Middle School Math I Curriculum:
First Quarter
QUANTITIES (These standards should be addressed throughout the curriculum in each unit of study.)
Reason quantitatively and use units to solve problems.
- N.Q.1: Use units as a way to understand problems and to guide the solution of multi-step problems; choose and interpret units consistently in formulas; choose and interpret the scale and the origin in graphs and data displays.
- N.Q.2: Define appropriate quantities for the purpose of descriptive modeling.
- N.Q.3: Choose a level of accuracy appropriate to
limitations on measurement when reporting quantities.
THE REAL NUMBER SYSTEM
Know that there are numbers that are not rational, and approximate them by rational numbers. - 8.NS.1: Know that numbers that are not rational are called irrational. Understand informally that every number has a decimal expansion; for rational numbers show that the decimal expansion repeats eventually, and convert a decimal expansion which repeats eventually into a rational number.
- 8.NS.2: Use rational approximations of irrational numbers to
compare the size of irrational numbers, locate them approximately on a
number line diagram, and estimate the value of expressions (e.g., π2).
For example, by truncating the decimal expansion of √2, show that √2 is
between 1 and 2, then between 1.4 and 1.5, and explain how to continue on
to get better approximations.
Extend the properties of exponents to rational exponents. - N.RN.1: Explain how the definition of the meaning of rational exponents follows from extend the propertieso f interger exponents to thos values, allowing for a notation for radicals in terms of rational exponents. For example, we define 51/3 to be the cube root of 5 because we want (51/3)3 = 5(1/3)3 to hold, so (51/3)3 must equal 5.
- N.RN.2: Rewrite expressions involving radicals and rational
exponents using the properties of exponents. Note: At
this level, focus on fractional exponents with a numerator of 1.
EXPRESSIONS AND EQUATIONS
Expressions and Equations Work with radicals and integer exponents. - 8.EE.1: Know and apply the properties of integer exponents to generate equivalent numerical expressions. For example, 32 × 3–5 = 3–3 = 1/33 = 1/27.
- 8.EE.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. For example, estimate the population of the United States as 3 times 108 and the population of the world as 7 times 109, and determine that the world population is more than 20 times larger.
- 8.EE.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.
Analyze and solve linear equations and pairs of simultaneous linear equations. - 8.EE.7: Solve linear equations in one variable.
8.EE.7a: Give examples of linear equations in one variable with one solution, infinitely many solutions, or no solutions. Show which of these possibilities is the case by successively transforming the given equation into simpler forms, until an equivalent equation of the form x = a, a = a, or a = b results (where a and b are different numbers).
8.EE.7b: Solve linear equations with rational number coefficients, including equations whose solutions require expanding expressions using the distributive property and collecting like terms.
SYSTEMS OF LINEAR EQUATIONS AND INEQUALITIES
Create equations that describe numbers or relationships. - A.CED.1: Create equations and inequalities in one variable and use them to solve problems. Include equations arising from linear and quadratic functions, and simple rational and exponential functions. Note: At this level, focus on linear and exponential functions.
- A.CED.4: Rearrange formulas to highlight a quantity of
interest, using the same reasoning as in solving equations. For example,
rearrange Ohm’s law V = IR to highlight resistance R. Note: At this
level, limit to formulas that are linear in the variable of interest, or
to formulas involving squared or cubed variables.
REASONING WITH EQUATIONS AND INEQUALITIES
Understand solving equations as a process of reasoning and explain the reasoning. - A-REI.1: Explain each step in solving a simple equation as following from the equality of numbers asserted at the previous step, starting from the assumption that the original equation has a solution. Construct a viable argument to justify a solution method.
Solve equations and inequalities in one variable.- A.REI.3: Solve linear equations and inequalities in one variable, including equations with coefficients represented by letters.
INTERPRETING CATEGORICAL AND QUANTITATIVE DATA
Summarize, represent, and interpret data on a single count or measurement variable.- S.ID.1: Represent data with plots on the real number line (dot plots, histograms, and box plots).
- S.ID.2: Use statistics appropriate to the shape of the data distribution to compare center (median, mean) and spread (interquartile range, standard deviation) of two or more different data sets.
- S.ID.3: Interpret differences in shape, center, and spread in
the context of the data sets, accounting for possible effects of extreme
data points (outliers).
STATISTICS AND PROBABILITY
Investigate patterns of association in bivariate data. - 8.SP.1: Construct and interpret scatter plots for bivariate measurement data to investigate patterns of association between two quantities. Describe patterns such as clustering, outliers, positive or negative association, linear association, and nonlinear association.
