Lesson GuideAllow students who have a clear understanding of the content thus far in the unit to work on Gallery problems of their choosing. You can then use this time to provide additional help to students who need review of the unit's concepts or to assist students who may have fallen behind on work.Gallery DescriptionsBuilding BridgesStudents will examine a pattern and use expressions to show how to continue the pattern.Patterns in a TableStudents will complete a table by noticing relationships within the table and using those relationships to fill in empty cells in the table.Expressions for Perimeter and AreaStudents will write equivalent expressions for the perimeters and areas of various rectangles.Multiplication TableStudents will complete an unusual multiplication table by writing the algebraic expression that results from multiplying the terms given in the top row by the ones given in the left column.Garden BedsStudents will find the number of square tiles needed to pave around various configurations of rectangular garden beds. Then, students will write an algebraic equation to represent the number of square tiles needed to go around any number of plants in a single row.Telephone TreeStudents will solve problems about a telephone tree and use expressions to show the number of calls completed after a given number of rounds of calling.Stacks of DVDsStudents will write an expression to describe the width of a stack of DVDs, and then they will evaluate the expression for different numbers of DVD cases and boxed sets.Exponent Card SortStudents will complete a card sort that will give them practice working with exponents. Then they will use a set of blank cards to complete sets that purposely have one or two representations missing.Matching Words and ExpressionsStudents will match a verbal statement with its expression in this card sort.Investigating Factors and MultiplesStudents will investigate an interesting property of numbers involving the greatest common factor and the least common multiple.Fourth RockStudents will solve a problem about how long it will take for two imaginary planets in an imaginary solar system to align so that they are at their closest distance from each other.Factors of a NumberStudents will decide whether a mathematical claim about factors and multiples is true or false based on given criteria.Common FactorsStudents will look at two unknown numbers with a greatest common factor of 20 and determine what other factors must be common to the two unknown numbers. Students will use their answer to make a generalization.History of VariablesStudents will research the history of variables. When were they first used? Where were they first used? Who used them?Create a VideoStudents will use their creative powers to produce a video about expressions.

Students play an Expressions Game in which they describe expressions to their partners using the vocabulary of expressions: term, coefficient, exponent, constant, and variable. Their partners try to write the correct expressions based on the descriptions.Key ConceptsMathematical expressions have parts, and these parts have names. These names allow us to communicate with others in a precise way.A variable is a symbol (usually a letter) in an expression that can be replaced by a number.A term is a number, a variable, or a product of numbers and variables. Terms are separated by the operator symbols + (plus) and – (minus).A coefficient is a symbol (usually a number) that multiplies the variable in an algebraic expression.An exponent tells how many copies of a number or variable are multiplied together.A constant is a number. In an expression, it can be a constant term or a constant coefficient. In the expression 2x + 3, 2 is a constant coefficient and 3 is a constant term.Goals and Learning ObjectivesIdentify parts of an expression using appropriate mathematical vocabulary.Write expressions that fit specific descriptions (for example, the expression is the sum of two terms each with a different variable).

Students express the lengths of trains as algebraic expressions and then substitute numbers for letters to find the actual lengths of the trains.Key ConceptsAn algebraic expression can be written to represent a problem situation. More than one algebraic expression may represent the same problem situation. These algebraic expressions have the same value and are equivalent.To evaluate an algebraic expression, a specific value for each variable is substituted in the expression, and then all the calculations are completed using the order of operations to get a single value.Goals and Learning ObjectivesEvaluate expressions for the given values of the variables.

Getting Started

Type of Unit: Introduction

Prior Knowledge

Students should be able to:

Solve and write numerical equations for whole number addition, subtraction, multiplication, and division problems.
Use parentheses to evaluate numerical expressions.
Identify and use the properties of operations.

Lesson Flow

In this unit, students are introduced to the rituals and routines that build a successful classroom math community and they are introduced to the basic features of the digital course that they will use throughout the year.

An introductory card sort activity matches students with their partner for the week. Then over the course of the week, students learn about the lesson routines: Opening, Work Time, Ways of Thinking, Apply the Learning, Summary of the Math, and Reflection. Students learn how to present their work to the class, the importance of taking responsibility for their own learning, and how to effectively participate in the classroom math community.

Students then work on Gallery problems to further explore the program’s technology resources and tools and learn how to organize their work.

The mathematical work of the unit focuses on numerical expressions, including card sort activities in which students identify equivalent expressions and match an expression card to a word card that describes its meaning. Students use the properties of operations to identify equivalent expressions and to find unknown values in equations.

