enVision, Mathematics, Grade 7

In this Grade 7 lesson from enVision Mathematics Chapter 1, students learn how integers and their opposites relate to each other, including how opposite quantities combine to make zero and how absolute value represents distance from zero. Using real-world contexts like temperature changes, bank deposits and withdrawals, and vertical movement, students practice representing situations with positive and negative integers on a number line.

In this Grade 7 enVision Mathematics lesson from Chapter 1, students learn to recognize rational numbers and convert them from fraction form to decimal form by dividing the numerator by the denominator. The lesson covers the difference between terminating decimals, where the remainder reaches zero, and repeating decimals, where a digit or block of digits repeats infinitely and is written using bar notation. Students practice identifying whether a given decimal is rational based on whether it terminates or repeats.

In Lesson 3 of Chapter 1 from enVision Mathematics Grade 7, students learn how to add integers, including adding two negative integers, adding integers with different signs, and identifying additive inverses. The lesson connects integer addition to absolute value, teaching students to find the sum or difference of absolute values and determine the correct sign of the result. Real-world contexts like temperature changes and score adjustments help students build and apply these skills.

In this Grade 7 lesson from enVision Mathematics Chapter 1, students learn how to subtract integers by applying the additive inverse property, rewriting subtraction expressions such as a − b as equivalent addition expressions a + (−b). The lesson covers subtracting positive integers, integers with different signs, and negative integers, using number lines to model each case. Real-world contexts like football yardage and temperature changes help students connect integer subtraction to practical problem-solving.

In this Grade 7 lesson from enVision Mathematics Chapter 1, students learn how to add and subtract rational numbers, including mixed numbers and decimals with different signs, using the same rules that apply to integers. Students apply concepts such as absolute value, additive inverses, and the Commutative Property to solve problems and find distances between rational numbers on a number line.

In this Grade 7 lesson from enVision Mathematics, students learn how to multiply integers by exploring how the signs of factors determine the sign of a product. The lesson covers three key rules: positive times negative equals negative, negative times positive equals negative, and negative times negative equals positive. Students use number lines and properties of operations to model and justify integer multiplication in real-world contexts such as board games, balloon elevation, and coastal erosion.

In this Grade 7 lesson from enVision Mathematics, students learn how to multiply rational numbers, including fractions, decimals, and mixed numbers that are positive or negative. The lesson covers the sign rules for multiplication — positive times negative yields a negative product, and negative times negative yields a positive product — extending the integer multiplication rules to all rational numbers. Students apply these skills through real-world contexts such as changes in elevation and bank account balances.

In this Grade 7 enVision Mathematics lesson from Chapter 1: Integers and Rational Numbers, students learn how to divide integers by applying the inverse relationship between multiplication and division. The lesson covers the sign rules for integer division — that dividing integers with the same sign produces a positive quotient while dividing integers with different signs produces a negative quotient — and extends this understanding to equivalent quotients such as −(p/q), −p/q, and p/−q. Real-world contexts like drilling depth and scuba diving help students connect these rules to practical problem solving.

In this Grade 7 enVision Mathematics lesson from Chapter 1, students learn how to divide rational numbers — including fractions, mixed numbers, and decimals — using multiplicative inverses and reciprocals. The lesson extends integer division sign rules (same signs yield a positive quotient, different signs yield a negative quotient) to all rational numbers, with students rewriting division expressions as multiplication by the reciprocal to find quotients. Real-world contexts such as glacier retreat, water drainage rates, and submarine depth changes help students apply these skills across positive and negative rational number scenarios.

In this Grade 7 lesson from enVision Mathematics Chapter 1, students learn how to solve multi-step problems involving rational numbers by selecting appropriate operations such as addition, subtraction, multiplication, and division. The lesson applies the Distributive Property and other properties of operations to real-world contexts including temperature change, elevation, and financial calculations. Students practice deciding which rational number operations to use based on problem structure and reasoning.

In this Grade 7 lesson from enVision Mathematics Chapter 2, students learn how to find and apply unit rates by dividing ratios to a per-unit value, and use equivalent ratios to compare rates and solve multi-step problems. Students practice calculating unit rates in real-world contexts such as hourly pay, consumption over time, and speed, then use those rates to make comparisons and draw conclusions. The lesson builds foundational understanding of how ratios, rates, and unit rates are interconnected tools for proportional reasoning.

In this Grade 7 enVision Mathematics lesson from Chapter 2, students learn how to find unit rates when the given ratio contains fractions or mixed numbers by dividing the fractions and scaling to a denominator of 1. Through real-world problems involving cycling speeds, lawn mowing areas, and map distances, students practice dividing a fraction by a fraction to express rates such as miles per hour or square feet per hour. The lesson builds proportional reasoning skills that help students compare rates and solve multi-step problems involving ratios of fractions.

