LET Elementary Mathematics — Measurement and Problem SolvingStudy Notes

The metric (SI) system is built on a base-10 structure, which means all conversions involve powers of 10 and can be executed by shifting the decimal point. This elegance makes metric conversion more accessible for students than English unit conversions. The system uses three base units: the **meter (m)** for length, the **gram (g)** for mass, and the **liter (L)** for capacity. Around each base unit, a set of prefixes scales the unit up or down: **Key Prefixes and Their Multipliers:** - **kilo (k):** 1,000 times the base unit - **hecto (h):** 100 times the base unit - **deka (da):** 10 times the base unit - **Base unit:** 1 (meter, gram, or liter) - **deci (d):** 0.1 (one-tenth) of the base unit - **centi (c):** 0.01 (one-hundredth) of the base unit - **milli (m):** 0.001 (one-thousandth) of the base unit **The Conversion Principle:** When moving from a **larger unit to a smaller unit** (e.g., meters to centimeters), you **multiply** because you need more of the smaller units. When moving from a **smaller unit to a larger unit** (e.g., millimeters to meters), you **divide** because you need fewer of the larger units. Alternatively, think of shifting the decimal point: moving down the prefix ladder shifts the decimal right (multiply), and moving up shifts it left (divide). **Essential Metric Facts for Teachers:** Length conversions: - 1 kilometer (km) = 1,000 meters (m) - 1 meter (m) = 100 centimeters (cm) - 1 centimeter (cm) = 10 millimeters (mm) - Therefore: 1 m = 1,000 mm and 1 km = 100,000 cm Mass conversions: - 1 kilogram (kg) = 1,000 grams (g) - 1 gram (g) = 1,000 milligrams (mg) - 1 metric ton (t) = 1,000 kilograms (kg) Capacity conversions: - 1 liter (L) = 1,000 milliliters (mL) - 1 kiloliter (kL) = 1,000 liters (L) - 1 milliliter (mL) = 1 cubic centimeter (cm³) **Why This Matters in the Classroom:** When teaching metric conversions to Grade 3–4 students, emphasize the prefix meaning and the decimal shift rather than memorizing isolated facts. A student who understands that "centi" means one-hundredth can figure out that 1 meter = 100 centimeters without rote memory. This conceptual approach builds mathematical reasoning, a key goal in the DepEd curriculum. **Worked Example 1: Converting 5.2 km to meters** - Understand: We are converting kilometers (larger) to meters (smaller), so we multiply. - Plan: Use the factor 1 km = 1,000 m. - Carry out: 5.2 km × 1,000 = 5,200 m. - Look back: The numerical value increased (5.2 became 5,200), which is correct because meters are smaller units. ✓ **Worked Example 2: Converting 350 centimeters to meters** - Understand: We are converting centimeters (smaller) to meters (larger), so we divide. - Plan: Use the factor 1 m = 100 cm, so 1 cm = 0.01 m. - Carry out: 350 cm ÷ 100 = 3.5 m (or 350 × 0.01 = 3.5 m). - Look back: The numerical value decreased (350 became 3.5), which is correct because meters are larger units. ✓ **Worked Example 3: Converting 2,450 grams to kilograms** - Understand: We are converting grams (smaller) to kilograms (larger), so we divide. - Plan: Use the factor 1 kg = 1,000 g. - Carry out: 2,450 g ÷ 1,000 = 2.45 kg. - Look back: A typical Grade 5 student weighs about 35–40 kg; 2.45 kg (about 5.4 pounds) might be a large textbook or a small sack of rice, which is reasonable. ✓ **Worked Example 4: Converting 0.75 liters to milliliters** - Understand: We are converting liters (larger) to milliliters (smaller), so we multiply. - Plan: Use the factor 1 L = 1,000 mL. - Carry out: 0.75 L × 1,000 = 750 mL. - Look back: A 0.75 L bottle of water is a common drink size; 750 mL is a reasonable glass-filling amount. ✓

The English (customary or imperial) system is **not base-10**, so each conversion factor is arbitrary and must be memorized separately. This system remains in use in some contexts (aviation, some U.S.-sourced materials), and the LET expects familiarity. The three main categories are length, weight (or force), and capacity. **English Length Conversions:** - 1 foot (ft) = 12 inches (in) - 1 yard (yd) = 3 feet (ft) - 1 mile (mi) = 5,280 feet (ft) - Derived: 1 yard = 36 inches; 1 mile = 1,760 yards **English Weight (Mass) Conversions:** - 1 pound (lb) = 16 ounces (oz) - 1 short ton (ton) = 2,000 pounds (lb) **English Capacity Conversions:** - 1 gallon (gal) = 4 quarts (qt) - 1 quart (qt) = 2 pints (pt) - 1 pint (pt) = 2 cups (cup) - Derived: 1 gallon = 8 pints = 16 cups **Why the Metric System Is Easier for Students:** When teaching, point out that converting 5 feet to inches is straightforward (5 × 12 = 60), but the factor 12 is arbitrary and hard to remember, whereas converting 5 meters to centimeters is always a