consolidates and expands students’ previous work with generating equivalent
expressions and solving equations and inequalities. By the end of this unit
students should fluently solve multistep problems posed with positive and
negative rational numbers in any form (whole numbers, fractions, and
decimals), using tools strategically. This work is the culmination of many
progressions of learning in arithmetic, problem solving and mathematical
practices. In solving word problems
leading to one-variable equations of the form px
+ q = r and p(x + q)
= r, as well as px + q < r and px + q > r, students should solve these equations
fluently. This will require fluency with rational number arithmetic, as well
as fluency to some extent with applying properties of operations to rewrite
linear expressions with rational coefficients. Students use the properties of operations
and the relationships between addition and subtraction, and multiplication
and division, as they formulate expressions, equations, and inequalities in
one variable and use these to solve problems. They solve real‐life and mathematical problems using numerical
and algebraic expressions, equations, and inequalities. At the end of the
unit students’ work with expressions and equations is applied to finding
unknown angles in a figure. By using
facts about supplementary, complementary, vertical, and adjacent angles,
students are able to write and solve simple equations to solve for the
unknown angle. The work and required
fluency expectations in this unit are a major capstone leading to the
mathematical development necessary to perform operations with linear
equations in Grade 8.
MP.1 Make sense of problems and persevere
in solving them.
MP.2 Reason abstractly and quantitatively.
MP.3 Construct viable arguments and
critique the reasoning of others.
MP.4 Model with mathematics.
MP.5 Use appropriate tools strategically.
MP.6 Attend to precision.
MP.7 Look for and make use of structure.
MP.8 Look for and express regularity in
EE: Expressions and Equations
A. Use properties of operations to
generate equivalent expressions
Apply properties of operations as strategies to add,
subtract, factor, and expand linear expressions with rational coefficients
to include multiple grouping symbols (e.g.,
parentheses, brackets, and braces).
*I can combine like terms with rational
*I can factor and expand linear expressions with
rational coefficients using the distributive property.
*I can apply properties of operations as
strategies to add, subtract, factor, and expand linear expressions with
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. For example, a + 0.05a = 1.05a means that
“increase by 5%” is the same as “multiply by 1.05.”
*I can write equivalent expressions with
fractions, decimals, and integers.
*I can rewrite an expression in an equivalent
form in order to provide insight about how quantities are related in a
real-life and mathematical problems using numerical and algebraic
expressions and equations.
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
*I can convert between numerical forms as
*I can 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
*I can apply properties of operations to
calculate with numbers in any form.
*I can assess the reasonableness of answers
using mental computation and estimation strategies.
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.
a. Solve word problems
leading to equations of the form and where , , and 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?
b. Solve word problems
leading to inequalities of the form or , where , , and 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.
*For an algebraic equation in the form and , I
can fluently solve with speed and accuracy and identify the sequence of operations used to solve.
*I can solve word problems leading to equations
of the form and .
*I can compare an algebraic solution to an
arithmetic solution by identifying the sequence of the operations used in
real-life and mathematical problems involving angle measure, area, surface
area, and volume.
Know the formulas for the area and circumference of a
circle and use them to solve problems; give an informal derivation of the
relationship between the circumference and area of a circle.
*I can determine the parts of a circle including
radius, diameter, area, circumference, center and chord.
*I can identify .
*I can recognize and use the formulas for area
and circumference of a circle.
*I can find the circumference of a circle, given
the area of the circle.
*I can justify that can be derived from the circumference and
diameter of a circle.
*I can apply the circumference or area formulas
to solve mathematical and real-world problems.
*I can justify the formulas for area and
circumference of a circle and how they relate to .
*I can informally derive the relationship
between circumference and area of a circle.
Use facts about supplementary, complementary, vertical,
and adjacent angles in a multi-step problem to write and solve simple
equations for an unknown angle in a figure.
*I can identify and recognize types of angles:
supplementary, complementary, vertical, adjacent.
*I can determine complements and supplements of
a given angle.
*I can determine unknown angle measures by
writing and solving algebraic equations based on relationships between
Solve real-world and mathematical problems involving
area, volume and surface area of two- and three-dimensional objects
composed of triangles, quadrilaterals, polygons, cubes, and right prisms. (Pyramids limited to surface area only.)
*I can determine the formulas for area and
volume of triangles, quadrilaterals, polygons, cubes and right prisms.
*I can determine the procedure for finding
surface area for two- and three- dimensional objects composed of triangles,
quadrilaterals, polygons, cubes and right prisms.
*I can determine when to use these formulas
(area, volume, surface area) in real-world and math problems for two- and
three- dimensional objects composed of triangles, quadrilaterals, polygons,
cubes and right prisms.
*Variables can be used to represent numbers
in any type of mathematical problem.
*Understand the difference between an
expression and an equation.
*Expressions can be manipulated to suit a
particular purpose to solve problems efficiently.
*Mathematical expressions, equations,
inequalities, and graphs are used to represent and solve real-world and
*Constructing simple equations and
inequalities to solve real life word problems is a necessary concept.
*Writing and solving real-life and
mathematical problems involving simple equations for an unknown angle in a
figure helps students as they engage in higher geometry concepts.
*Reason about relationships among
two-dimensional figures, which leads to gaining familiarity with the
relationships between angles formed by intersecting lines.
*Geometry and spatial sense offer ways to
interpret and reflect on our physical environment.
*Analyzing geometric relationships develops
reasoning and justification skills.
*How can I apply the order of operations and
the fundamentals of algebra to solve problems involving equations and
*How can I justify that multiple
representations in the context of a problem are equivalent expressions?
*How do I assess the reasonableness of my
*How can I use and relate facts about
special pairs of angles to write and solve simple equations involving unknown
*What is the total number of degrees in
supplementary and complementary angles?
*What is the relationship between vertical
and adjacent angles?
*When and how are expressions, equations,
and inequalities applied to real world situations?
*What are some possible real-life situations
to which there may be more than one solution?
*How does the ongoing use of decimals apply
to real-life situations?
*How can geometry be used to solve problems
about real-world situations, spatial relationships, and logical reasoning?