In this Grade 7 enVision Mathematics lesson from Chapter 2, students learn how to identify proportional relationships by testing whether ratios between two quantities are equivalent. Using ratio tables and unit rates, students practice recognizing when relationships are proportional and when they are not, then apply proportions as equations to solve real-world problems. This lesson builds foundational skills in proportional reasoning that support more advanced algebra concepts.

In this Grade 7 lesson from enVision Mathematics Chapter 2, students learn how to identify the constant of proportionality and use it to write equations of the form y = kx to represent proportional relationships. Students practice finding the constant of proportionality from tables, applying the equation to solve real-world problems, and determining whether a given situation can be modeled with a proportional equation.

In this Grade 7 enVision Mathematics lesson from Chapter 2, students learn to graph proportional relationships on a coordinate plane and identify key features, including that the graph must be a straight line passing through the origin. Students practice finding the constant of proportionality from ordered pairs, interpreting specific points on a graph, and writing equations of the form y = kx. The lesson also distinguishes proportional from non-proportional relationships by examining whether a line is straight and passes through the origin.

In this Grade 7 enVision Mathematics lesson from Chapter 2, students learn how to determine whether a relationship between two quantities is proportional or additive and apply that reasoning to solve real-world problems. Using ratio tables, equations, and diagrams, students practice identifying constant multiples, writing proportional equations, and recognizing when proportional reasoning does not apply. The lesson builds fluency with proportional relationships using contexts such as card collections, subscription billing, and flower arrangements.

In Grade 7 enVision Mathematics Chapter 3, Lesson 1, students learn how to find and analyze percents of numbers by setting up equivalent ratios and solving for unknown values. The lesson covers three types of percent calculations: standard percents, percents greater than 100%, and percents less than 1%. Students apply these skills to real-world contexts such as calculating tips, battery life, and distances.

In this Grade 7 enVision Mathematics lesson from Chapter 3, students learn how to use percent proportions in the form part/whole = p/100 to solve three types of percent problems: finding the percent, finding the part, and finding the whole. Through real-world examples like calculating shooting percentages and room dimensions, students apply proportional reasoning to translate between ratios and percents. This lesson builds the foundational skill of writing and solving percent proportions that students will use throughout the chapter on analyzing and solving percent problems.

In this Grade 7 lesson from enVision Mathematics Chapter 3, students learn to represent and solve percent problems using the percent equation, part = percent × whole. They practice finding the percent, the part, and the whole across real-world contexts such as meal tax, sales commission, and weight comparisons, connecting the percent equation to proportional reasoning and the constant of proportionality. This lesson builds students' ability to set up and solve all three forms of the percent equation fluently.

In this Grade 7 enVision Mathematics lesson from Chapter 3, students learn how to calculate percent change (both percent increase and percent decrease) and percent error using the percent equation. They apply the formula — amount of change divided by the original amount — to real-world scenarios such as population growth, website traffic, and estimation accuracy. Students also explore why the same percent change produces different absolute changes when applied to different original values.

In this Grade 7 lesson from enVision Mathematics Chapter 3, students learn how to solve percent markup and markdown problems using the percent equation. They calculate the percent markup by finding the difference between selling price and cost, then dividing by the original cost, and apply the same approach in reverse to find marked-down sale prices and final costs including sales tax. The lesson builds directly on students' understanding of percent increase and decrease within real-world retail and shopping contexts.

In this Grade 7 lesson from enVision Mathematics Chapter 3, students learn to apply the simple interest formula by using the percent equation to find interest earned, the interest rate, or the principal. The lesson covers how simple interest represents a proportional relationship between yearly interest and principal, with examples involving savings accounts and loans. Students practice solving for each variable in the relationship: interest amount equals interest rate times principal, extended over multiple years.

In this Grade 7 enVision Mathematics lesson from Chapter 4, students learn how to write algebraic expressions with variables, constants, and coefficients to represent real-world situations, then evaluate those expressions by substituting given values. The lesson covers translating word problems into expressions such as 20 − (2/5)d and (1/3)m − 32.5, and calculating results using rational numbers including fractions and decimals.

In this Grade 7 enVision Mathematics lesson from Chapter 4, students learn how to generate equivalent expressions using the Distributive Property, Commutative Property, and Associative Property. Students practice rewriting algebraic expressions by expanding, rearranging, and combining like terms with rational coefficients. The lesson also covers how to identify whether two expressions are equivalent by applying these properties of operations.

