algebra 1 semester 1 rubric

Algebra 1 Semester 1 RubricUnit 0: Pre-Algebra Skills

Above and BeyondMeets Requirements

Needs ReviewNever Learned/ Needs Re-teaching

Number properties and types of number

You know the sets of numbers and number properties. You can recognize any number property in context. You know why having names for these sets and properties is useful.

You have all the number sets and number properties memorized and you can rattle off the differences between them. You can recognize any number property in context.

You may know some of the number sets and properties by rote, but you don’t understand the differences between them

Like most students (including me in high school) you have no idea why you should memorize all these words.

Fraction Operations (rational numbers)

You can reduce, add, subtract, multiply and divide fractional and mixed number expressions with no hesitation. You have taken the time to understand why we need common denominators to add or subtract fractions. You may be able to intuit why division uses the reciprocal.

You can reduce, add, subtract, multiply and divide fractional expressions with no hesitation. You understand what the GCF and LCM are and why they are useful in working with fractions.

You can perform 3 or 4 of the operations (reducing, adding, subtracting, multiplying or dividing) fractions. You may forget the need for a common denominator and/or may forget to use the reciprocal in division. You hesitate but can chug through rules by rote most of the time.

You can perform one or two of the operations, but may have a lot of fear and anxiety built up around fractions (just like about 70% or 80% of adults). Don’t panic. Almost all of my students have come in belonging to this category and moved to “above and beyond” by the end of the course.

Negative Number Operations (integers,) Absolute Value and Cartesian grid

You can add, subtract, multiply and divide negative numbers. You can understand application problems involving negative numbers and you understand how and why the rules work as they do. You understand what absolute value is and you plot all points flawlessly on the Cartesian grid

You can add, subtract, multiply and divide negative numbers including decimals and fractions. You may stumble with word problems but can usually figure them out. You know the definition of absolute value. You can plot points on the Cartesian grid.

You can add, subtract, multiply and divide integers (meaning no decimals or fractions) though you may have the rules memorized by rote and you confuse them. You know absolute value makes things positive. You can usually graph points but you may mix up x and y.

You mix up the negative rules (ex: writes that a negative times a negative is negative.) You have trouble with addition and subtraction- when the values go up and when they go down. You may be intimidated by word problems. You’re not sure what absolute value is. You have trouble plotting points correctly.

Order of Operations and Distributive Property

You have the order of operations memorized and find the correct answer every time. You can play with the order of operations to solve

You have the order of operations memorized and find the correct answer most of the time. You can correctly do and undo the

You know of the order of operations, but sometimes forget the “left to right” rule and do additions before subtractions. You

You confuse the order or were taught PEMDAS or BEDMAS and think that multiplication has to come before division or vice

puzzles and work backwards through problems. You know why the distributive property works.

distributive property, but you may not be able to explain why it works.

may have trouble recognizing multiplication and division operators or implied parentheses.

versa. These are flawed teaching techniques that confuse many students. You never remember to distribute.

Exponents and Roots

You know the exponent rules and you can prove them all. You know how to take square roots and you can apply some rules of exponents to help simplify sq roots.

You know the exponent rules that allow us to combine exponent expressions and you know how to take square roots and what square roots are for.

You know what exponents and square roots are, but you may only know two or three of the rules for combining exponent expressions.

You might know what an exponent and a square root is, but you don’t know any rules that help us work with exponents.

Unit 1: Introduction to Algebra



Variable basics: writing expressions and formulas

You understand that there are several types of variables and each serves a unique purpose. You can write expressions and formulas for everyday occurrences or from scenarios and stories that can help you understand a problem in a new way.

You know the names for different types of variables and their definitions. You can write expressions and formulas when given verbal sentences or word problems. You understand what a variable is.

You don’t understand the differences between types of variables, but you recognize that variables aren’t always representing just one unknown number. You can write expressions but may confuse the operations or the order of operations.

You may know what one definition of a variable but you think of it only as an “unknown” number. You have trouble turning a verbal sentence into an expression and you mix up operations or order of operations.

Combining expressions

You understand that we can add and subtract things that are of the same type (like terms). You can deduce that when we multiply things of the same type, exponent rules apply.

You understand that we can add and subtract things that are of the same type. You think we can multiply and divide things of the same type (like terms) too but you’re not sure how to write it.

You know that we can add or subtract things of the same type, but sometimes you have trouble recognizing what “same type” means.

You add things of different types together often. You’re not sure why 2x+3x doesn’t equal 5x2. You combine expressions incorrectly and you’re not sure why it’s incorrect.

Evaluating expressions and formulas

You can write an expression to fit a situation, stick in appropriate numbers, apply the correct order of operations and then interpret your answers as reasonable or unreasonable.

