A. Explain the Distinction Between Factor and Term

ALGEBRA p. 19

7.1 Recognize and use appropriate concepts, procedures, definitions, and properties to simplify expressions and solve equations.

High School / Community College / EWU

a. Explain the distinction between factor and term.

Core 1
Define Term
Core 2
Core 3
It is not in our curriculum, not specifically addressed
It is asked to simplify and asked to solve, but no questions specifically stating the difference of the two.
Comes up in our discussions, but is not explicitly part of an assignment
Integrated 3
List the terms in each of the following:
(a) 
(b) 
(c) 
(d) 
(e) 
I address this component by using examples where factors indicate products and terms are expressions that are added or subtracted (e.g., in the expression , the terms are ,,; the factors of the term are and , the factors of the term are 2 and x, and the factors of the term are -1
and 5)
Algebra 2, CPM
List the factors of . Name a terms of
The multiplication table below has factors along the top row and left column. Their product is where the row and column intersect. With your team complete the table with all the factors and products.
Multiply /

/ =
Algebra 2
Verbal explanation, examples of the two and how they are different, but I don’t specifically address this.
has two terms, but has one term
Explain the difference.
Algebra 2/Trigonometry
Identify the terms:
(a) 
(b) 
(c) 
In-class a distinction is made between the two.
e.g., has 3 factors and 2 terms
In class: Binomials have factors and are one term. But, when expanded, they have 3 terms.
(e.g., )
Course not indicated
Given identify the leading term, the constant term, and write in its linear factorization form.
Core 4
It is not in our curriculum.
Precalculus
Given
(a)  State how many terms there are.
(b)  Completely factor: How many factors are there?
(c)  What is the difference between #1 and #2? / SFCC – Beginning & Intermediate Algebra
SCC – Intermediate Algebra
It is not in our curriculum. / Basic Algebra
All constant terms in an algebraic expression are coefficients, but not all coefficients are constant terms. Explain.
Intermediate Algebra
Give an example of an algebraic expression in which
(a)  2x is a factor
(b)  2x is a term

b. Explain the distinction between expression and equation.

Core 1
Core 2
Expand the exponential expression
Rewrite the exponential expression:
Core 3
It is not in our curriculum, not specifically addressed
Discussed in class the difference between and .
Integrated 3
Which of the following are equations?
(a) 
(b) 
(c) 
(d) 
Algebra 2, CPM
Discuss: is an equation, while is an expression
Build the following on your expression mat and simplify:
Build the following equation on your equation mat and solve for x:
Algebra 2
I use the terminology in class and work with examples such as and . Have students compare and contrast and go over vocabulary. I don’t specifically assess this.
3x+4 is an expression and is an equation
Algebra 2/Trigonometry
Core 4
It is not in our curriculum.
Precalculus
What is the difference between and ?
If implies , does imply ? Why or why not?
Only mentioned indirectly, such as factor: and solve: . / SFCC – Beginning & Intermediate Algebra
SCC – Intermediate Algebra
It is not in our curriculum. / Basic Algebra
Explain the difference between “seven more than double a number” and “seven more than double a number is 15.”
Intermediate Algebra
Explain how the following are different:
and .
Explain how the following are different:
and .

c. Explain the distinction between simplify and solve.

Core 1
Discussed in class to simplify or solve and .
Core 2
Core 3
It is not in our curriculum.
Integrated 3
Simplify:
(a) 
(b) 
(c) 
Solve:
Problems such as:
Solve:
(a) 
(b) 
(c) 
Simplify:
(a) 
(b) 
Algebra 2, CPM
Simplify the expression:
Solve:
Build the following on your expression mat and simplify:
Build the following equation on your equation mat and solve for x:
Algebra 2
I use the terminology in class: Solving goes with equations only! Show examples of solve, simplify, and evaluate
I don’t specifically assess this.
I don’t specifically show examples. Our use of the term is to solve for equation and evaluate for expression. Explain the difference.
Algebra 2/Trigonometry
In class discuss vs.
Core 4
It is not in our curriculum.
Precalculus
What is the difference between the two following?
·  Simplify and solve
·  Factor and solve / SFCC – Beginning & Intermediate Algebra
SCC – Intermediate Algebra
It is not in our curriculum. / Basic Algebra
Which of the following can be solved? Explain.
(a) 
(b) 
(c) 
(d) 
Intermediate Algebra
Determine whether each of the following can be simplified or solved: and . Explain how you know.

d. Know what it means to have a solution to an equation.

