ACT Tutoring Session Notes
Session Overview
Main Topics Covered
- •Factoring quadratic expressions (trinomials with leading coefficients ≠ 1)
- •Right triangle geometry (Pythagorean Theorem)
- •Circle formulas (area and circumference)
- •Probability (single events, compound events, with/without replacement)
- •Counting principle problems
- •Test-taking strategies and timing for new ACT format
Key Concepts Taught
- •Trial and error approach to factoring complex trinomials
- •Identifying factorable vs. non-factorable expressions
- •Proper placement of values in Pythagorean Theorem
- •Probability operations: "OR" means add, "AND" means multiply
- •Strategic test-taking and time management
Detailed Content
1. Factoring Quadratic Expressions
Key Challenge
Factoring trinomials when the leading coefficient is not 1
Example Problem:
Factor 9x² - x - 56
Step-by-Step Process:
- 1.Identify what the first terms must multiply to (9 in this case)
- 2.Identify what the last terms must multiply to (-56)
- 3.Use trial and error to find the correct combination
- 4.Check work by expanding (FOIL method)
Important Notes
- • Requires significant trial and error
- • Need to consider both positive and negative combinations
- • Position of numbers matters significantly
- • Always verify by expanding back out
Successful Example:
Problem: 7x² - 32x - 60
Solution: (7x + 10)(x - 6)
Strategy: Find two numbers that multiply to -60 and work with the coefficient 7
Greatest Common Factor (GCF) Strategy:
Problem: 4x² + 8x - 12
First step: Factor out GCF of 4
Result: 4(x² + 2x - 3)
Then factor: 4(x + 3)(x - 1)
2. Pythagorean Theorem
Formula:
a² + b² = c²
- • a and b = legs of right triangle
- • c = hypotenuse (longest side)
Critical Rule
Always identify which value is the hypotenuse before plugging in numbers
Example Problem:
Given: One leg = 40, hypotenuse = 50
Setup: 40² + b² = 50²
Calculation: 1,600 + b² = 2,500
Solution: b² = 900, so b = 30
Common Mistake to Avoid:
Placing values in wrong positions (confusing legs with hypotenuse)
3. Circle Formulas
Area Formula:
A = πr²
Circumference Formula:
C = 2πr
Key Points:
- • Always use radius (r), not diameter
- • Diameter = 2 × radius
- • Check answer choices before calculating to see if π should remain in answer
Example Problem (Question 26):
Given: Diameter = 6
Radius = 3
Circumference = 2π(3) = 6π (not 18.84...)
Strategy: Leave π in answer when answer choices show π
Concentric Circles Problem (Question 29):
Concentric = circles with same center
Find difference in circumferences
Circle 1: C = 2π(5) = 10π
Circle 2: C = 2π(6) = 12π
Difference: 12π - 10π = 2π
Area of Shaded Region (Question 42):
Strategy: Find area of outer circle minus area of inner circle
Outer circle (r=10): A = 100π
Inner circle (r=6): A = 36π
Shaded region: 100π - 36π = 64π
4. Probability
Fundamental Rule:
- • Bottom number = total possibilities
- • Top number = favorable outcomes
Basic Example:
16 bills total, 5 are $20 bills
P(selecting $20 bill) = 5/16
"OR" vs "AND" Operations
OR = ADD (easier to achieve)
Example: 3 red, 4 yellow, 3 green marbles (10 total)
P(red OR yellow) = 3/10 + 4/10 = 7/10
AND = MULTIPLY (harder to achieve)
P(red AND then yellow) = 3/10 × 4/10 = 12/100
Logic: Multiple requirements make success harder, so probability gets smaller
With vs. Without Replacement
With Replacement:
P(red) AND P(yellow) = 3/10 × 4/10
Denominator stays same
Without Replacement:
P(red) AND P(yellow) = 3/10 × 4/9
Reduce both numerator and denominator
Assume previous event succeeded
Example: Two greens in a row (without replacement)
First green: 3/10
Second green: 2/9 (one green removed, total reduced)
Combined: 3/10 × 2/9 = 6/90 = 1/15
5. Counting Principle
Strategy:
Draw blanks for each choice, fill in number of options, then multiply
Example Problem (Question 3):
Noses: 4 options
Lips: 3 options
Wigs: 2 options
Total combinations: 4 × 3 × 2 = 24
Example Problem (Question 44):
Letters (3 positions): 26 × 26 × 26
Digits (3 positions): 10 × 10 × 10
Total: 26³ × 10³
Common Mistake (Question 51):
Don't assume 5 × 5 × 5 for all positions
Some positions may have only 1 option