factor of 100 and the logic is transparent. This comparison builds appreciation for the metric system's design and justifies its use in the DepEd K–12 curriculum. **Worked Example 1: Converting 3.5 pounds to ounces** - Understand: We are converting pounds (larger) to ounces (smaller), so we multiply. - Plan: Use the factor 1 lb = 16 oz. - Carry out: 3.5 lb × 16 = 56 oz. - Look back: A small bag of rice weighs a few kilograms (about 5–10 pounds), so 56 ounces (about 1.75 pounds) might be a small snack pack. Reasonable. ✓ **Worked Example 2: Converting 72 inches to feet** - Understand: We are converting inches (smaller) to feet (larger), so we divide. - Plan: Use the factor 1 ft = 12 in. - Carry out: 72 in ÷ 12 = 6 ft. - Look back: 6 feet is a tall adult height (about 1.8 m), which is reasonable. ✓ **Worked Example 3: Converting 2 gallons to cups** - Understand: We are converting gallons (larger) to cups (smaller), so we multiply. - Plan: Use the factor 1 gal = 16 cups (or 1 gal = 4 qt and 1 qt = 2 pt and 1 pt = 2 cups, giving 4 × 2 × 2 = 16). - Carry out: 2 gal × 16 = 32 cups. - Look back: A typical drinking cup is 8 ounces (0.5 pint); 32 cups of liquid is a large volume for a household, suitable for filling a bathtub or large container. Reasonable. ✓

The Philippines officially uses the metric system, but global trade, imported materials, and occasional references in English texts mean students and teachers encounter English units. Converting between systems requires bridge factors—experimentally determined equivalences that link the two systems. These bridge factors are **approximations**, not exact, so they are given to you; you do not derive them. **Key Bridge Factors (Memorize These):** - **Length:** 1 inch (in) = 2.54 centimeters (cm) [exact by definition] - **Mass:** 1 kilogram (kg) ≈ 2.2 pounds (lb) [approximate] - **Distance:** 1 mile (mi) ≈ 1.6 kilometers (km) [approximate] - **Capacity:** 1 liter (L) ≈ 1.06 quarts (qt) [approximate]; for practical purposes, 1 L ≈ 1 qt - **Temperature:** Special formulas (covered separately) **Why These Matter:** When a student's parent gives a height in feet and inches, or a teacher imports a lesson plan with English measurements, these bridge factors allow quick conversions without error. On the LET, a single cross-system conversion might appear, and the examiners expect you to know these four facts. **Dimensional Analysis for Cross-System Conversions:** The technique remains the same: multiply by a unit-fraction (a ratio equal to 1) that contains the bridge factor, so the original unit cancels and the target unit remains. **Worked Example 1: Convert 6 feet to centimeters** - Understand: We need to go from feet (English) to centimeters (metric). We cannot do this directly; we must go through inches. - Plan: Feet → inches → centimeters. - Carry out: - 6 ft × 12 in/ft = 72 in. - 72 in × 2.54 cm/in = 182.88 cm. - Alternatively, combine: 6 ft × (12 in/ft) × (2.54 cm/in) = 6 × 12 × 2.54 = 182.88 cm. - Look back: A person 6 feet tall is approximately 1.83 meters, or 183 cm. Our answer 182.88 cm rounds to 183 cm. ✓ **Worked Example 2: Convert 150 pounds to kilograms** - Understand: We are converting pounds (English) to kilograms (metric). - Plan: Use the bridge factor 1 kg ≈ 2.2 lb, rearranged to 1 lb ≈ 1/2.2 kg ≈ 0.45 kg. - Carry out: 150 lb × (1 kg / 2.2 lb) = 150 / 2.2 ≈ 68.18 kg. - Look back: An average adult weighs about 70 kg; 150 pounds is a reasonable adult weight (about 5'8", medium build). ✓ **Worked Example 3: Convert 50 miles per hour (mph) to kilometers per hour (km/h)** - Understand: We are converting a speed in English units to metric. - Plan: Use the bridge factor 1 mi ≈ 1.6 km. - Carry out: 50 mi/h × 1.6 km/mi = 80 km/h. - Look back: 50 mph is a typical highway speed in the United States; 80 km/h is the standard highway limit in the Philippines, so the conversion is reasonable. ✓ **Worked Example 4: Convert 2 quarts to liters** - Understand: We are converting quarts (English) to liters (metric). - Plan: Use the bridge factor 1 L ≈ 1.06 qt, rearranged to 1 qt ≈ 1/1.06 L ≈ 0.94 L. - Carry out: 2 qt × (1 L / 1.06 qt) = 2 / 1.06 ≈ 1.89 L. - Alternatively, using the practical approximation 1 L ≈ 1 qt: 2 qt ≈ 2 L (less precise but good for estimation). - Look back: A 2-quart liquid container is common in kitchens and roughly fills two 1-liter bottles. ✓ **Teaching Tip:** When students encounter cross-system conversions, remind them that bridge factors are provided on the LET (or, in real-world contexts, found in references). The core skill is recognizing which bridge factor to use and setting up the dimensional analysis correctly. Encourage estimation before computing to catch magnitude errors early.

