Expressions & Equations

Expressions & Equations: 6th GradeErin Craig, Chelsea Keen, Krista Milroy, Becki Schwindt, & Joana Wu

Strand 1Apply and extend previous understandings of arithmetic to algebraic expressions.

Progressions strand iK-5 Grade6th Grade7th-8th GradeSince Kindergarten, students have been writing numerical expressions such as: 2 + 3 3 + 6 + 7 4 x (2 x 3) 8 x 5 + 8 x 2

Students begin working with the Order of Operations in 3rd grade.

In 5th grade, they write and interpret numerical expressions using brackets and parentheses.

In Grade 5 they used whole number exponents to express powers of 10

In Grade 6 they start to incorporate whole number exponents into numerical expressions, for example when they describe a square with side length 50 feet as having an area of 50 square feet

They use the any order, any grouping property (combination of the commutative and associative properties) to see the expression 7 + 6 + 3 as (7 + 3) + 6 allowing them to quickly evaluate it.

Start working systematically with the square root and cube root symbols

Work with estimates of very large and very small quantities

Move toward an understanding of the idea of a function

In 7th grade, students will solve numeric and algebraic expressions, equations, and inequalities with rational numbers applying the properties of operations.

In 8th grade, students will be working with integer exponents.

Apply and extend previous understandings of arithmetic to algebraic expressions

Progressions strand iK-5 Grade6th Grade7th-8th GradeAs they start to solve word problems algebraically, students use more complex expressions. As word problems get more complex, students nd greater benet in representing the problem algebraically by choosing variables to represent quantities, rather than attempting a direct numerical solution, since the former approach provides general methods and relieves demands on working memoryAs students move from numerical to algebraic work, the multiplication and division symbols x and are replaced by the conventions of algebraic notationstudents learn to use a dot for multiplication, or 3x instead of 3 x x

As students start to build a unified notion of the concept of function, they are able to compare proportional relationships in different ways

By 8th grade, students have the tools to solve an equation which has a general linear expression on each side of the equal signApply and extend previous understandings of arithmetic to algebraic expressions

Lloyd, G., Herbel-Eisenmann, B., & Star, J.R. (2011). Developing essential understanding of expressions, equations, and functions for teaching mathematics in grades 6-8. Reston, VA: The National Council of Teachers of Mathematics, Inc.3 + 3 + 3 + 3 + 3 + 3 + 33 x 78 + 8 + 8 + 8 + 8 Multiplication is Shorthand for Repeated Addition 8 x 52 + 2 + 2 + 2 2 x 43 x 3 x 3 x 3 x 3 x 3 x 3 378 x 8 x 8 x 8 x 8 Exponents are Shorthand for Repeated Multiplication 85 2 x 2 x 2 x 2 24Vocabulary: Numeric expression, Repeated multiplication, Power, Exponent, Base, Squared, CubedBegin with what students know and what is concrete. Squares (draw arrays) and (cubes make models)Discuss with your group.. Is (2+5)3 = 2 + 5 3 6.EE.A.1 Write and evaluate numerical expressions involving whole-number exponents.

Literature Connection

Write an expression that is equivalent to 64 using each of the following numbers and symbols once in the expression. (_2 means to the exponent of 2)

http://www.smarterbalanced.org 7 7 7 _2 + ( )In the end students will be expected to answer questions such as

Connections: Students will continue to use skills for evaluating numeric expressions using the order of operations. Students will continue to apply knowledge of operations with whole numbers.Common Mistakes/Misconceptions: Students often will evaluate 8 as 8 * 2 instead of 8 * 8. Students often get confused by the vocabulary. May have trouble understanding which base the exponent applies toEssential Questions:Why do people use exponents? What do they mean?When evaluating expressions, why do we evaluate exponents before we add and subtract?What exponent is described by each of the following words? Square? Cube?Why is something raised to the second power referred to as being squared? What about cubed?6.EE.A.2Write, read, and evaluate expressions in which letters stand for numbers.Growing Staircases

http://insidemathematics.org/index.php/6th-grade

6.EE.A.2aWrite expressions that record operations with numbers and letters standing for numbers.

