Download Mapping Common Core State Standard Clusters and Ohio`s Grade

Compiled by Kathleen Carpenter, Warren County ESC Nov. 2011
Mapping Common Core State Standard Clusters and
Ohio’s Grade Level Indicators:
Grade 8 Mathematics
Common Core Domain and
Standard
The Number System: Know
that there are numbers that
are not rational, and
approximate them by
rational numbers.
Expressions and Equations:
Work with radicals and
integer exponents.
Common Core Cluster Standards
Ohio Grade Level Indicator
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.
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.
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.
2. Use square root and cube root symbols to represent solutions to
equations of the form x2 = p and x3 = p, where p is a positive rational
number. Evaluate square roots of small perfect squares and cube
roots of small perfect cubes. Know that √2 is irrational.
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 × 108 and
the population of the world as 7 × 109, and determine that the world
population is more than 20 times larger.
NO2. Recognize that natural numbers, whole
numbers, integers, rational numbers and
irrational numbers are subsets of the real number
system.
1
NO3. Apply order of operations to simplify
expressions and perform computations involving
integer exponents and radicals.
NO7. Find the square root of perfect squares, and
approximate the square root of non-perfect
squares as consecutive integers between which
the root lies; e.g.,√130 is between 11 and 12.
Compiled by Kathleen Carpenter, Warren County ESC Nov. 2011
Common Core Domain and
Standard
Expressions and Equations:
Understand the
connections between
proportional relationships,
lines, and linear equations.
Expressions and Equations:
Analyze and solve linear
equations and pairs of
simultaneous linear
equations.
Common Core Cluster Standards
Ohio Grade Level Indicator
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.
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.
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.
7. Solve linear equations in one variable.
a. 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).
b. Solve linear equations with rational number coefficients, including
equations whose solutions require expanding expressions using the
distributive property and collecting like terms.
8. Analyze and solve pairs of simultaneous linear equations.
a. Understand that solutions to a system of two linear equations in
two variables correspond to points of intersection of their graphs,
because points of intersection satisfy both equations simultaneously.
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
NO1. Use scientific notation to express large
numbers and small numbers between 0 and 1.
NO8. Add, subtract, multiply, divide and compare
numbers written in scientific notation.
2
G4. Represent and analyze shapes using
coordinate geometry; e.g., given three vertices
and the type of quadrilateral, find the
coordinates of the fourth vertex.
PFA9. Solve linear equations and inequalities
graphically, symbolically and using technology.
PFA10. Solve 2 by 2 systems of linear equations
graphically and by simple substitution.
PFA11. Interpret the meaning of the solution of a
2 by 2 system of equations; i.e., point, line, no
solution.
Compiled by Kathleen Carpenter, Warren County ESC Nov. 2011
Common Core Domain and
Standard
Functions: Define, evaluate,
and compare functions.
Common Core Cluster Standards
Ohio Grade Level Indicator
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.
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.
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.
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.
Functions: Use functions to
model relationships
between quantities.
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.
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.
3
PFA1. Relate the various representations of a
relationship; i.e., relate a table to graph,
description and symbolic form.
PFA7. Use symbolic algebra (equations and
inequalities), graphs and tables to represent
situations and solve problems.
PFA3. Identify functions as linear or nonlinear
based on information given in a table, graph or
equation.
PFA6. Describe the relationship between the
graph of a line and its equation, including being
able to explain the meaning of slope as a
constant rate of change and y-intercept in realworld problems.
PFA4. Extend the uses of variables to include
covariants where y depends on x.
Compiled by Kathleen Carpenter, Warren County ESC Nov. 2011
Common Core Domain and
Standard
Geometry: Understand
congruence and similarity
using physical models,
transparencies, or
geometry software.
Geometry: Understand and
apply the Pythagorean
Theorem
Common Core Cluster Standards
Ohio Grade Level Indicator
1. Verify experimentally the properties of rotations, reflections, and
translations:
a. Lines are taken to lines, and line segments to line segments of the
same length.
b. Angles are taken to angles of the same measure.
c. Parallel lines are taken to parallel lines.
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.
3. Describe the effect of dilations, translations, rotations, and
reflections on two-dimensional figures using coordinates.
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 twodimensional figures, describe a sequence that exhibits the similarity
between them.
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.
6. Explain a proof of the Pythagorean Theorem and its converse.
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. Apply the Pythagorean Theorem to find the distance between two
points in a coordinate system.
4
G5. Draw the results of translations, reflections,
rotations and dilations of objects in the
coordinate plane, and determine properties that
remain fixed; e.g., lengths of sides remain the
same under translations.
G3. Use proportions in several forms to solve
problems involving similar figures (part-to-part,
part-to-whole, corresponding sides between
figures).
G2. Recognize the angles formed and the
relationship between the angles when two lines
intersect and when parallel lines are cut by a
transversal.
Compiled by Kathleen Carpenter, Warren County ESC Nov. 2011
Common Core Domain and
Standard
Geometry: Solve real-world
and mathematical problems
involving volume of
cylinders, cones and
spheres.
