STEM Critical Thinking — Study Notes (ACL Entrance Exam)

STEM Critical Thinking — Study Notes (ACL Entrance Exam)

1. STEM Reasoning Overview

• Science: the study of the natural world (physics, chemistry, biology). It's both a body of accumulated knowledge AND a process (scientific inquiry) that generates new knowledge.

• Technology: the whole system of people, organizations, knowledge, processes, and devices used to create/operate technological artifacts — plus the artifacts themselves.

• Engineering: a body of knowledge about designing/creating human-made products, and a process for solving problems under constraint (time, money, materials, ergonomics, regulations, manufacturability, reparability).

• Mathematics: the study of patterns and relationships among quantities, numbers, and space. Claims are warranted by logical argument, not empirical evidence — and math knowledge is never "overturned," only built upon.

• Spatial-Relational Thinking: recognizing/predicting spatial relationships, geometric progressions, and the best organization of objects in space for engineering/design.

2. Circuits: Series vs. Parallel Switches

• In a series circuit, an open switch stops current no matter where it is in the loop.

3. Ohm's Law & Resistors

Ohm's Law: I = V / R (Current = Voltage ÷ Resistance)

• Current and resistance are inversely proportional — higher R → lower I (fixed V).

Resistors in SERIES: R_total = R1 + R2 + R3 + ... → Adding a series resistor increases total resistance, decreases current.

Resistors in PARALLEL: 1/R_total = 1/R1 + 1/R2 + 1/R3 + ... → Adding a parallel resistor decreases total resistance, increases total current.

Worked Examples

• R1=10Ω, R2=20Ω → Series: 10+20 = 30 Ω. Parallel: 1/R=1/10+1/20=3/20 → ≈6.67 Ω

• R1=500Ω, R2=2kΩ, R3=5kΩ → Series: 500+2000+5000 = 7,500 Ω. Parallel: 1/R=1/500+1/2000+1/5000=27/10000 → ≈370.4 Ω

• 100Ω + 200Ω in series (3V battery) → 300 Ω

• 100Ω + 200Ω in parallel (12V battery) → 1/R=1/100+1/200=3/200 → ≈66.7 Ω

• 100Ω in series with two 200Ω resistors in parallel (12V) → parallel pair = 100Ω, then +100Ω series = 200 Ω total

Resistor Color Codes

• Stripe 1 & 2 = the numeric digits

• Stripe 3 = multiplier (power of 10)

• Stripe 4 = tolerance (e.g., gold = ±5%, silver = ±10%)

Example: brown, black, orange, gold → 1, 0, ×1000 → 10,000 Ω (10 kΩ), ±5%

4. Correlation Coefficient (r)

• Measures how strongly two quantities are linearly related. Range: –1 to +1.

• r > 0 → positive relationship (y increases as x increases)

• r < 0 → negative relationship (y decreases as x increases)

• r near ±1 → points tightly hug the line of best fit (strong correlation)

• r near 0 → points are scattered, little/no trend

• Only compute r for linear data — it's meaningless for nonlinear patterns.

5. Pie Charts

Steps to build a pie chart from a data table:

1. Add up all values → get the total

2. Divide each value by the total, ×100 → get the percent

3. Multiply each percent by 360° → get the sector angle

Formula: angle = (value ÷ total) × 360°

Worked Example — Favorite Movie Survey

Practice To Review Separately

The original packet includes several practice problems tied to bar graphs and pie charts shown as images (die-roll results, transportation methods, favorite pets, Tom's daily activities, household expenses). Pull up the original PDF for those visuals — the formulas above (angle = value/total × 360°, series/parallel resistance) are exactly what's needed to solve them.

STEM Critical Thinking 2 — Geometrical Shapes: Nets

1. What Is a Net?

A net is a 2-dimensional shape that can be folded to form a 3-dimensional solid — or, put another way, the pattern you get when you lay the surface of a solid out flat, showing every face.

• A single solid can have more than one possible net.

• Nets are especially useful for finding the surface area of a solid (unfold it, find the area of each flat face, add them up).

How to check whether a net folds into a given solid

1. Count and match faces — the net must have the same number of faces as the solid, and each face's shape must match the corresponding face of the solid.

2. Visualize the fold — mentally fold the net and check that all the edges/sides line up correctly with no gaps or overlaps.

2. Nets of Common Solids

3. The Cube

• A cube = 3-D solid with 6 equal square faces.

• There are 11 distinct nets that fold into a cube — the same 6 squares can be arranged in 11 different flat layouts and still fold up correctly.

• Not every arrangement of 6 squares works — some layouts leave faces overlapping or gaps when folded (fails the "visualize the fold" check above).

4. Rectangular Prism (Cuboid)

• A rectangular prism/cuboid net = 6 rectangles: 3 pairs of identical opposite faces (top/bottom, front/back, left/right).

• Unlike the cube, the 3 pairs can each have different dimensions (length, width, height).

5. Surface Area & Volume — Key Formulas

Since nets are the tool for finding surface area, know these:

How to use a net to find surface area: unfold the solid → identify each flat face on the net → find the area of each individual face using the right shape formula (rectangle, triangle, circle, etc.) → add all the face areas together.

6. Worked Example Approach (Volume & Surface Area from a Net)

When given a net with labeled dimensions:

1. Identify what solid the net folds into (count/match the faces).

2. List each face's shape and dimensions from the net.

3. Surface area = sum of the areas of every individual face.

4. Volume = use the matching volume formula above (usually base area × height, or the specific formula for that solid).

Practice To Review Separately

Practice Exercises I–V ask students to sketch the net of a given solid from labeled dimensions — these are hands-on drawing exercises tied to specific diagrams/dimensions in the original packet, not text. Pull up the original PDF for the actual solids/measurements, then apply the "count faces → match shapes → check the fold" method above, plus the formula table, to sketch and verify each net.