How can you teach order of operations with real-world examples involving large numbers like 143, 305, and 70?
Executive summary
Teach order of operations using real-world contexts like shopping totals, recipe scaling and crowd calculations to make rules meaningful; examples that scale to large numbers (e.g., “3 × (14,3? — available sources do not mention 143 specifically)” are not directly in the provided reporting, but sources show using word problems with large addends such as 18,932 and 921 to illustrate grouping and scaling [1]. Multiple curricula and teacher-guides recommend framing multiplication as “scaling” and parentheses as “packaging” so students understand why 3 × (18932 + 921) is interpreted as three times the whole sum without computing the sum first [1] [2].
1. Make the abstract concrete with everyday story problems
Real-life situations—splitting pizza costs, back-to-school shopping totals, or balancing a check register—translate symbols into decisions students face, and reinforce the need for a consistent procedure; Texas Gateway uses examples like balancing a bank register to show why order matters [3] and Cuemath frames a 5‑pizza, $20 example to show parentheses and division in context [4]. These scenarios let teachers replace contrived strings of operators with meaningful combinations: total cost, then per-person split; or scale a recipe then apply a discount [4] [5].
2. Use “packaging” and “scaling” language for large numbers
Sources recommend explaining parentheses as a way to “package” quantities and multiplication as scaling that package. Teacher-made materials explicitly present expressions such as 3 × (18932 + 921) to show students that the multiplier applies to the entire packaged sum, so you don’t need to compute the internal sum first to understand what the expression means [1]. This framing helps students work with very large addends without stumbling over sheer arithmetic.
3. Design classroom tasks that require choosing order — not just following PEMDAS
Several sources warn that mnemonics like PEMDAS can create misconceptions (multiplication-before-division or addition-before-subtraction). ElementaryMath points out that multiplication/division and addition/subtraction operate left-to-right as pairs, and that grouping symbols must be emphasized to override default priorities [2]. Tasks should therefore include expressions where students must decide whether parentheses change the natural left-to-right pairing, e.g., compare (A + B) × C versus A + B × C using story contexts like batching supplies for 143 or 305 people (note: these specific large-number scenarios are suggested by pedagogy but not present verbatim in the available sources; available sources do not mention 143, 305, and 70 together) [2] [1].
4. Start with realistic scaffolded examples, then scale up
Math-Drills and Texas Gateway recommend beginning with familiar small numbers or day-count examples, then generalize the structure to larger values so students master the pattern before arithmetic complexity overwhelms them [6] [3]. For instance: “You make 70 batches of cookies; each batch uses (2 + 3) cups of flour—do you multiply first or add first?” The suggested sequence is: concrete story → symbolic expression → evaluate with given numbers → explain the rule [6] [3].
5. Use calculators and spreadsheets intentionally to expose conventions and limits
Wikipedia and other references note that calculators, programming languages and spreadsheets sometimes differ in how they parse expressions and exponents; this can be a teachable moment about convention versus tool behavior [7]. Presenting students with the same expression on paper and in a spreadsheet (or different calculators) shows why parentheses are the reliable way to make your intent explicit [7].
6. Anticipate and address common misconceptions with contrasting cases
Provide paired problems that differ only by parentheses so students see different outcomes: e.g., 2 + 3 × 10 versus (2 + 3) × 10. ElementaryMath emphasizes using such contrasts to teach why grouping symbols change results and why “more powerful” operations have priority [2]. Teacher-sourced worksheets show how everyday scenarios (pancake batches, pizza splits) produce different numeric answers when order is ignored [1] [4].
7. Assessment and extension: ask for expression writing and justification
Beyond evaluating expressions, have students write expressions from word problems and justify the grouping they chose. TeachersPayTeachers materials emphasize having learners translate “add 8 and 7, then multiply by 2” into 2 × (8 + 7) and to reason about why that grouping matters; asking students to do the same with larger numbers tests conceptual understanding more than arithmetic speed [1].
Limitations and disagreements in the sources: PEMDAS-style mnemonics are widely used but criticized for encouraging the false idea that multiplication always precedes division or addition always precedes subtraction; ElementaryMath explicitly warns about that misconception [2]. Conventions in software and calculators can also differ [7]. The specific large-number triple 143, 305, and 70 is not present in the supplied reporting; available sources do not mention those exact figures together (available sources do not mention 143, 305, and 70 together).