- 8.SP.2: Know that straight lines are widely used to model relationships between two quantitative variables. For scatter plots that suggest a linear association, informally fit a straight line, and informally assess the model fit by judging the closeness of the data points to the line.
- 8.SP.3: Use the equation of a linear model to solve problems in the context of bivariate measurement data, interpreting the slope and intercept. For example, in a linear model for a biology experiment, interpret a slope of 1.5 cm/hr as meaning that an additional hour of sunlight each day is associated with an additional 1.5 cm in mature plant height.
- 8.SP.4: Understand that patterns of association can also be
seen in bivariate categorical data by displaying frequencies and relative
frequencies in a two-way table. Construct and interpret a two-way table
summarizing data on two categorical variables collected from the same
subjects. Use relative frequencies calculated for rows or columns to
describe possible association between the two variables. For example,
collect data from students in your class on whether or not they have a
curfew on school nights and whether or not they have assigned chores at
home. Is there evidence that those who have a curfew also tend to have
chores?
Second Quarter:
FUNCTION INTRODUCTION
Define, evaluate, and compare functions - 8.F.1: Understand that a function is a rule that assigns to each input exactly one output. The graph of a function is the set of ordered pairs consisting of an input and the corresponding output.
- 8.F.2: Compare properties of two functions each represented in a different way (algebraically, graphically, numerically in tables, or by verbal descriptions). For example, given a linear function represented by a table of values and a linear function represented by an algebraic expression, determine which function has the greater rate of change.
- 8.F.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. For example, the function A = s2 giving the area of a square as a function of its side length is not linear because its graph contains the points (1,1), (2,4) and (3,9), which are not on a straight line.
Use functions to model relationships between quantities.- 8.F.4: Construct a function to model a linear relationship between two quantities. Determine the rate of change and initial value of the function from a description of a relationship or from two (x, y) values, including reading these from a table or from a graph. Interpret the rate of change and initial value of a linear function in terms of the situation it models, and in terms of its graph or a table of values.
- 8.F.5: Describe qualitatively the
functional relationship between two quantities by analyzing a graph
(e.g., where the function is increasing or decreasing, linear or
nonlinear). Sketch a graph that exhibits the qualitative features of a
function that has been described verbally.
INTERPRETING FUNCTIONS
Understand the concept of a function and use function notation. - F.IF.1: Understand that a function from one set (called the domain) to another set (called the range) assigns to each element of the domain exactly one element of the range. If f is a function and x is an element of its domain, then f(x) denotes the output of f corresponding to the input x. The graph of f is the graph of the equation y = f(x).
- F.IF.2: Use function notation, evaluate functions for inputs in their domains, and interpret statements that use function notation in terms of a context.
- F.IF.3: Recognize that sequences are functions, sometimes defined recursively, whose domain is a subset of the integers. For example, the Fibonacci sequence is defined recursively by f(0) = f(1) = 1, f(n+1) = f(n) + f(n-1) for n ≥ 1.
Interpret functions that arise in applications in terms of the context.- F.IF.4: For a function that models a relationship between two quantities, interpret key features of graphs and tables in terms of the quantities, and sketch graphs showing key features given a verbal description of the relationship. Key features include: intercepts; intervals where the function is increasing, decreasing, positive, or negative; relative maximums and minimums; symmetries; end behavior; and periodicity.
- F.IF.5: Relate the domain of a function to its graph and, where applicable, to the quantitative relationship it describes. For example, if the function h(n) gives the number of person-hours it takes to assemble n engines in a factory, then the positive integers would be an appropriate domain for the function.
- F.IF.6: Calculate and interpret the
average rate of change of a function (presented symbolically or as a
table) over a specified interval. Estimate the rate of change from a
graph.
Analyze functions using different representations. - F.IF.7: Graph functions expressed
symbolically and show key features of the graph, by hand in simple cases
and using technology for more complicated cases.
a. Graph linear and quadratic functions and show intercepts, maxima, and minima. - F.IF.9: Compare properties of two
functions each represented in a different way (algebraically,
graphically, numerically in tables, or by verbal descriptions). For
example, given a graph of one quadratic function and an algebraic
expression for another, say which has the larger maximum.
BUILDING FUNCTIONS
Build a function that models a relationship between two quantities. - F.BF.1: Write a function that describes a relationship between two quantities.
- a. Determine an explicit
expression, a recursive process, or steps for calculation from a context.
b. Combine standard function types using arithmetic operations. For example, build a function that models the temperature of a cooling body by adding a constant function to a decaying exponential, and relate these functions to the model. - F.BF.2: Write arithmetic and
geometric sequences both recursively and with an explicit formula, use
them to model situations, and translate between the two forms.
