Algebraic Reasoning

Type of Unit: Concept

Prior Knowledge

Students should be able to:

Add, subtract, multiply, and divide rational numbers.
Evaluate expressions for a value of a variable.
Use the distributive property to generate equivalent expressions including combining like terms.
Understand solving an equation or inequality as a process of answering a question: which values from a specified set, if any, make the equation or inequality true?
Write and solve equations of the form x+p=q and px=q for cases in which p, q, and x are non-negative rational numbers.
Understand and graph solutions to inequalities x<c or x>c.
Use equations, tables, and graphs to represent the relationship between two variables.
Relate fractions, decimals, and percents.
Solve percent problems included those involving percent of increase or percent of decrease.

Lesson Flow

This unit covers all of the Common Core State Standards for Expressions and Equations in Grade 7. Students extend what they learned in Grade 6 about evaluating expressions and using properties to write equivalent expressions. They write, evaluate, and simplify expressions that now contain both positive and negative rational numbers. They write algebraic expressions for problem situations and discuss how different equivalent expressions can be used to represent different ways of solving the same problem. They make connections between various forms of rational numbers. Students apply what they learned in Grade 6 about solving equations such as x+2=6 or 3x=12 to solving equations such as 3x+6=12 and 3(x−2)=12. Students solve these equations using formal algebraic methods. The numbers in these equations can now be rational numbers. They use estimation and mental math to estimate solutions. They learn how solving linear inequalities differs from solving linear equations and then they solve and graph linear inequalities such as −3x+4<12. Students use inequalities to solve real-world problems, solving the problem first by arithmetic and then by writing and solving an inequality. They see that the solution of the algebraic inequality may differ from the solution to the problem.

Students match equations such as 3x − 50 = 90 and 3(x − 50) = 90 to real-world and mathematical situations. They identify the steps needed to solve these equations.Key ConceptsStudents solve equations such as 3x − 50 = 90 by using first the addition property and then the multiplication property of equality.Students also solve equations such as 3(x − 50) = 90. Equations with parentheses were introduced in the Challenge Problem of Lesson 6. Now, in this lesson, students use two methods to solve the equation. First method: use the multiplication property of equality and then the addition property of equality; second method: use the distributive property to eliminate the parentheses, then use the addition property of equality, and then the multiplication property of equality.Goals and Learning ObjectivesMatch equations to problems.Solve two-step equations.

Students work with a partner to revise their work on the Self Check. Students work with their partner to do activities that involve using expressions and equations to solve problems.Key ConceptsStudents will use what they have learned so far in this unit about writing expressions as well as writing and using equations to solve problems.Goals and Learning ObjectivesUse expressions and equations to solve problems.

Students solve real-world problems by writing and solving equations. Students estimate the solution and determine if the estimate is reasonable before finding the exact solution. They write the solution as a complete sentence.Students complete a Self Check.Key ConceptsStudents solve real-world problems by first estimating the solution and assessing the reasonableness of the solution. Next, they write an equation to solve the problem and then use the properties of equality to solve the equation. Students write the solution to the problem as a complete sentence.Goals and Learning ObjectivesWrite equations to solve multi-step real-life problems involving rational numbers.Solve equations using addition, subtraction, multiplication, and division of rational numbers.Use estimations strategies to estimate the solution and determine if the estimate is reasonable.Write the solution as a complete sentence.

Putting Math to Work

Type of Unit: Problem Solving

Prior Knowledge

Students should be able to:

Solve problems involving all four operations with rational numbers.
Write ratios and rates.
Write and solve proportions.
Solve problems involving scale.
Write and solve equations to represent problem situations.
Create and interpret maps, graphs, and diagrams.
Use multiple representations (i.e., tables, graphs, and equations) to represent problem situations.
Calculate area and volume.
Solve problems involving linear measurement.

Lesson Flow

Students apply and integrate math concepts they have previously learned to solve mathematical and real-world problems using a variety of strategies. Students have opportunities to explore four real-world situations involving problem solving in a variety of contexts, complete a project of their choice, and work through a series of Gallery problems.

First, students utilize their spatial reasoning and visualization skills to find the least number of cubes needed to construct a structure when given the front and side views. Then, students select a project to complete as they work through this unit to refine their problem-solving skills. Students explore the relationship between flapping frequency, amplitude, and cruising speed to calculate the Strouhal number of a variety of flying and swimming animals. After that, students explore the volume of the Great Lakes, applying strategies for solving volume problems and analyzing diagrams. Next, students graphically represent a virtual journey through the locks of the Welland Canal, estimating the amount of drop through each lock and the distance traveled. Students have a day in class to work on their projects with their group.

Then, students have two days to explore Gallery problems of their choosing. Finally, students present their projects to the class.

Students explore the relationship between the flapping frequency, the amplitude, and the cruising speeds of a variety of animals to calculate their Strouhal numbers.Key ConceptsStudents are expected to use the mathematical skills they have acquired in previous lessons or in previous math courses. The lessons in this unit focus on developing and refining problem-solving skills. Students will:Try a variety of strategies to approaching different types of problems.Devise a problem-solving plan and implement their plan systematically.Become aware that problems can be solved in more than one way.See the value of approaching problems in a systematic manner.Communicate their approaches with precision and articulate why their strategies and solutions are reasonable.Make connections between previous learning and real-world problems.Create efficacy and confidence in solving challenging problems in a real-world setting.Goals and Learning ObjectivesAnalyze the relationship between the variables in an equation.Write formulas to show how variables relate.Communicate findings using multiple representations including tables, charts, graphs, and equations.

