Entrance check
Three private responses in Top Hat.
MATH 130 · Friday, August 28, 2026
Factor First
A focused bridge from algebra review to rational expressions.
Diagnose→Factor→Apply→Practice
Today’s flow
One chain. Five moves.
Factor toolbox
GCF, squares, and simple trinomials.
Rational bridge
Restrictions and safe cancellation.
Practice + exit
Work your lane and show one result.
Top Hat · Entrance check
Join now. Work alone.
Open Top Hat
Choose MATH 130 and today’s Entrance Check.
Use paper
Write enough work to diagnose your first step.
Submit privately
No leaderboard. We will use the class pattern.
Do not use notes, ALEKS, or a calculator.
Entrance check · 1 of 3
Positive exponents only
x5 ÷ x8
Ax3
B1/x3
Cx13
D1/x13
Answer: B. Subtract exponents: 5 − 8 = −3, then move to the denominator.
Entrance check · 2 of 3
Interval notation
{ x | −2 < x ≤ 4 }
A[−2, 4]
B(−2, 4]
C[−2, 4)
D(−2, 4)
Answer: B. −2 is excluded; 4 is included.
Entrance check · 3 of 3
Factor completely
6x3 − 9x2
A3x2(2x − 3)
B3x(2x2 − 3x)
Cx2(6x − 9)
D3x2(2x + 3)
Answer: A. It removes the full numerical and variable GCF.
Entrance results · 37 responses
Today’s route: factor first
Quotient exponents · 81%
30 of 37 correct.
Give one-sentence feedback; no repair.
Interval notation · 89%
33 of 37 correct.
No repair; Appendix B stays closed.
Complete factoring · 73%
27 of 37 correct.
Proceed, with extra attention to the full GCF.
Core target
Every rational expression begins here.
Factor completely→State restrictions→Cancel factors→Find the LCD
We will practice the first three moves, then add the fourth if the class is ready.
Factor type 1
Start with the GCF
- 1Find the GCF of every term.
- 2Factor it out.
- 3Check by distributing.
6x3 − 9x2
3x2(2x − 3)
Your turn · 90 seconds
Factor completely
8x4 − 12x3
Work alone for 45 seconds. Then compare one line with a partner.
4x3(2x − 3). Distribute to verify.
Factor type 2
Difference of squares
Pattern
a2 − b2
= (a − b)(a + b)
= (a − b)(a + b)
Two perfect squares with subtraction.
Try it
9x2 − 16
(3x − 4)(3x + 4)
Factor type 3
For x² + bx + c
Find p and q
p · q = c
p + q = b
(x + p)(x + q)
Try it
x2 + 9x + 20
(x + 4)(x + 5)
Apply the factor toolbox
Restrictions survive cancellation
- 1Factor numerator.
- 2Factor denominator.
- 3Write original restrictions.
- 4Cancel common factors.
(x2 − 9)/(x2 + 6x + 9)
(x+3)(x−3) / [(x+3)(x+3)]
= (x−3)/(x+3), x ≠ −3
= (x−3)/(x+3), x ≠ −3
Your turn · 3 minutes
Simplify and state restrictions
(x2 − 25)/(x2 + 2x − 15)
Write the restriction before you cancel.
(x − 5)/(x − 3), with x ≠ −5, 3.
Guided workshop · 13 minutes
Work the lane ALEKS gives you
Lane A · Repair
Negative exponents → interval notation → union/intersection.
Finish one current topic or write one precise question.
Lane B · Core
GCF → simple factoring → restrictions → simplify/multiply rational expressions.
Show a fully factored line.
Lane C · Extend
LCD with shared factors → add unlike rational expressions.
Explain why no factor is duplicated.
13:00
Whole-class debrief · 5 minutes
Which error cost the most?
Incomplete factoring
A GCF remains inside the parentheses.
Canceling terms
Something separated by + or − is crossed out.
Lost restriction
A canceled denominator factor is forgotten.
Repair sentence: Factor completely first. Record restrictions from the original denominator. Cancel only complete factors.
Optional extension
Do not duplicate a shared factor
1/(x2 − 4) + 1/(x2 + x − 6)
Factor both denominators. What is the least common denominator?
LCD = (x − 2)(x + 2)(x + 3)
because both denominators already share one factor of (x − 2).
because both denominators already share one factor of (x − 2).
Top Hat · Exit check
Show the complete chain
Question 1
Simplify and preserve every original restriction.
Question 2
Identify your next-step blocker so the next class can respond.
Exit check · 1 of 2
Simplify + keep restrictions
(x2 − 4)/(x2 + x − 6)
A(x+2)/(x+3), x≠−3
B(x+2)/(x+3), x≠2,−3
C(x−2)/(x−3), x≠2,3
D1, x≠2,−3
Answer: B. The canceled factor still contributes x ≠ 2.
Exit check · 2 of 2
Where is your next blocker?
AChoosing a factor method
BFactoring completely
CFinding restrictions
DCanceling only factors
Optional E: I am ready for LCDs and unlike denominators.
Before Wednesday
Finish the current review modules
R.1–R.3
Complete the assigned modular ALEKS learning.
R.4–R.5
Complete the assigned modular ALEKS learning.
Quiz 1
Canvas date: Sep. 2 at 2:10 PM. Confirm content in Canvas or class.
When stuck, bring one written line and one precise question.
Appendix A · five-minute repair
Negative exponents
Combine first. Then move.
xm/xn = xm−n
A negative result moves across the fraction bar.
Try it
y−3/y5
y−8 = 1/y8
Appendix B · five-minute repair
Interval notation
Translate each endpoint
Open endpoint
< or > → ( )
Not included.
Closed endpoint
≤ or ≥ → [ ]
Included.
−3 ≤ x < 7 → [−3, 7)