In this Grade 7 enVision Mathematics lesson from Chapter 4, students learn how to simplify algebraic expressions by combining like terms using the Commutative and Associative Properties of operations. The lesson covers combining like terms with integer coefficients, rational coefficients (including fractions and decimals), and expressions with two variables. Students practice identifying and grouping like terms to write equivalent expressions in simplest form.

In this Grade 7 enVision Mathematics lesson from Chapter 4, students learn how to expand algebraic expressions by applying the Distributive Property to multiply a factor outside parentheses by each term inside, including expressions with decimals, fractions, and multiple variables. Students practice expanding and simplifying expressions such as 1.5(b + 2.5) and -1/3(2 - 3x + 3), working with both numeric and variable coefficients. The lesson builds understanding of how expanded and factored forms of an expression are equivalent in value.

In this Grade 7 lesson from enVision Mathematics Chapter 4, students learn how to factor algebraic expressions using the Greatest Common Factor (GCF) and the Distributive Property. The lesson covers factoring two- and three-term expressions, including those with negative coefficients, and shows how factoring produces an equivalent expression in the form of a product of factors. Students also practice checking their work by applying the Distributive Property in reverse to verify factored results.

In this Grade 7 enVision Mathematics lesson from Chapter 4, students learn how to add algebraic expressions by identifying and combining like terms using the Commutative and Associative Properties. The lesson covers adding expressions with one or more variables, including those with rational number coefficients such as decimals and fractions. Students apply these skills to real-world contexts like calculating total costs and profits represented by multi-term expressions.

In this Grade 7 enVision Mathematics lesson from Chapter 4, students learn how to subtract algebraic expressions by applying the Distributive Property, the Commutative Property, and the concept of adding the opposite to simplify multi-term expressions. Students practice distributing a negative sign across parentheses, combining like terms, and working with rational and decimal coefficients in real-world contexts such as calculating border areas and comparing membership costs. By the end of the lesson, students can fluently subtract more complex expressions involving multiple variables and fractional or decimal terms.

In Grade 7 enVision Mathematics Chapter 4, Lesson 8, students learn how to write and analyze equivalent expressions to reveal relationships between quantities. Using the Distributive Property and combining like terms, they interpret what each term or factor represents in real-world contexts such as percent increase, tiled frames, and rectangular areas. Students practice rewriting expressions like 4(s + 1) and 1.25p to show that different forms can describe the same mathematical situation.

In this Grade 7 lesson from enVision Mathematics Chapter 5, students learn how to write two-step equations using two operations — such as multiplication and addition, or division and subtraction — to represent real-world situations. Students practice translating word problems into algebraic equations involving a variable, using tools like bar diagrams to model relationships between quantities. The lesson builds foundational skills for solving problems with equations and inequalities throughout the chapter.

In this Grade 7 enVision Mathematics lesson from Chapter 5, students learn how to solve two-step equations by applying the properties of equality — including the Subtraction, Addition, and Division Properties — to isolate a variable in two sequential steps. The lesson connects algebraic and arithmetic solution methods and uses real-world contexts such as movie ticket costs and gift card balances to build understanding. Students also compare one-step and two-step equation strategies to see how inverse operations work consistently across both.

In this Grade 7 enVision Mathematics lesson from Chapter 5, students learn how to apply the Distributive Property to solve equations in the form p(x + q) = r, including those with negative and rational coefficients. Students practice expanding expressions such as -5(s + 30) and ¼(p + 258) to isolate the variable and find the solution. The lesson builds algebraic reasoning by connecting area models to equation setup and reinforces how distributing a factor simplifies multi-step equations.

In this Grade 7 lesson from enVision Mathematics Chapter 5, students learn to solve one-variable inequalities using the Addition and Subtraction Properties of Inequality, applying the same inverse operations used to solve equations while maintaining the inequality relationship. Students write, solve, and graph inequalities on a number line using real-world contexts such as baggage weight limits and temperature changes. The lesson also addresses inequalities with rational numbers, including decimals and fractions.

In this Grade 7 enVision Mathematics lesson from Chapter 5, students learn to solve inequalities using the Multiplication and Division Properties of Inequality, including how to isolate the variable with inverse operations. A key focus is understanding why multiplying or dividing both sides of an inequality by a negative value reverses the inequality symbol. Students also practice graphing solution sets on a number line and applying these skills to real-world problems.

In this Grade 7 enVision Mathematics lesson from Chapter 5, students learn how to write and solve two-step inequalities using the Addition, Subtraction, Multiplication, and Division Properties of Inequality, including the rule that the inequality symbol reverses when multiplying or dividing by a negative number. Students practice translating real-world situations into inequalities of the form ax + b ≤ c or ax + b > c and graphing their solutions on a number line. Unlike two-step equations, students discover that two-step inequalities produce a range of solutions rather than a single value.