SAMPLE PROBLEMSExpress the calculation Subtract y from 5 as 5 yWrite an expression that is equivalent to 64 using each of the following numbers and symbols once in the expression. 7, 7, 7, 2 - as exponent, +, , and ( )

6.EE.A.2bcoefficientvariables6.EE.A.2cEvaluate expressions at specific values of their variables. Include expressions that arise from formulas used in real-world problems. Perform arithmetic operations, including those involving whole-number exponents, in the conventional order when there are no parentheses to specify a particular order.The Truth About PEDMAS Approaches to teaching order of operations do not involve students developing significant understanding of why order should be followed (no transfer)Developing the hierarchy-of-operators triangle and its application encourages conceptual thinking and understanding

Mathematics Teaching in the Middle School Vol. 16, No. 7, March 2011Writing a PEMDAS StoryGoal is to identify the basic operations, show the need for establishing an order to approach expressions with multiple operations, and explore possible problems that might arise w/out such an orderPEMDAS story activity helps students realize that numbers and manipulation of quantities in math are often descriptions of real-world events

Mathematics Teaching in the Middle School Vol. 5, No. 9, May 20006.EE.A.3Apply the properties of operations to generate equivalent expressions.

6.EE.A.3 example 1

6.ee.a.3 example 2

misconceptionsThinking that subtraction and division are commutativeThinking that subtraction and division are associativeWhen speaking, mixing up the order of the problemExample: the problem is 15 3 and the student verbally says, Three divided by 15.6.EE.A.4

THE STANDARD:Identify when two expressions are equivalent (i.e. when the two expressions name the same number regardless of which value is substituted into them).

Example:y + y + y = 3y6.EE.A.4 - misconceptionsThe equal sign means you have to solve or do an operation.The answer has to go on the right.Simplify means making the problem easier.Errors such as n 2 + 5 as n 7 (instead of n + 3) OR 5x x = 5 Ask students to prove it what if x=1? Not thinking in a relational manner.

Percent of students at each grade level who provided each type of equal sign definition as their best definition (n = 375)Best DefinitionGrade 6Grade 7Grade 8Relational293646Operational585245Other798No response / Dont know631Source: Knuth, E., Alibali, M., Hattikudur, S., McNeil, N., Stephens, A. (2008). The importance of equal sign understanding in the middle grades. Mathematics Teaching in the Middle School.6.EE.A.4

TEACHING STRATEGIES:Goal is to develop Conceptual Understanding and Relational ThinkingTrue/False and Open Sentences with the use of variablesHave students write their own relationships___ + ___ = ___ + ___Have students look at simplified equations that have errors and fix the mistakes explaining their thinkingExplain how to fix this simplification. Give reasons.(2x +1) (x + 6) = 2x + 1 x + 6

Visual Representations such as:Algebra Balance ScalesWeight Logic RelationTake the visual representations and progress to having the students write equations to explain the relationships

Image Source: Van de Walle, J., Karp, K., Bay-Williams, J. (2013).6.EE.A.4 - GAMESWhats in the Bag?Equation War

Online Tools Algebra ScalesAssessmentWhich of the following expressions are equivalent? Why? If an expression has no match, write 2 equivalent expressions to match it.a. 2(x + 4)b. 8 + 2xc. 2x + 4d. 3(x + 4) (4 + x)e. x + 4

Source: Illustrative Mathematics

misconceptionsMisunderstanding/misreading of the expression. For example, knowing the operations that are being referenced with notation like 4x, 3(x + 2y) is critical. The fact that x means xxx, means x times x times x, not 3x or 3 times x4x means 4 times x or x+x+x+x, not forty-something. When evaluating 4x when x = 7, substitution does not result in the expression meaning 47. Use of the x notation as both the variable and the operation of multiplication can complicate this understanding.

http://katm.org/wp/wp-content/uploads/2012/06/6FlipBookedited22.pdf

Strand 2Reason about and solve one-variable equations and inequalities.

