Algebra Launchpad Slides TSI Math Diagnostic Lab
30-Minute Sprint
Algebra Foundations
Algebra Launchpad
High-Yield TSI Strategies: Linear equations, expressions, and the art of solving smarter, not longer.
Warm-Up
5 Minutes
Core Strategy
15 Minutes
Practice & Check
10 Minutes
Step 1
Warm-Up Diagnostic (5 min)
5:00 on timer
Problem A
If \( 3(2x - 4) = 4x + 6 \), what is the value of \( x \)?
Target Check: Expand parentheses and isolate the variable in under 60 seconds.
Problem B
Solve the inequality: \( 5 - 2y \le 13 \). Which direction does the sign flip?
Watch Out: Dividing or multiplying by a negative reverses inequality direction.
Work independently on page 1 of your student packet, then be ready to defend your answer!
Strategy Toolkit
Two Game-Changing TSI Tactics
Work Smarter
Tactic 1: Backsolve
When answers are numbers and algebra looks messy, test the middle choice (usually B or C).
Plug answer directly into the prompt.
Too high? Eliminate smaller/larger options.
Best for: Value of \( x \), age problems, rate puzzles.
Tactic 2: Pick Easy Numbers
When choices have variables (e.g. \( 2x + 4 \)), replace variables with friendly numbers like 2 or 3.
Avoid 0 and 1 if they create duplicate answers.
Calculate target, then test answer choices.
Best for: Equivalent expressions, radical simplifications.
Rule of Thumb: The TSI test writers test your efficiency as much as your calculation!
Model Problem
Backsolve in Action
Small Group Walkthrough
TSI Sample Item
A car rental company charges $35 per day plus $0.20 per mile. If Marcus paid $71 for a one-day rental, how many miles did he drive?
A) 120 miles
B) 160 miles
C) 180 miles (Test First!)
D) 210 miles
Step-by-Step Backsolving
1. Test Choice C (\( m = 180 \)):
\( \text{Cost} = 35 + 0.20(180) = 35 + 36 = \$71 \)
2. Compare to target:
Match! $71 equals target. Done in 25 seconds without solving equations!
3. What if it was too small?
If the cost was under $71, we'd eliminate A and B immediately and test D.
Key Insight: Standard algebra \( 35 + 0.20m = 71 \) works, but arithmetic backsolving prevents algebraic errors under pressure!
Guided Practice
Partner Challenge: 10 Minutes
Pair & Share
Challenge 1
Which expression is equivalent to \( (3x^2 - 4x + 7) - (x^2 - 6x - 2) \)?
Strategy: Distribute the negative sign or test \( x = 2 \).
Challenge 2
If \( \frac{2x - 3}{5} = \frac{x + 4}{3} \), what is the value of \( x \)?
Strategy: Cross-multiply or backsolve with whole number choices.
Challenge 3
For what values of \( k \) does the system \( 2x + 4y = 10 \) and \( 4x + ky = 20 \) have infinitely many solutions?
Strategy: Check proportional coefficients (multiplied by 2).
Track your chosen strategy on page 2 of your packet.
Explain your thinking to your partner!
Wrap-Up
Exit Check (5 Minutes)
Individual Work
TSI Mastery Habits
1 Scan answers before solving: Are they numbers or algebraic expressions?
2 Double check negatives: Watch out for distribution and sign flips.
3 Check reasonableness: Does the answer make sense in the context?
Complete the Exit Ticket
Flip to the back page of your worksheet. Complete the 2 exit ticket questions independently without partner help.
Turn in your packet to the facilitator before transitioning to the next station.
"Confidence in math comes from having more than one way to find the truth."
