Algebra Play · Lesson 10 of 10
Chapter Summary and Practice
“Review the chapter's algebraic tools through mixed examples, error analysis, proof, modelling, and cumulative practice.”
• Connect the chapter's tricks, pyramids, grids, divisibility claims, and story problems through common algebraic ideas. • Choose a suitable variable and translate a new situation into expressions or equations. • Distinguish experimental evidence from a general algebraic justification. • Select efficient methods such as forward calculation, backward reasoning, place-value expansion, or equation solving. • Solve mixed problems and explain why each method works.
This chapter has used algebra in several different-looking settings: number tricks, dates, pyramids, calendars, digit puzzles, divisibility claims, business costs, and stories. The surface details change, but the core process is the same: represent a quantity, express the relationships, simplify or solve, and interpret the result.
The Big Ideas of Algebra Play
| Situation | Main algebraic idea |
|---|---|
| Think-of-a-number trick | Track an arbitrary x and watch variable terms cancel. |
| Date trick | Use place value to encode M and D. |
| Number pyramid | Translate repeated addition into expressions. |
| Calendar grid | Use fixed positional differences such as +1 and +7. |
| Largest product | Use place value and inequality comparisons. |
| Divisibility trick | Factor expanded digit expressions. |
| Story problem | Translate verbal conditions into equations. |
| Repeated process | Use a recurrence or work backwards. |
One important habit runs through all of these: do not calculate blindly. Before manipulating symbols, say what each symbol represents and what each operation means in the situation.
From Examples to Proof
Examples help us notice a pattern. If several reverse-number differences are divisible by 9, we may conjecture that this is always true. But the proof comes from writing ab=10a+b and ba=10b+a, then showing their difference has a factor of 9.
A mathematical statement suggested by patterns or examples but not yet justified for all relevant cases.
A chain of symbolic reasoning that explains why a statement follows from stated assumptions.
Problem
A student checks 23, 41, and 72 and says, 'The difference between a two-digit number and its reverse is always 27.' What is wrong?
- 1.The examples do not all even give the same difference: 41−14=27, but 72−27=45.
- 2.The correct general form is 9 times the digit difference.
- 3.If the digit difference is 3, the number difference is 27; if it is 5, the number difference is 45.
- 4.The algebraic statement is broader and accurate.
Choosing the Right Representation
Some problems become simple when one variable is chosen. Others need two. A calendar block can be described entirely from its top-left entry a because the other positions are fixed offsets. A heads-and-legs problem naturally begins with two counts, though one equation can be used to eliminate one of them.
- Use one variable when all unknown quantities can be expressed from one starting quantity.
- Use two variables when two independent quantities are being counted or compared.
- Use place-value expressions when digits are rearranged.
- Use a recurrence or backward steps when the same operation repeats.
- Use a direct formula when a repeated structure has already been generalised.
Mixed Worked Examples
Problem
A trick says: multiply a number by 7, add 35, divide by 7, subtract the original. Explain the result.
- 1.Let the original number be x.
- 2.The expression is (7x+35)/7−x.
- 3.This becomes x+5−x=5.
- 4.The result is always 5 because the x terms cancel.
Problem
A 2×2 calendar block totals 100. Find the entries.
- 1.Use 4a+16=100.
- 2.Then 4a=84, so a=21.
- 3.The block is 21,22,28,29.
- 4.Check the sum and confirm the positions form a valid calendar block.
Problem
Find the top of a four-row pyramid with bottom 2,6,5,4.
- 1.Use a+3b+3c+d.
- 2.Top = 2+18+15+4=39.
- 3.The coefficient pattern avoids filling all intermediate boxes.
Problem
Show that 352352 is divisible by 7,11,13.
- 1.352352=1001×352.
- 2.1001=7×11×13.
- 3.Therefore 352352 contains all three factors.
- 4.Dividing successively by 7,11,13 leaves 352.
Problem
Daily fixed cost is ₹3000, variable cost ₹20 per item, and selling price ₹70. How many items must be sold for ₹2000 profit?
- 1.Profit per item before fixed cost is 70−20=₹50.
- 2.Use 50n−3000=2000.
- 3.Then 50n=5000, so n=100.
- 4.Check: revenue ₹7000, total cost ₹5000, profit ₹2000.
Spot the Mistake
Error analysis is useful because many algebra mistakes are not arithmetic errors—they are modelling errors. A correct calculation based on a wrong equation still gives a wrong solution.
| Mistake | Why it fails | Repair |
|---|---|---|
| Treating abc as a×b×c | Digits indicate place value. | Write 100a+10b+c. |
| Using 5(d+6) for mother's future age when she is 5d now | Six years are added after the present age is known. | Use 5d+6. |
| Assuming a calendar total divisible by 4 is automatically valid | The derived a may not fit a real block. | Solve for a and check the calendar. |
| Testing three examples and calling it a proof | Unseen cases remain. | Use a general variable or factorisation. |
| Subtracting a fixed cost once per item | Fixed cost is paid once per day. | Separate fixed and variable costs. |
Create, Explain, and Generalise
The strongest way to finish this chapter is to create a puzzle of your own. A good algebra puzzle has a clear rule, a predictable structure, and an explanation that works beyond one example.
Create either a number trick, a calendar trick, a pyramid puzzle, or a digit-divisibility claim. Provide one numerical example, then give an algebraic explanation showing when and why it works.
Do not hide assumptions. State whether digits are distinct or positive, whether dates must be valid, whether quantities must be whole numbers, and whether a fee is applied before or after doubling.
Cumulative Quiz
Quiz
Which expression proves that a double-add-divide trick can end at a constant?
In the date trick F=100M+165+D, what should be done first to decode?
What is the top of a three-row pyramid with bottom a,b,c?
Which is the sum of a 2×2 calendar block with top-left a?
For positive p<q<r, which product is largest?
A two-digit number plus its reverse is always divisible by what?
If a mother is 5d years old now, what is her age six years later?
Which equation represents profit with price p, quantity n, fixed cost F, and variable cost v per item?
Cumulative Practice
Practice Problems
- Prove that 'multiply by 4, add 28, divide by 4, subtract the original' always gives 7.
- Using the original date trick, decode a final result of 1176 and check whether the date is valid.
- Find the top of a four-row pyramid with bottom 9,4,7,2.
- A 2×2 calendar block sums to 92. Find its four dates.
- Using digits 2,5,8, find the largest product of a two-digit number and a one-digit number and justify the arrangement.
- Prove algebraically that a two-digit number and its reverse have a sum divisible by 11.
- Show that the sum of abc,bca,cab is divisible by 37.
- A farm has 36 animals, all chickens or goats, and 100 legs. Find each number.
- A shop has fixed daily cost ₹2400 and variable cost ₹30 per item. At ₹70 selling price, how many items are needed for ₹1600 profit?
- A repeated process doubles a quantity and subtracts 5 each round. Starting from 9, find the amount after four rounds.
- Explain why the fraction made from the first n odd numbers over the next n odd numbers is always 1/3.
- Design one original algebra trick or puzzle and provide a general explanation.
(4x+28)/4−x=x+7−x=7.
Chapter Reflection
Algebra is not only a collection of procedures for finding x. It is a language for describing structure. When a trick always works, algebra explains the invariance. When a pattern seems true, algebra can prove or correct the conjecture. When a story contains several conditions, algebra organises them into relationships that can be solved.
The most useful question to carry forward is: 'What stays the same, what changes, and how can I represent that relationship?' That question connects every puzzle in this chapter.
Key Takeaways
• Algebra models numerical situations and reveals the structure behind tricks and puzzles. • Variables, place value, equations, and factorisation are different tools for different kinds of relationships. • A few successful examples are evidence; a general derivation is a justification. • Efficient problem solving often depends on choosing the right representation or reasoning direction. • A complete solution states assumptions, solves accurately, and checks the result in its original context. • The chapter's deeper goal is to explain and create patterns, not merely calculate answers.
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Algebra in Money, Patterns, and Repeated Processes
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