Expressions using Letter-Numbers · Lesson 11 of 11
Chapter Summary and Practice
“Connect the whole chapter, explain the rules, and apply expressions to mixed situations and patterns.”
• Connect letter-numbers, expressions, formulas, substitution, and simplification. • Translate situations into expressions and interpret the meanings and units. • Collect like terms and handle multiplication and subtraction of brackets correctly. • Explain equivalence and use counterexamples to detect invalid claims. • Apply occurrence, grid, calendar, matchstick, and rope formulas with their stated conditions. • Solve mixed problems and check both numerical results and reasoning.
The Chapter at a Glance
This chapter uses letters to describe number relationships, calculate quantities, and explain patterns. The central habit is to understand a situation before choosing operations. Once an expression is written, arithmetic rules let us evaluate it for given values or simplify it into an equivalent form.
| Idea | Meaning or rule | Example |
|---|---|---|
| Letter-number | A letter represents a defined numerical quantity. | a may represent an age in years. |
| Expression and formula | An expression gives a calculation; a formula states a general relationship. | a + 3; s = a + 3 |
| Substitution | Replace every letter by its assigned value, keeping operations. | For a = −4, 10 − a = 14. |
| Like terms | Same letter part: combine numerical coefficients. | 5c + 3c = 8c |
| Unlike terms | Different letter parts remain separate without further information. | 18c + 11d |
| Distribution | Multiply every term of a bracket. | 4(x + y) = 4x + 4y |
| Subtracted bracket | Change every sign in the removed expression. | A − (B − C) = A − B + C |
| Equivalence | Same value for every allowed assignment. | 3 + 2(y − 1) = 2y + 1 |
| Repeating positions | Cycle length and offset describe occurrences. | A: 3n − 2; B: 3n − 1; C: 3n |
| Four-column grid | Count completed rows, then the column. | 4(r − 1) + c |
| Calendar patterns | Horizontal offsets are 1; vertical offsets are 7. | Diagonals 2a + 8; cross 5a |
| Growing figures | Initial count plus repeated additions; respect sharing. | Triangle sticks 2y + 1; square sticks 3w + 1 |
| X and rope counts | State the arrangement and the counting convention. | X squares 4n + 1; illustrated rope pieces r + 2 |
From a Situation to a Formula
Begin by identifying the quantity being requested. Define each letter, attach the correct units, and describe the operations in words. Distinguish one-time contributions from contributions repeated for each item, day, or travel leg. This prevents many errors before algebra begins.
| Situation | Expression or formula | Meaning of letters |
|---|---|---|
| Age gap of 3 years | s = a + 3; a = s − 3 | a and s are the two ages in years. |
| Separate L shapes | M = 2n | n is the number of separate Ls. |
| Coconuts and jaggery | 35c + 60j | c counts coconuts; j is kilograms of jaggery. |
| Note collection | 100x + 20y + 5z | x, y, z count ₹100, ₹20, ₹5 notes. |
| Mill run | 10 + 8y | One start-up; y kilograms. Time in seconds. |
| Regular perimeters | 3a, 5a, 6a | a is side length for triangle, pentagon, hexagon. |
| Rectangle perimeter | 2l + 2b | l and b are length and breadth. |
| Rental after refund | 34x + 65y | x chairs and y tables. Amount in rupees. |
| Flower customers | p + q + r | Three separate groups of customers. |
| Ten snail cycles | 10(u − d) | Signed change, with u up and d down each cycle. |
| Three cycling weeks | 105 + 21z | z is weekly increase in daily kilometres. |
| Four-leg train journey | 4t + 6 | t minutes per leg plus three 2-minute waits. |
Problem
A machine starts once in 12 seconds and processes each item in 5 seconds. Write the time for n items and find it for n = 8.
- 1.Define n as the number of items and T as total time in seconds.
- 2.The start-up happens once; processing is repeated n times. Therefore T = 12 + 5n.
- 3.For n = 8, T = 12 + 40 = 52 seconds. The expression 17n would repeat the start-up for each item and model a different procedure.
The numbers in a formula are not interchangeable labels. In 35c + 60j, the rate 35 belongs to coconuts and 60 to kilograms of jaggery. In p + q + r, the letters count people in separate groups. Understanding what each contribution counts gives a reliable reason for the operations.
