LET Secondary Mathematics — Measurement and Problem SolvingRevision Notes

Formulas

Exam Tips

• Draw or mentally visualize the prefix ladder. Count how many steps you move, then multiply or divide by 10 for each step.
• For area: remember 'square the conversion factor.' For volume: 'cube the conversion factor.'
• Watch for 'metric ton' — it is 1,000 kg, not 1,000 g.
• LET items often express a measurement in one unit and ask for an equivalent in a non-adjacent unit — practice multi-step conversions.

Key Points

• The metric system is base-10: moving from a larger unit to a smaller unit MULTIPLIES by a power of 10; moving from a smaller unit to a larger unit DIVIDES by a power of 10.
• The prefix ladder (largest to smallest): kilo (k) = 1,000 → hecto (h) = 100 → deka (da) = 10 → BASE = 1 → deci (d) = 0.1 → centi (c) = 0.01 → milli (m) = 0.001.
• Base units: meter (m) for length, gram (g) for mass, liter (L) for capacity.
• Key length conversions: 1 km = 1,000 m; 1 m = 100 cm; 1 cm = 10 mm.
• Key mass conversions: 1 kg = 1,000 g; 1 g = 1,000 mg; 1 metric ton = 1,000 kg.
• Key capacity conversions: 1 L = 1,000 mL; 1 kL = 1,000 L; 1 mL = 1 cm³.
• Shifting the decimal point left = dividing (going up the ladder); shifting right = multiplying (going down the ladder).
• Derived units: area uses squared units (m², cm²); volume uses cubic units (m³, cm³).

Definitions

Section Title

The Metric (SI) System

Common Mistakes

• Forgetting that area conversions square the linear factor: 1 m² = 10,000 cm² (not 100 cm²), because 1 m = 100 cm and 100² = 10,000.
• Volume conversions cube the linear factor: 1 m³ = 1,000,000 cm³ (not 1,000 cm³).
• Confusing mL and L: writing 1.5 L as 15 mL instead of 1,500 mL.
• Mixing up 'kilo' (×1,000) and 'milli' (÷1,000) — they are inverses of each other.

Formulas

Exam Tips

• Memorize the key English facts as a short list: 12 in = 1 ft; 3 ft = 1 yd; 5,280 ft = 1 mi; 16 oz = 1 lb; 2,000 lb = 1 ton; 4 qt = 1 gal.
• Always set up the unit fraction so the unit you want to eliminate is in the denominator.
• For multi-step English conversions (e.g., miles to inches), chain the fractions: × (5,280 ft/1 mi) × (12 in/1 ft).

Key Points

• The English system is NOT base-10; every conversion factor must be memorized separately.
• Length: 1 foot (ft) = 12 inches (in); 1 yard (yd) = 3 ft = 36 in; 1 mile = 5,280 ft = 1,760 yd.
• Weight: 1 pound (lb) = 16 ounces (oz); 1 ton = 2,000 lb.
• Capacity: 1 gallon (gal) = 4 quarts (qt); 1 quart = 2 pints (pt); 1 pint = 2 cups.
• Use dimensional analysis (unit-fraction method) to convert: multiply by a fraction equal to 1 whose numerator and denominator are equivalent quantities in different units, arranged so the unwanted unit cancels.

Definitions

Section Title

The English (Customary) System

Common Mistakes

• Inverting the unit fraction: writing (1 ft / 12 in) when you need (12 in / 1 ft) to cancel feet.
• Confusing weight 'ton' (2,000 lb, English) with 'metric ton' (1,000 kg, SI).
• Thinking that 1 yd = 36 in instead of calculating: 1 yd = 3 ft = 3 × 12 = 36 in.
• Mixing fluid capacity 'ounces' (fl oz) with weight 'ounces' (oz).

Formulas

Exam Tips

• Write the bridge fact as a fraction with the unit you want in the numerator BEFORE computing.
• For LET multiple-choice, check if the answer choices are approximate — this confirms that you should use the approximate bridge factor.
• Common LET cross-system item: a person's height in feet/inches converted to meters and centimeters.

Key Points

• Bridge facts to memorize: 1 inch = 2.54 cm (exact); 1 kg ≈ 2.2 lb; 1 mile ≈ 1.6 km; 1 liter ≈ 1.06 quarts.
• Use dimensional analysis with the bridge fact as the unit fraction.
• For length: metric-to-English uses 1 in = 2.54 cm; English-to-metric uses the same fact inverted.
• For mass: 1 kg ≈ 2.2 lb is the standard approximation in LET problems.
• Temperature conversion between Celsius and Fahrenheit uses dedicated formulas (see next section).

Definitions

Section Title

Cross-System Conversions (Metric ↔ English)

Common Mistakes

• Using 2.54 as a multiplier when converting cm to in (should divide): 30 cm ÷ 2.54 ≈ 11.81 in, NOT 30 × 2.54.
• Rounding the bridge factor too early and accumulating error in multi-step problems.
• Forgetting that 1 kg ≈ 2.2 lb is an approximation — LET answers based on this will be approximate.