Common Core Cluster Standards
Ohio Grade Level Indicator
9. Know the formulas for the volumes of cones, cylinders, and
spheres and use them to solve real-world and mathematical
problems.
M4. Derive formulas for surface area and volume
and justify them using geometric models and
common materials. For example, find:
a. the surface area of a cylinder as a function of
its height and radius;
b. that the volume of a pyramid (or cone) is onethird of the volume of a prism (or cylinder) with
the same base area and height.
M9. Demonstrate understanding of the concepts
of perimeter, circumference and area by using
established formulas for triangles, quadrilaterals,
and circles to determine the surface area and
volume of prisms, pyramids, cylinders, spheres
and cones. (Note: Only volume should be
calculated for spheres and cones.)
Statistics and Probability:
Investigate patterns of
association in bivariate
data.
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.
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.
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
5
M10. Use conventional formulas to find the
surface area and volume of prisms, pyramids and
cylinders and the volume of spheres and cones to
a specified level of precision.
D1. Use, create and interpret scatterplots and
other types of graphs as appropriate.
D6. Make conjectures about possible relationship
in a scatterplot and approximate line of best fit.
Compiled by Kathleen Carpenter, Warren County ESC Nov. 2011
Common Core Domain and
Standard
Common Core Cluster Standards
Ohio Grade Level Indicator
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.
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?
Grade 8 Ohio Grade Level Indicators that “Do Not Match” the Common Core:
Number, Number Sense and Operations:
Measurement:
4. Explain and use the inverse and identity properties and use
inverse relationships (addition/subtraction, multiplication/division,
squaring/square roots) in problem solving situations.
5. Determine when an estimate is sufficient and when an exact
answer is needed in problem situations, and evaluate estimates in
relation to actual answers; e.g., very close, less than, greater than.
6. Estimate, compute and solve problems involving rational
numbers, including ratio, proportion and percent, and judge the
reasonableness of solutions.
1. Compare and order the relative size of common U.S. customary
units and metric units; e.g., mile and kilometer, gallon and liter,
pound and kilogram.
2. Use proportional relationships and formulas to convert units from
one measurement system to another; e.g., degrees Fahrenheit to
degrees Celsius.
3. Use appropriate levels of precision when calculating with
measurements.
5. Determine surface area for pyramids by analyzing their parts.
6. Solve and determine the reasonableness of the results for
problems involving rates and derived measurements, such as
velocity and density, using formulas, models and graphs.
6
Compiled by Kathleen Carpenter, Warren County ESC Nov. 2011
Data Analysis and Probability:
7. Apply proportional reasoning to solve problems involving indirect
measurements or rates.
8. Find the sum of the interior and exterior angles of regular convex
polygons with and without measuring the angles with a protractor.
2. Evaluate different graphical representations of the same data to
determine which is the most appropriate representation for an
identified purpose; e.g., line graph for change over time, circle
graph for part-to-whole comparison, scatterplot for relationship
between two variants.
3. Differentiate between discrete and continuous data and
appropriate ways to represent each.
4. Compare two sets of data using measures of center (mean, mode,
median) and measures of spread (range, quartiles, interquartile
range, percentiles).
5. Explain the mean's sensitivity to extremes and its use in
comparison with the median and mode.
7. Identify different ways of selecting samples, such as survey
response, random sample, representative sample and convenience
sample.
8. Describe how the relative size of a sample compared to the target
population affects the validity of predictions.
9. Construct convincing arguments based on analysis of data and
interpretation of graphs.
10. Calculate the number of possible outcomes for a situation,
recognizing and accounting for when items may occur more than
once or when order is important.
11. Demonstrate an understanding that the probability of either of
two disjoint events occurring can be found by adding the
probabilities for each and that the probability of one independent
event following another can be found by multiplying the
probabilities.
Geometry and Spatial Sense:
1. Make and test conjectures about characteristics and properties
(e.g., sides, angles, symmetry) of two-dimensional figures and
three-dimensional objects.
6. Draw nets for a variety of prisms, pyramids, cylinders and cones.
Patterns, Functions, Algebra:
2. Generalize patterns and sequences by describing how to find the
nth term.
5. Use physical models to add and subtract monomials and
polynomials, and to multiply a polynomial by a monomial.
8. Write, simplify and evaluate algebraic expressions (including
formulas) to generalize situations and solve problems.
12. Solve simple quadratic equations graphically; e.g., y = x2 – 16.
13. Compute and interpret slope, midpoint and distance given a set
of ordered pairs.
14. Differentiate and explain types of changes in mathematical
relationships, such as linear vs. nonlinear, continuous vs.
noncontinuous, direct variation vs. inverse variation.
15. Describe and compare how changes in an equation affects the
related graphs; e.g., for a linear equation changing the coefficient of
x affects the slope and changing the constant affects the intercepts.
16. Use graphing calculators or computers to analyze change; e.g.,
interest compounded over time as a nonlinear growth pattern.
7