Time is measured in units that do not follow a single base: seconds, minutes, and hours use base-60, while days and weeks use base-7, and months and years vary. This mixed-base system requires careful attention to avoid errors, especially when adding, subtracting, or converting durations. **Essential Time Conversion Facts:** - 1 minute = 60 seconds - 1 hour = 60 minutes - 1 day = 24 hours - 1 week = 7 days - 1 month = 28, 29, 30, or 31 days (depending on the month and leap years) - 1 year = 365 days (or 366 in a leap year) **Clock Time vs. Duration:** - **Clock time** (also called absolute time) refers to a specific moment in a 12-hour or 24-hour cycle: 2:45 PM, 14:30 (24-hour format). - **Duration** (also called elapsed time) is a span of time: "the meeting lasted 90 minutes" or "the film runs for 2 hours 15 minutes." The LET often tests duration problems, where you must convert a duration to a single unit before adding, subtracting, or comparing. Clock-time problems require you to "wrap" the arithmetic: when you add 45 minutes to 11:30 AM, you get 12:15 PM (the hour wraps at 12, not 13). **Converting Durations to a Single Unit:** To add or subtract durations in mixed units, convert all to the smallest unit mentioned, perform the arithmetic, then convert back to a readable form. **Worked Example 1: Adding durations with mixed units** A lesson runs for 40 minutes, followed by a 10-minute break, then another lesson of 35 minutes. What is the total time? - Convert all to minutes: 40 + 10 + 35 = 85 minutes. - Convert back to hours and minutes: 85 ÷ 60 = 1 remainder 25, so 1 hour 25 minutes. - Verification: 1 × 60 + 25 = 85 minutes. ✓ **Worked Example 2: Adding clock time and duration** A film screening starts at 2:45 PM and runs for 135 minutes. What time does it end? - Convert 135 minutes to hours and minutes: 135 ÷ 60 = 2 remainder 15, so 2 hours 15 minutes. - Add to the start time: - 2:45 PM + 2 hours = 4:45 PM. - 4:45 PM + 15 minutes = 5:00 PM. - The film ends at 5:00 PM. - Verification: From 2:45 PM to 5:00 PM is 2 hours 15 minutes, or 135 minutes. ✓ **Worked Example 3: Subtracting clock times to find duration** School starts at 8:30 AM and ends at 3:15 PM. How many hours and minutes is the school day? - From 8:30 AM to 3:15 PM: - From 8:30 AM to 12:30 PM (noon + 30 min) is 4 hours. - From 12:30 PM to 3:15 PM is 2 hours 45 minutes. - Total: 4 hours + 2 hours 45 minutes = 6 hours 45 minutes. - Verification: 8:30 AM + 6 hours 45 minutes = 3:15 PM. ✓ **Worked Example 4: Duration in different units** A project takes 2 days and 18 hours. How many hours is this? - Convert days to hours: 2 days × 24 hours/day = 48 hours. - Total: 48 + 18 = 66 hours. - Verification: 66 hours ÷ 24 = 2 remainder 18, or 2 days 18 hours. ✓ **Worked Example 5: Elapsed time with multiple segments** A teacher conducts three class periods of 50 minutes each, with two 10-minute breaks in between. If the first class starts at 9:00 AM, when does the final period end? - Total class time: 3 × 50 = 150 minutes. - Total break time: 2 × 10 = 20 minutes. - Total elapsed time: 150 + 20 = 170 minutes = 2 hours 50 minutes. - End time: 9:00 AM + 2 hours 50 minutes = 11:50 AM. **Common Mistake:** Students often add hours and minutes separately without carrying: "1 hour 45 minutes + 2 hours 30 minutes = 3 hours 75 minutes" instead of converting 75 minutes to 1 hour 15 minutes and arriving at 4 hours 15 minutes. Teach students to convert to a single unit (all minutes) when adding or subtracting mixed-unit durations. **Practical Teaching Note:** In the Grade 1–6 classroom, time is taught progressively: Grade 1 learns hour and half-hour, Grade 2 introduces quarter-hour and five-minute intervals, and Grade 3–4 master minute-level accuracy. By Grade 5, students should handle duration problems and 24-hour format. The LET assumes Grade 6 fluency, including mental arithmetic with clock times. Practice with familiar school schedules (dismissal time, duration of assemblies, recess length) to anchor the concept.

The Philippines uses the **Celsius scale** (°C), named after Anders Celsius. Water freezes at 0°C and boils at 100°C at sea level. The scale is divided into 100 equal intervals, which is why it is sometimes called the centigrade scale. However, the LET expects teachers to know the **Fahrenheit scale** (°F) as well, because it appears in international materials and historical texts. The Fahrenheit scale, used primarily in the United States, places the freezing point of water at 32°F and the boiling point at 212°F, giving 180 intervals between them. **Temperature Conversion Formulas:** Celsius to Fahrenheit: **F = (9/5)C + 32** or **F = 1.8C + 32** Fahrenheit to Celsius: **C = (5/9)(F − 32)** or **C = (F − 32) / 1.8** These formulas account for two differences between the scales: 1. The zero points are offset: Celsius's 0°C corresponds to Fahrenheit's 32°F. 2. The interval sizes differ: a 1°C change equals a 9/5 = 1.8°F change. **Key Reference Points:** - Water freezes: 0°C = 32°F - Room temperature: 20–25°C = 68–77°F (approximately) - Normal human body temperature: 37°C = 98.6°F - Water boils: 100°C = 212°F - Comfortable outdoor temperature (tropical): 28–32°C = 82–90°F **Why This Matters for Filipino Teachers:** In the Philippines, temperature is reported in Celsius, and students learn Celsius in Grade 1–6. However, when teaching weather, health (fever), or cooking, awareness of both scales adds depth. On the LET, a temperature-conversion item might appear, and you are expected to apply the formula accurately and estimate reasonableness. **Worked Example 1: Convert 25°C to Fahrenheit** - Use the formula F = (9/5)C + 32. - F = (9/5)(25) + 32. - (9/5) × 25 = 9 × 25 / 5 = 225 / 5 = 45. - F = 45 + 32 = 77°F. - Look back: 25°C is a pleasant tropical room temperature, and 77°F is indeed comfortable. ✓ **Worked Example 2: Convert 98.6°F to Celsius** - Use the formula C = (5/9)(F − 32). - First subtract: 98.6 − 32 = 66.6. - Then multiply: (5/9) × 66.6 = 5 × 66.6 / 9 = 333 / 9 = 37. - C = 37°C. - Look back: 37°C is normal human body temperature in Celsius. ✓ **Worked Example 3: Convert −10°C to Fahrenheit** - Use the formula F = (9/5)C + 32. - F = (9/5)(−10) + 32 = −18 + 32 = 14°F. - Look back: −10°C is a cold winter day, and 14°F is indeed below freezing. ✓ **Worked Example 4: Convert 32°F to Celsius** - Use the formula C = (5/9)(F − 32). - C = (5/9)(32 − 32) = (5/9)(0) = 0°C. - Look back: 32°F is the freezing point of water, and 0°C is the freezing point in Celsius. ✓ **Common Mistake:** Students often forget to subtract 32 before multiplying by 5/9 when converting Fahrenheit to Celsius, or forget to add 32 after multiplying by 9/5 when converting Celsius to Fahrenheit. Emphasize the order of operations: **"First adjust for the offset (32), then scale for the interval difference."** **Classroom Connection:** When teaching weather or health topics in Grade 4–5, encourage students to learn the reference points (freezing at 0°C/32°F, boiling at 100°C/212°F, body temp at 37°C/98.6°F) rather than the formulas. For routine LET questions, knowing these anchors often allows quick estimation without full calculation.