Formulas

Exam Tips

• Memory trick: °F → °C: subtract 32 first, then multiply by 5/9. °C → °F: multiply by 9/5 first, then add 32.
• Use the three anchor points (0°C/32°F, 37°C/98.6°F, 100°C/212°F) to verify your formula application.
• If the LET answer choices are integers, your computation result should round cleanly — recheck arithmetic if it does not.

Key Points

• The Philippines officially uses Celsius (°C); Fahrenheit (°F) is used in the US and in some medical/science contexts.
• Water freezes at 0°C = 32°F and boils at 100°C = 212°F — use these as checkpoint values.
• Normal human body temperature: 37°C = 98.6°F.
• Room temperature range in the Philippines: approximately 25–32°C (77–89.6°F).
• Two formulas handle all temperature conversion needs on the LET.

Definitions

Section Title

Temperature Conversion

Common Mistakes

• Applying the Celsius-to-Fahrenheit formula backward when the question asks for Fahrenheit from Celsius.
• Forgetting to subtract 32 FIRST in the Fahrenheit-to-Celsius formula.
• Computing (9/5)(25) as (9 × 25)/5 = 225/5 = 45 correctly, but then writing 45 + 32 as 66 instead of 77.

Formulas

Exam Tips

• For multi-session problems (e.g., 3 class periods of 50 minutes with 2 breaks of 10 minutes), compute session total and break total separately, then add.
• Convert everything to minutes for complex elapsed-time problems, then convert the answer back.
• Read carefully: 'How long does it take?' asks for elapsed time; 'What time does it end?' asks for a clock reading.

Key Points

• Time uses mixed bases: 60 s = 1 min; 60 min = 1 hr; 24 hr = 1 day; 7 days = 1 week; 365 days = 1 year (366 in a leap year).
• Always convert all durations to a single unit before adding or subtracting.
• Clock arithmetic wraps: after 12:00 comes 12:01 (12-hour); after 23:59 comes 0:00 (24-hour).
• To add time: add hours and minutes separately; if minutes ≥ 60, carry 1 hour and subtract 60 from minutes.
• To find elapsed time: subtract start time from end time, borrowing 60 minutes if needed.
• 135 minutes = 2 hours 15 minutes (divide minutes by 60; quotient = hours, remainder = minutes).

Definitions

Section Title

Measurement of Time

Common Mistakes

• Adding 2:45 + 2:15 as 4:60 instead of carrying: 4:60 = 5:00.
• Confusing AM and PM: crossing noon or midnight resets the hour.
• Subtracting time without borrowing 60 minutes when the ending minutes are less than the starting minutes.

Formulas

Exam Tips

• Memorize the 'DST triangle': write D at the top; S and T at the bottom. Cover the unknown to see the formula.
• For fuel/consumption problems, use A = R × T: fuel = consumption rate × distance (or time).
• Combined work: if both rates are simple fractions, find a common denominator, add, then invert.

Key Points

• Core relationship: Distance = Speed × Time (D = S × T).
• Rearranges to: Speed = D ÷ T and Time = D ÷ S.
• Units must match: if speed is in km/h, distance must be in km and time in hours.
• To convert km/h to m/s: multiply by 1,000 m per km and divide by 3,600 s per hour → divide by 3.6.
• Rate problems generalize D = S × T to Amount = Rate × Time (A = R × T).
• Work problems: if one worker completes a job in n hours, their rate is 1/n of the job per hour.
• Combined work rate: add individual rates, then take the reciprocal to find combined time.

Definitions

Section Title

Speed, Rate, and the Distance Formula

Common Mistakes

• Adding times instead of rates in work problems (e.g., adding 4 hr + 6 hr = 10 hr instead of adding 1/4 + 1/6).
• Not matching units: using speed in km/h with time in minutes — always convert to the same time unit.
• Applying km/h to m/s conversion incorrectly: multiplying by 3.6 instead of dividing.

Formulas

Exam Tips

• Key word clues: 'fence' or 'border' → perimeter; 'tile,' 'paint,' or 'carpet' → area; 'fill' or 'pour' → volume.
• After finding perimeter or area, check: is the unit linear, square, or cubic? Mismatched units signal an error.
• For tiles: convert all dimensions to the same unit (usually cm) before dividing areas.

Key Points

• Perimeter (P): total length around a figure — used for fencing, framing, bordering. Unit: linear (m, cm, ft).
• Area (A): amount of surface covered — used for tiling, painting, carpeting. Unit: square (m², cm², ft²).
• Volume (V): amount of space inside a 3D figure — used for filling, capacity problems. Unit: cubic (m³, cm³) or liters.
• Rectangle: P = 2(l + w); A = l × w.
• Square: P = 4s; A = s².
• Triangle: P = a + b + c; A = (1/2) × base × height.
• Rectangular prism: V = l × w × h.
• Number of tiles = total floor area ÷ area of one tile (ensure same units).
• Cost of fencing = perimeter × price per unit length; cost of tiles = number of tiles × price per tile.