In this Grade 7 enVision Mathematics lesson from Chapter 5, students learn to solve multi-step inequalities by applying the Distributive Property, combining like terms, and using inverse operations, including reversing the inequality sign when multiplying or dividing by a negative number. Students practice writing, solving, and graphing solutions to inequalities on a number line using real-world contexts. The lesson builds on multi-step equation skills and extends them to inequalities involving expressions like 3(x + 2) + 13 > 55.

In this Grade 7 enVision Mathematics lesson, students learn to distinguish between a population and a sample, and determine whether a sample is representative of a population. The lesson covers key concepts including random sampling, representative samples, and how to generate random samples by assigning numbers to population members. Students also explore how multiple random samples drawn from the same population can vary while still reflecting the broader group.

In this Grade 7 lesson from enVision Mathematics Chapter 6, students learn how to draw both qualitative and quantitative inferences about a population using sample data displayed in dot plots and box plots. Students practice calculating and interpreting measures such as mean, median, and range to determine whether an inference is valid, and use proportional reasoning to make numerical estimates about a population based on sample results.

In this Grade 7 lesson from enVision Mathematics Chapter 6, students learn how to make comparative inferences about two populations by analyzing box plots alongside measures of center and variability, including median and interquartile range. Students practice interpreting differences in medians and spread between two data sets to draw conclusions about the populations they represent. This lesson builds on sampling concepts to help seventh graders reason about real-world comparisons such as homework time and plant growth across different groups.

In this Grade 7 enVision Mathematics lesson from Chapter 6, students learn how to make comparative inferences about two populations using dot plots, measures of center (mean, median, mode), and measures of variability (range, interquartile range, and mean absolute deviation). Students practice interpreting and comparing data sets to draw conclusions, such as determining whether one group generally outperformed another, and use statistical measures like the mean and MAD to support or refine those inferences. The lesson also guides students in applying population data to make predictions about individual cases.

In this Grade 7 lesson from enVision Mathematics Chapter 7, students learn what probability is and how to express it as a ratio or percent on a scale from 0 to 1. Using spinners and number cubes, students practice describing the likelihood of outcomes as impossible, unlikely, likely, or certain, and determine whether a spinner is fair based on equally likely outcomes.

In this Grade 7 lesson from enVision Mathematics Chapter 7, students learn how to determine theoretical probability using the formula P(event) = number of favorable outcomes divided by total number of possible outcomes. Students apply this concept to real-world scenarios involving spinners, tile bags, and number cubes to calculate probabilities and use proportional reasoning to make predictions about likely outcomes.

In this Grade 7 lesson from enVision Mathematics, students learn to determine experimental probability by calculating relative frequency as the ratio of favorable outcomes to total trials. They compare experimental probability to theoretical probability, explore why the two may differ, and use proportional reasoning to make predictions based on experimental data. The lesson builds on prior knowledge of theoretical probability within Chapter 7's broader study of probability.

In this Grade 7 lesson from enVision Mathematics Chapter 7, students learn how to develop and use probability models by identifying sample spaces and calculating theoretical and experimental probabilities of events. They practice listing all possible outcomes, assigning probabilities to each event, and using proportional reasoning to make estimates based on experimental data. The lesson also distinguishes between theoretical probability and experimental probability using real-world scenarios involving marble draws.

In this Grade 7 lesson from enVision Mathematics, students learn how to identify the sample space of compound events — situations involving two or more combined events — using organized lists, tables, and tree diagrams. The lesson walks through examples such as mapping gender combinations for a three-child family, letter-number spinner pairings, and coin-and-number-cube outcomes to systematically count all possible results. By the end, students can confidently represent and count outcomes in a sample space for any compound event.

In this Grade 7 lesson from enVision Mathematics Chapter 7, students learn how to find the probability of compound events by identifying all possible outcomes in a sample space. Students practice using organized lists, tables, and tree diagrams to calculate the ratio of favorable outcomes to total equally likely outcomes. Real-world scenarios involving spinners, coins, and number cubes help students compare probabilities and determine which option in a multi-step event is most likely to occur.

In this Grade 7 lesson from enVision Mathematics Chapter 7, students learn how to design and run simulations to approximate the probability of compound events. Using tools such as spinners, coins, and random number generators, students model real-world probability situations by developing probability models, matching outcomes to simulation tools, and calculating experimental probabilities from trial results. The lesson builds students' understanding of how simulations can estimate probabilities when theoretical calculations are complex.