Algebra Launchpad Student Packet TSI Math Prep Small Group Intervention
Algebra Foundations Launchpad
Session Time
30 Minutes
Name:
Date:
Period:
1
Diagnostic Warm-Up (5 Minutes)
Work independently • Show all steps
Problem 1: Solve for \( x \)
\( 3(2x - 4) = 4x + 6 \)
Show work:
Problem 2: Solve the inequality
\( 5 - 2y \le 13 \)
Show work:
TSI High-Yield Strategy Toolbox
Tactic A: Backsolve
If choices are numbers, plug choice B or C into the equation. Adjust higher or lower based on the result.
Tactic B: Number Substitution
If choices have variables, plug in simple test numbers (like \( x = 2 \)) into both the problem and each answer choice.
2
Guided Practice: Authentic TSI Items
Identify strategy used: [ ] Backsolve [ ] Algebra [ ] Substitute
Question 1 TSI Standard: Linear Modeling
A mobile repair service charges a flat diagnostic fee of $45 plus $28 per hour of labor. If a customer received a total bill of $157, how many hours of labor were billed?
A) 3 hours
B) 4 hours
C) 5 hours
D) 6 hours
Work / Backsolve check:
Question 2 TSI Standard: Polynomial Operations
Which expression is equivalent to \( (3x^2 - 4x + 7) - (x^2 - 6x - 2) \) for all real values of \( x \)?
A) \( 2x^2 - 10x + 5 \)
B) \( 2x^2 + 2x + 9 \)
C) \( 2x^2 - 10x + 9 \)
D) \( 4x^2 + 2x + 5 \)
Work / Test \( x = 2 \):
TSI Math Intervention • Algebra Foundations Page 1 of 2
Guided Practice & Exit Check Algebra Foundations Launchpad
Page 2 of 2
Question 3 TSI Standard: Rational Linear Equations
What is the value of \( x \) that satisfies the equation \( \frac{2x - 3}{5} = \frac{x + 4}{3} \)?
A) \( x = 7 \)
B) \( x = 17 \)
C) \( x = 23 \)
D) \( x = 29 \)
Cross-multiply or backsolve:
Question 4 TSI Standard: Systems of Linear Equations
In the system of equations below, \( k \) is a constant:
\( 2x + 4y = 10 \)
If the system has infinitely many solutions, what is the value of \( k \)?
A) \( k = 2 \)
B) \( k = 4 \)
C) \( k = 8 \)
Algebra Launchpad Facilitator Guide Teacher Facilitator Guide TSI Math Intervention (Small Group)
Algebra Launchpad Pacing & Coaching
Target Level
9th Grade • 30 Min
TSI Benchmark Focus Algebraic Reasoning (Linear Equations, Inequalities, Equivalent Expressions)
Core Objective Solve multi-step linear items using both algebraic methods and test-taking heuristics.
Group Size & Setup 3–6 students with printed student packets and shared slide visual.
Minute-by-Minute Session Roadmap
0:00–5:00
Warm-Up Diagnostic (Slide 2 & Packet Page 1)
Students attempt Warm-Up 1 & 2 silently (3 min). Conduct rapid 2-minute whiteboard or verbal check: "Who got \( x = 9 \)? Who remembered to flip the sign on the inequality?"
5:00–15:00
Core Strategy Spotlight (Slides 3 & 4)
Teach Backsolving (starting with choice C to halve the search space) and Number Picking (\( x = 2 \)). Model the car rental problem on Slide 4. Emphasize that TSI assesses rapid problem solving, not page-long derivations.
15:00–25:00
Guided Coaching & Paired Practice (Slide 5 & Items 1–4)
Students partner up to complete Questions 1 through 4. Facilitator circulates with probing questions: "What strategy did you pick? Can you test choice B before you solve algebraically?"
25:00–30:00
Individual Exit Ticket & Wrap-Up (Slide 6 & Packet Page 2)
Students complete the 2-question exit ticket completely independently (4 min). Collect packets for diagnostic tallying. 1 min key takeaway review.
Teacher Facilitation Scripts & Probing Prompts
Prompting the "TSI Mindset":
"Before you write out 5 lines of algebra, look at the choices. Are they numbers? What happens if you plug in choice C?"