You can take any number, stick it into an algebraic expression and follow the correct order of operations to arrive at a numerical answer.

You understand how to plug in a number for a variable, but you sometimes forget to put in parentheses or you don’t follow the correct order of operations and you arrive at incorrect answers sometimes.

You may always forget the parentheses thus leading to incorrect order of operations. You may not understand why we want to stick in numbers for variables.

Writing equations

You can write an equation from a

You can take a scenario, write an

You know that an equation is when we

You’re not sure what the difference is

scenario or word problem and you see that we do this so that we can find the value of an unknown variable.

expression and set it equal to a number to create an equation.

have an expression set equal to a number. You’re not sure why we want to do this though or how to write one.

between an expression and an equation and you’re not sure how to write an equation when given a scenario.

Solving equations with guess and check or backtracking

You know that an equation is shorthand for saying “find what this variable is when you know that if this stuff is done to it, it equals this number” and you know that you can “undo” the stuff done to the variable to find its value.

You know that an equation is shorthand for saying “find what this variable is when you know that if this stuff is done to it, it equals this number.” If you think carefully, you know how to approximate the number and then plug it in to see if you’re right.

You can sometimes figure out what a variable is by just plugging in numbers and seeing what works, but you’re not sure which numbers to try first and you end up trying lots of numbers before you find the one that works. You probably find it tedious and frustrating.

When you see an equation, you’re not sure what to do with it. You may know that you’re supposed to find the number that the variable represents, but you’re not sure how.

Unit 2: Solving Equations



Solving simple equations with “balancing”

You can solve for a variable by performing inverse operations to both sides of an equation and you understand why we need to follow the reverse order of operations to do so.

You can find the value of a variable by performing inverse operations to both sides of an equation until the variable is isolated. You follow correct backwards order of operations.

You understand that you need to get the variable by itself to find its value, but you may get confused as to whether to add and subtract first, or multiply and divide first. You aren’t always successful.

You may know you want to find what value the variable equals, but you don’t know how to do it or you are using guess and check which fails to work once the equations get complicated.

Solving equations with like terms

You understand that when there’s more than one variable of the same type in an equation, it represents the same number. We should be able to find the value of that number if we combine like terms and solve like we did before.

You can find the value of a variable by combining like terms first, then performing inverse operations to both sides of an equation.

You know that you need to get the variable by itself, but there are two or three of the same variable on one side and this causes trouble. Sometimes you combine variables incorrectly or you mix up order of operations.

You don’t know how to combine the variables if there’s more than one of the same type and you can’t see how to solve for the value of a variable if there’s more than one of them.

Solving equation

You can see that this isn’t any different

You can use inverse operations to group

You know that you need to get the

You don’t know what that variable is

s with variables on both sides.

from what we’ve already been doing. Solving equations is about one idea- do reverse operations to find the value of an unknown and you can do those operations on variables and numbers alike.

the variables on one side of the equation, the numbers on the other and you can solve for the variable.

variables together somehow, but you don’t know how. You may you mix up the order of operations or aren’t sure how to combine like terms, or isolate a variable on one side just to have variables still left on the other.

doing on both sides of an equation and you have no idea what you’re supposed to do about it. How can you find the value of a variable if it equals a form of itself? Isn’t this an impossible equation?

Solving equations involving the distributive property

You have realized that as long as you follow reverse order of operations, you can solve this however you please. You can distribute first and solve for the variable, or you can get rid of the junk in front of the parentheses first then solve. You can solve these problems in a variety of ways.

You always distribute first to get rid of parentheses, then combine like terms, then solve for the variable by performing inverse operations on both sides.

You know that you can use the distributive property to simplify the expression, but you’re not sure where to go from there or you forget how to do the distributive property. Once the parentheses are gone though, you usually know what to do.

You can’t remember how to get rid of the parentheses so you try doing stuff to both sides but can’t ever get the variable by itself. You may just not know where to start or what you’re supposed to do with this mess.

Word problems

You read the problem and you identify what it is you want to know. You can write an equation and solve it, then you can understand the answer in the context of the original problem and your answer makes sense. You can extend this to everyday problems, not just textbook problems.

You can most of the time read the problem, rewrite it as an equation, then solve the equation for the variable. You have the correct number.

You often have trouble turning the words into an equation. You may be confused about which words mean which operation. Once it’s set up though, you can solve the problem just fine. You may not be sure what the answer means.

You have no idea how to turn words into an equation. You may be able to tell what the variable should represent, but you can’t set up the equation or you can’t see how these words even represent and equation at all.

Rearranging Formulas

You understand that this is no different from what we were doing before and you isolate the indicated variable with ease. Furthermore you recognize that this process makes formulas easier to use as you can manipulate them to give you any information you want.