The class reviews the properties of operations. The use of “ask myself” questions to make sense of problems and persevere is modeled. Students review things to do when they feel stuck on a problem. Finally, students use the properties of operations to evaluate expressions.Key ConceptsStudents use the properties of operations to justify whether two expressions are equivalent.Goals and Learning ObjectivesTo start to work on a problem, make sense of the problem by using “ask myself” questions.Persevere in solving a problem even when feeling stuck.Use the properties of operations to evaluate expressions.

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 explore the effects of wind on a plane's time and distance and represent these situations using algebraic expressions and equations. They use terms with positive, negative, and zero coefficients.Key ConceptsIn this lesson, students show what they remember from Grade 6 about writing expressions and solving one-step equations. They use what they learned earlier in Grade 7 about adding and subtracting integers. They extend these concepts to write and interpret an expression with a negative coefficient.Goals and Learning ObjectivesReview addition and subtraction of integers.Review the relationship between distance, time, and speed.Write an algebraic expression for distance in terms of time, t.Write a term with a negative coefficient.Review solving a one-step equation using the multiplication property of equality.

Students write expressions for geometric situations. They examine how different equivalent expressions can show different ways of thinking about the same problem.Key ConceptsStudents use their previous knowledge of how to find the perimeter and area of squares and rectangles. They write algebraic expressions for the perimeter and area of geometric figures. They examine how equivalent expressions, used to represent a problem situation, give clues to the approach the writer of the expression used to solve the problem. In the Challenge Problem, they use the distributive property to find the solution.ELL: For ELLs, access prior knowledge by writing the words area and perimeter on the board. Have students create concept maps associated with area and perimeter. Record students' responses on large poster paper that you can display in the room. The goal is to generate a list of words that students can use as a reference.Goals and Learning ObjectivesAccess prior knowledge of how to find the perimeter and area of squares and rectangles.Write algebraic expressions for finding perimeter or area of figures.Identify equivalent expressions.

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.

Working With Rational Numbers

Type of Unit: Concept

Prior Knowledge

Students should be able to:

Compare and order positive and negative numbers and place them on a number line.
Understand the concepts of opposites absolute value.

Lesson Flow

The unit begins with students using a balloon model to informally explore adding and subtracting integers. With the model, adding or removing heat represents adding or subtracting positive integers, and adding or removing weight represents adding or subtracting negative integers.

Students then move from the balloon model to a number line model for adding and subtracting integers, eventually extending the addition and subtraction rules from integers to all rational numbers. Number lines and multiplication patterns are used to find products of rational numbers. The relationship between multiplication and division is used to understand how to divide rational numbers. Properties of addition are briefly reviewed, then used to prove rules for addition, subtraction, multiplication, and division.

This unit includes problems with real-world contexts, formative assessment lessons, and Gallery problems.

Magic or math? Marko the Magnificent has one more trick up his sleeve, see if he impresses you with his math magic. Use your mental math skills to calculate your answer then see if you can figure out the secret to this trick.

Here is a site that clearly and thoroughly explains many topics related to algebraic expressions. There are example problems solved, problems for the student to attempt, and answers to the student problems. Point this site out to students who have been absent or who need additional instruction on this or many other topics.

This site contains activities that will help students learn concepts dealing with expressions, equations, and the coordinate plane.

This task assumes students are familiar with mixing problems. This approach brings out different issues than simply asking students to solve a mixing problem, which they can often set up using patterns rather than thinking about the meaning of each part of the equations.

The problem deals with a rational expression which is built up from operations arising naturally in a context: adding the volumes of the fertilizer and the water, and dividing the volume of the fertilizer by the resulting sum. Thus it encourages students to see the expression as having meaning in terms of numbers and operations, rather than as an abstract arrangement of symbols.

Visualize a word problem in multiple ways to develop your problem solving skills. This video focuses on converting a problem with fractions into a one-variable equation and alternately modeling it using unit blocks.

This task requires students to answer questions about the structure of an expression.

This task compares the usefulness of different forms of a quadratic expression. Students have to choose which form most easily provides information about the maximum value, the zeros and the vertical intercept of a quadratic expression in the context of a real world situation. Rather than just manipulating one form into the other, students can make sense out of the structure of the expressions.

This question provides students with an opportunity to see expressions as constructed out of a sequence of operations: first taking the square root of n, then dividing the result of that operation into s. Students studying statistics encounter the expression in this question as the standard deviation of a sampling distribution with samples of size n when the distribution from which the sample is taken has standard deviation s.