Build new functions from existing functions. - F.BF.3: Identify the effect on the graph of replacing f(x) by f(x) + k, k f(x), f(kx), and f(x + k) for specific values of k (both positive and negative); find the value of k given the graphs. Experiment with cases and illustrate an explanation of the effects on the graph using technology. Include recognizing even and odd functions from their graphs and algebraic expressions for them.
LINEAR, QUADRATIC, AND EXPONENTIAL MODELS
Construct and compare linear and exponential models and solve problems.- F.LE.1: Distinguish between
situations that can be modeled with linear functions and with
exponential functions
a. Prove that linear functions grow by equal differences over equal intervals, and that exponential functions grow by equal factors over equal intervals.
b. Recognize situations in which one quantity changes at a constant rate per unit interval relative to another. - F.LE.2: Construct linear and exponential functions, including arithmetic and geometric sequences, given a graph, a description of a relationship, or two input-output pairs (include reading these from a table).
- F.LE.5: Interpret the parameters in
a linear or exponential function in terms of a context.
EXPRESSIONS AND EQUATIONS
Expressions and equations work with radicals and integer exponents.
8.EE.5: Graph proportional relationships, interpreting the unit rate as the slope of the graph. Compare two different proportional relationships represented in different ways. For example, compare a distance-time graph to a distance-time equation to determine which of two moving objects has greater speed. - 8.EE.6: Use similar triangles to
explain why the slope m is the same between any two distinct points on a
non-vertical line in the coordinate plane; derive the equation y = mx
for a line through the origin and the equation y = mx + b for a line
intercepting the vertical axis at b.
SEEING STRUCTURE IN EXPRESSIONS
Interpret the structure of expressions.
A.SSE.1: Interpret expressions that represent a quantity in terms of its context.
a. Interpret parts of an expression, such as terms, factors, and coefficients.b. Interpret complicated expressions by viewing one or more of their parts as a single entity. For example, interpret P(1+r)n as the product of P and a factor not depending on P.
CREATING EQUATIONS
Create equations that describe numbers or relationships. - A-CED.2: Create
equations in two or more variables to represent relationships between
quantities; graph equations on coordinate axes with labels and scales.
REASONING WITH EQUATIONS AND INEQUALITIES
Represent and solve equations and inequalities graphically. - A.REI.10: Understand that the graph
of an equation in two variables is the set of all its solutions plotted
in the coordinate plane, often forming a curve (which could be a line).
INTERPRETING CATEGORICAL AND QUANTITATIVE DATA
Summarize, represent, and interpret data on two categorical and quantitative variables. - S.ID.5: Summarize categorical data for two categories in two-way frequency tables. Interpret relative frequencies in the context of the data (including joint, marginal, and conditional relative frequencies). Recognize possible associations and trends in the data.
- S.ID.6: Represent data on two quantitative
variables on a scatter plot, and describe how the variables are related.
a. Fit a function to the data; use functions fitted to data to solve problems in the context of the data. Use given functions or choose a function suggested by the context. Emphasize linear and exponential models.
b. Informally assess the fit of a function by plotting and analyzing residuals.
c. Fit a linear function for a scatter plot that suggests a linear association.
- Interpret linear models.
Third Quarter
EXPRESSIONS AND EQUATIONS
Analyze and solve linear equations and pairs of simultaneous linear equations.
- 8.EE.8: Analyze and solve pairs of simultaneous linear equations.
b. Solve systems of two linear equations in two variables algebraically, and estimate solutions by graphing the equations. Solve simple cases by inspection. For example, 3x + 2y = 5 and 3x + 2y = 6 have no solution because 3x + 2y cannot simultaneously be 5 and 6.
c. Solve real-world and mathematical problems leading to two linear equations in two variables. For example, given coordinates for two pairs of points, determine whether the line through the first pair of points intersects the line through the second pair.
CREATING EQUATIONS
Create equations that describe numbers or relationships.
A.CED.2: Create equations in two or more variables to represent relationships between quantities; graph equations on coordinate axes with labels and scales.
A.CED.3: Represent constraints by equations or inequalities, and by systems of equations and/or inequalities, and interpret solutions as viable or non- viable options in a modeling context. For example, represent inequalities describing nutritional and cost constraints on combinations of different foods.
REASONING WITH EQUATIONS AND INEQUALITIES
Solve systems of equations.
- A.REI.5: Prove that, given a system of two equations in two variables, replacing one equation by the sum of that equation and a multiple of the other produces a system with the same solutions.