You can see that it wants you to get a variable by itself and you know to perform inverse operations to both sides to get it by itself.

You know you need to isolate the indicated variable, but all the other variables are confusing and you’re not sure how to get rid of them or how to combine them. You know to perform inverse operations, but you just seem to get a mess when you do it.

You stare blankly at the tangle of variables and you pull out all your hair. Don’t worry, most people want to do this. You maybe see why the original formula is valuable but you have no idea why you’d want to mess with it.

Solving You know how to You know that You can solve the You don’t see that

and graphing inequalities

solve inequalities and you know why we flip the inequality sign when you multiply or divide by a negative. Also, you realize that with inequalities, isolating the variable gives you an infinite number of answers while equalities only gave you one and that’s why we plot the solution on number lines.

solving inequalities is the same as solving equalities as long as you remember to reverse the inequality sign when you multiply or divide by a negative. You can plot the solution set on a number line.

inequality most of the time but you may forget to flip the inequality sign when multiplying or dividing by a negative. You can plot the solution set but you mix up the solid dot with the hollow one and you’re not sure what the point is.

solving equalities and inequalities uses the same process. You’re confused about what inequalities are or how they’re different from equalities. You don’t know what a solution to an inequality means.

Unit 3: Linear Functions



Rates You know that a rate relates two variables each of which stands for a whole group of numbers. You can

You know that variables can represent unknown things but also groups of unknown

You know what a rate is and you can work with some rates but others may confuse you.

You’re not sure what a rate is. It may be confusing that they can be written as a fraction

write equations using rates and you can use rates to find information you may want to know. For example if you know how many miles you’ve driven, you can figure out how much time has passed.

things. You know that a rate relates two groups of things. For example “miles per hour” relates the miles traveled “m” to hours that have passed “h”. You know how to write simple equations using rates.

You have difficulty remembering how to turn a rate into an equation.

and as an equation. You may not understand why rates should involve variables at all.

Slopes

You can do everything in the “meets requirements” box and you can look at real world problems and situations to determine what slope means in context. You can use slopes to help you figure out new information and where points further down the line will fall.

You can use a rate equation to make a table that can be plotted on a grid. You can see that the rate determines how steep the line is. You know that the rate (slope) tells us how much “up” to go and how much “over” to go to get to the next point. You can go backwards and find rates when given graphed points.

You can use a rate equation to make a table that can be plotted on a grid. You can see that lines emerge and they have different “steepnesses”. You may also know that slope has something to do with “steepness” but you’re not sure how that relates to the rate equation or how to use this information.

You have trouble finding slopes because you’re not sure how they’re related to rates. You may know that slope is “rise over run” but you don’t know how this relates to anything real. You may mix up which number goes in the top of the fraction and which goes in the bottom.

Starting points and y-intercept

When given a situation, you can write a rate equation that describes the situation but you also note that the rate equation doesn’t match up to the scenario because of a “starting point”. You can write rate equations now that involve starting points, and you can plot the points that satisfy these equations on a Cartesian grid. You may also be able to look at a graphed line and write the equation and situation that matches that line.

You know that a rate equation looks like: dependent variable=rate x independent variable. When there’s a starting point that’s not zero, you can add it on to the right side of the equation. You can graph the result by plotting points.

You can see that adding something to the right-hand side of a rate equation moves the graph of a line up and down. You’re not sure what this starting point means in context though. You may have trouble writing equations from situations that involve starting points because when you write a normal rate equation, points that satisfy it don’t make sense.

You’re not sure what that extra number added on means. You may get confused about which number in the equation of a line is the slope and which is the starting point (y-intercept).

Graphing equations of lines

You can graph any line without plotting points. You can rearrange any linear equation so that it can

You can graph any line given to you without plotting points. You can rearrange any linear

You can graph equations of lines that are in y=mx+b form. You’re not sure what to do if

You can’t graph lines without plotting points first.

easily be graphed by solving for y. You may know how to write the equation in point-slope form so that it is graphable without an intercept.

equation so that it can easily be graphed by solving for y.

they’re in a different form.

Writing equations of lines

You can do everything in the “meets requirements” box and you know that equations of lines represent real-world rate situations. You can take real world data and find lines that fit it and you can use the equations of those lines to make predictions.

You’re comfortable writing equations of lines when you’re given a graph, when you’re given a slope and a y-intercept or when you’re given two points. You are comfortable manipulating the equations of lines algebraically.

You can find equations of lines by looking at graphs because you can find the slope and y-intercept. When given a point and a slope, you may not know to plug it in to find the info you want and you have no idea what to do with two points.

You may know that a line is in y=mx+b form, but you’re not sure what the symbols mean, you mix them up, or you know what the m and the b are but you don’t know what the x and the y represent.

