Strand: EXPRESSIONS AND EQUATIONS (6.EE)

Apply and extend previous understandings of arithmetic to algebraic expressions involving exponents and variables (Standards 6.EE.1–4). They reason about and solve one-variable equations and inequalities (Standards 6.EE.5–8). Represent and analyze quantitative relationships between dependent and independent variables in a real-world context (Standard 6.EE.9).
• Algebra Four
This lesson contains a game activity designed to help students practice solving algebraic equations.
• All, Some, or None?
This is a rich discussion question that spotlights the different possibilities for an equation when a variable is involved. Its important to encourage students to understand what a solution to an equation is and what it means to solve an equation before they learn general and efficient procedures.
• Anna in D.C.
The purpose of this task is to give students an opportunity to solve a multi-step percentage problem that can be approached in many ways.
• Balancing Scales To Solve Equations
The focus of this lesson and interactive is balancing algebraic equations. Students engage with the Annenberg interactive to solve 3 balance problems. The classroom activity then asks students to change balance problems into algebraic equations where variables represent unknown amounts. NOTE: You have to create a Free PBS Account to view this web page, but it is easy to do and worth the effort.
• Busy Day
This problem can be solved using simple arithmetical reasoning but is also naturally represented by a simple equation. Thus, it provides a good entry point for students into representing quantities in contexts with variables and expressions and building equations that reflect the relationships presented in the context.
• Chapter 6 - Mathematical Foundations (UMSMP)
This is Chapter 6 of the Utah Middle School Math: Grade 6 textbook. It provides a Mathematical Foundation for Expressions and Equations.
• Chapter 6 - Student Workbook (UMSMP)
This is Chapter 6 of the Utah Middle School Math: Grade 6 student workbook. It covers the following topics: Expressions and Equations.
• Chocolate Bar Sales
In this task students use different representations to analyze the relationship between two quantities and to solve a real world problem. The situation presented provides a good opportunity to make connections between the information provided by tables, graphs and equations.
• Commutative and Associative Equations
This lesson focuses on how to rearrange and combine parts of algebraic expressions by using the commutative and associative properties of addition. NOTE: You have to create a Free PBS Account to view this web page, but it is easy to do and worth the effort.
• Determining Surface Area with Unit Blocks, Rulers, and Nets
In this video students are shown how to calculate the surface area of a prism. The classroom activity in the lesson requires that students apply this knowledge and measure the surface areas of real 3-Dl objects. NOTE: You have to create a Free PBS Account to view this web page, but it is easy to do and worth the effort.
• Distance to School
This task asks students to find equivalent expressions by visualizing a familiar activity involving distance.
• Distributive Property with Variables
Algebra tiles are used to generate equivalent expressions using the distributive property in this instructional video. The classroom activity asks student to further explore the distributive property. NOTE: You have to create a Free PBS Account to view this web page, but it is easy to do and worth the effort.
In this problem students have to transform expressions using the distributive, commutative and associative properties to decide which expressions are equivalent.
• Equivalent Expressions with the Distributive Property
This animated Math Shorts video explains how the distributive property can help students model and create equivalent expressions. In the accompanying classroom activity, students play a quick game where they identify common factors within an expression and work on a series of problems that expand their understanding of how to apply the distributive property. While the problems begin with whole number expressions, students soon work toward algebraic notation and eventually develop the idea that ax + bx can be rewritten as x(a + b). NOTE: You have to create a Free PBS Account to view this web page, but it is easy to do and worth the effort.
• Exponent Experimentation 1
The purpose of this task is to give students experience working with exponential expressions and to promote making use of structure to compare exponential expressions.
• Exponent Experimentation 2
The purpose of this task is to give students experience experimenting with equivalent numerical expressions. This work supports fluency because students practice working with operations, decomposing numbers, and recognizing perfect squares and perfect cubes.
• Exponent Experimentation 3
The purpose of this task is to give students experience working with exponential expressions and with what is meant by a solution to an equation.
• Expressions and Equations (6.EE) - 6th Grade Core Guide
The Utah State Board of Education (USBE) and educators around the state of Utah developed these guides for Mathematics Grade 6 - Expressions and Equations.
