles in triangles and extends to calculating the sum of interior and exterior angles of polygons.

In this Grade 7 Pengi Math lesson from Chapter 7, students learn to identify the essential parts of a circle — including the radius, diameter, and center — and explore the relationship between radius and diameter. Students then apply these concepts to calculate the circumference of a circle using either the radius or diameter, and extend their skills to find the perimeter of semicircles and partial circular regions.

In this Grade 7 Pengi Math lesson from Chapter 7: 2D Geometry and Measurement, students derive and apply the formula for the area of a circle, practicing conversion between diameter and radius to find exact and approximate areas. The lesson extends to calculating the area of semicircles and quadrants before applying these skills to real-world problems involving circular area.

In this Grade 7 lesson from Pengi Math Chapter 7, students learn to decompose composite figures into rectangles, triangles, and semicircles to calculate perimeter and area. They apply both addition (decomposition) and subtraction strategies to find the area of irregular figures, and use grid methods to estimate measurements. This lesson builds essential 2D geometry and measurement skills for working with real-world shapes.

In this Grade 7 lesson from Pengi Math Chapter 8, students learn to identify similar figures and their corresponding parts, apply the AA Similarity Criterion to triangles, and solve for unknown sides using proportions. The lesson also covers how scale factor relates to perimeter and area ratios, and introduces indirect measurement techniques such as shadow problems to apply similarity in real-world contexts.

In this Grade 7 Pengi Math lesson from Chapter 8, students learn to identify and define polyhedra, prisms, and pyramids, including their bases, faces, edges, and vertices. Students then apply these concepts to calculate the total surface area of prisms and explore cross-sections formed by slicing 3D figures.

In this Grade 7 Pengi Math lesson from Chapter 8, students apply the general volume formula V = Bh to calculate the volume of prisms and cylinders, including using the area of a circular base multiplied by height. Students also practice solving for missing dimensions such as height or radius when given a volume, and work through problems involving partial cylinders and unit conversions.

In this Grade 7 Pengi Math lesson from Chapter 9: Statistics, students learn to distinguish between a population and a sample, and explore why representative samples are essential for making valid inferences. The lesson covers methods for generating simple random samples to minimize bias and introduces the concept of sampling variability — why different random samples can produce different results.

In this Grade 7 Pengi Math lesson from Chapter 9, students learn to interpret the shape of data distributions by identifying clusters, peaks, gaps, and outliers in dot plots. They create and analyze double dot plots to compare two populations on the same scale, informally assessing the degree of visual overlap between distributions. The lesson builds skills in comparing the centers and spreads of two data sets using visual reasoning.

In this Grade 7 Pengi Math lesson from Chapter 9, students learn to compare two populations using measures of center (mean and median) and measures of variability (MAD and IQR). Students practice choosing the appropriate measure of center based on whether a distribution is skewed or symmetrical, and calculate Mean Absolute Deviation to evaluate consistency. They also express the difference between population centers as a multiple of a variability measure to draw meaningful statistical inferences.

In this Grade 7 lesson from Pengi Math Chapter 9, students learn to construct and interpret side-by-side box plots to compare two populations. They analyze variability using interquartile range (IQR), whisker length, and box height to draw informal comparative inferences. Students then use these statistical measures to make predictions about populations based on their box plot analysis.

In this Grade 7 Pengi Math lesson from Chapter 10, students learn to understand probability as a number between 0 and 1, ranging from impossible to certain. The lesson introduces key vocabulary including experiment, outcome, sample space, and event, while students practice distinguishing between equally likely and unequally likely outcomes. Fairness in probability contexts is also explored as part of this foundational introduction to probability models.

In this Grade 7 Pengi Math lesson, students learn to calculate both theoretical probability using sample spaces and uniform probability models and experimental probability from observed trial data. They compare the two types of probability, exploring discrepancies through the Law of Large Numbers, and apply probability models to predict the approximate frequency of future events.

In this Grade 7 Pengi Math lesson from Chapter 10, students learn to distinguish between simple events and compound events, then represent sample spaces using organized lists and tables, including area models. Students practice calculating probabilities of compound events by identifying and counting favorable outcomes within these structured representations. The lesson also introduces the concept of independent events as it applies to compound probability.

In this Grade 7 lesson from Pengi Math Chapter 10, students learn how to construct tree diagrams to organize and visualize the outcomes of multistep experiments and compound events. Using tree diagrams, they identify total and favorable outcomes to calculate the probability of specific compound events. The lesson also applies these skills to real-world scenarios such as clothing combinations and menu options.

In this Grade 7 Pengi Math lesson from Chapter 10, students learn how to design simulations to model compound events and generate experimental frequencies. They explore tools such as coins for binary outcomes, spinners with proportional angles for specific probabilities, and random number generators for more complex experiments. This lesson builds practical understanding of how simulations represent real-world probability scenarios.