Algebraic language

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:

• a) Double the number of gems.

• b) Triple the gems plus $50$ extra points.

• c) The number of gems they would have if they lost $5$ gems.

• 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) The total price of buying two hoodies.

• c) The total price of buying one hoodie and a pair of trousers.

We want to fence off an area for a music festival that is square in shape. We do not know the length of the side yet, so we will call it $x$.

• a) Write an algebraic expression for the perimeter of the area (the sum of all its sides).

• b) Write an algebraic expression for the area of the area.

In a popular adventure video game, players earn different scores depending on their actions. If we let $x$ be the number of gold chests found and $y$ be the number of side quests completed, express the following situations in algebraic language:

• Triple the number of gold chests found.

• The number of side quests plus five extra bonus quests.

• Double the sum of the gold chests and side quests.

• Half of the chests found after losing three of them in a battle.

• The total points if each chest is worth 100 points and each quest is worth 50 points.

A group of friends wants to buy a video games console as a gift for a classmate. The console costs $C$ euros. They decide that two of the friends, who are siblings, will each pay only half of what each of the others pays.

1. a) If $x$ is the amount of money contributed by each of the non-sibling friends, and there are a total of $6$ friends (including the two siblings), express the total cost $C$ of the console algebraically.
2. b) If the cost of the console is $300$ euros, calculate how much money each of the siblings must contribute.

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 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 is twice Ioana's age. How old is her brother?

• d) Her mother is three times the age Ioana will be in $1$ year's time.

During a review activity, the teacher asks students to translate the following sentence into algebraic language: "Three times the predecessor of a number $x$, increased by twice the successor of that number $x$".

Two students hand in different proposals:

• Cian writes: $3 \times x - 1 + 2 \times x + 1$

• Aoife writes: $3 \times (x - 1) + 2 \times (x + 1)$

1. a) Analyze Cian's proposal: identify the two mistakes made by not using parentheses and explain what everyday sentence would correspond word-for-word to the expression he wrote.
2. b) Expand and simplify Aoife's expression by applying the distributive property and grouping like terms to obtain a reduced expression of the form $a \times x + b$.
3. c) Reason and justify: could it be possible that, with $x$ being an integer, the final value of Aoife's expression is equal to $40$? Set the reduced expression equal to $40$, solve for $x$, and justify your conclusion by analyzing whether the result belongs to the set of integers.

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):

• a) $3x + 5$ when $x = 2$

• b) $10 - 2n$ when $n = 3$

• c) $a^2 + 1$ when $a = 5$

• d) $\frac{x}{2} + 7$ when $x = 10$

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 $1$):

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:

• $x + y$

• $2x + 3y$

• $x - 10$

• $0.85 \times x$

• $3(x + y)$

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 have?

• 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 ($n=1$) seats 6 people, two tables ($n=2$) seat 10 people and three tables ($n=3$) seat 14 people:

• Create an algebraic expression to calculate the number of people who can sit as a function of the number of tables $n$.

• Calculate how many people can 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 previous one to save material. Look at the following design sequence:

If we call $n$ the number of flowerbeds we want to place in a row:

1. a) Find an algebraic expression to calculate the number of tiles needed for $n$ flowerbeds.
2. b) Use this expression to calculate how many tiles would be needed for a row of $25$ flowerbeds.

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. Think of any number ($n$).

2. Add 5 to it.

3. Multiply the result by 2.

4. Subtract 10 from the value obtained.

5. Divide the result by 2.

6. 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 everything by $3$.

5. Subtract twice the number you originally thought of.

1. a) Translate each step of the trick into algebraic language and write the final simplified expression.
2. 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.

In a recreational mathematics workshop, the following number magic trick is proposed:

1. Think of any integer.

2. Add $5$ to it.

3. Multiply the result obtained by $2$.

4. Subtract $6$ from it.

5. Divide the result by $2$.

6. Subtract the number you had thought of at the beginning.

The final result is always $2$, no matter the chosen number!

To convince the class that this is not a coincidence, two approaches are discussed:

• Method A (Empirical verification): Test the trick with five very different numbers (a single-digit number, a three-digit number, zero, a negative number, and a fraction).

• Method B (Algebraic proof): Call the initial number $n$, algebraically chain all the operations into a single expression, and simplify it step by step.

1. a) Fully develop Method B: write the resulting algebraic expression after each of the 6 instructions and simplify it to prove that the final result is $2$.
2. b) Choose the method and justify: compare both approaches, explaining why Method A, despite testing many cases, does not prove that the trick always works, and what definitive advantage the algebraic language of Method B provides.