PrepShorts · Study sheet · Class 7 Mathematics · Chapter 7, Finding the Unknown
Chapter 7 · Finding the Unknown
Generating an equation from a situation
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Every equation so far has arrived already written down. Out in the world nobody hands you one.
The idea
The hard half of a word problem was never the solving; it was deciding what the letter is going to stand for. This chapter makes that choice visible by printing three students' attempts at one party budget side by side — and the striking thing is that the three take different routes, two of them different equations and one needing no letter at all, and all of them are correct. There is no one right equation for a situation. What makes a modelling right is that every line of it can be read back into the story, and once you accept that, the reverse job becomes obvious too: hand someone an equation and ask what story it could have come from.
What you should be able to do
- Choose what an unknown letter will denote, and state that choice in words before writing anything
- Build an expression for a situation one described step at a time
- Show that two different countings of the same pattern give the same expression
- Produce two different but equally correct equations for one situation, and reconcile their answers
- Read a solved value back into the situation and answer the question that was actually asked
- Reduce a two-unknown situation to one unknown by expressing one in terms of the other
- Write several equations with a stated solution, and build a chain of equations that all share it
- Given a bare equation, invent a situation it could model
Words to know
| Term | Definition in one line | First introduced |
|---|---|---|
| unknown | the quantity a situation does not hand you, and which you name with a letter | printed throughout Part II, §7.2, pp.174–180 |
| letter-number | a letter used in place of a number whose value is not yet known | printed in Part II, §7.2, p.179 |
| expression | a formula built from numbers, letters and operations | printed in Part II, §7.2, pp.175, 178 |
| equation | a written line asserting that two expressions name the same number | printed in Part II, §7.1, p.167 |
| solution | a value of the letter for which the assertion comes out true | printed in Part II, §7.2, pp.179–180 |
| position number | the place an arrangement occupies in a growing pattern; the book writes Step k here | printed in Part II, §7.1, pp.166–167; the tile pattern of p.174 uses Step instead |
| fixed amount | a charge that does not change with how many items are bought | printed in Part II, §7.2, p.175 |
| cost per unit | the charge attaching to each single item | printed in Part II, §7.2, p.180 |
| modelling (algebraically) | turning a described situation into an equation | printed as algebraically modeling in Part II, §7.2, p.178 |
| chains of equations | a run of equations, each got from the one above by one operation applied to each side | printed in Part II, §7.2, p.180 |
| word problem | the usual name elsewhere for a situation given in prose | not printed in this chapter — all 28 printed pages, 164 to 191, were read; the bold subheading here is "Solving Problems" |
| naming the unknown | deciding, in words, what the letter will stand for before writing the equation | the explanation's phrase for the step the chapter demonstrates three times on pp.175–176 without labelling it |
Where people slip up
- "Every problem has one right equation." The single most important correction in this topic, and the chapter builds a whole page around it. Mahesh's 25p + 50 = 500 and Srikanth's 25(f + 5) = 450 are different equations from the same story, and Fatima solved it without writing either.
- "The letter always stands for the thing the question asks for." Mahesh's p is the number of people, but the question asks for friends. His answer, 18, is not the answer to the question until 5 is taken off it. Reading the value back into the story is a separate step and it is where marks are lost.
- "A word problem is solved by hunting for keywords." Nothing on pp.174–180 works that way. Fatima's diagram comes before any letter, and Riyaz's table builds the expression one instruction at a time.
- "Two unknowns need two letters." Example 12 writes two letters, notices it cannot proceed, and goes back to one. Expressing Ramesh's count as y + 30 is the whole move. This is a genuine limitation of Class 7, stated by the chapter itself, not a shortcut.
- "Counting a pattern a different way gives a different formula." It gives a different-looking one. Method 1 reaches k + k + k + 1 and Method 2 reaches k + (2k + 1); both simplify to 3k + 1. That they must agree is the point.
- "Generating an equation is a game with no content." The chains on p.180 are the reason it is not: every equation in a chain has the same solution, so the reader can read a solution off the bottom line and know it holds all the way up. That is the earlier property looked at from the other end.
- "3k + 1 = 100 obviously has an answer because 100 is a nice number." It does here, but the parallel question in the chapter's exercises deliberately mixes reachable and unreachable targets (Part II, the exercise block following §7.4, p.188, question 16). Reachability has to be checked, not assumed.
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Worked answers to this chapter’s exercises · this video explains Figure it Out · 2 Q1, Figure it Out · 2 Q3, End-of-Chapter Figure it Out Q2, End-of-Chapter Figure it Out Q6, End-of-Chapter Figure it Out Q7, End-of-Chapter Figure it Out Q8, End-of-Chapter Figure it Out Q9, End-of-Chapter Figure it Out Q12, End-of-Chapter Figure it Out Q13, End-of-Chapter Figure it Out Q14, End-of-Chapter Figure it Out Q20
Transcript1,440 words
So far, every equation has arrived already written down. Somebody hands you five x minus four equals seven, and you solve it. Out in the world, nobody hands you the equation. You get a situation, and the hard half is not the solving. The hard half is deciding what the letter is going to stand for. And that choice is not forced. Two people can look at one situation, choose differently, and write two different equations.
Both correct. This video is about that. Start with something you can see. Here is a shape of square tiles, growing one step at a time. A row across, with one tile hanging below the middle. Step one: four tiles. Step two: seven. Step three: ten. How do you count it without counting? One way: cut it into three equal runs, and one left over. At step one that is one and one and one, and one. Four.
At step three, three and three and three, and one. Ten. So at step k it is k and k and k, and one. But that is not the only way to cut it up. Somebody else sees a short arm, and everything else. At step one, one and three. At step two, two and five. At step three, three and seven. The arm is just k. The rest goes up by two each time, starting at three.