Derived units are formed by combining base units (meter, gram, liter) through multiplication or division. The key insight is that **the conversion factor for a derived unit is not the same as for the base unit**. This is the single most common error in measurement: applying a linear conversion factor (like 100 cm/m) directly to an area or volume, without squaring or cubing it. **Area (Length Squared):** Area is always a length times a length, so the units are squared. If 1 meter = 100 centimeters, then 1 square meter (m²) = 100 × 100 = 10,000 square centimeters (cm²). The conversion factor is **100² = 10,000**, not 100. **Key Area Conversions:** - 1 m² = 10,000 cm² - 1 km² = 1,000,000 m² (because 1 km = 1,000 m, so 1,000² = 1,000,000) - 1 cm² = 100 mm² **Worked Example 1: Convert 5 m² to cm²** - Understand: We are converting square meters to square centimeters. - Plan: Use the factor 1 m² = 10,000 cm² (not 100). - Carry out: 5 m² × 10,000 = 50,000 cm². - Look back: A 5 m² room is small but plausible; 50,000 cm² is many small squares, which is consistent. ✓ **Worked Example 2: Convert 8 km² to m²** - Understand: We are converting square kilometers to square meters. - Plan: Use the factor 1 km² = 1,000,000 m² (because (1,000)² = 1,000,000). - Carry out: 8 km² × 1,000,000 = 8,000,000 m². - Look back: 8 km² is a large area, roughly 8 square kilometers; 8 million square meters is consistent. ✓ **Volume (Length Cubed):** Volume is a length times a length times a length, so the units are cubed. If 1 meter = 100 centimeters, then 1 cubic meter (m³) = 100 × 100 × 100 = 1,000,000 cubic centimeters (cm³). The conversion factor is **100³ = 1,000,000**, not 100. **Key Volume Conversions:** - 1 m³ = 1,000,000 cm³ (also written as 1 m³ = 10⁶ cm³) - 1 cm³ = 1,000 mm³ - 1 m³ = 1,000 liters (L) [This is exact: 1 L = 1,000 cm³, so 1 m³ = 1,000,000 cm³ = 1,000 L] - 1 kilometer = 1,000 meters, so 1 km³ = 1,000,000,000 m³ (one billion cubic meters) **Worked Example 3: Convert 2 m³ to cm³** - Understand: We are converting cubic meters to cubic centimeters. - Plan: Use the factor 1 m³ = 1,000,000 cm³ (not 100). - Carry out: 2 m³ × 1,000,000 = 2,000,000 cm³. - Look back: A 2 m³ container (like a small swimming pool section) is substantial; 2 million tiny cubic centimeters is consistent. ✓ **Worked Example 4: Convert 5,000,000 cm³ to m³** - Understand: We are converting cubic centimeters to cubic meters. - Plan: Divide by 1,000,000 (the same as multiplying by 1 m³ / 1,000,000 cm³). - Carry out: 5,000,000 cm³ ÷ 1,000,000 = 5 m³. - Look back: Reasonable. ✓ **Worked Example 5: Convert 3 m³ to liters** - Understand: We are converting cubic meters to liters. - Plan: Use the equivalence 1 m³ = 1,000 L. - Carry out: 3 m³ × 1,000 = 3,000 L. - Look back: A 3,000-liter tank is large (perhaps for a community water supply), which matches a 3 m³ volume. ✓ **Speed (Length Divided by Time):** Speed relates distance to time. Common units are kilometers per hour (km/h) and meters per second (m/s). To convert between them, you must convert both the distance and the time units. **Key Speed Conversions:** - 1 km/h = (1,000 m) / (3,600 s) = 1,000/3,600 m/s = 0.2778 m/s (approximately 0.28 m/s or 5/18 m/s) - 1 m/s = 3.6 km/h (the reciprocal of the above) - **Shortcut:** Divide km/h by 3.6 to get m/s, or multiply m/s by 3.6 to get km/h. **Worked Example 6: Convert 72 km/h to m/s** - Understand: We are converting a speed from km/h to m/s. - Plan: Use the shortcut 1 km/h = 1/3.6 m/s, or directly set up the conversion. - Carry out (shortcut): 72 ÷ 3.6 = 20 m/s. - Carry out (direct): 72 km/h × (1,000 m/km) × (1 h / 3,600 s) = 72 × 1,000 / 3,600 = 72,000 / 3,600 = 20 m/s. - Look back: 72 km/h is a highway speed; 20 m/s is about 20 meters per second, which translates to covering a 20-meter distance (like the length of a small classroom) in one second—a fast speed consistent with 72 km/h. ✓ **Worked Example 7: Convert 15 m/s to km/h** - Understand: We are converting m/s to km/h. - Plan: Use the shortcut 1 m/s = 3.6 km/h. - Carry out: 15 × 3.6 = 54 km/h. - Look back: 15 m/s is a fairly quick speed; 54 km/h is a moderate city/town speed. Reasonable. ✓ **Common Mistake for Derived Units:** When converting area, students often forget to square the linear conversion factor. For example, they might convert 5 m² to cm² by multiplying by 100 (the linear factor) instead of 10,000 (the squared factor), arriving at 500 cm² instead of 50,000 cm². **Always square the factor for area, cube it for volume.** **Practical Classroom Connection:** In Grade 4–5, students learn area (square units) and volume (cubic units) through hands-on activities: tiling a floor, filling containers, and building blocks. By Grade 5–6, they apply this to conversion problems. The LET assumes Grade 6 proficiency: students can convert between area units and understand why the conversion factor changes.