Core 1
Core 2


Use graphs to solve this system. Describe two other methods for solving this system of equations. Use each method to solve the system of equations. Compare your solutions.
Core 3
represents the monthly cost for a shoe store. What values of x satisfy , and what do those values tell about the store’s monthly business costs?
Systems of Equations: Situations are described that a student would have to write equations for this system.



Graph, produce a table, and solve systems of equations and inequalities.



Find the zeros of the function and verify by graphing
Integrated 3
Give a solution and show that is indeed a solution for .
What does a solution to a linear function look like? Ordered pair.
Is a solution of ?
Mark said that & are solutions to . Is he correct?
The Intermediate Algebra example is similar to ones we would use.
Algebra 2, CPM
Is a solution to the equation ? Justify your answer.
With your partner discuss what you think a solution to an equation is. Write down a description of what you and your partner agree on.
Algebra 2
Tell whether each # is a solution to . ,





Check
Also, expand in class.
Algebra 2/Trigonometry
To solve the equation a student divides both sides of the equation by x and solves the equation . The resulting solution satisfies the original equation. Is there an error? Explain.
Is a factor of ?
Course not indicated
Solve . (Eliminate extraneous solutions.)
Core 4
Use symbolic reasoning to solve each equation for . Check your solutions by substitution in the original equation. If no solution exists, explain why.
(a) 
(b) 
(c) 
(d) 
(see attached sheet):Suppose daily operating profit in dollars, for a movie theater, is a function of the number of tickets sold with the rule for .
·  What is and what does it mean?
·  Explain why the profit function has an inverse.
·  . Rewrite this equation in terms of.
·  Verify that and describe what this equation means in context.
·  Find a rule for and then verify that
Precalculus
Verify that are solutions to .
Is a solution to . Why or why not?
Find all real numbers x for which
Find all zeros of the function
Solve:
Find all real numbers x, such that: . What does our solution mean in context with the graph? / SFCC – Beginning & Intermediate Algebra
SCC – Intermediate Algebra
I address this component with a different type of problems from those shown. For example:
/ Basic Algebra
Compare the solution, or solution set, of each pair of equations. Explain what a solution of an equation is.
/
/
/
Intermediate Algebra
Justin claimed that 4 and 2 are solutions to . Without solving it yourself, explain how you know he is right or wrong.
Algebra Concepts
Verify that is a solution to the equation
Precalculus I
Verify that and are solutions to the equation
For what values of does the equation have a real solution?

e. Use properties of equality to solve an equation through a series of equivalent equations.

Core 1
Core 2
Solve the following equation for x:
Core 3
Does for any value of x, regardless of specific values of a and b?
Write and solve equations and inequalities to help you answer the following:
(a)  For what numbers of tickets sold will the concession operation break even?
(b)  For what numbers of tickets sold will concession profit be equal to ticket sale profit? (Previous infocontext pg. 235)
Here is a solution of an inequality that leads to an incorrect result. Show with function tables and/or graphs that the proposed solution is not correct.



Solve the following linear inequalities. Justify each step of your solution process.
Integrated 3
Similar to Basic Algebra Example at EWU
Solve the following equations:
(a) 
(b) 
(c) 
Algebra 2, CPM
Solve:
Build each equation below (given in text) and determine if it has a solution for x. If it has a solution, find it. If it does not have a solution explain why not (create zeros, flip tiles, balance by removing identical value tiles from both sides)
Algebra 2
Solved the following:
(a) 
(b) 
(c) 
(d) 
Same as above (#d).
Solve:
(a) 
(b) 
Algebra 2/Trigonometry
Solve the following:
(a) 
(b) 
(c) 
(d) 
Core 4
Solve each equation:
(a) 
(b) 
(c) 
(d) 
Precalculus
Solve: where .
Solve the following equation / SFCC – Beginning & Intermediate Algebra
SCC – Intermediate Algebra
Solve: / Basic Algebra
Solve the following equation in more than one way.

Intermediate Algebra
Solve . Justify each step by stating the property you used. Solve . Justify each step by stating the property you used. Should be quadratic
Algebra Concepts
Solve
Precalculus
Solve the following:
(a) 
(b) 

f. Use appropriate properties to simplify an expression, resulting in an equivalent expression.

Core 1
Core 2
Use the properties of exponents and the relationship between exponential and radical expressions to help complete the following tasks:
Determine whether the following statements are true or false. If a statement is false, rewrite the right side of the equation to make a true statement.
(a) 
(b) 
(c) 
(d) 
(e) 
(f) 
(g) 
(h) 
Write each of these exponential expressions in another equivalent form:
(a) 
(b) 
(c) 
Rewrite these exponential expressions in equivalent radical form:
(a) 
(b) 
Core 3
Which of the following pairs of equations and inequalities are equivalent? Explain your reasoning in each case.
(a)  and
(b)  and
(c)  and
Write each of the following in at least 3 different but equal forms:
(a) 
(b) 
(c) 
Is equivalent to ?
Is equivalent to ?
Why is not equivalent to ?
Integrated 3
Similar to Basic Algebra example at EWU
Simplify:
(a) 
(b) 
Algebra 2, CPM
Simplify:
Simplify the algebraic expressions using legal moves on your expression comparison mat. (create zeros, flip tiles, balance by removing identical value tiles from both sides.
Algebra 2
Simplify the following expressions:
(a) 
(b) 
(c) 
Algebra 2/Trigonometry
Put the following quadratic functions in standard form:
(a) 
(b) 
Course not indicated
Simplify to a basic trigonometric function:
Core 4
Prove that is equivalent to each of the following expressions:
(a) 
(b) 
(c) 
Factor each polynomial:
(a) 
(b) 
(c) 
Precalculus
Verify.
Prove:
Prove:
Prove the algebraic identity by starting with the LHS expression and supplying a sequence of equivalent expressions that ends with the RHS expression:
/ SFCC – Beginning & Intermediate Algebra
SCC – Intermediate Algebra
Add or subtract: / Basic Algebra
In each pair of expressions, simplify if possible. Also, explain how each expression is simplified differently. (Assume no denominators have a value of zero.)
(a)  and
(b)  and
(c)  and
(d)  and
T The pool border problem…
Intermediate Algebra
Simplify . Demonstrate that your final expression is equivalent to the original expression.
Algebra Concepts
Consider the work below in simplifying.

Explain why the restriction is necessary.
Precalculus I
Simplify the following:
(a) 
(b) 
Find the domain of

g. Recognize the equivalence between expressions with rational exponents and radicals.

Core 2
Rewrite in radical form.
Core 3
It is not in our curriculum, mainly in Course 2.
Integrated 3
Similar to Intermediate Algebra Example at EWU
Write in radical form:
Simplify with positive exponents:
(a) 
(b) 
(c) 
Algebra 2, CPM
Show 2 steps to simplify each of the following expressions and then calculate the value:
What happens when an exponent is a fraction? Consider this as you answer the questions below.
a)  Calculate with your scientific calculator. What is the result? Also use your calculator to find and. What effect does having ½ in the exponent appear to have?
b)  Based on your observation in part (a), predict the value of and. Then confirm your prediction with your calculator.
Algebra 2
Rewrite in simplest radical form:
Rewrite using rational exponents:
Write as a radical expression:
Algebra 2/Trigonometry
Fill in the missing form of the expression.
Radical: Rational Exponent:
___
____
___
____
Course not indicated
Write the expression as a sum or difference of logarithms: