Algebraic Language
A magician asks a student: "Think of a number. Add 4. Triple the result. Subtract 12. Divide by 3." Then she says, "You're thinking of... your original number!" She is right every time, no matter what number you pick. Magic? No, algebra. By the end of this page you will be able to prove the trick yourself.
Words to Algebra
A variable is a letter that stands for a number we don't know yet (or a number that can change). An algebraic expression combines numbers, variables, and operations, like $3x + 50$.
To translate a long phrase, split it into pieces, translate each piece, and then put the pieces together, like following a recipe one step at a time.
| Words | Algebra |
|---|---|
| sum, plus, more than, increased by | $+$ |
| difference, minus, less than, decreased by | $-$ |
| times, product, twice ($\times 2$), triple ($\times 3$) | multiply |
| half, quotient, split into equal parts | divide, or a fraction |
| the square of, the cube of | $x^2$, $x^3$ |
| $10\%$ of $x$ | $0.10x$ |
In the world of video games, we sometimes use formulas to calculate scores. Translate the following sentences into algebraic language, calling $x$ the number of gems a…
In the world of video games, we sometimes use formulas to calculate scores. Translate the following sentences into algebraic language, calling $x$ the number of gems a player has collected:
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a) Twice the number of gems.
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b) Three times 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 of the gems they have.
In a clothing store, a hoodie costs $s$ dollars. If a pair of pants costs $12$ dollars less than the hoodie, express in algebraic language: a) The price of the pants. b)…
In a clothing store, a hoodie costs $s$ dollars. If a pair of pants costs $12$ dollars less than the hoodie, express in algebraic language:
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a) The price of the pants.
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b) The total price of buying two hoodies.
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c) The total price of buying a hoodie and a pair of pants.
In a popular adventure video game, players get different scores depending on their actions. If we call $x$ the number of gold chests found and $y$ the number of side…
In a popular adventure video game, players get different scores depending on their actions. If we call $x$ the number of gold chests found and $y$ the number of side quests completed, express the following situations in algebraic language:
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Three times the gold chests found.
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The number of side quests plus five extra bonus quests.
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Twice the sum of the gold chests and the side quests.
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Half of the chests found after losing three of them in a battle.
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The total points if each chest is worth 100 points and each quest is worth 50 points.
Next, Previous, and Consecutive Numbers
If a number is $n$, the next number is $n + 1$ and the previous number is $n - 1$. It doesn't matter which number $n$ is: the rule always works.
Ages work the same way. If Mia is $x$ years old now, in 8 years she will be $x + 8$, and 3 years ago she was $x - 3$. Moving forward in time adds, and moving back subtracts.
If a problem tells you the first number is $x - 2$, build the others from it: the next two are $x - 1$ and $x$.
We call $x$ Mia's current age. Express algebraically the ages of her family: a) The age Mia will be in $8$ years. b) The age Mia was $3$ years ago. c) Her brother is…
We call $x$ Mia's current age. Express algebraically the ages of her family:
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a) The age Mia will be in $8$ years.
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b) The age Mia was $3$ years ago.
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c) Her brother is twice Mia's age. What age is the brother?
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d) Her mother is three times the age Mia will be in $1$ years.
If $n$ represents any integer, write the algebraic expression for: a) The next number after $n$. b) The previous number before $n$. c) Three times the next number after…
If $n$ represents any integer, write the algebraic expression for:
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a) The next number after $n$.
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b) The previous number before $n$.
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c) Three times the next number after $n$.
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d) The sum of a number and its previous number.
Express the following complex sentences in algebraic language, using a single variable $x$ for the unknown number: a) Three times the difference between the square of a…
Express the following complex sentences in algebraic language, using a single variable $x$ for the unknown number:
- a) Three times the difference between the square of a number and its half.
- b) The sum of three consecutive natural numbers, knowing that the first of them is $x-2$.
- c) The final price of a hoodie that cost $x$ dollars, if first a $10\%$ percent increase is applied for shipping costs and then a discount of $5$ dollars for a coupon.
- d) The difference between the cube of the next number and twice its previous number.
Reading Algebra Backward
You also need to go the other way: from an expression to a sentence in plain English. It is the same skill, in reverse. Let $x$ be the price of a movie ticket and $y$ the price of a popcorn.
| Expression | In plain English |
|---|---|
| $x + y$ | one ticket and one popcorn |
| $2x + y$ | two tickets and one popcorn |
| $x - 3$ | the ticket price after a $\$3$ coupon |
| $0.75x$ | $75\%$ of the ticket price (that is, $25\%$ off) |
| $2(x + y)$ | two friends, each buying a ticket and a popcorn |
When an expression has a number and a variable part, ask what each piece means. A skate park charges $C = 3h + 5$ dollars, where $h$ is the number of hours you skate. The $3$ multiplies $h$, so it is the price per hour ($\$3$ each hour). The $5$ doesn't change with $h$, so it is a fixed fee ($\$5$ to get in).