George Polya's four-step heuristic is the framework every mathematics educator must teach and model. It transforms a confusing, multi-step problem into a systematic process where each step has a clear purpose. The LET tests not only whether you can solve a problem, but whether you recognize and apply Polya's structure. In the DepEd K–12 curriculum, Polya's heuristic underpins the "mathematics as problem-solving" philosophy, and it appears explicitly in the curriculum guides for Grades 5 and above. **The Four Steps:** **Step 1: Understand the Problem** Before attempting any calculation, read carefully and identify: - **What is given?** (the known information and conditions) - **What is asked?** (the goal or what you must find) - **What constraints or conditions apply?** (time limits, maximum/minimum, relationships between quantities) - **What are the units?** (ensure you know whether the answer should be in meters, hours, dollars, etc.) Restate the problem in your own words or, for complex problems, draw a diagram or make a list of known and unknown quantities. This step prevents misinterpretation and careless errors. **Worked Example of Step 1:** *Problem: "A school bus travels 180 kilometers in 3 hours. At the same average speed, how far will it travel in 5 hours?"* - Given: distance = 180 km, time = 3 hours; same speed applies. - Asked: distance traveled in 5 hours. - Constraint: the speed is constant. - Units: distance in kilometers, time in hours, answer in kilometers. - Restatement: "Find the constant speed, then use it to find distance for the new time." **Step 2: Devise a Plan (Choose a Strategy)** Select an appropriate problem-solving strategy based on the problem type. Common strategies include: - **Draw a diagram or picture:** useful for geometry, motion, arrangement, and spatial problems. - **Work backward:** useful when the end state is known and you must find the beginning (e.g., age problems, money-left problems). - **Look for a pattern:** useful for sequences, repeating structures, and number puzzles. - **Guess, check, and revise:** useful for constrained problems where the solution space is small. - **Write an equation:** useful for translating word problems into algebraic relationships. - **Make an organized list or table:** useful for counting problems, combinations, and systematic exploration. - **Solve a simpler version:** useful for large, complex problems; solve a smaller version first to understand the pattern. - **Break into smaller subproblems:** useful for multi-step problems; solve each part, then combine. For the bus example, the strategy is "write an equation" or "find a rate and use it." **Step 3: Carry Out the Plan** Execute the steps carefully, showing all work. Check each step before moving to the next. If the plan stalls or leads to nonsense, **pause and reconsider the strategy**—do not just power through to an answer. **For the bus example:** - Find the constant speed: speed = distance ÷ time = 180 km ÷ 3 hours = 60 km/h. - Use the speed to find distance for 5 hours: distance = speed × time = 60 km/h × 5 hours = 300 km. **Step 4: Look Back (Verify and Reflect)** Before selecting your final answer: - **Check the units:** The answer should have the expected units (km, not hours; dollars, not pounds). - **Verify by working backward:** If you found distance to be 300 km, does 300 km at 60 km/h take 5 hours? Yes: 300 ÷ 60 = 5. ✓ - **Test reasonableness:** In 3 hours the bus traveled 180 km; in 5 hours (more than 3), it should travel more than 180 km. 300 km is indeed more, so the answer is reasonable. ✓ - **Reflect:** Were there alternative strategies? (Yes, you could have set up a proportion: 180/3 = x/5, giving x = 300. The answer is consistent.) Did the solution method teach you something about rate problems? (Yes: the rate is the key, once found, the rest follows.) This reflection step is where deep learning happens and where the Board expects teachers to model mathematical thinking. **Why Polya Matters for the LET:** The LET includes "non-routine" problem-solving items—scenarios that do not fit a standard formula and require reasoning. The Board's rubric for these items explicitly values the **choice of strategy** and the **documentation of reasoning**, not just the final answer. If you show Polya's four steps clearly, even if you make an arithmetic error, you demonstrate competence in problem-solving methodology. Conversely, an answer with no shown work—even if correct—earns few points because the examiner cannot assess your reasoning. **Common Mistake:** Students (and some teachers) jump to arithmetic without understanding the problem or choosing a strategy. They guess at operations, hope for the best, and often arrive at plausible-looking but incorrect answers. The LET rewards the systematic approach; clarity of reasoning is often worth more than speed. **Teaching Reflection (RA 7836 and Pedagogical Competence):** Under RA 7836 (Code of Ethics for Professional Teachers), a teacher must "demonstrate competence in... the subject matter" and employ "appropriate teaching methodologies." When you teach problem-solving, model Polya's heuristic explicitly: read aloud, restate in your own words, draw diagrams, articulate the chosen strategy, execute step-by-step, verify. This transparent modeling is how students internalize the method and develop independence.