Definitions

Section Title

Area, Volume, Perimeter — Choosing the Right Measure

Common Mistakes

• Using area formula when the problem asks for perimeter (e.g., computing l × w to fence a lot instead of 2(l + w)).
• Mixing units when dividing to find the number of tiles (e.g., floor area in m² divided by tile area in cm²).
• Forgetting to convert between m and cm before computing tile count — always use the same unit.

Exam Tips

• LET items may describe a teacher's actions and ask which step of Polya's heuristic is being demonstrated — know the steps by name.
• LET items may ask: 'Which strategy is MOST appropriate for this problem?' Know the match: diagram for geometry, equation for quantitative relationships, work backward for known-end problems.
• Always allocate time to Look Back on the actual exam — it catches wrong units and arithmetic errors before you finalize your choice.
• Practice narrating your solution in Polya's four steps — this is also a skill you will demonstrate in the actual teaching field.

Key Points

• George Polya's framework is the standard tested heuristic on the LET for non-routine problems.
• Step 1 — UNDERSTAND: Identify what is given (data), what is asked (goal), and any conditions. Restate in your own words. Note units.
• Step 2 — DEVISE A PLAN: Choose a strategy. Common strategies: draw a diagram, work backward, find a pattern, guess-and-check, write an equation, make a table, solve a simpler problem.
• Step 3 — CARRY OUT THE PLAN: Execute steps carefully, one at a time. Check each computation. If a strategy fails, return to Step 2.
• Step 4 — LOOK BACK: Verify the answer against the original conditions. Check units. Ask: Is the answer reasonable?
• The LET tests recognition of the four steps AND identification of the appropriate strategy for a given problem type.
• Non-routine problems have no single formula; they require combining strategies and careful logical reasoning.

Definitions

Section Title

Polya's Four-Step Problem-Solving Heuristic

Common Mistakes

• Skipping Step 1 (Understand) and jumping to computation — this is the root cause of most wrong answers on non-routine LET items.
• Confusing the PLAN (strategy choice) with the EXECUTION — the plan is deciding WHAT to do; carrying out is actually DOING it.
• Not checking units in the Look Back step — an answer in the wrong unit is a wrong answer.
• Abandoning a strategy after one difficulty instead of returning to Step 2 and selecting a different strategy.

Exam Tips

• For LET strategy-identification items: look for the key noun in the problem. 'Arrangement' or 'position' → diagram. 'Left over' or 'remaining' → work backward. 'How many ways' → organized list.
• The equation strategy is safest when the problem states exact numerical relationships between quantities.
• Example: 'A class of 30 has 6 more girls than boys. How many boys?' → let boys = b; b + (b + 6) = 30; 2b = 24; b = 12. Check: 12 + 18 = 30 ✓ and 18 − 12 = 6 ✓.

Key Points

• Draw a diagram: geometry, distance, arrangement, and any problem where a visual clarifies relationships.
• Work backward: when the end state is known and the initial value is unknown (e.g., money remaining, final score).
• Find a pattern: sequences, tiling, and number puzzles — identify the rule, extend it.
• Guess, check, and revise: constrained problems with a small number of possible answers; systematic guessing.
• Write an equation: quantitative relationships between unknowns (connects to algebra — most reliable strategy for LET).
• Make an organized list or table: counting problems, combinations, schedules — ensures no case is missed.
• Solve a simpler version: tackle a scaled-down version first, identify the method, then apply it to the original.

Definitions

Section Title

Problem-Solving Strategies — Quick Reference

Common Mistakes

• Applying guess-and-check without recording attempts — unrecorded guessing wastes time and misses patterns.
• Using 'draw a diagram' for non-spatial problems where an equation would be faster.
• Selecting a strategy that worked on a previous problem and forcing it on a different problem type.

Exam Tips

• Before computing, mentally estimate the answer's order of magnitude. This takes 5 seconds and eliminates wrong-unit distractors.
• If your computed answer is an 'ugly' number (e.g., 14,732.8) and all choices are round numbers, recheck your unit conversions.
• The LET will include 'trap' distractors that result from applying a linear conversion factor to an area or volume — reasonableness checking catches these.

Key Points

• Every measurement is an approximation — its precision is limited by the smallest unit of the measuring tool.
• Reporting 3.5 m implies precision to the nearest 0.1 m; reporting 3.50 m implies precision to the nearest 0.01 m.
• Reasonableness checking: before finalizing an answer, estimate the expected magnitude and reject options that are wildly off.
• A classroom is a few meters wide (not centimeters, not kilometers). An adult's mass is tens of kilograms (not grams, not metric tons).
• Round only at the FINAL step of a multi-step computation to minimize accumulated rounding error.
• Pairing estimation with Polya's Look Back step catches most unit and magnitude mistakes.

Definitions

Section Title

Reasonableness, Estimation, and Precision

Common Mistakes

• Choosing an answer with the right digits but the wrong magnitude (e.g., selecting 152.4 mm when the answer is 152.4 cm).
• Rounding intermediate values and then reporting the final answer as if it were exact.
• Not applying reasonableness: accepting that a classroom is '500 cm wide' without pausing to check (500 cm = 5 m — actually reasonable; 500 mm = 50 cm — too narrow; 500 m — absurd).