In a sporting goods store, a soccer jersey costs $x$ dollars and a pair of shorts costs $y$ dollars. Translate from algebraic language to ordinary language what the…
In a sporting goods store, a soccer jersey costs $x$ dollars and a pair of shorts costs $y$ dollars. Translate from algebraic language to ordinary language what the following expressions mean:
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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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$3(x + y)$
A TikTok* content creator analyzes their monthly income. If $v$ represents the number of videos published in a month, their earnings in dollars are defined by the…
A TikTok content creator analyzes their monthly income. If $v$ represents the number of videos published in a month, their earnings in dollars are defined by the algebraic expression: $G = 0,5 \cdot v + 12$.
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What do the numbers $0,5$ and $12$ represent in this context?
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What will their earnings be if they have managed to publish a total of $v = 40$ videos this month?
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If next month they decide to publish twice as many videos as this month, write the new expression for their earnings.
Monomials: Coefficient, Variable Part, Degree
A monomial is a single term: a number, a variable, or a product of a number and variables, like $-5x^2$ or $7ab$. A monomial has three parts.
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The coefficient is the number multiplied by the variables. If you don't see a number, the coefficient is $1$ (and for $-y$ it is $-1$).
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The variable part is the letters with their exponents. Some problems call this the literal part.
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The degree is the sum of the exponents of the variables. A letter with no written exponent has exponent $1$. A plain number has degree $0$.
Copy the following table into your notebook and complete the information about these monomials (remember that if there is no number in front of the letter, the…
Copy the following table into your notebook and complete the information about these monomials (remember that if there is no number in front of the letter, the coefficient is $1$):
| Monomial | Coefficient | Literal part | Degree |
|---|---|---|---|
| $7a^2$ | |||
| $-5x$ | |||
| $x^3y$ | |||
| $\frac{1}{2}b$ |
Copy and complete the following table about monomials in your notebook. Remember that if the coefficient does not appear, it is understood to be one. | Monomial |…
Copy and complete the following table about monomials in your notebook. Remember that if the coefficient does not appear, it is understood to be one.
| Monomial | Coefficient | Literal part | Degree |
|---|---|---|---|
| $-5x^2$ | |||
| $ab^3c$ | |||
| $\frac{3}{4}x^4y$ | |||
| $-z$ | |||
| $12$ |
Evaluating Expressions
To evaluate an expression, replace each variable with its value (substitute) and follow the order of operations. Between a number and a letter there is an invisible multiplication: $3x$ means $3 \times x$.
Two expressions that give the same values
Sometimes two expressions that look different always give the same answer. Let's evaluate $Q = \dfrac{x^2 - 9}{x + 3}$.
| $x$ | $x^2 - 9$ | $x + 3$ | $Q$ |
|---|---|---|---|
| $0$ | $-9$ | $3$ | $-3$ |
| $3$ | $0$ | $6$ | $0$ |
| $7$ | $40$ | $10$ | $4$ |
Look at the pattern: $-3, 0, 4$ are exactly $x - 3$ for $x = 0, 3, 7$. Evaluating a few values and comparing is a great way to guess a simpler equivalent expression. Then check that your guess works for more values.
What about $x = -3$? The bottom is $-3 + 3 = 0$, and the top is $9 - 9 = 0$. A fraction means division, and you can't divide by zero. Think about why: $12 \div 3 = 4$ because $4 \times 3 = 12$. But $12 \div 0$ would need a number that times $0$ gives $12$, and there isn't one. So any fraction whose denominator becomes $0$ has no value.
Calculate the numerical value of the following expressions for the given value (remember that between a number and a letter there is always an invisible multiplication):…
Calculate the numerical value of the following expressions for the given value (remember that between a number and a letter there is always an invisible multiplication):
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a) $3x + 5$ when $x = 2$
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b) $10 - 2n$ when $n = 3$
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c) $a^2 + 1$ when $a = 5$
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d) $\frac{x}{2} + 7$ when $x = 10$
Consider the algebraic expression $P(x) = \frac{x^2 - 4}{x + 2}$. a) Calculate the numerical value of $P(x)$ for $x = 0, x = 2$ and $x = 10$. b) Observing the results,…
Consider the algebraic expression $P(x) = \frac{x^2 - 4}{x + 2}$.
- a) Calculate the numerical value of $P(x)$ for $x = 0, x = 2$ and $x = 10$.
- b) Observing the results, could you propose a much simpler expression that is equivalent to $P(x)$ for any value of $x$ (except for $x = -2$)?
- c) Why do you think the numerical value of $P(x)$ cannot be calculated for $x = -2$? Reason your answer based on what you know about fractions.
Perimeter and Formulas
The perimeter of a shape is the distance around it: add up all the sides. If the sides are expressions, you add expressions. A square with side $x$ has perimeter $x + x + x + x = 4x$.
When you add expressions, you can combine like terms: terms with the same variable part. $3x + 2x = 5x$, but $3x + 2$ can't be combined because one has $x$ and the other doesn't.
A border around a rectangle
Now a garden that is $l$ feet long and $w$ feet wide gets a path $1$ ft wide all the way around. The path adds $1$ ft on each side, so it adds $2$ ft to the length and $2$ ft to the width.
The outer rectangle is $(l + 2)$ by $(w + 2)$. Its perimeter is $2(l + 2) + 2(w + 2) = 2l + 2w + 8$. The area of the path is the outer area minus the inner area.
We want to fence an enclosure for a music festival that has the shape of a square. We do not know how long the side is yet, so we will call it $x$. a) Write an algebraic…
We want to fence an enclosure for a music festival that has the shape of a square. We do not know how long the side is yet, so we will call it $x$.
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a) Write an algebraic expression for the perimeter of the enclosure (the sum of all its sides).
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b) Write an algebraic expression for the area of the enclosure.
Calculate the perimeter of the following pickleball court whose dimensions are given in feet by algebraic expressions. Once the simplified expression is obtained,…
Calculate the perimeter of the following pickleball court whose dimensions are given in feet by algebraic expressions. Once the simplified expression is obtained, calculate its numerical value for $x = 4$ feet.
In the design of a modern swimming pool, a rectangular water area of length $l$ and width $a$ is to be surrounded by a wooden border $2$ feet wide all around its…
In the design of a modern swimming pool, a rectangular water area of length $l$ and width $a$ is to be surrounded by a wooden border $2$ feet wide all around its perimeter.
- a) Write the algebraic expression that represents the total area occupied by the wood (the border).
- b) Simplify the expression obtained in the previous section.
- c) What would be the numerical value of that area if the pool is $10$ feet long and $5$ feet wide?
Word Problems: Build the Expression in Parts
Many real problems have several people or several stages. The trick is the same as before: break the story into parts, write an expression for each part, then combine them.
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Total = the sum of the parts.
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Cost = number of items $\times$ price of each.
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Distance = rate $\times$ time ($d = rt$).
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If a problem gives you a time in minutes and the other time in hours, first convert to one unit. For example, $30$ minutes $= 0.5$ hour and $15$ minutes $= 0.25$ hour.
When a problem tells you the total, set the expression equal to that number and undo the operation. Three friends split a $\$45$ bill equally, paying $x$ each, so $3x = 45$. Divide both sides by $3$: $x = 15$.
Comparing amounts works the same way. If Priya has $p$ followers, "three times as many" is $3p$ and "200 more than that" is $3p + 200$.
Sophia, Brandon, and Chloe are comparing how many followers they have on social media. Brandon has twice as many followers as Sophia. Chloe has 500 fewer followers than…
Sophia, Brandon, and Chloe are comparing how many followers they have on social media. Brandon has twice as many followers as Sophia. Chloe has 500 fewer followers than Brandon. If we call $s$ the number of followers Sophia has:
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Express in algebraic language the followers Brandon has.
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Express in algebraic language the followers Chloe has.
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Write an expression that represents the sum of the followers of the three friends.
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If Sophia has $s = 1200$ followers, how many followers do they have among the three of them?
A group of friends wants to give a video game console to a classmate. The console costs $C$ dollars. They decide that two of the friends, who are brothers, will only pay…
A group of friends wants to give a video game console to a classmate. The console costs $C$ dollars. They decide that two of the friends, who are brothers, will only pay half of what each of the rest pays.
- a) If $x$ is the amount of money contributed by each of the friends who are not brothers, and there is a total of $6$ friends (including the two brothers), express algebraically the total cost $C$ of the console.
- b) If the cost of the console is $300$ dollars, calculate how much money each of the brothers must contribute.
In a triathlon competition, an athlete runs at a constant speed of $v$ mph. a) Express algebraically the distance they travel in a time of $t$ hours and $15$ minutes.…
In a triathlon competition, an athlete runs at a constant speed of $v$ mph.
- a) Express algebraically the distance they travel in a time of $t$ hours and $15$ minutes. (Remember to express the time in a single unit).