Evaluate or Simplify: Choose the Task
Evaluation and simplification are connected but different tasks. Evaluation needs assigned letter values and produces a numerical value. Simplification uses arithmetic rules to rewrite an expression before, or after, values are supplied. It may leave letters in the answer.
Problem
Simplify 2(2x − 3) + 8x + 12, then find its value when x = −1.
- 1.Distribute 2: 4x − 6 + 8x + 12.
- 2.Collect x terms and constants: (4 + 8)x + (−6 + 12) = 12x + 6.
- 3.For x = −1, substitute with the sign visible: 12 × (−1) + 6 = −6.
- 4.Directly, 2(−2 − 3) − 8 + 12 = −10 − 8 + 12 = −6, confirming the calculation.
Problem
Simplify (7x − 2y + 5) − (3x + 4y − 6).
- 1.The second whole expression is being removed, so change all its signs.
- 2.Write 7x − 2y + 5 − 3x − 4y + 6.
- 3.Collect terms: (7 − 3)x + (−2 − 4)y + (5 + 6) = 4x − 6y + 11.
The hidden coefficient of x is 1 and of −x is −1. Constants form their own group. A subtracted bracket changes even an internal negative to positive. A multiplier outside a bracket reaches every internal term. These rules explain the work; they are more dependable than guessing which symbols to combine.
Explain Equivalence and Pattern Rules
A valid transformation explains why two expressions agree for every allowed value. One or several matching numerical cases cannot alone establish that general result. Conversely, one correct counterexample can establish that two expressions are not equivalent.
Problem
Compare 3(x + 2) with 3x + 2, and then with 3x + 6.
- 1.At x = 0, 3(x + 2) gives 6, while 3x + 2 gives 2. This counterexample shows the first pair is not equivalent.
- 2.Distribution gives 3(x + 2) = 3x + 3 × 2 = 3x + 6.
- 3.The second equality holds for every allowed x because the same valid rule applies regardless of its value.
| Pattern | General rule | Condition to remember |
|---|---|---|
| A-B-C border | A: 3n − 2; B: 3n − 1; C: 3n | n counts appearances of one chosen design. |
| Red-yellow-green-yellow | Red: 4n − 3; green: 4n − 1; yellow: 2n | Yellow occurs twice per four-position cycle. |
| Four-column grid | 4(r − 1) + c | r starts at 1; c is 1, 2, 3, or 4. |
| Two-by-two calendar | Both diagonals: 2a + 8 | a is the top-left entry; the block is full. |
| Calendar cross | Total: 5a | a is the centre, with all four neighbours. |
| Triangle chain | 3 + 2(y − 1) = 2y + 1 | Each new triangle shares one side. |
| Connected square row | 4 + 3(w − 1) = 3w + 1 | Each new square shares one side. |
| Growing X squares | 4n + 1 | One centre plus n on each of four arms. |
| X vertex occurrences | 16n + 4 | Count four corners per square, including repeats. |
| Illustrated rope pattern | r + 2 pieces | r bends give r + 1 distinct interior cut points. |
Problem
Find the entry at row 9, column 2 of the four-column grid, the diagonal sum of a calendar block starting at 11, and sticks in 9 connected triangles.
- 1.Grid: eight complete rows precede row 9, so the entry is 4 × 8 + 2 = 34.
- 2.Calendar: the entries are 11, 12 above 18, 19. Both diagonals total 2 × 11 + 8 = 30.
- 3.Triangle chain: one starting triangle and eight additions give 3 + 2 × 8 = 19 sticks, agreeing with 2 × 9 + 1.
Check the Meaning as Well as the Arithmetic
A calculation can be arithmetically correct and still answer the wrong question. Counting both kinds of flower purchases is different from counting customers. Counting three intermediate stations is different from counting four travel legs. Counting four corners of every square is different from counting shared geometric points only once.
A good final check asks whether the formula matches the arrangement and whether the answer has the required unit. Try a small case you can count directly. Verify that a growing formula reproduces its first step, and that a cycle formula uses positions starting from 1. These checks connect symbols back to the original problem.
Preserve signs, input order, units, and counting conventions. Do not infer equivalence from one matching value, apply a shared-side rule to separate figures, or use a rope rule for a different folding arrangement. For repeated-change models, check that the stated cycles can actually continue.
Check Your Understanding
Use the ideas from this lesson to choose an answer. Explain your choice to yourself before opening the explanations below.
