Topic: Expressions and Equations |
Common Core Mathematics 6-8 |
| Cluster Apply and extend previous understandings of arithmetic to algebraic
expressions. |
| | Grade 6 |
| | | 6.EE.1. | Write and evaluate numerical expressions involving whole-number
exponents. |
| | | 6.EE.2. | Write, read, and evaluate expressions in which letters stand for
numbers. |
| | | 6.EE.2.a. | Write expressions that record operations with numbers and with
letters standing for numbers. For example, express the calculation
“Subtract y from 5” as 5 – y. |
| | | 6.EE.2.b. | Identify parts of an expression using mathematical terms (sum,
term, product, factor, quotient, coefficient); view one or more
parts of an expression as a single entity. For example, describe the
expression 2 (8 + 7) as a product of two factors; view (8 + 7) as both
a single entity and a sum of two terms. |
| | | 6.EE.2.c. | Evaluate expressions at specific values of their variables. Include
expressions that arise from formulas used in real-world problems.
Perform arithmetic operations, including those involving wholenumber
exponents, in the conventional order when there are no
parentheses to specify a particular order (Order of Operations).
For example, use the formulas V = s3 and A = 6 s2 to find the volume
and surface area of a cube with sides of length s = 1/2. |
| | | 6.EE.3. | Apply the properties of operations to generate equivalent expressions.
For example, apply the distributive property to the expression 3 (2 + x) to
produce the equivalent expression 6 + 3x; apply the distributive property
to the expression 24x + 18y to produce the equivalent expression
6 (4x + 3y); apply properties of operations to y + y + y to produce the
equivalent expression 3y. |
| | | 6.EE.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). For example, the expressions y + y + y and 3y
are equivalent because they name the same number regardless of which
number y stands for. |
| Cluster Reason about and solve one-variable equations and inequalities. |
| | Grade 6 |
| | | 6.EE.8. | 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. |
| | | 6.EE.6. | Use variables to represent numbers and write expressions when
solving a real-world or mathematical problem; understand that a
variable can represent an unknown number, or, depending on the
purpose at hand, any number in a specified set. |
| | | 6.EE.7. | Solve real-world and mathematical problems by writing and solving
equations of the form x + p = q and px = q for cases in which p, q and
x are all nonnegative rational numbers. |
| | | 6.EE.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. |
| Cluster Represent and analyze quantitative relationships between
dependent and independent variables. |
| | Grade 6 |
| | | 6.EE.9. | Use variables to represent two quantities in a real-world problem that
change in relationship to one another; write an equation to express
one quantity, thought of as the dependent variable, in terms of the
other quantity, thought of as the independent variable. Analyze the
relationship between the dependent and independent variables using
graphs and tables, and relate these to the equation. For example, in a
problem involving motion at constant speed, list and graph ordered pairs
of distances and times, and write the equation d = 65t to represent the
relationship between distance and time. |
| Cluster Solve real-world and mathematical problems involving area, surface
area, and volume. |
| | Grade 6 |
| | | 6.G.2. | Find the volume of a right rectangular prism with fractional edge
lengths by packing it with unit cubes of the appropriate unit fraction
edge lengths, and show that the volume is the same as would be
found by multiplying the edge lengths of the prism. Apply the
formulas V = l w h and V = b h to find volumes of right rectangular
prisms with fractional edge lengths in the context of solving real-world
and mathematical problems. |
| | | 6.G.3. | Draw polygons in the coordinate plane given coordinates for the
vertices; use coordinates to find the length of a side joining points with
the same first coordinate or the same second coordinate. Apply these
techniques in the context of solving real-world and mathematical
problems. |
| | | 6.G.4. | Represent three-dimensional figures using nets made up of rectangles
and triangles, and use the nets to find the surface area of these
figures. Apply these techniques in the context of solving real-world
and mathematical problems. |
| Cluster Use properties of operations to generate equivalent expressions. |
| | Grade 7 |
| | | 7.EE.1. | Apply properties of operations as strategies to add, subtract, factor,
and expand linear expressions with rational coefficients. |
| Cluster Solve real-life and mathematical problems using numerical and
algebraic expressions and equations. |
| | Grade 7 |
| | | 7.EE.3. | Solve multi-step real-life and mathematical problems posed with
positive and negative rational numbers in any form (whole numbers,
fractions, and decimals), using tools strategically. Apply properties of
operations to calculate with numbers in any form; convert between
forms as appropriate; and assess the reasonableness of answers using
mental computation and estimation strategies. For example: If a woman
making $25 an hour gets a 10% raise, she will make an additional 1/10 of
her salary an hour, or $2.50, for a new salary of $27.50. If you want to place
a towel bar 9 3/4 inches long in the center of a door that is 27 1/2 inches
wide, you will need to place the bar about 9 inches from each edge; this
estimate can be used as a check on the exact computation. |
| | | 7.EE.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. |
| | | 7.EE.4.a. | Solve word problems leading to equations of the form px + q = r
and p(x + q) = r, where p, q, and r are specific rational numbers.
Solve equations of these forms fluently. Compare an algebraic
solution to an arithmetic solution, identifying the sequence of the
operations used in each approach. For example, the perimeter of a
rectangle is 54 cm. Its length is 6 cm. What is its width? |
| | | 7.EE.4.b. | Solve word problems leading to inequalities of the form px + q > r
or px + q < r, where p, q, and r are specific rational numbers. Graph
the solution set of the inequality and interpret it in the context of
the problem. For example: As a salesperson, you are paid $50 per
week plus $3 per sale. This week you want your pay to be at least
$100. Write an inequality for the number of sales you need to make,
and describe the solutions. |
| Cluster Work with radicals and integer exponents. |
| | Grade 8 |
| | | 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.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 ã2 is irrational. |
| | | 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 × 108 and the population
of the world as 7 × 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. |
| Cluster Understand the connections between proportional relationships,
lines, and linear equations. |
| | Grade 8 |
| | | 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. |
| Cluster Analyze and solve linear equations and pairs of simultaneous linear
equations. |
| | Grade 8 |
| | | 8.EE.7. | Solve linear equations in one variable. |
| | | 8.EE.7.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). |
| | | 8.EE.8. | Analyze and solve pairs of simultaneous linear equations. |
| | | 8.EE.8.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. |
| | | 8.EE.8.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. |