- A.REI.6: Solve systems of linear equations exactly and approximately (e.g., with graphs), focusing on pairs of linear equations in two variables.
- A.REI.11: Explain why the x-coordinates of the points where the graphs of the equations y = f(x) and y = g(x) intersect are the solutions of the equation f(x) = g(x); find the solutions approximately, e.g., using technology to graph the functions, make tables of values, or find successive approximations. Include cases where f(x) and/or g(x) are linear, polynomial, rational, absolute value, exponential, and logarithmic functions.
- A.REI.12: Graph the solutions to a
linear inequality in two variables as a half- plane (excluding the
boundary in the case of a strict inequality), and graph the solution set
to a system of linear inequalities in two variables as the intersection
of the corresponding half-planes.
ARITHMETIC WITH POLYNOMIALS AND RATIONAL EXPRESSIONS
Perform arithmetic operations on polynomials.
- A.APR.1: Understand that polynomials
form a system analogous to the integers, namely, they are closed under
the operations of addition, subtraction, and multiplication; add,
subtract, and multiply polynomials.
SEEING STRUCTURE IN EXPRESSIONS
Interpret the structure of expressions.
- A.SSE.2: Use the structure of an expression to identify ways to rewrite it. For example, see x4 – y4 as (x2)2 – (y2)2, thus recognizing it as a difference of squares that can be factored as (x2 – y2)(x2 + y2).
- A.SSE.3: Choose and produce an equivalent form of an expression to reveal and explain properties of the quantity represented by the expression.
INTERPRETING FUNCTIONS
Interpret functions that arise in applications in terms of the context.
- F.IF.4: For a function that models a relationship between two quantities, interpret key features of graphs and tables in terms of the quantities, and sketch graphs showing key features given a verbal description of the relationship. Key features include: intercepts; intervals where the function is increasing, decreasing, positive, or negative; relative maximums and minimums; symmetries; end behavior; and periodicity.
- F.IF.7: Graph functions expressed symbolically and show key features of the graph, by hand in simple cases and using technology for more complicated cases.
- F.IF.8: Write a function defined by an expression in different but equivalent forms to reveal and explain different properties of the function.
- F.IF.9: Compare properties of two
functions each represented in a different way (algebraically,
graphically, numerically in tables, or by verbal descriptions). For
example, given a graph of one quadratic function and an algebraic
expression for another, say which has the larger maximum.
LINEAR, QUADRATIC, AND EXPONENTIAL MODELS
Construct and compare linear and exponential models and solve problems.
- F.LE.3: Observe using graphs and
tables that a quantity increasing exponentially eventually exceeds a
quantity increasing linearly, quadratically, or (more generally) as a
polynomial function.
GEOMETRY
Understand and apply the Pythagorean Theorem.
- 8.G.6: Explain a proof of the Pythagorean Theorem and its converse.
- 8.G.7: Apply the Pythagorean Theorem to determine unknown side lengths in right triangles in real-world and mathematical problems in two and three dimensions.
- 8.G.8: Apply the Pythagorean Theorem to find the distance between two points in a coordinate system.
Fourth Quarter
CREATING EQUATIONS
Create equations that describe numbers or relationships.
- A.CED.2: Create equations in two
or more variables to represent relationships between quantities; graph
equations on coordinate axes with labels and scales.
REASONING WITH EQUATIONS AND INEQUALITIES
Represent and solve equations and inequalities graphically.
- A.REI.10: Understand that the graph
of an equation in two variables is the set of all its solutions plotted
in the coordinate plane, often forming a curve (which could be a
line).
BUILDING FUNCTIONS
Build a function that models a relationship between two quantities.
- F.BF.1: Write a function that describes a relationship between two quantities.
b. Combine standard function types using arithmetic operations. For example, build a function that models the temperature of a cooling body by adding a constant function to a decaying exponential, and relate these functions to the model.
- F.BF.2: Write arithmetic and geometric sequences both recursively and with an explicit formula, use them to model situations, and translate between the two forms.
- F.BF.3: Identify the effect on
the graph of replacing f(x) by f(x) + k, k f(x), f(kx), and f(x +
k) for specific values of k (both positive and negative); find the
value of k given the graphs. Experiment with cases and illustrate an
explanation of the effects on the graph using technology. Include
recognizing even and odd functions from their graphs and algebraic
expressions for them.
INTERPRETING FUNCTIONS
Interpret functions that arise in applications in terms of the context.