- b) If the athlete decides to increase their speed by $2$ mph during the second half of the time, write the expression for the total distance travelled in a total time of $t$ hours.
- c) Calculate the total distance from part b) if $v = 12$ mph and $t = 2$ hours.
Patterns: Find the Rule
In many problems a figure grows step by step, and you need an expression that works for any step number $n$, not just the first few.
There is a second way to see it, by counting and then fixing the overlap. $n$ separate squares need $4n$ sticks. But each pair of neighbors shares one stick, and there are $n - 1$ shared sticks. So the total is $4n - (n - 1) = 4n - n + 1 = 3n + 1$. Both methods agree.
Use this "count everything, then subtract what is shared" idea whenever shapes in a pattern touch each other, such as tiles or tables placed side by side.
Always read the figure to see how many tiles each pair really shares. If it is a different number than $2$, use that number in the subtraction. Rows of hexagonal tables for a school party work the same way. Each table has $6$ seats, one on each side. When two tables are pushed together, the side they touch is blocked on both tables, so $2$ seats are lost for each join. That gives $6n - 2(n - 1) = 4n + 2$ seats for $n$ tables.
Observe the following series of figures formed by orange dots. If we call $n$ the position of the figure (Figure 1, Figure 2...): a) How many dots will Figure 4 have? b)…
Observe the following series of figures formed by orange dots. If we call $n$ 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 there are in Figure $n$.
Observe the following series of hexagonal tables placed in a line for a school celebration. Each hexagon represents a table where students sit. If 6 people fit at one…
Observe the following series of hexagonal tables placed in a line for a school celebration. Each hexagon represents a table where students sit.
If 6 people fit at one table ($n=1$), 10 people fit at two tables ($n=2$) and 14 people fit at three tables ($n=3$):
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Create an algebraic expression that allows calculating the number of people who can sit depending on the number of tables $n$.
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Calculate how many people will be able to sit if 15 tables are joined together.
In a garden design company, hexagonal tiles are used to surround flowerbeds. If we place the flowerbeds in a line, each additional flowerbed shares some tiles with the…
In a garden design company, hexagonal tiles are used to surround flowerbeds. If we place the flowerbeds in a line, each additional flowerbed shares some tiles with the previous one to save material. Observe the following design sequence:
Patterns with Squares
Not every pattern goes up by the same amount each time. Look at square arrays of dots.
The numbers $1, 4, 9, 16, \ldots$ don't go up by a constant amount (the jumps are $3, 5, 7, \ldots$). Instead, level $n$ is an $n$ by $n$ square, so it has $n \times n = n^2$ items. This is the rule for a square pattern.
When boxes are stacked into a pyramid, the total for a pyramid with $n$ levels is the running total of the square numbers. Each new total is the previous total plus the new level $n^2$.
| Levels in the pyramid | Boxes on the new level | Total boxes |
|---|---|---|
| $1$ | $1^2 = 1$ | $1$ |
| $2$ | $2^2 = 4$ | $1 + 4 = 5$ |
| $3$ | $3^2 = 9$ | $5 + 9 = 14$ |
| $4$ | $4^2 = 16$ | $14 + 16 = 30$ |
| $5$ | $5^2 = 25$ | $30 + 25 = 55$ |
To find how many more boxes you need to go from a 4-level pyramid to a 5-level pyramid, you just need the new level: $5^2 = 25$.
In a logistics warehouse, boxes are stacked forming square-based pyramidal structures. On the highest level (level 1) there is $1$ box. On level 2 (below the first)…
In a logistics warehouse, boxes are stacked forming square-based pyramidal structures. On the highest level (level 1) there is $1$ box. On level 2 (below the first) there are $4$ boxes forming a $2 \times 2$ square. On level 3 there are $9$ boxes forming a $3 \times 3$ square.
Why the Magic Trick Works
Now you can take apart the trick from the start. Instead of a number, we follow a box labeled $n$ through each step.
Parentheses matter in step 3: "times 3" applies to the whole result, so it is $3(n + 4) = 3n + 12$. After subtracting $12$ we have $3n$, and dividing by $3$ leaves $n$.
Some tricks always give the same answer no matter what number you start with, because the $n$'s cancel. For example: think of $n$, add $6$, subtract the number you started with. We get $n + 6 - n = 6$. The result is always $6$ and doesn't depend on $n$ at all.
Back to the beginning: the magician's trick works because $[3(n + 4) - 12] \div 3 = n$, so she can name your number every time.
Emma proposes a magic trick to her brother using algebraic language. Follow the steps and then explain algebraically why the same result is always obtained: 1. Think of…
Emma proposes 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 of 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 thought of at the beginning.
- a) Translate each step of the trick into algebraic language and write the simplified final 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.