The67Sprint — LearningMap: Expressions using Letter-Numbers

🔤 Letter-Numbers & Expressions

Letters like a, s, n, c are letter-numbers — placeholders for unknown or changing values, not abbreviations for objects
An algebraic expression combines numbers, letter-numbers, and operations (+, −, ×, ÷)
Terms are the parts of an expression separated by a "+" sign — subtracting is really adding a negative
The × sign is dropped as shorthand: 4 × n is written simply as 4n

🧩 Like, Unlike & Combining Terms

Like terms share the exact same letter-numbers (5c and 10c) and can be combined into 15c
Unlike terms have different letter-numbers (18c and 11d) and can never be merged into one term
Swapping and grouping terms never changes an expression's value — but the sign must travel with its term

Like Terms Unlike Terms

🧮 Brackets & Distribution

A number outside a bracket multiplies every term inside: 6(p + 2) = 6p + 12
A negative sign outside a bracket flips the sign of every term inside when removed
Example: 5 − (2 − 6z) becomes 5 − 2 + 6z = 3 + 6z

Golden Rule

Only like terms combine. 3a + 2b can never become 5 or 5ab — a and b are different unknowns and the expression is already in simplest form.

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Quick Recall — Sprint Facts

Substitution If a = 23, then a + 3 = 26 — replace the letter-number with its value

Square Perimeter Side q → perimeter is 4q

Matchstick "L"s n shapes need 2n matchsticks

Sign Reversal 100 − (15 + 56) = 100 − 15 − 56

Commutative Rule Swapping and grouping terms never changes the value

Calendar Grid Bottom-middle cell w → cell above is w − 7

🔑 Key Terms to Know

Letter-Numbers

Letters acting as dynamic placeholders for numerical values that can change

Algebraic Expression

A phrase combining numbers, letter-numbers, and operations to describe a pattern

Substitution

Replacing a letter-number with a numeric value to evaluate the expression

Terms

Individual parts of an expression separated by addition or subtraction

Like Terms

Terms with identical letter-numbers — their coefficients can combine

Unlike Terms

Terms with different letter-numbers — they can never be merged

🏆 Olympiad & Challenger Edge

The Cryptic Number Machine

Inputs a=4, b=1 give output 5; a=10, b=3 give output 31. The hidden rule is ab + 1 — the machine multiplies the inputs and adds one.

The Recursive Rope Fold

A rope folded r times then cut yields r + 2 pieces. For 10 folds: 10 + 2 = 12 pieces.

Complex Modeling — The Snail Problem

A snail climbs u cm by day but slips d cm by night. Total progress after 10 days models as 10(u − d).

Advanced Polynomial Subtraction

Subtracting 9a − 6b + 14 from 6a + 9b − 18 simplifies fully to −3a + 15b − 32.

❌ 3d = 36?

Wrong. 3d means 3 × d, not digit concatenation. If d = 6, then 3d = 18, not 36.

❌ Forcing unlike terms together

3a + 2b is already fully simplified — it can never become 5 or 5ab since a and b are different unknowns.

❌ Nested negative sign failure

5 − (2 − 6z) is NOT 3 − 6z. The negative must flip both inner terms: 5 − 2 + 6z = 3 + 6z.

❌ "Subtract A from B" order confusion

This always means B − A, never A − B. Writing it backwards is a common mark-losing mistake.

❌ Calendar grid direction errors

If the bottom-middle cell is w, the top-left cell is w − 8, not w − 7 — moving diagonally combines both the "up a row" and "left one" steps.

🛒 Shopping Bills

Ketaki buys c coconuts at ₹35 each and j kg jaggery at ₹60/kg — total cost is 35c + 60j.

🏪 Commercial Rentals

Renting x chairs and y tables with refunds simplifies (40x + 75y) − (6x + 10y) to 34x + 65y.

📝 Quiz Scoring Systems

Charu's score from p correct answers and q penalties models as 7p − 3q.

📅 Calendar Magic

In a calendar's 2×3 grid, if the bottom-middle date is w, the date directly above is always w − 7.

🚨 Skipping grouping steps

Always show (5y + 5y) + (x + x) before writing 10y + 2x — skipping this costs marks.

🚨 Undefined variables

Always state what a letter-number represents — e.g. "Let x be the cost of one notebook" — before using it.

🚨 Incomplete final form

4x + 4y − y is not a finished answer — simplify fully to 4x + 3y.

🚨 Expression vs. equation

An expression never has an "=" sign — don't confuse the two or treat one as the other.

🚨 Losing the sign on a term

In 13 − 2x, the term is −2x, not 2x. Always rewrite subtraction as addition first — 13 + (−2x) — so the negative sign can't get left behind.

Logic Chain — How a Real-World Scenario Becomes an Expression

Step 1

Reading the real-world scenario connects to identifying which quantities are unknown or changing

Step 2

Assigning letter-numbers to those unknowns connects to translating the situation into math operations

Step 3

Forming the expression and expanding any brackets connects to collecting all like terms together

Step 4

Grouping like terms leads to a simplified expression ready for substitution and final evaluation

🦉

Mentor Byte

"Stop treating letters as magical objects — they are mathematical shorthand for everyday situations, obeying the exact same rules as regular numbers. The biggest hurdle is sign errors: train yourself to circle the negative sign together with the term it belongs to, before you move anything around the page."

💛 Parent Tip

Try the "Box Method" at home — call each letter-number an empty box waiting for a number to be dropped inside. Ask your child to physically circle a term's sign with it before rearranging anything, until it becomes automatic.