- F.IF.4: For a function that models a relationship between two quantities, interpret key features of graphs and tables in terms of the quantities, and sketch graphs showing key features given a verbal description of the relationship. Key features include: intercepts; intervals where the function is increasing, decreasing, positive, or negative; relative maximums and minimums; symmetries; end behavior; and periodicity.
- F.IF.5: Relate the domain of a function to its graph and, where applicable, to the quantitative relationship it describes. For example, if the function h(n) gives the number of person-hours it takes to assemble n engines in a factory, then the positive integers would be an appropriate domain for the function.
- F.IF.6: Calculate and interpret the average rate of change of a function (presented symbolically or as a table) over a specified interval. Estimate the rate of change from a graph.
- F.IF.7: Graph functions expressed symbolically and show key features of the graph, by hand in simple cases and using technology for more complicated cases.
Analyze functions using different representations.
- F.IF.8: Write a function defined by an expression in different but equivalent forms to reveal and explain different properties of the function.
- F.IF.9: Compare properties of two
functions each represented in a different way (algebraically,
graphically, numerically in tables, or by verbal descriptions). For
example, given a graph of one quadratic function and an algebraic
expression for another, say which has the larger maximum
LINEAR, QUADRATIC, AND EXPONENTIAL MODELS
Construct and compare linear and exponential models and solve problems.
- F.LE.1: Distinguish between situations that can be modeled with linear functions and with exponential functions
c. Recognize situations in which a quantity grows or decays by a constant percent rate per unit interval relative to another.
- F.LE.2: Construct linear and exponential functions, including arithmetic and geometric sequences, given a graph, a description of a relationship, or two input-output pairs (include reading these from a table).
- F.LE.3: Observe using graphs and tables that a quantity increasing exponentially eventually exceeds a quantity increasing linearly, quadratically, or (more generally) as a polynomial function.
- F.LE.5: Interpret the parameters
in a linear or exponential function in terms of a context.
GEOMETRIC MEASUREMENT AND DIMENSION
Explain volume formulas and use them to solve problems.
- G.GMD.1: Give an informal argument for the formulas for the circumference of a circle, area of a circle, volume of a cylinder, pyramid, and cone. Use dissection arguments, Cavalieri’s principle, and informal limit arguments.
- G.GMD.3: Use volume formulas for cylinders, pyramids, cones, and spheres to solve problems.
- 8.G.9: Know the formulas for the volumes of cones, cylinders, and spheres and use them to solve real-world and mathematical problems.
Experiment with transformations in the plane.
- G.CO.1: Know precise definitions
of angle, circle, perpendicular line, parallel line, and line segment,
based on the undefined notions of point, line, distance along a line,
and distance around a circular arc.
EXPRESSING GEOMETRIC PROPERTIES WITH EQUATIONS
Use coordinates to prove simple geometric theorems algebraically.
- G.GPE.4: Use coordinates to prove simple geometric theorems algebraically. For example, prove or disprove that a figure defined by four given points in the coordinate plane is a rectangle; prove or disprove that the point (1, √3) lies on the circle centered at the origin and containing the point (0, 2). Note: Conics is not the focus at this level, therefore the last example is not appropriate here.
- G.GPE.5: Prove the slope criteria for parallel and perpendicular lines and use them to solve geometric problems (e.g., find the equation of a line parallel or perpendicular to a given line that passes through a given point).
- G.GPE.6: Find the point on a directed line segment between two given points that partitions the segment in a given ratio.
- G.GPE.7: Use coordinates to
compute perimeters of polygons and areas of triangles and rectangles,
e.g., using the distance formula.
GEOMETRY
Understand congruence and similarity using physical models, transparencies, or geometry software.
- 8.G.1: Verify experimentally the properties of rotations, reflections, and translations:
b. Angles are taken to angles of the same measure.
c. Parallel lines are taken to parallel lines.
- 8.G.2: Understand that a two-dimensional figure is congruent to another if the second can be obtained from the first by a sequence of rotations, reflections, and translations; given two congruent figures, describe a sequence that exhibits the congruence between them.
- 8.G.3: Describe the effect of dilations, translations, rotations, and reflections on two-dimensional figures using coordinates.
- 8.G.4: Understand that a two-dimensional figure is similar to another if the second can be obtained from the first by a sequence of rotations, reflections, translations, and dilations; given two similar two-dimensional figures, describe a sequence that exhibits the similarity between them.
- 8.G.5: Use informal arguments to
establish facts about the angle sum and exterior angle of triangles,
about the angles created when parallel lines are cut by a transversal,
and the angle-angle criterion for similarity of triangles. For
example, arrange three copies of the same triangle so that the sum of
the three angles appears to form a line, and give an argument in terms
of transversals why this is so.