Progressions Strand iiReason about and solve one-variable equations and inequalitiesK-5 Grade6th Grade7th-8th GradeStudents have been writing numerical equations and simple equations involving one operation with a variableStudents start the systematic study of equations and inequalities and methods of solving them

As word problems grow more complex in grades 6-7, analogous arithmetical and algebraic solutions show the connection between the procedures of solving equations and the reasoning behind those procedures6.EE.B.5Understand 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? Use substitution to determine whether a given number in a specified set makes an equation or inequality true.

Example: Billy has 27 marbles and his friend gave Billy all of his marbles, now Billy has 100 marbles, how many marbles did his friend give him? Students write the equation: 27 + n = 100

http://www.brainpop.com/math/dataanalysis/inequalities/preview.weml

Students are being asked to solve equations, based on prior knowledge and reasoning, requiring them to understand the meaning of the equation as well demonstrate their thinking with the use of a scale or model to demonstrate understanding of the question being asked.

www.commoncorestandards.org

Research Algebraic thinking is present at every grade level progressing from patterns to generalizations.At middle school level, algebra becomes more abstract and symbolic in comparison to K-5 levels.Methods used to compute and the structures of our number system should be generalized, for example a+b = b+aAll areas of mathematics involve generalizing and formalizing connecting all content areas of mathematics together in some way.

Van de Walle, NCTM, CCSSO 2010AssessmentSample assessment question:

Which of the following is a solution to n 5 = 19?n=10n=11n=24n=16Now, can you write a problem with possible solutions like this one?

FormulasA formula is an algebraic rule for evaluating some quantity. A formula is a statement.Example 7.Here is the formula for the area A of a rectangle whose base is b and whose height is h.

6.ee.b.6Variables are tools for expressing mathematical ideas clearly and concisely. They are the basis for the transition from arithmetic to algebra and have many different meanings, depending on context and purpose.Understanding Algebra requires knowing what variables are and using them as tools to indicate relationship. Students need to learn both to use that language to show their ideas and to understand the meaning of someone else's mathematical representation

Using variables permits writing expressions whose values are not known or vary under different circumstances. Traditionally schools have taken to the unknown meaning of variables although the idea of a changing quantity is more powerful for understanding relationships mathematically and in real world situations.

Varying (functions, linear relationships)R=d/t25-29

The language and use of Mathematics: The meaning and use of variable Glenda Lappan NCTM news bulletin, January 2000 Hisaki is making sugar cookies for a school bake sale. He has3 1/2 cups of sugar. The recipe calls for 3/4 cup of sugar for onebatch of cookies. Which equation can be used to find b, thetotal number of batches of sugar cookies Hisaki can make?A 3 x = bB 3 = bC 3 + b = D 3 b = Let b represent any number.Use the numbers, operation symbols, and letter below to create an expression that represents the following:

5 more than the product of 3 and the number b(Not all objects will be used)

5 b + x http://www.smarterbalanced.orgRicardo has 8 pet mice. He keeps them in two cages that are connected so that the mice can go back and forth between the cages. One of the cages is blue and the other is green. Show all the ways that 8 mice can be in the two cages. Stevens, C. Ana Developing Students Understandings of Variables Mathematics teaching in the Middle School September 2005 Vol. 11 No 2

g+b=8Misconceptions:

Students view variables as abbreviations or labels rather than letters that stand for quantitiesAssign values to letters based on their positions in the alphabetUnable to operate with algebraic letters as varying quantities rather than specific valuesDifferent letter within a number sentence must represent different numerical valuesa=ac=r Always SometimesNeverStevens, C. Ana Developing Students Understandings of Variables Mathematics teaching in the Middle School Setpember 2005 Vol. 11 No 2 6.EE.B.7 Solve real-world and mathematical problems by writing and solving equations of the form x + p = q and px = q for cases in which p, q, and x are all nonnegative rational numbers.