Correct: 1 × 1 × 1 × 10 × 10 × 10 × 10
6. Test-Taking Strategies
Timing Strategy for New ACT Format:
- • Total: 50 minutes for 45 questions
- • Goal: Reach question 20 in 15-20 minutes
- • Remaining time: 30-35 minutes for questions 21-45 (slightly over 1 min/question)
"Rounds" Strategy for Questions 21-45:
- 1.Round 1: Skim and complete all questions you definitely know how to do
- 2.Round 2: Tackle questions where you're uncertain but can attempt something
- 3.Round 3: Guess or struggle through remaining difficult questions
Rationale: Ensures you see all questions and maximize points on accessible problems
General Test-Taking Tips:
- Write everything down - even on digital test, use scratch paper extensively
- Read slowly and carefully - better to read 1-2 times slowly than panic-read 10 times
- Annotate as you go - circle important numbers and values
- Check answer choices early - see if π should stay in answer, check units, etc.
- Verify before bubbling - reread question one last time before selecting answer
- Watch for eye movement - on digital test, eyes moving between screen and paper can cause errors
- Break down complex questions - don't expect one-step solutions
Calculator Tips:
- • Use fraction simplification feature (MATH → FRAC on TI-84, or automatic on Casio)
- • Find π button (2nd + key on TI-84)
- • Use parentheses when entering fractions to avoid order of operations errors
Morning of Test:
- • Complete 1 English passage and ~10 math questions as warm-up
- • Don't grade them - just get mentally engaged
- • Prevents silly mistakes from not being warmed up
Homework & Action Items
Specific Assignments
- Complete full-length practice test in new ACT format (focus on math section)
- Time yourself on at least the first 20 questions to practice pacing
- Review math formula sheet provided
- Practice additional problems from 'types of math questions' document as needed
- Work with math teacher for additional support during the week
Practice Problems to Complete
- • Additional factoring problems (with focus on GCF first)
- • Probability problems with "and" vs "or" scenarios
- • Counting principle problems
- • Circle area and circumference problems
Topics to Review Before Test
- • Slope formula and writing equations from graphs (student confirmed comfortable)
- • Triangle angle relationships (student confirmed comfortable)
- • Factoring strategies (especially identifying when to use GCF)
- • Probability operations (OR = add, AND = multiply)
- • Pythagorean Theorem (careful placement of values)
Student Progress
Areas of Strength
- • Slope formula and graphing equations
- • Basic triangle concepts and angle relationships
- • Circle formulas (area and circumference)
- • Pythagorean Theorem execution
- • Basic probability concepts
- • Counting principle problems
- • Good habit of writing work down
- • Calculator proficiency
Topics Needing More Practice
- • Factoring complex trinomials (may skip on test if time-consuming)
- • Reading comprehension of word problems - tendency to rush
- • Careful value placement in formulas
- • Checking what question actually asks for
- • SOH-CAH-TOA (trigonometry) - needs independent practice
- • Managing test anxiety
Student Confidence Level
• Current score range: 19-22 on math section
• Goal: Maximize score by focusing on first 20 questions with high accuracy
• This is intended to be student's final ACT attempt
• Student is taking test digitally (has taken it twice before in digital format)
Tutor Observations
- • Student benefits from being reminded to slow down and read carefully
- • Strong foundational understanding but sometimes rushes through problems
- • Good self-awareness about which topics need more work
- • Realistic about time management and willing to strategically skip difficult problems
Next Session
Final session before test
Test Date
This weekend (Saturday)
Additional Resources
Formula sheet, practice test, reference docs
Student should focus on building confidence, practicing careful reading, and maximizing performance on accessible questions rather than attempting every problem on the test.