• Families of Triangles
The purpose of this task is to introduce students to the idea of a relationship between two quantities by using a familiar geometic context. In order to benefit from this task, students should have already developed and become comfortable with a formula for the area of a triangle. The focus of this task should be on noticing the relationship between height and area and creating a graphical and algebraic representation of this relationship, not on understanding the meaning behind the geometric terms.
• Finding Patterns to Make Predictions
This activity asks students to identify and contemplate mathematical patterns that we see around us. They are asked to represent them in a table and predict the pattern to the 7th, 9th, and nth terms. NOTE: You have to create a Free PBS Account to view this web page, but it is easy to do and worth the effort.
• Firefighter Allocation
In this task students are asked to write an equation to solve a real-world problem. There are two natural approaches to this task. In the first approach, students have to notice that even though there is one variable, namely the number of firefighters, it is used in two different places. In the other approach, students can find the total cost per firefighter and then write the equation
This particular task could be used for instruction or assessment. The context lends itself to the use of inequalities, so it could also be used to introduce inequalities.
The purpose of this task is for students to solve a contextual problem where there is a multiplicative relationship between several quantities in the context. These relationships can either be represented in a ratio table or with a linear equation.
• Grade 6 Math Module 4: Expressions and Equations (EngageNY)
In Module 4, Expressions and Equations, students extend their arithmetic work to include using letters to represent numbers in order to understand that letters are simply "stand-ins" for numbers and that arithmetic is carried out exactly as it is with numbers. Students explore operations in terms of verbal expressions and determine that arithmetic properties hold true with expressions because nothing has changedthey are still doing arithmetic with numbers. Students determine that letters are used to represent specific but unknown numbers and are used to make statements or identities that are true for all numbers or a range of numbers.
In order to assist educators with the implementation of the Common Core, the New York State Education Department provides curricular modules in Pre-K-Grade 12 English Language Arts and Mathematics that schools and districts can adopt or adapt for local purposes.
• Grade 6 Unit 3: Expressions (Georgia Standards)
In this unit students will represent repeated multiplication with exponents. Evaluate expressions containing exponents to solve mathematical and real world problems. Translate verbal phrases and situations into algebraic expressions. Identify the parts of a given expression. Use the properties to identify equivalent expressions. Use the properties and mathematical models to generate equivalent expressions.
• Grade 6 Unit 4: One Step Equations and Inequalities (Georgia Standards)
In this unit students will: Determine if an equation or inequality is appropriate for a given situation. Solve mathematical and real-world problems with equations. Represent real-world situations as inequalities. Interpret the solutions to equations and inequalities. Represent the solutions to inequalities on a number line. Analyze the relationship between dependent and independent variables through the use of tables, equations and graphs.
• Graphing Inequalities: Fractions and Decimals from 0 to 1
In this interactive, use logic to solve three riddles involving high-jump performers in a flea circus. Then, using knowledge of inequalities, place the fleas in the appropriate range on a vertical number line. NOTE: You have to create a Free PBS Account to view this web page, but it is easy to do and worth the effort.
• Graphing Inequalities: Fractions, Mixed Numbers, and Decimals
In this interactive, use logic to solve three riddles involving a jumping frog competition. Then, using knowledge of inequalities and rational numbers, place the frogs in the correct range from 0 to 5 on a number line. NOTE: You have to create a Free PBS Account to view this web page, but it is easy to do and worth the effort.
• Graphing Inequalities: Rational Numbers
In this interactive, use logic to solve three riddles involving a jumping wallaby competition. Then, using knowledge of inequalities and rational numbers, place the wallabies on the correct range from 5 to 5 on the number line. NOTE: You have to create a Free PBS Account to view this web page, but it is easy to do and worth the effort.
• Height Requirements
The goal of this task is to express constraints from a real world context using one or more inequalities. In addition to writing inequalities, students also display the numbers (heights) satisfying the inequalities on a number line.
• Introducing Equivalent Expressions 1
The purpose of this series of two tasks is to highlight that while a numerical expression can only take one value, an algebraic expression can take many different values depending on the value of the variable. All work is done out of a context with positive whole numbers, so that discussion can focus on the shift from numerical to algebraic expressions.
• Introducing Equivalent Expressions 2
The purpose of this series of two tasks is to highlight that while a numerical expression can only take one value, an algebraic expression can take many different values depending on the value of the variable.
• IXL Game: Algebra: Evaluate expressions
This game is designed to help sixth graders understand how to evaluate expressions involving integers. This is just one of many online games that supports the Utah Math core. Note: The IXL site requires subscription for unlimited use.
• Linear Inequalities
This online tutorial is designed to help the student to understand the vocabulary of inequalities and then use addition, subtraction, multiplication and division to solve linear inequalities.
• Log Ride
In this instructional task students are given two inequalities, one as a formula and one in words, and a set of possible solutions. They have to decide which of the given numbers actually solve the inequalitie
• Make Use of Structure
The purpose of this task is to help students reason about the meaning of equations and the solution of an equation, and to give them an opportunity to make connections with operations with fractions and decimals.
• Morning Walk
This task presents a straight forward question that can be solved using an equation in one variable. The numbers are complicated enough so that it is natural to set up an equation rather than solve the problem in one's head.
• Order of Operations: PEMDAS
A Flocabulary rap song instructs students on the order of operations and then they apply that knowledge to the classroom activity. NOTE: You have to create a Free PBS Account to view this web page, but it is easy to do and worth the effort.
• Pennies to Heaven
The goal of this task is to give students a context to investigate large numbers and measurements. Students need to fluently convert units with very large numbers in order to successfully complete this task.
• Reciprocity
The purpose of this task is to help students understand why dividing by a fraction gives the same result as multiplying by its reciprocal. This is accomplished by writing the division equation along with related multiplication equations and diagrams showing the situation for several different contexts.
• Rectangle Perimeter 1
This tasks gives a verbal description for computing the perimeter of a rectangle and asks the students to find an expression for this perimeter. They then have to use the expression to evaluate the perimeter for specific values of the two variables.
• Rectangle Perimeter 2
This task is a natural follow up for task Rectangle Perimeter 1. After thinking about and using one specific expression for the perimeter of a rectangle, students now extend their thinking to equivalent expressions for the same quantity.
• Rectangle Perimeter 3
The purpose of this task is to ask students to write expressions and to consider what it means for two expressions to be equivalent.
• Seven to the What?!?
Students are asked to find the last digit and the last two digits of 7 to the 2011th power. So the purpose of this task is to give students an opportunity to practice working with positive integer exponents.
• Sierpinski's Carpet
The purpose of this task is to help motivate the usefulness of exponential notation in a geometric context and to give students an opportunity to see that sometimes it is easier to write a number as a numeric expression rather than evaluating the expression.
• The Djinnis Offer
The purpose of this task is to introduce the idea of exponential growth and then connect that growth to expressions involving exponents. It illustrates well how fast exponential expressions grow.
• Triangular Tables
This task provides a good opportunity for group work and class discussions where students generate and compare equivalent expressions.
• Watch Out for Parentheses
This problem asks the student to evaluate three numerical expressions that contain the same integers yet have differing results due to placement of parentheses.
• Which Goes with Which?
Sometimes when algebraic equations are introduced in grade 6, students are just asked to write an equation for a problem that they could easily solve numerically. This task takes the focus off of the answer and offers an opportunity to focus on, for example, how we write the product of 2 and an unknown quantity.

http://www.uen.org - in partnership with Utah State Board of Education (USBE) and Utah System of Higher Education (USHE).  Send questions or comments to USBE Specialist - Lindsey Henderson and see the Mathematics - Secondary website. For general questions about Utah's Core Standards contact the Director - Jennifer Throndsen .

These materials have been produced by and for the teachers of the State of Utah. Copies of these materials may be freely reproduced for teacher and classroom use. When distributing these materials, credit should be given to Utah State Board of Education. These materials may not be published, in whole or part, or in any other format, without the written permission of the Utah State Board of Education, 250 East 500 South, PO Box 144200, Salt Lake City, Utah 84114-4200.