Checking Sign Errors in Inequalities:
"You divided both sides by -2. What physical motion must your pencil make with that inequality symbol right now?"
Tiered Differentiation Framework
Support / Intervention
Have students underline the "per hour" / "rate" term in word problems.
For Question 2, provide colored highlighters to match like terms: \( x^2 \), \( x \), and constants.
Enforce starting with Choice C for all backsolving items.
Extension / Mastery
Challenge students to solve Question 3 both algebraically and by backsolving, timing both ways.
Ask students to write an alternate question that would produce "No solution" for Question 4.
Require verbal justification of why distractors were designed.
Algebra Launchpad Solutions Key Answer Key & Solutions TSI Math Intervention
Algebra Launchpad Solutions Key
Quick Reference
100% Verified Solutions
1
Warm-Up Diagnostic Solutions
Warm-Up 1: \( 3(2x - 4) = 4x + 6 \) x = 9
1. Distribute: \( 6x - 12 = 4x + 6 \)
2. Subtract \( 4x \): \( 2x - 12 = 6 \)
3. Add 12 & divide by 2: \( 2x = 18 \implies x = 9 \)
Warm-Up 2: \( 5 - 2y \le 13 \) y ≥ -4
1. Subtract 5: \( -2y \le 8 \)
2. Divide by -2 (Flip inequality!): \( y \ge -4 \)
Error Alert: Failing to flip ≤ to ≥ is the #1 TSI trap.
2
Guided Practice Solutions & Dual Methods
Question 1 (Car/Repair Cost) B (4 hours)
Algebra: \( 45 + 28h = 157 \implies 28h = 112 \implies h = 4 \).
TSI Shortcut (Backsolve): Test middle choice C (5 hrs): \( 45 + 28(5) = 185 \) (too high!). Test B (4 hrs): \( 45 + 112 = 157 \) (Exact match!).
Question 2 (Polynomials) B (\( 2x^2+2x+9 \))
Algebra: Distribute negative: \( 3x^2 - 4x + 7 - x^2 + 6x + 2 = 2x^2 + 2x + 9 \).
TSI Shortcut (Pick \( x=2 \)): Expression gives \( (12-8+7) - (4-12-2) = 11 - (-10) = 21 \). Choice B gives \( 2(4)+2(2)+9 = 8+4+9 = 21 \). Matches!
Question 3 (Rational Equation) D (\( x = 29 \))
Algebra: Cross multiply: \( 3(2x - 3) = 5(x + 4) \implies 6x - 9 = 5x + 20 \implies x = 29 \).
TSI Shortcut: Plug in \( x=29 \): LHS \( \frac{2(29)-3}{5} = \frac{55}{5} = 11 \). RHS \( \frac{29+4}{3} = \frac{33}{3} = 11 \). True statement!
Question 4 (Infinite Solutions) C (\( k = 8 \))
Algebra: Infinitely many solutions requires proportional coefficients: \( \frac{2}{4} = \frac{4}{k} = \frac{10}{20} \implies \frac{1}{2} = \frac{4}{k} \implies k = 8 \).
TSI Insight: Observe bottom equation is top equation multiplied by 2: \( 2(4y) = 8y \), so \( k = 8 \). Done in 5 seconds.
3
Exit Ticket Mastery Solutions
Exit Question 1 A (\( x \ge -3 \))
\( 4 - 3x + 6 \le 19 \)
\( 10 - 3x \le 19 \implies -3x \le 9 \)
Divide by -3 & flip sign: \( x \ge -3 \)
Common distractor picked: B (\( x \le -3 \)) if sign un-flipped.
Exit Question 2 C (35)
1. \( \frac{x}{4} + 7 = 12 \implies \frac{x}{4} = 5 \implies x = 20 \)
2. TSI Trap check: Question asks for \( 2x - 5 \), NOT \( x \)!
