HCPSS Curriculum Framework
Common Core 7
Unit 3: Expressions and Equations
Overview (Big Ideas):
In this unit, students use the arithmetic of rational numbers as they formulate expressions and
equations in one variable and use these equations to solve problems. Students will understand
and use properties of operations to generate equivalent expressions. Students use and solve reallife mathematical problems using numerical and algebraic expressions and equations. Also,
students will use variables to represent quantities in a real-world or mathematical problem to
construct and solve simple equations and inequalities in one and two step. In gaining the
understanding of solving the one and two-step equations and inequalities, students will reason
about the quantity of their solutions. (adapted from CCSC Grade 7 page 46)
Enduring Understandings:
1. Expressions are powerful tools for exploring, reasoning about, and representing situations.
2. Two or more expressions may be equivalent, even when their symbolic forms differ. A
relatively small number of symbolic transformations can be applied to expressions to yield
equivalent expressions.
3. Variables have many different meanings, depending on context and purpose.
4. Using variables permits writing expressions whose values are not known or vary under
different circumstances.
5. Using variables permits representing varying quantities. This use of variables is particularly
important in studying relationships between varying quantities.
6. The equals sign can indicate that two expressions are equivalent. It is often important to find
the value(s) of a variable for which two expressions represent the same quantity. Finding the
value(s) of a variable for which two expressions represent the same quantity is know as
solving an equation.
7. An inequality is another way to describe a relationship between expressions; instead of
showing that the values of two expressions are equal, inequalities indicate that the value of
one expression is greater than (or greater than or equal to) the value of the other expression.
8. In solving an inequality, multiplying or dividing both expressions by a negative number
reverses the sign that indicates the relationships between the two expressions.
9. The equals sign can be used in defining or giving a name to an expression or function rule.
10. Functions provide a tool for describing how variables change together. Using a function in
this way is called modeling, and the function is called a model.
11. Functions can be represented in multiple ways—in algebraic symbols, graphs, verbal
descriptions, tables, and so on—and these representations, and the links among them, are
useful in analyzing patterns of change.
12. One important way of describing functions is by identifying the rate at which the variables
change together. It is useful to group functions into families with similar patterns of change
because these functions, and the situations that they model, share certain general
characteristics.
13. Some representations of a function may be more useful than others, depending on how they
are used.
Source:
Lloyd, G., Herbel-Eisenmann, B., & Star, J.R. (2011). Developing essential understanding of
expressions, equations, and functions for teaching mathematics in grades 6-8. Reston, VA:
The National Council of Teachers of Mathematics, Inc.
Essential Questions:
o Are students able to utilize properties of operations in order to rewrite expressions in a
different form?
o Can students factor by using division to express a linear expression by its factors and then
also expand by using multiplication to rewrite the factors in a linear expression as a
product?
o Do students understand linear expression terminology: sum, difference, term, product,
factor, quotient, and coefficient?
o Can students fluently construct and solve two-step equations?
o Do students use all four inequality symbols and related terminology correctly?
o Are students able to solve inequalities and determine the solution set?
o Can students solve real-world mathematical problems using numerical and algebraic
expressions and equations?
Curriculum Standards:
Use properties of operations to generate equivalent expressions.
1. Apply properties of operations as strategies to add, subtract, factor, and expand linear
expressions with rational coefficients.
2. Understand that rewriting an expression in different forms in a problem context can shed
light on the problem of 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.”
Solve real-life and mathematical problems using numerical and algebraic expressions and
equations.
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.
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.
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?
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.
Common Misconceptions:
o Students may want to solve two-step equations or inequalities by multiply or dividing
first to eliminate the term that is being multiplied or divided by the variable before adding
or subtracting.
o When solving equations and inequalities, they may use the inverse operation on only one
side and not the other or they may use the same operation rather than the inverse.
o Students may also have difficulty multiplying a fraction through the use of reciprocals.
o When solving inequalities that have a negative coefficient with the variable or the
variable is being divided by a negative number, students may forget to reverse the
direction of the inequality symbol when multiplying or dividing by a negative number.
o Students might confuse the symbolic components of the graph for an inequality,
specifically open circle vs. closed circle and the direction of shading. Similarly, they
might forget that an open circle represents a value that is NOT actually part of the
solution set.
o Students may not realize that solutions to equations and inequalities are not just integers,
but also can be fractions and decimals they just need to make sure that their solution
makes sense in the context of the problem.