Quiz
Which is an algebraic expression?
If x = −3, what is 2x + 5?
Which expression correctly describes 13 less than twice n?
What is 7a − 2a + 3b − b?
Simplify 8x − (2x − 3) + 12.
Which method establishes that 2(x + 4) and 2x + 8 are equivalent?
A number machine uses ab + 1. What output comes from a = 3, b = 2?
At which position is the twentieth C in an A-B-C border?
What is the sum of a complete calendar cross centred at 18?
How many sticks make 12 connected squares in one row?
4x + 3 contains a letter-number and operations.
Practice Problems
- Shabnam is 3 years older than Aftab. Define letters and give formulas in both directions. Find both ages if Aftab is 16.
- Write formulas for n separate Ls and a square of side q cm. Evaluate for n = 9 and q = 8.
- Find the total value of 4 ₹100 notes, 3 ₹20 notes, and 8 ₹5 notes.
- Write the cost for c coconuts at ₹35 and j kilograms of jaggery at ₹60. Evaluate for c = 6, j = 2.
- A mill takes 10 seconds to start once and 8 seconds per kilogram. Write the time for y kg and evaluate for y = 9.
- Evaluate 3(m + 1) for m = −6 and h − (3 − n) for h = 5, n = 6.
- Simplify 3a + 9b − 6 + 8a − 4b − 7a + 16.
- Simplify 3(3a − 3b) − 8a − 4b − 16.
- Simplify 2(2x − 3) + 8x + 12 and 8x − (2x − 3) + 12.
- Simplify 8h − (5 + 7h) + 9.
- Simplify 23 + 4(6m − 3n) − 8n − 3m − 18.
- Add 4d − 7c + 9 and 8c − 11 + 9d.
- Subtract 9a − 6b + 14 from 6a + 9b − 18.
- A round total is 7p − 3q. Explain p and q, find its value for p = 4, q = 1, and interpret q = 0.
- Show 5u and 5 + u are not equivalent. Is one counterexample sufficient?
- Correct 2y + (3y − 6) = −y + 6.
- A machine gives outputs 5, 7, 18 for inputs (5,2), (8,1), (9,11). Propose a simple rule and check it.
- Starting at w + 2, subtract 5 then multiply the result by 3. Find the final expression.
- There are p customers buying only one flower type, q buying only the other, and r buying both. How many one-per-customer gifts are needed?
- For ten complete snail cycles, write the signed change in terms of u and d. Explain the case d > u.
- Give the three-week cycling total for daily distances 5, 5 + z, 5 + 2z km in successive seven-day weeks. Evaluate for z = 1.
- A train has three intermediate 2-minute stops and equal t-minute travel legs. Write its total and evaluate for t = 6.
- In an A-B-C border, find the designs at positions 99, 122, 148 and the position of the twentieth B.
- In red-yellow-green-yellow repetition, find the colours at positions 90, 190, 343. Give each colour’s occurrence formula.
- Locate 124, 147, 201 in the four-column grid. Find its entry at row 15, column 4.
- Write all entries in a two-by-three calendar block with bottom middle w.
- Prove both diagonals in a two-by-two calendar block beginning at a total 2a + 8.
- Write the five entries in a calendar cross centred at a and simplify the sum.
- Derive the stick formula for y connected triangles and calculate step 20.
- Derive the stick formula for w connected squares and calculate w = 20.
- Count X-pattern squares at step 10 and vertex occurrences when every square contributes four corners.
- In the illustrated rope arrangement, derive the pieces after r bends and calculate r = 10.
- Explain why 3j + 6k + 9h + 12 and 3(j + 2k + 3h + 4) are equivalent.
- Invent a short situation with a fixed part and a repeated part. Give a formula and test one value.
Let a and s be the ages. s = a + 3 and a = s − 3. If a = 16, s = 19 years.
Key Takeaways
• Define letters, quantities, and units before writing a model. • Use arithmetic rules to evaluate and simplify expressions accurately. • Collect like signed terms and keep unlike terms separate. • Explain equivalence through valid transformations and detect false claims with counterexamples. • Derive pattern formulas from cycles, offsets, shared parts, or repeated additions. • Check small cases, reference positions, and counting conventions to connect an answer back to its situation.
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Matchstick Patterns and Growing Figures
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