Beyond Polya's four-step framework, specific strategies address common problem types. Mastery of these strategies—and knowing when to apply each—is the hallmark of a strong problem-solver and the focus of LET items that ask, "Which strategy would be most efficient?" **Strategy 1: Draw a Diagram or Picture** Useful for: geometry, spatial relationships, motion problems, and any situation where visualization clarifies the logic. *Example: A farmer has a rectangular lot 30 meters by 20 meters. She wants to plant a border of flowers around the perimeter. If each flower needs a 1-meter space, how many flowers can she plant?* - **Diagram approach:** Sketch the rectangle. Mark the perimeter. Visualize the 1-meter spaces around the border. - **Calculation:** Perimeter = 2(30 + 20) = 100 meters. Number of flowers = 100 ÷ 1 = 100 flowers. - **Why diagrams work:** They prevent misinterpretation (e.g., confusing perimeter with area) and make the spatial arrangement clear. **Strategy 2: Work Backward** Useful for: age problems, money-remaining problems, and any scenario where the end state is known. *Example: After spending 150 pesos at the store and giving 50 pesos to a friend, Jose has 200 pesos left. How much money did he start with?* - **Backward approach:** Start with 200 pesos (final amount). Add back the 50 pesos given to the friend: 200 + 50 = 250. Add back the 150 spent: 250 + 150 = 400 pesos. - **Verification (forward):** Start with 400. Spend 150: 400 − 150 = 250. Give away 50: 250 − 50 = 200. ✓ - **Why backward works:** It reverses the operations naturally, avoiding the confusion of setting up an equation. **Strategy 3: Look for a Pattern** Useful for: sequences, repeating arrangements, number puzzles, and optimization. *Example: A stack of tiles has 1 tile in the first row, 3 in the second, 5 in the third, and so on. How many tiles are in the 10th row?* - **Pattern:** The rows follow the sequence 1, 3, 5, 7, ... (odd numbers). - **Pattern rule:** The nth row has 2n − 1 tiles. - **For the 10th row:** 2(10) − 1 = 19 tiles. - **Why patterns work:** Once the rule is identified, the answer follows without exhaustive counting. **Strategy 4: Guess, Check, and Revise** Useful for: constrained problems, problems with small solution spaces, trial-and-error scenarios. *Example: A number is between 1 and 20. When divided by 5, it leaves a remainder of 2. What is the number?* - **Guess:** Try 7. 7 ÷ 5 = 1 remainder 2. ✓ - **Check:** Is 7 between 1 and 20? Yes. Does 7 ÷ 5 leave remainder 2? Yes. - **Other solutions:** 12, 17 also work (12 ÷ 5 = 2 remainder 2; 17 ÷ 5 = 3 remainder 2). - **Why guessing works:** For problems with limited choices, systematic guessing is efficient and intuitive. **Strategy 5: Write an Equation** Useful for: algebraic problems, rate problems, and quantitative relationships. *Example: A jeepney charges 10 pesos for the first kilometer and 2 pesos for each additional kilometer. If a trip costs 34 pesos, how far did the jeepney travel?* - **Equation setup:** Let d = distance in kilometers. Cost = 10 + 2(d − 1) = 34. - **Solve:** 10 + 2d − 2 = 34. 8 + 2d = 34. 2d = 26. d = 13 kilometers. - **Verification:** Cost = 10 + 2(13 − 1) = 10 + 24 = 34. ✓ - **Why equations work:** They translate the verbal relationships into mathematical form, which can be manipulated systematically. **Strategy 6: Make an Organized List or Table** Useful for: counting problems, combinations, systematic exploration. *Example: A small sari-sari store has 3 types of bread (white, brown, whole wheat) and 2 types of spread (butter, jam). How many different sandwiches can be made using one bread and one spread?* - **Organized list:** - White + Butter, White + Jam - Brown + Butter, Brown + Jam - Whole Wheat + Butter, Whole Wheat + Jam - **Count:** 6 sandwiches. - **Why lists work:** They prevent double-counting and ensure all cases are considered. **Strategy 7: Solve a Simpler Version** Useful for: complex problems, recursive structures, and problems where the pattern is not obvious at full scale. *Example: How many times do the hour and minute hands of a clock overlap in a 24-hour period?* - **Simpler version:** How many times do they overlap in 12 hours? (Observe or calculate: 11 times, because the minute hand "laps" the hour hand once per hour, but at the 11:00 to 12:00 boundary they separate without overlapping.) - **Scale up:** In 24 hours (two 12-hour periods), they overlap 2 × 11 = 22 times. - **Why simplifying works:** Understanding the 12-hour case clarifies the pattern for 24 hours. **Strategy 8: Break into Subproblems** Useful for: multi-step problems where each step builds on the previous. *Example: A book costs 300 pesos. A pencil costs one-fifth the price of a book. If a student buys 2 books and 3 pencils, and pays with a 1,000-peso note, how much change does he receive?* - **Subproblem 1:** Find the cost of a pencil. 300 ÷ 5 = 60 pesos. - **Subproblem 2:** Find the cost of 2 books. 2 × 300 = 600 pesos. - **Subproblem 3:** Find the cost of 3 pencils. 3 × 60 = 180 pesos. - **Subproblem 4:** Find the total cost. 600 + 180 = 780 pesos. - **Subproblem 5:** Find the change. 1,000 − 780 = 220 pesos. - **Why subproblems work:** Breaking a complex problem into smaller steps reduces overwhelm and allows verification at each stage. **Rate, Speed, and Work Problems (Application of Strategy 5: Write an Equation)** These are among the most common LET problem types. The core relationship is **amount = rate × time**, which can be rearranged: - **rate = amount ÷ time** - **time = amount ÷ rate** *Distance-Speed-Time Example:* A car travels 240 kilometers in 4 hours. What is its average speed? - Equation: speed = distance ÷ time = 240 ÷ 4 = 60 km/h. *Work Rate Example:* One worker can paint a room in 6 hours. Another worker can paint the same room in 4 hours. If they work together, how long will it take to paint the room? - Worker A's rate: 1/6 of the room per hour. - Worker B's rate: 1/4 of the room per hour. - Combined rate: 1/6 + 1/4 = 2/12 + 3/12 = 5/12 of the room per hour. - Time to paint the room: 1 ÷ (5/12) = 12/5 = 2.4 hours (2 hours 24 minutes). - **Key insight:** Add the rates, not the times. If you said "6 + 4 = 10 hours," you'd be wrong because the workers are working simultaneously, not sequentially. *Tank-Filling Example:* A pipe fills a tank in 8 hours. A drain empties the same tank in 10 hours. If both the pipe and the drain are open, how long will it take to fill the tank? - Filling rate: 1/8 of the tank per hour. - Draining rate: 1/10 of the tank per hour (negative). - Net filling rate: 1/8 − 1/10 = 5/40 − 4/40 = 1/40 of the tank per hour. - Time to fill: 1 ÷ (1/40) = 40 hours. - **Verification:** In 40 hours, the pipe would fill 40 × (1/8) = 5 full tanks; the drain would empty 40 × (1/10) = 4 tanks. Net result: 5 − 4 = 1 tank. ✓ **Practical Classroom Context (DepEd K–12 BEC):** In Grade 5–6, problem-solving is embedded in all content strands: number and algebra (equations), measurement (rate, capacity), geometry (area, perimeter), and data handling (tables, patterns). The LET expects you to teach these strategies explicitly and model them for students. When a student asks, "How do I solve this?" the answer is not "Just do these steps," but "What strategy could help? Shall we draw a picture? Work backward? Make a table?" This inquiry approach, grounded in Polya's heuristic, develops independent thinkers.