Students create a bar graph showing the Strouhal numbers for a variety of birds and bats and use their graph and other data to compare the Strouhal numbers of the different animals to analyze variation and to make predictions.Key ConceptsStudents are expected to use the mathematical skills they have acquired in previous lessons or in previous math courses. The lessons in this unit focus on developing and refining problem-solving skills. Students will:Try a variety of strategies to approaching different types of problems.Devise a problem-solving plan and implement their plan systematically.Become aware that problems can be solved in more than one way.See the value of approaching problems in a systematic manner.Communicate their approaches with precision and articulate why their strategies and solutions are reasonable.Make connections between previous learning and real-world problems.Create efficacy and confidence in solving challenging problems in a real-world setting.Goals and Learning ObjectivesAnalyze the relationship among the variables in an equation.Write formulas to show how variables relate.Calculate ranges of Strouhal numbers and use these ranges to make predictions.Communicate findings using multiple representations including tables, charts, graphs, and equations.Create bar graphs.

How much water is in the Great Lakes? Students read and interpret a diagram that shows physical features of the Great Lakes and answer questions based on the diagram. They find the volume of each of the Great Lakes, as well as all five lakes combined, and make a bar graph to represent the volumes.Key ConceptsStudents are expected to use the mathematical skills they have acquired in previous lessons or in previous math courses. The lessons in this unit focus on developing and refining problem-solving skills.Students will:Try a variety of strategies to approaching different types of problems.Devise a problem-solving plan and implement their plan systematically.Become aware that problems can be solved in more than one way.See the value of approaching problems in a systematic manner.Communicate their approaches with precision and articulate why their strategies and solutions are reasonable.Make connections between previous learning and real-world problems.Create efficacy and confidence in solving challenging problems in a real-world setting.Goals and Learning ObjectivesRead and interpret graphs and diagrams.Solve problems involving volume.

Students first create a diagram that represents the distance a ship drops in each of a series of locks. Students create their diagrams based on a video of an actual ship traveling through the locks. Students need to use contextual clues in order to determine the relative drops in each of the locks.Key ConceptsStudents are expected to use the mathematical skills they have acquired in previous lessons or in previous math courses. The lessons in this unit focus on developing and refining problem-solving skills.Students will:Try a variety of strategies to approaching different types of problems.Devise a problem-solving plan and implement their plan systematically.Become aware that problems can be solved in more than one way.See the value of approaching problems in a systematic manner.Communicate their approaches with precision and articulate why their strategies and solutions are reasonable.Make connections between previous learning and real-world problems.Create efficacy and confidence in solving challenging problems in a real-world setting.Goals and Learning ObjectivesRead and interpret maps, graphs, and diagrams.Solve problems that involve linear measurement.Estimate length.Critique a diagram.

Students critique the diagrams of other students from the previous lesson and receive feedback about their own diagrams. Students revise their diagrams from the first part of the lesson based on the feedback they receive.Key ConceptsStudents are expected to use the mathematical skills they have acquired in previous lessons or in previous math courses. The lessons in this unit focus on developing and refining problem-solving skills. Students will:Try a variety of strategies to approaching different types of problems.Devise a problem-solving plan and implement their plan systematically.Become aware that problems can be solved in more than one way.See the value of approaching problems in a systematic manner.Communicate their approaches with precision and articulate why their strategies and solutions are reasonable.Make connections between previous learning and real-world problems.Create efficacy and confidence in solving challenging problems in a real-world setting.Goals and Learning ObjectivesRead and interpret maps, graphs, and diagrams.Solve problems that involve linear measurement.Estimate length.Critique a diagram.SWD: Some students with disabilities will benefit from a preview of the goals in each lesson. Students can highlight the critical features and/or concepts and will help them to pay close attention to salient information. Students need to know their goal is to develop and refine their problem solving skills.

The intent of clarifying statements is to provide additional guidance for educators to communicate the intent of the standard to support the future development of curricular resources and assessments aligned to the 2021 math standards.  Clarifying statements can be in the form of succinct sentences or paragraphs that attend to one of four types of clarifications: (1) Student Experiences; (2) Examples; (3) Boundaries; and (4) Connection to Math Practices.

Using paper, paper clips and tape, student teams design flying/falling devices to stay in the air as long as possible and land as close as possible to a given target. Student teams use the steps of the engineering design process to guide them through the initial conception, evaluation, testing and re-design stages. The activity culminates with a classroom competition and scoring to evaluate how each team's design performed.

Students create equations that have solutions to ordered pairs of an image on a graph. First students create an image on a graph and identify the ordered pairs for all the points of the image. Next, students create equations so that the x and y values of the ordered pairs are solutions to the equations.

This word problem is based estimating the height of a person over time. Note that there is a significant amount of rounding in the final answer. This is because people almost never report their heights more precisely than the closest half-inch. If we assume that the heights reported in the task stem are rounded to the nearest half-inch, then we should report the heights given in the solution at the same level of precision.

The purpose of this task is to present students with a context that can naturally be represented with an inequality and to explore the relationship between the context and the mathematical representation of that context; thus, this is an intended as an instructional task.

This lesson unit is intended to help teachers assess how well students are able to: form and solve linear equations involving factorizing and using the distributive law. In particular, this unit aims to help teachers identify and assist students who have difficulties in: using variables to represent quantities in a real-world or mathematical problem and solving word problems leading to equations of the form px + q = r and p(x + q) = r.

This lesson unit is intended to help teachers assess how well students are able to understand and use directed numbers in context. It is intended to help identify and aid students who have difficulties in ordering, comparing, adding, and subtracting positive and negative integers. Particular attention is paid to the use of negative numbers on number lines to explore the structures: starting temperature + change in temperature = final temperature final temperature Đ change in temperature = starting temperature final temperature Đ starting temperature = change in temperature.