So the rest is two k plus one, and the whole is k plus, bracket, two k plus one. Two people, two expressions, one shape. Tidy both up and they are the same: three k plus one. They had to be. They are counting the same tiles. Now watch what that buys you. Which step uses exactly one hundred tiles? Counting your way there would take a while. Instead: three k plus one equals one hundred.
Take one off each side, divide each side by three. k is thirty three. Check it. Three thirty threes is ninety nine, plus one is one hundred. And notice what else falls out. Step thirty two uses ninety seven, step thirty four uses a hundred and three. So no step uses ninety nine. The formula answers questions you did not ask it. Now a situation with no picture in it at all.
You are ordering plates of snacks for a party. Each plate costs twenty five. Delivery is a flat fifty, however many. Your family is five, including you, and everyone gets a plate. You have five hundred to spend. How many friends can you invite? Before any letter, one person draws it: a box for snacks, a box for delivery, a total. Take the fifty off the five hundred. That leaves four hundred and fifty for plates.
Four hundred and fifty divided by twenty five is eighteen plates. Take off the five family: thirteen friends. No letter anywhere, and that is a complete solution. Now two other people do write letters, and they choose differently. The first says: let p be the total number of people. Each eats a plate, so snacks cost twenty five p, and delivery is fifty. Twenty five p, plus fifty, equals five hundred.
The second says: let f be the number of friends. The family is five more, so there are f plus five people, and delivery comes off the budget first. Twenty five, bracket f plus five, equals four hundred and fifty. Those are not the same equation, and neither was got from the other. Both of them are right. Solve them and something worth noticing happens. The first gives p equals eighteen. The second gives f equals thirteen.
Different answers, same party, and neither is wrong. They differ by five, because one letter counted the family in and the other did not. But the question asked for friends. Eighteen is not the answer to that until you take five off. And the two are not interchangeable. Put eighteen into the friends equation and you get five hundred and seventy five. Put thirteen into the people equation and you get three hundred and seventy five.
Reading the value back into the story is a separate step, and it is the one that gets skipped. Sometimes the letter is not a how many. It is a when. Two people are saving. One has four thousand, and adds six fifty a month. The other has five thousand and fifty, and adds five hundred a month. The second is ahead. The first is catching up. When are they level?
Let m be the number of months. Four thousand plus six fifty m, equals five thousand and fifty plus five hundred m. The letter is on both sides, so take five hundred m off each side. Then take four thousand off each side. A hundred and fifty m equals one thousand and fifty, so m is seven. Check it: both have eight thousand five hundred and fifty, and the gap closes by a hundred and fifty every month.
Here is one where the situation arrives as instructions. Think of a number. Take away three. Multiply by four. Add eight. Someone does that and announces twenty four. What did they start with? Do not guess. Build the expression one instruction at a time. The number: x. Take three: x minus three. Multiply by four — all of it, both pieces — four x minus twelve. Add eight: four x minus four.
So four x minus four equals twenty four, and four x minus four is four lots of x minus one. Four lots of something is twenty four, so that something is six, and x is seven. And you have more than an answer. A quarter of whatever they announce, plus one, is what they started with. Now one that looks like it needs two letters. Sixty marbles between two people. One has thirty more than the other.
Call them x and y. x plus y is sixty, and x is y plus thirty. Two letters, two equations, no way forward. So choose again. Call the smaller pile y. Then the bigger pile is not a new letter. It is y plus thirty. y, plus y plus thirty, equals sixty. Two y plus thirty equals sixty. Two y is thirty, so y is fifteen, and the other pile is forty five.
They make sixty and differ by thirty. Choosing the letter well was the whole job. Now run the machine backwards. Instead of solving an equation, write one with a given answer. Say the answer must be five. y plus one equals six works. So does three y equals fifteen. They look nothing alike, and both are satisfied by five and nothing else. Now build a chain, starting from three y equals fifteen.
Add six to each side: three y plus six equals twenty one. Divide each side by three: y plus two equals seven. Take two off each side: y equals five. Every line in that chain has the same answer as every other. So you can read it off the bottom and know it holds all the way up, without solving one of them. Which makes a trick possible. Think of any number, and keep it to yourself.
Double it. Add ten. Halve the result. Take away the number you first thought of. Then add three. You have eight. I know you do. Watch it in symbols. Your number is x. Double it, two x. Add ten, two x plus ten. Halve it: x plus five. And there it is — one x, on its own. Take away the number you thought of, and the x is gone. Five. Add three. Eight.
Every x lands on eight, negative or fraction alike, because after the halving there is exactly one of it to take away. One last job, and it is the first one turned round. Here is a bare equation. A hundred x, plus seventy five, equals two hundred and fifty. No situation, no words. What could it have come from? Read the pieces. The seventy five does not change, whatever x is.
The hundred is a count of something, and x is what each one costs. So: a delivery charge of seventy five, a hundred items at x each, a bill of two fifty. You can finish it. Take the seventy five off, a hundred x is a hundred and seventy five, so each item costs one and three quarters. The point is not the answer. It is that the letter meant something you chose, and every line reads back into a story.
That is what makes a modelling right. Not that it is the only one.
Where this fits
Taken from the notes each video was made from, not from the reading order — these are the ideas this one rests on and the ones that later rest on it.
Builds on
- Isolating the unknown, step by stepClass 7 · Ch 7, Finding the Unknown
- An equation is a claim of equality, not an instruction to computeClass 7 · Ch 7, Finding the Unknown
- Writing a situation as an expression before computing itClass 7 · Ch 2, Arithmetic Expressions
Either side of this one
- A pinch of history: where solving for an unknown came fromClass 7 · Ch 7, Finding the Unknown