Parallel, PerpendicularHorizontal and Vertical lines

You know that parallel lines all share the same slopes but have different y-intercepts and you can find the equations of lines parallel to each other. You know perpendicular lines have slopes that are negative reciprocals of each other and you can find equations of lines perpendicular to each other. You can find the equation of any vertical or horizontal line and you know why these equations are work the way they do.

You know that parallel lines all share the same slopes but have different y-intercepts. You know perpendicular lines have slopes that are negative reciprocals of each other. You can find the equation of any vertical or horizontal line and you know.

You may know the slope relationships of parallel and perpendicular lines but you’re not sure how to write the equations for these lines. You know what vertical and horizontal lines look like, but you may mix up their equations with each other.

You may not be sure what parallel, perpendicular, horizontal and vertical mean. You don’t know what the equations of these types of lines have in common with each other.

Graphing linear inequalities

You can graph linear inequalities correctly every time. You also can interpret the meaning of these graphs in the context of word problems and you know that any point in the shaded region satisfies the inequality.

You can take a linear inequality, graph the line correctly with a solid or dotted line then you shade the appropriate portion of the graph.

You sometimes mix up when to use a dotted line and when to use a solid line. Every once in a while you shade the wrong region of the graph.

You’re not sure where to start graphing inequalities. You may be able to graph the line indicated in the equation but you don’t know where to go from there.

Absolute value graphs

You can graph an absolute value equation and you can write the equation for a graphed absolute value function. You

You can graph an absolute value equation and you can write the equation for a graphed absolute

You may know that an absolute value equation, when graphed, looks like a “V”. You get confused though

You’re not sure what an absolute value equation should look like when graphed. You sometimes graph it

understand why it looks the way it does and why the slopes on either side of the “V” are opposites of each other. You can move the vertex of an absolute function around by manipulating its equation.

value function. You can move the vertex of an absolute function around by manipulating its equation.

when there are numbers multiplying the absolute value or are added to it in the equation. You can graph it sometimes. You may not be able to write the equation of a graphed absolute value function

as a line because the equation looks like the equation of a line.

Unit 4: Systems of Linear Equations



Solving systems of equations graphically

You can graph two lines and find the point where they intersect. You understand that this is the point that makes both equations true and this point, when plugged back in will satisfy both equations.

You can graph two lines and find the point where they intersect (if it exists) whether or not the intersection point is depicted in the graphing window.

You are still having trouble graphing lines that aren’t in y=mx+b form and you’re not sure how to find the intersection point if it’s not shown in the graphing window.

You don’t know how to find the point of intersection when you’re given the equations of two or more lines.

Solving systems of equations algebraically

You can solve systems of equations with substitution and elimination, you know when it is most efficient to use which method, and you can check your work. You also know why both methods are valid and what number properties they represent.

You can solve systems of equations with substitution and elimination. You may prefer one method of the other so sometimes you do more math than you need to. You can check your work to make sure it’s correct.

You may be able to solve systems with one method but not the other. This makes solving some systems very tedious and you arrive at incorrect answers sometimes.

You may be able to solve some systems with guess and check or intuition, but you can’t or don’t know how to get the right answer algebraically. You may confuse the two methods leading to incorrect answers.

Solving word problems with linear systems

You can differentiate between word problems that require a system of equations to solve and those that require only one equation. You can

You can take a word problem, write two separate equations based on that word problem, and solve the system of equations to yield a

You know that the word problem has two equations embedded in it but you’re confused about which information to use

You read the word problem and write one equation and try to solve it, or you’re not sure how to write even one equation from the

then write the equations and solve the system algebraically. You can interpret your answer in the context of the original problem.

correct answer.

in which equation, or you can’t solve the system of equations that you created out of the word problem. Sometimes it works and sometimes it doesn’t.

information in the word problem. You may be able to solve it with guess and check.

Solving systems of inequalities

You may be able to write your own linear inequality system from a word problem or situation. You can graph the two inequalities, correctly use solid or dotted lines, and shade the region that satisfies both inequalities. You can interpret the meaning of the solution set in the context of a real- world problem.

You can graph the two inequalities and you correctly use solid or dotted lines. You shade the correct regions of the graph and indicate where they overlap identifying the solution to the system of inequalities. You can pick a point and test it make sure you’ve shaded the correct region.

You can graph the two inequalities but sometimes you mix up when to use solid or dotted lines. You may have forgotten to flip the inequality sign when multiplying or dividing by a negative to isolate y and end up with the wrong line graphed. You don’t always shade the correct portions of the graph.

You may be able to graph one or both of the lines indicated in the equation of the inequality but you’re not sure how the inequality symbols should affect the graph.

algebra 1 semester 1 rubric

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