Every measurement is approximate, limited by the precision of the measuring tool and the care of the measurer. Precision refers to **how finely the measurement is specified**; accuracy refers to **how close the measurement is to the true value**. On the LET, precision and accuracy are tested indirectly through reasonableness checking and through understanding that not all digits in a result are equally trustworthy. **Precision and Significant Figures:** When a length is reported as 3.5 meters, the notation implies precision to the nearest tenth of a meter (±0.05 m). The digits 3 and 5 are **significant figures**, and in this case there are 2 significant figures. Reporting the same length as 3.50 meters claims precision to the nearest centimeter (±0.005 m), with 3 significant figures. The trailing zero signals precision. For the LET, the key insight is: **report your final answer with precision matching the given data**. If two lengths are given to the nearest centimeter, do not report the sum to the nearest millimeter; it is false precision. *Example:* A student measures a desk as 0.75 m long (2 significant figures in the measurement) and 0.50 m wide (2 significant figures). The area is 0.75 × 0.50 = 0.375 m². Strictly, we should report this as 0.38 m² (2 significant figures, matching the input data). Reporting 0.375 m² or 0.3750 m² overstates the precision. **Rounding: When and How** Round only at the **end** of a computation, not at intermediate steps, to avoid accumulating rounding error. *Wrong approach:* A student calculates 123.456 × 2.34. At the first step, she rounds 123.456 to 123.5 (one fewer decimal place), then multiplies: 123.5 × 2.34 = 289.79. The intermediate rounding introduced error. *Right approach:* Keep all decimal places until the final step. 123.456 × 2.34 = 288.88704. Then round to the appropriate precision (say, 288.89 if two decimal places are needed). **Estimation and Reasonableness Checking** Before selecting an answer, **estimate** what the answer should be, then check whether the computed result is close. *Example:* A student computes the perimeter of a rectangular lot as 140 meters, with sides 35 meters and 40 meters. - **Estimate:** Perimeter ≈ 2(35 + 40) ≈ 2 × 75 ≈ 150 meters. The computed 140 is off, suggesting an arithmetic error. (Correct: 2(35 + 40) = 2 × 75 = 150.) - **Lesson:** Estimation catches errors and builds number sense. *Context checks:* A classroom is a few meters wide, not a few centimeters or a few kilometers. An adult weighs tens of kilograms, not grams or tons. A school day lasts a few hours, not a few minutes or days. These quick reasonableness checks are invaluable on the LET. **Common Measurement Errors and How to Avoid Them** 1. **Forgetting units:** A student computes 5 m × 10 m = 50 and forgets to write m². The answer is wrong because units are lost. - *Prevention:* Carry units through every step. Units cancel, just like numbers. 2. **Wrong conversion factor:** A student converts 3 m² to cm² as 3 × 100 = 300 cm² instead of 3 × 10,000 = 30,000 cm². - *Prevention:* Remember that area conversion factors are squared (100² = 10,000), not linear (100). 3. **Mismatching measurement type to the problem:** A fencing problem asks for perimeter (length units), not area (square units). A tiling problem asks for area, not perimeter. - *Prevention:* Read carefully and identify whether the problem requires length, area, volume, or capacity. Match the measurement to the question. 4. **Rounding at intermediate steps:** A student rounds 12.7 to 13 partway through a calculation, accumulating error. - *Prevention:* Keep full precision throughout; round only the final answer. 5. **Ignoring the context:** A student solves a problem mathematically but the answer is nonsensical (e.g., a negative time, a person weighing 0.5 grams, a distance of 0.001 millimeters). - *Prevention:* Always step back and ask, "Does this make sense in the real world?" **Worked Example: Combining Measurement, Conversion, and Reasonableness** *Problem:* A rectangular plot of land measures 50 feet by 30 feet. What is its area in square meters? (1 foot ≈ 0.3048 meters.) - **Estimate:** 50 feet ≈ 15 meters, 30 feet ≈ 9 meters. Area ≈ 15 × 9 ≈ 135 square meters. - **Carry out:** - Convert sides: 50 ft × 0.3048 m/ft = 15.24 m; 30 ft × 0.3048 m/ft = 9.144 m. - Area: 15.24 × 9.144 = 139.36 m². - **Check units:** The calculation yields square meters, as required. ✓ - **Check reasonableness:** The computed 139.36 m² is close to the estimate of 135 m², so the answer is reasonable. ✓ - **Report:** 139.4 m² (three significant figures, matching the input data precision). **Classroom Application (DepEd K–12 BEC):** In Grade 3–4, students measure objects using non-standard units (hand spans, paces) and standard tools (meter sticks, measuring cups). Naturally, measurements vary slightly depending on technique. By Grade 5–6, students compare measurements, recognize that all measurements are approximate, and learn to record measurements to the nearest marking on the tool. On the LET, the expectation is that you understand and teach this uncertainty, and that you model estimation as a check on computed answers.

