Algebraic language
A magician asks a spectator: "think of a number, add 5, multiply by 2, subtract 10 and divide by 2". The spectator ends up with their original number, whatever it was. Magic? No: algebra. By the end of this section you will be able to prove it yourself.
Sentences with several steps
For long sentences, translate in parts, from the inside out, just as you follow a recipe step by step.
In the world of video games, we sometimes use formulas to calculate scores. Translate the following phrases into algebraic language, letting $x$ be the number of gems a…
In the world of video games, we sometimes use formulas to calculate scores. Translate the following phrases into algebraic language, letting $x$ be the number of gems a player has obtained:
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a) Double the number of gems.
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b) Triple the gems plus $50$ extra points.
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c) The number of gems they would have if they lost $5$ gems.
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d) Half the gems they have.
In a clothes shop, a hoodie costs $s$ euros. If a pair of trousers costs $12$ euros less than the hoodie, express in algebraic language: a) The price of the trousers. b)…
In a clothes shop, a hoodie costs $s$ euros. If a pair of trousers costs $12$ euros less than the hoodie, express in algebraic language:
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a) The price of the trousers.
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b) The total price of buying two hoodies.
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c) The total price of buying one hoodie and a pair of trousers.
The next and the previous
If a number is $n$, the next one is $n + 1$ and the previous one is $n - 1$. It does not matter which number $n$ is: the rule always works.
If $n$ represents any integer, write the algebraic expression for: a) The number after $n$. b) The number before $n$. c) Triple the number after $n$. d) The sum of a…
If $n$ represents any integer, write the algebraic expression for:
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a) The number after $n$.
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b) The number before $n$.
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c) Triple the number after $n$.
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d) The sum of a number and the number before it.
Let $x$ be Ioana's current age. Express her family's ages algebraically: a) The age Ioana will be in $8$ years' time. b) The age Ioana was $3$ years ago. c) Her brother…
Let $x$ be Ioana's current age. Express her family's ages algebraically:
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a) The age Ioana will be in $8$ years' time.
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b) The age Ioana was $3$ years ago.
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c) Her brother is twice Ioana's age. How old is her brother?
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d) Her mother is three times the age Ioana will be in $1$ year's time.
Reading algebra in reverse
You also need to be able to go in the opposite direction: from the expression to the sentence in plain English. It is the same skill, but the other way round.
| Expression | In plain language |
|---|---|
| $x + y$ | What a T-shirt and a pair of trousers cost together |
| $2x + 3y$ | Twice the T-shirt plus three times the trousers |
| $0,85 \cdot x$ | 85 % of the price of the T-shirt (a 15 % discount) |
| $3(x + y)$ | Three times what the two items cost together |
In a sports shop, a football shirt costs $x$ euros and a pair of shorts costs $y$ euros. Translate the following expressions from algebraic language into plain English:…
In a sports shop, a football shirt costs $x$ euros and a pair of shorts costs $y$ euros. Translate the following expressions from algebraic language into plain English:
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$x + y$
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$2x + 3y$
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$x - 10$
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$0,85 \cdot x$
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$3(x + y)$
Patterns: find the rule
In many problems, a figure grows step by step and you have to find the expression that works for any step $n$, not just the first few.
Look at the following sequence of figures formed by orange dots. If we let $n$ be the position of the figure (Figure 1, Figure 2...): a) How many dots will Figure 4…
Look at the following sequence of figures formed by orange dots. If we let $n$ be the position of the figure (Figure 1, Figure 2...):
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a) How many dots will Figure 4 have?
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b) Write an algebraic expression that tells us how many dots are in Figure $n$.
Look at the following sequence of hexagonal tables placed in a line for a school celebration. Each hexagon represents a table where students sit. If one table ($n=1$)…
Look at the following sequence of hexagonal tables placed in a line for a school celebration. Each hexagon represents a table where students sit.
If one table ($n=1$) seats 6 people, two tables ($n=2$) seat 10 people and three tables ($n=3$) seat 14 people:
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Create an algebraic expression to calculate the number of people who can sit as a function of the number of tables $n$.
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Calculate how many people can sit if 15 tables are joined together.
The magician's trick, explained
Now you can take apart the trick from the beginning. Follow the steps with a box $n$ instead of a number:
Valentina suggests a magic trick to her brother using algebraic language. Follow the steps and then explain algebraically why the same result is always obtained: 1.…
Valentina suggests a magic trick to her brother using algebraic language. Follow the steps and then explain algebraically why the same result is always obtained:
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Think of any number ($n$).
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Add 5 to it.
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Multiply the result by 2.
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Subtract 10 from the value obtained.
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Divide the result by 2.
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What number do you get compared to the initial number? Write the simplified expression for the process.
A magician asks a spectator to follow these instructions: 1. Think of any natural number $n$. 2. Multiply it by $6$. 3. Add $12$ to the previous result. 4. Divide…
A magician asks a spectator to follow these instructions:
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Think of any natural number $n$.
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Multiply it by $6$.
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Add $12$ to the previous result.
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Divide everything by $3$.
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Subtract twice the number you originally thought of.
- a) Translate each step of the trick into algebraic language and write the final simplified expression.
- b) What is the final result of the trick? Does it depend on the number the spectator thought of? Justify your answer using algebraic language.