Download HCPSS Curriculum Framework Common Core 8 Unit 2: Expressions

HCPSS Curriculum Framework
Common Core 8
Unit 2: Expressions and Equations
Overview (Big Ideas):
In this unit, students use linear equations and systems of linear equations to represent, analyze,
and solve a variety of problems. Students recognize equations for proportions (y/x = m) as
special linear equations (y = mx + b), understanding that the constant of proportionality (m) is the
slope, and the graphs are lines through the origin. They understand that the slope (m) of a line is
a constant rate of change, so that if the input or x-coordinate changes by an amount A, the output
or y-coordinate changes by the amount mA. Students also use a linear equation to describe the
association between two quantities in bivariate data (such as arm span vs. height for students in a
classroom). At this grade, fitting the model, and assessing its fit to the data are done informally.
Interpreting the model in the context of the data requires students to express a relationship
between the two quantities in question and to interpret components of the relationship (such as
slope and y-intercept) in terms of the situation.
Students strategically choose and efficiently implement procedures to solve linear equations in
one variable, understanding that when they use the properties of equality and the concept of
logical equivalence, they maintain the solutions of the original equation. Students solve systems
of two linear equations in two variables and relate the systems to pairs of lines in the plane; these
intersect, are parallel, or are the same line. Students use linear equations, systems of linear
equations, linear functions, and their understanding of slope of a line to analyze situations and
solve problems. (CCSS Grade 8 page 53)
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 recognize the properties of integer exponents to compare large and
small numbers?
o How do radicals and exponents influence one’s understanding of other content, such as
geometry and science?
o How can proportional relationships be used to represent authentic situations in life and
solve actual problems?
o Can students relate and compare graphic, symbolic, and numerical representations of
proportional relationships?
o In what way(s) do proportional relationships relate to functions and functional
relationships?
o Are students able to calculate the slope of a line graphically, apply direct variation,
differentiate between zero slope and undefined slope, and understand that similar right
triangles can be used to establish that slope is a constant for a non-vertical line?
o Do students have the knowledge to solve multi-step equations using simple cases by
inspection, one solution, infinitely many solutions, or no solution?
o Are students able to solve systems of linear equations numerically, graphically, or
algebraically using substitution or elimination?
o Do students have the ability to discuss efficient solution methods when solving a system
of equations?
o Can students use a system of equations to solve real-world problems and interpret the
solution in the context of the problem?
Curriculum Standards:
Work with radicals and integer exponents.
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
x 2  p and x 3  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 the world population is
more than 20 times larger.
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.
Understand the connections between proportional relationships, lines, and linear
equations.
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.
Analyze and solve linear equations and pairs of simultaneous linear equations.
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 have no solutions 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.
Common Misconceptions:
o Often students might misunderstand the relative size of numbers.
o Students may confuse the operations for the properties of integer exponents. There is a
tendency to memorize rules rather than internalize the concepts behind the laws of
exponents.
o Students struggle with the appropriate application of exponents to coefficients.
o Students may incorrectly assume that the value of a number is negative when its exponent
is negative.
o Students often confuse power of a power and product of powers.
o As a result of weaknesses with fractions, students struggle with the simplification of
negative exponents. Many students do not fully grasp that division is defined as
multiplication by a reciprocal. As a result, students memorize rules for negative
exponents without conceptualizing the reason behind the rules.
o When simplifying using the quotient of powers rule, subtraction mistakes are often made.
o Students may forget that each rational number has a negative square root, as well as a
principal (positive) square root.
o Students may easily confuse the radical symbol with the long division symbol.
o Students may miscount the decimal places for a value in decimal notation when
expressing the same value in scientific notation.
o Students may mistakenly believe that a slope of zero is the same as “no slope” and then
confuse a horizontal line (slope of zero) with a vertical line (undefined slope).
o Students may interchange the meanings of x (independent variable) and y (dependent
variable), particularly when graphing the line of an equation.
o Students often struggle with the difference between combining like terms on one side of
an equation and eliminating a variable on one side of an equation.