Many practical LET problems hinge on recognizing whether a task requires **length** (perimeter, for fencing), **area** (for tiling or painting), or **volume/capacity** (for filling). Confusing these is a frequent error, and the LET tests this distinction explicitly. **Fencing Problems: Perimeter (Length Units)** Fencing goes around the **boundary** of a lot, so you need the **perimeter**. Perimeter is a one-dimensional measure in units like meters or feet. **Formula:** For a rectangle, Perimeter = 2(length + width). For a circle, Perimeter (circumference) = 2πr or πd. *Example:* A rectangular lot measures 20 meters by 15 meters. Fencing costs ₱120 per meter. What is the total cost to fence the lot? - Perimeter = 2(20 + 15) = 2 × 35 = 70 meters. - Cost = 70 × 120 = ₱8,400. - **Note:** If the problem also mentioned planting grass (which covers area), you would compute area = 20 × 15 = 300 m². But for fencing, you use perimeter. **Tiling or Painting Problems: Area (Square Units)** Tiling a floor or painting a wall covers a **surface**, so you need the **area**. Area is a two-dimensional measure in units like square meters or square centimeters. **Formula:** For a rectangle, Area = length × width. For a circle, Area = πr². *Example:* A classroom floor measures 6 meters by 5 meters. Tiles are 30 cm × 30 cm. How many tiles are needed to cover the floor? - Area of floor = 6 × 5 = 30 m². - Area of one tile = 30 cm × 30 cm = 0.3 m × 0.3 m = 0.09 m². - Number of tiles = 30 ÷ 0.09 = 333.33, so 334 tiles (rounding up because you need whole tiles). - **Alternative method:** 30 m² = 300,000 cm²; one tile = 900 cm²; tiles needed = 300,000 ÷ 900 = 333.33 ≈ 334. Same answer. ✓ **Filling Problems: Volume or Capacity (Cubic or Liter Units)** Filling a container (tank, pool, glass) requires knowing how much **three-dimensional space** the container holds, so you need **volume** or **capacity**. These are measured in cubic meters (m³), liters (L), milliliters (mL), gallons, or similar units. **Formula:** For a rectangular tank, Volume = length × width × height. For a cylinder, Volume = πr²h. *Example:* A water tank is 2 meters long, 1.5 meters wide, and 1 meter deep. How many liters of water will it hold when full? - Volume = 2 × 1.5 × 1 = 3 m³. - Convert to liters: 3 m³ × 1,000 L/m³ = 3,000 liters. *Example:* A glass holds 250 milliliters. How many glasses can be filled from a tank holding 3,000 liters? - 3,000 L = 3,000 × 1,000 = 3,000,000 mL. - Number of glasses = 3,000,000 ÷ 250 = 12,000 glasses. **Distinguishing Between These Measures: A Summary Table** | Task | Measure | Formula | Units | |---|---|---|---| | Fencing a lot | Perimeter | 2(l + w) | meters, feet | | Tiling a floor | Area | l × w | m², cm², ft² | | Painting a wall | Area | l × w | m², cm², ft² | | Filling a tank | Volume/Capacity | l × w × h | m³, L, mL, gallons | | Walking around a track | Perimeter | 2πr (for a circle) | meters, miles | | Carpeting a room | Area | l × w | m², ft² | | Pouring concrete for a driveway | Volume | l × w × h | m³, cubic feet | **Worked Example 1: Multi-step fencing and seeding problem** A farmer wants to fence a rectangular plot 40 meters by 30 meters. Fencing costs ₱80 per meter. He also wants to seed grass on the interior. Grass seed costs ₱500 per square meter. What is the total cost? - **Understand:** We need fencing cost (perimeter) and seeding cost (area). - **Plan:** Compute perimeter, multiply by fence cost. Compute area, multiply by seed cost. Add. - **Carry out:** - Fencing: Perimeter = 2(40 + 30) = 2 × 70 = 140 meters. Cost = 140 × 80 = ₱11,200. - Seeding: Area = 40 × 30 = 1,200 m². Cost = 1,200 × 500 = ₱600,000. - Total = ₱11,200 + ₱600,000 = ₱611,200. - **Look back:** Fencing cost is much less than seeding cost because perimeter (140 m) is much less than area (1,200 m²). The answer is reasonable. ✓ **Worked Example 2: Capacity and counting problem** A school cafeteria has a water tank holding 5,000 liters. Each student receives 250 milliliters of drinking water daily. If there are 800 students, will one tank be enough for one day? - **Understand:** Total water needed is 800 students × 250 mL/student. Check if this fits in 5,000 liters. - **Plan:** Compute total need in consistent units (liters or mL), compare to tank capacity. - **Carry out:** - Total need: 800 × 250 mL = 200,000 mL = 200 liters. - Tank capacity: 5,000 liters. - 200 liters < 5,000 liters, so yes, one tank is enough. - **Look back:** 200,000 mL ÷ 1,000 = 200 L is correct. 200 is much less than 5,000, so the tank is more than sufficient (it could serve about 25 days). ✓ **Worked Example 3: Area with unit conversion** A painter needs to paint two walls of a room. Wall 1 is 4 meters long and 3 meters high. Wall 2 is 5 meters long and 3 meters high. Paint covers 10 square meters per liter. How many liters are needed? - **Understand:** We need the total area to be painted, then divide by the coverage rate. - **Plan:** Compute area of each wall, add, divide by coverage rate. - **Carry out:** - Wall 1: 4 × 3 = 12 m². - Wall 2: 5 × 3 = 15 m². - Total area: 12 + 15 = 27 m². - Liters needed: 27 m² ÷ 10 m²/liter = 2.7 liters. - **Look back:** A typical liter of paint covers about 10–12 m² (depending on the paint and surface), so 2.7 liters for 27 m² is reasonable. ✓ **Classroom Connection:** When teaching Grade 4–6 students, use these context-based problems frequently: fencing the school garden, tiling a classroom, filling the water tank. This grounds measurement in familiar tasks and helps students internalize when to use which measure. The DepEd curriculum emphasizes this practical application, and the LET tests it explicitly.