6.ee.b.7 - example

6.EE.B.8researchSymbolism, especially involving equality and variables, must be well understood conceptually for students to be successful in mathematics, particularly algebra. (p. 258)

Exploration of representations in a variety of ways strengthens understanding of the same relationship from different points of view.

Van de Walle, 2013Instructional ApproachThe skill of solving the equation must be developed conceptually before it is developed procedurally.Provide multiple situations in which students must determine if there is a single solution or multiple solutions, creating the need to use different types of equations.Provide practice in the use of positive and negative numbers.Provide practice in writing equations by giving students an equation and having them write a problem and by having them write an equation from a word problem.

CCSS, Arizona DOE, Ohio DOE, North Carolina DOE

Assessmentgrade6commoncoremath.wikispaces.hcpss.org10 divided by y is greater than or equal to 2

Misconceptions Strand iiMisunderstanding or misreading of the expressionLack of understanding of the operations being usedUse of x as both the variable and the operation of multiplication complicates understanding (e.g. 6x x 5 = 600)Multiplication symbolvariable

Strand 3Represent and analyze quantitative relationships between dependent and independent variables.

Progressions Strand iiiRepresent and analyze quantitative relationships between dependent and independent variablesK-5 Grade6th Grade7th-8th GradeLearning to use symbols, such as parentheses, to write equations

Write simple equations

Interpret numerical expressions without evaluating them

Generate two numerical patterns using two given rules

As they work with equations, students begin to develop a dynamic understanding of variables

Students can use tables and graphs to develop an appreciation of varying quantities

Begin the systematic study of equations and inequalities and how to solve them

This prepares students for work with functions in later grades

Arithmetical and algebraic solutions show the connection between procedures of solving equations and the reasoning behind the procedures

Strand 3Represent and analyze quantitative relationships between dependent and independent variables.

Progression:

3.OA.9 4.OA.C 5.OA.B 6.RP 8.F 6.EE.C 7.RP 8.EE.B6.EE.C.9Use variables to represent two quantities in a real-world problem that change in relationship to one another; write an equation to express one quantity, thought of as the dependent variable, in terms of the other quantity, thought of as the independent variable. Analyze the relationship between the dependent and independent variables using graphs and tables, and relate these to the equation.

Integrates with Science BIG TIME

6.EE.C.9 - MisconceptionsConfusion about which variable is the dependent and which variable is the independentD.R.Y. M.I.X.Students may think of the variable as a place holder for one exact number, rather than a representation of multiple, possible infinite values (Van de Walle, et al., 2013).

6.EE.C.9 Teaching StrategiesLook at the relationships between the dependent and independent variables.Sketch or look at the shape of graphs and describe the story they would tell.

Include fractions and decimals with the variables.

Use real-world examples and integrate different representations of data.

Make tables with input (independent variable) and output (dependent variable) to help develop thinking about the functional relationship. It is a good idea to include a column for my thinking as well.

Source: Markworth, K. (2012). Growing patterns: Seeing beyond counting.

Image Source: Van de Walle, J., et al., (2013).

AssessmentWe would like students to make connections across multiple representations. Students should be able to translate freely across a variety of representations. When given a representation to start with, students should be able to develop a different representation using the same information.Physical materials or drawingsTablesWordsSymbolsGraphs

Research sourcesCCSS for MathematicsUniversity of Arizona Progression DocumentZimba SchematicVan de WalleBrainPOPConnected MathematicsTeachers Pay TeachersIllustrative Mathematicswww.learnzillion.comwww.graniteschools.org

Examples: What is the relationship between the two variables? Write an expression that illustrates the relationship.

Use the graph below to describe the change in y as x increases by 1. Susan started with $1 in her savings. She plans to add $4 per week to her savings. Use an equation, table and graph to demonstrate the relationship between the number of weeks that pass and the amount in her savings account. o Language: Susan has $1 in her savings account. She is going to save $4 each week. o Equation: y = 4x + 1

o Table: o Graph: