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Number properties and operations
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TopicCommon coreNAEPMisconceptionauthor
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1. Number sense6.NS.C.6
Understand a rational number as a point on the number line. Extend number line diagrams and coordinate axes familiar from previous grades to represent points on the line and in the plane with negative number coordinates.		Use place value to model and describe integers and decimals
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Model or describe rational number or numerical relationships using number lines and diagrams Misunderstanding the connections among proportional relationships part/whole, part/part, whole/part by writing part (shaded)/part (not shaded) relationship instead of part/whole Ashlock (2006)
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Misunderstanding the connections among proportional relationships part/whole, part/part, whole/part by failing to realize all parts must be of equal size
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Write or rename rational numbers
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Recognize, translate or apply multiple representations of rational numbers (fractions, decimals, and percentages) in meaningful contexts.
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8.EE.A.4
Perform operations with numbers expressed in scientific notation, including problems where both decimal and scientific notation are used. Use scientific notation and choose units of appropriate size for measurements of very large or very small quantities (e.g., use millimeters per year for seafloor spreading). Interpret scientific notation that has been generated by technology.		Express or interpret numbers using scientific notation from real-life contexts.Fail to generalize patterns in powers.Slavit, D. (2006) p. 6
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Incorrect interpretation of the exponential expressionPinchback (1991)
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6.NS.C.7
Understand ordering and absolute value of rational numbers.		Find or model absolute value or apply to problem situations.
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Order or compare rational numbers (fractions, decimals, percents, or integers) using various models and representations (e.g., number line).
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8.EE.A.3
Use numbers expressed in the form of a single digit times an integer power of 10 to estimate very large or very small quantities, and to express how many times as much one is than the other.		Order or compare rational numbers including very large and small integers, and decimal and fractions close to zero. Ordering decimals incorrectly basing the value of a decimal on incorrect reasoning such as the number of digitsAshlock (2006)
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TopicCommon coreNAEPMisconceptionauthor
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3. Number Operations7.NS.A.1
Apply and extend previous understandings of addition and subtraction to add and subtract rational numbers; represent addition and subtraction on a horizontal or vertical number line diagram.		Perform computations with rational numbers Attempted to write a fraction in lower terms by estimating when you cannot divide evenly (e.g. 3/8 to 1/4)Ashlock (2006)
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Simplified the numerator but not the denominator
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Added/subtracted the numerators and added/subtracted the denominators (overgeneralized whole number operations)
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Identified a common denominator but failed to change the fractions into equivalent form
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Subtracted mixed numbers without necessary regrouping and instead subtracting the smaller number from the larger
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Regrouped incorrectly – either by mistakenly using base ten reasoning or some other error
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Multiplied fractions incorrectly by cross-multiplying (as if solving a proportion) and then applied an invented algorithm to get the answers (such as adding the sum of the two cross multiplications)
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Incorrectly changing a whole number into a fraction (e.g. changing the whole number 6 into 6/6 instead of 6/1)
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Divided fractions incorrectly by dividing the numerators and then dividing the denominators (overgeneralizing whole number operations)
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Applied only part of the “invert and multiply” algorithm – student may remember to multiply but forget to invert the second fractionBrown, G., & Quinn, R. J. (2006)
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Ignored place value concepts to put the decimal in the answerAshlock (2006)
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Placed the decimal point in the incorrect place when multiplying or dividing
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Not knowing what sign to put on the sum involving adding a positive and negative number
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Making the sum of two negatives a positive
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Not using place value concept to put the decimal in the correct place in the answer
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When subtracting - not regrouping but instead taking the smaller number minus the larger (reversing order)
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Placing the decimal point in the incorrect place when multiplying - multiple reasons
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6.NS.B.2
Fluently divide multi-digit numbers using the standard algorithm.		Describe the effect of multiplying and dividing by numbers including the effect of multiplying or dividing a rational number by:

Zero, or
• A number less than zero, or
• A number between zero and one
• One, or
• A number greater than one.
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6.NS.B.3
Fluently add, subtract, multiply, and divide multi-digit decimals using the standard algorithm for each operation.
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7.NS.A.2
Apply and extend previous understandings of multiplication and division and of fractions to multiply and divide rational numbers.
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6.NS.A.1
Interpret and compute quotients of fractions, and solve word problems involving division of fractions by fractions, e.g., by using visual fraction models and equations to represent the problem.		Interpret rational number operations and the relationships between them
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6.NS.C.8
Solve real-world and mathematical problems by graphing points in all four quadrants of the coordinate plane. Include use of coordinates and absolute value to find distances between points with the same first coordinate or the same second coordinate.		Solve application problems involving rational numbers and operations using exact answers or estimates as appropriate
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7.NS.A.3
Solve real-world and mathematical problems involving the four operations with rational numbers.
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TopicCommon coreNAEPMisconceptionauthor
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4. Ratios and proportional reasoning6.RP.A.3
Use ratio and rate reasoning to solve real-world and mathematical problems, e.g., by reasoning about tables of equivalent ratios, tape diagrams, double number line diagrams, or equations.		Use ratios to describe problem situationsHolding a rigid understanding of proportionality – absolute vs. relative
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6.RP.A.1
Understand the concept of a ratio and use ratio language to describe a ratio relationship between two quantities.		Inability to recognize ratios can represent the same units or different unitsSingh, P. (2000).
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7.RP.A.1
Compute unit rates associated with ratios of fractions, including ratios of lengths, areas and other quantities measured in like or different units.		Use fractions to represent and express ratios and proportions Confusion with ratios written as quotientsBush, S. (2011)
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6.RP.A.2
Understand the concept of a unit rate a/b associated with a ratio a:b with b is not equal to 0, and use rate language in the context of a ratio relationship.		Lack of recognition of the different ways a ratio can be written – equivalent ratios and complex ratiosOjose. B. (2015)
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7.RP.A.2
Recognize and represent proportional relationships between quantities.		Use proportional reasoning to model and solve problems (including rates and scaling) Inability to unitizeSingh, P. (2000).
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Inability to recognize that ratios are a relationship between two quantitiesDe Bock, D., Van Dooren, W., Verschaffel, L., & Janssens, D. (2002)
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7.RP.A.3
Use proportional relationships to solve multistep ratio and percent problems.		Solve problems involving percentages (including percent increase and decrease, interest rates, tax, discount, tips, or part/whole relationships)
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TopicCommon coreNAEPMisconceptionauthor
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5. Properties of number and operationsDescribe odd and even integers and how they behave under different operations
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6.NS.B.4
Find the greatest common factor of two whole numbers less than or equal to 100 and the least common multiple of two whole numbers less than or equal to 12. Use the distributive property to express a sum of two whole numbers 1–100 with a common factor as a multiple of a sum of two whole numbers with no common factor.		Recognize, find, or use factors, multiples or prime factorization.
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Recognize or use prime and composite numbers to solve problems
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Use divisibility or remainders in problem settings
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Apply basic properties of operations
Believing that the commutative and associative properties are true for subtraction and divisionWarren (2003)
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Not knowing what sign to put on the sum involving adding a positive and negative numberAshlock (2006)
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Making the sum of two negatives a positive
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Not using place value concept to put the decimal in the correct place in the answer
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When subtracting - not regrouping but instead taking the smaller number minus the larger (reversing order)
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Placing the decimal point in the incorrect place when multiplying - multiple reasons
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Using faulty reasoning for performing order of operations such as performing operations left to rightLinchevski, L., & Livneh, D. (1999)
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Algebra
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TopicCommon coreNAEPMisconceptionauthor
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1. Patterns, relationships, and functionsRecognize, describe, or extend numerical and geometric patterns using tables, graphs, words, or symbols. Difficulty depicting key aspects and relationships in patternScheuermann and van Garderen (2008)
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Generalize a pattern appearing in a numerical sequence, table or graph using words or symbols. Difficulty accurately representing a problem using graphic notation or they do not understand the purpose of representation
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Difficulty interpreting graph-scaleSarah B. Bush (2011)
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Analyze or create patterns, sequence, or linear functions given a rule Not understanding a linear function represents a rate of change or not connecting linearity and proportionalityLobato 2010
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Confusion representing linear and exponential functionsLA Kasmer, OK Kim (2012)
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8.F.A.3
Interpret the equation y = mx + b as defining a linear function, whose graph is a straight line; give examples of functions that are not linear.		Identify functions as linear or nonlinear or contrast distinguishing properties of function from tables, graphs, or equations. Lack of recognition of the inconsistency between the positive slope of the line and the negative slope in the equation.Kalchman, M., & Koedinger, K. R. (2005)
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8.F.A.2
Compare properties of two functions each represented in a different way (algebraically, graphically, numerically in tables, or by verbal descriptions).
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Lacking understanding of and not seeing the connection between multiple representations of functions
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Interpret the meaning of slope or intercepts in linear functions Misunderstanding the proportionality (or non-proportionality) of linear functions. Students often believe that linear functions are proportional simply because they increase (or decrease) at a constant rate.W Van Dooren, D De Bock, A Hessels, D Janssens (2004)
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TopicCommon coreNAEPMisconceptionauthor
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2. Algebraic representations
Translate between different representations of linear expression using symbols, graphs, tables diagrams, or written descriptions
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Analyze or interpret linear relationships expressed in symbols, graphs, tables, diagrams, or written descriptions
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Graph or interpret points represented by ordered pairs of numbers on a rectangular coordinate system
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Difficulty plotting points - reversed x- and y-coordinateSwafford and Langrall, 2000
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Solve problems involving coordinate pairs on the rectangular coordinate system Difficulty understanding the concept of the independent and dependent variables
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Identify or represent functional relationships in meaningful context including proportional, linear, and common nonlinear (e.g., compound interest, bacterial growth) in tables, graphs, words, or symbols.
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TopicCommon coreNAEPMisconceptionauthor
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3. Variables, expressions, and operationsWrite algebraic expressions, equations, or inequalities to represent a situation With expressions involving subtraction, students would incorrectly write an expression such as 4 - n instead of n - 4.M. M. Capraro and Joffrion, 2006
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7.EE.A.1
Apply properties of operations as strategies to add, subtract, factor, and expand linear expressions with rational coefficients.		Perform basic operations, using appropriate tools, on linear algebraic expressions (including grouping and order of multiple operations involving basic operations, exponents, roots, simplifying, and expanding).
Incorrectly viewing variables as labels, units or believing that the value of a variable has something to do with its position in the alphabetHerscovics, N., & Kieran, C. (1980)
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8.EE.A.1
Know and apply the properties of integer exponents to generate equivalent numerical expressions.		Believing that two variables in the same equation cannot represent the same valueMacGregor and Stacey (1997)
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Failing to understand that variables can represent varying quantitiesAsquith et al. (2007)
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Believing that the commutative and associative properties are true for subtraction and divisionLinchevski, L., & Livneh, D. (1999)
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TopicCommon coreNAEPMisconceptionauthor
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4. Equations and inequalities
7.EE.A.2
Understand that rewriting an expression in different forms in a problem context can shed light on the problem and how the quantities in it are related.		Solve linear equations or inequalities (e.g., ax +b = c or ac +b = cx +d or ax +b > c). Struggle with the multiple meanings and uses of variablesAsquith et al. (2007)
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8.EE.C.7
Solve linear equations in one variable.		Difficulty with the syntax of algebraic notationNovotna, J., & Hoch, M. (2008)
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8.EE.C.8
Analyze and solve pairs of simultaneous linear equations.
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6.EE.A.4
Identify when two expressions are equivalent (i.e., when the two expressions name the same number regardless of which value is substituted into them).		Interpret “=” as an equivalence between two expressions and use these interpretations to solve problems.
Believing that the equal sign means “the answer is” rather than expressing a relationshipBaroudi (2006)
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6.EE.B.5
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? Use substitution to determine whether a given number in a specified set makes an equation or inequality true.		L Linchevski, N Herscovics (1996)
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7.EE.B.4
Use variables to represent quantities in a real-world or mathematical problem, and construct simple equations and inequalities to solve problems by reasoning about the quantities.		Analyze situations or solve problems using linear equations and inequalities with rational coefficients symbolically or graphically (e.g., ax +b = c or ax +b = cx +b)
Not checking their solution or error in checking their solutionPerrenet & Wolters, 1994
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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.		Symbolic notationSwafford & Langrall (2000)
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6.EE.B.8
Write an inequality of the form x > c or x < c to represent a constraint or condition in a real-world or mathematical problem. Recognize that inequalities of the form x > c or x < c have infinitely many solutions; represent solutions of such inequalities on number line diagrams.		Reversal order errorSwan (2020)
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Combining (or not combining) like termAshlock (2006)
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Use and evaluate common formulas (e.g., relationship between a circle’s circumference and diameter [C = pi d], distance and time under constant speed).
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