Algebraic expressions
Your weekly pocket money is 5 €, but you don't know how much your grandmother will give you on Sunday. If you call what she gives you $x$, you can already write how much money you will have in total: $5 + x$. You don't know the number, but you already know the calculation. That is algebra.
Letters that hold a number
Think of the letter as a closed box: there is a number inside, but you can't see it. You can do calculations with the box without opening it.
In the world of video games, it is very common to translate situations into mathematical language. If we let $x$ be the number of gems you have in a chest, express the…
In the world of video games, it is very common to translate situations into mathematical language. If we let $x$ be the number of gems you have in a chest, express the following situations in algebraic language:
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Triple the number of gems you have.
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The number of gems you will have if you find 15 more.
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The number of gems you have left after spending half on a skin.
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Double your gems minus 5 that you owe to a friend.
Translating sentences into expressions
Each key word becomes an operation. Memorise these equivalents:
| Everyday language | Algebraic language |
|---|---|
| Twice a number | $2x$ |
| Three times a number | $3x$ |
| Half of a number | $\dfrac{x}{2}$ |
| A number plus 5 | $x + 5$ |
| 5 less than a number | $x - 5$ |
| The square of a number | $x^2$ |
The same happens in geometry: the perimeter of a rectangle is the sum of its four sides, even if you don't know the measurements.
Kenia and her brother Adrián are comparing the money they have saved for the upcoming festival. Kenia has $x$ euros. Express in algebraic language the amount of money…
Kenia and her brother Adrián are comparing the money they have saved for the upcoming festival. Kenia has $x$ euros. Express in algebraic language the amount of money Adrián has in each of these independent cases:
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Adrián has 12 € more than Kenia.
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Adrián has twice as much as Kenia.
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Adrián has 5 € less than three times what Kenia has.
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If they put both their money together and the town hall gives them an extra 20 €, they have 100 € in total.
In PE lessons, the teacher asks us to run around a rectangular training pitch. We do not know the exact measurements, but we know that the length is $L$ metres and the…
In PE lessons, the teacher asks us to run around a rectangular training pitch. We do not know the exact measurements, but we know that the length is $L$ metres and the width is $a$ metres.
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Write the algebraic expression for the perimeter of the pitch.
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If the teacher tells us that the length is twice the width ($L = 2a$), rewrite the expression for the perimeter using only the letter $a$.
The numerical value
The numerical value is the result of replacing the letter with a number. Imagine that the cost of organising a birthday party is given by the expression $15n + 50$,…
The numerical value is the result of replacing the letter with a number. Imagine that the cost of organising a birthday party is given by the expression $15n + 50$, where $n$ is the number of guests.
Calculate how much the party will cost if you decide to invite:
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$n = 10$ friends.
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$n = 20$ friends.
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$n = 25$ friends.
Calculate the numerical value of the following expressions for the given values. Be very careful with negative signs and remember the order of operations: $2x + 7$ for…
Calculate the numerical value of the following expressions for the given values. Be very careful with negative signs and remember the order of operations:
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$2x + 7$ for $x = -3$
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$x^2 - 5$ for $x = 4$
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$3(a + b)$ for $a = 2, b = 5$
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$\frac{x}{2} + 10$ for $x = 8$
Patterns that depend on $n$
When a figure grows following the same rule every time, you can describe any step at once with an expression in $n$. Try the slider and notice how the number of pieces changes:
With 1 square you need 4 matchsticks; with 2 squares, 7; with 3, 10. Each new square adds 3 matchsticks, and there is always one loose one at the start: the rule is $3n + 1$.
Look at the following series of figures made with wooden sticks. We want to know how many sticks we will need to build a row of $n$ squares. If for 1 square you use 4…
Look at the following series of figures made with wooden sticks. We want to know how many sticks we will need to build a row of $n$ squares.
If for 1 square you use 4 sticks, for 2 squares you use 7 sticks and for 3 squares you use 10 sticks:
- a) How many sticks would you add for each new square?
- b) Write the formula or algebraic expression that lets you calculate the total number of sticks for $n$ squares.
We are building a series of figures with square tiles following a certain pattern. Look at the first three figures in the series: a) If $n$ represents the figure number,…
We are building a series of figures with square tiles following a certain pattern. Look at the first three figures in the series:
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a) If $n$ represents the figure number, write an algebraic expression indicating how many square tiles figure $n$ will have.
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b) How many tiles will figure number $25$ have?
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c) Which figure number will have exactly $40$ tiles?
Parts of a monomial and like terms
The simplest expressions, with a single term, are called monomials. They have three parts:
Two monomials are like terms if they have the same literal part: $3x$ and $5x$ are; $3x$ and $3x^2$ are not. You can only add or subtract like terms, just as you can't add euros and gems:
$$3x + 5x - x = (3 + 5 - 1)x = 7x$$Identifying the parts of a monomial is essential to "speak" algebra. Complete the information for the following monomials mentally or in your notebook:…
Identifying the parts of a monomial is essential to "speak" algebra. Complete the information for the following monomials mentally or in your notebook:
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$5x^3 \rightarrow$ Coefficient: ___ , Literal part: ___ , Degree: ___
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$-ab^2 \rightarrow$ Coefficient: ___ , Literal part: ___ , Degree: ___
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$\frac{1}{2}y \rightarrow$ Coefficient: ___ , Literal part: ___ , Degree: ___
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$-7 \rightarrow$ Coefficient: ___ , Literal part: ___ , Degree: ___
To simplify expressions, we can only add or subtract like monomials (those with the same literal part). Fully simplify the following expressions: $3x + 5x - x$…
To simplify expressions, we can only add or subtract like monomials (those with the same literal part). Fully simplify the following expressions:
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$3x + 5x - x$
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$2a + 7b - a + 3b$
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$4x^2 + 2x^2 - 5x$
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$m + m + m + n + n$
Back to the start: If on Sunday your grandmother gives you 10 €, the unknown becomes $x = 10$ and the numerical value of your total money is $5 + 10 = 15$ €.