Algebraic language

Translate these word phrases into algebraic expressions using a single variable $x$:

1. a) The sum of a number and its half.
2. b) Three times the result of subtracting five from a number.
3. c) The square of the number preceding a given number.
4. d) The difference between double a number and the next number.

Match each phrase with its corresponding algebraic expression:

1. Double a number plus five units.                      a) $x - 5$

2. A number decreased by five units.                         b) $2x + 5$

3. One fifth of a number.                                           c) $x^5$

4. A number raised to the fifth power.                           d) $\frac{x}{5}$

In a video game shop, the cost of renting a console is a fixed fee of $10$ euros plus $2$ euros per hour of use.

1. a) Write the algebraic expression for the total cost for $h$ hours.
2. b) How much will Aoife pay if she rents the console for $3$ hours?

Cian rents a shared electric scooter in Dublin. The company charges a fee of $1$ € for the initial unlock plus $0.25$ € for each minute of the ride. To calculate the cost in euros of a ride of $m$ minutes, Cian writes the expression $(1 + 0.25) \times m$ in his notebook.

1. Identify the conceptual error that Cian made when setting up this expression.

2. Explain in your own words why the fixed term should never be multiplied by the time variable in this type of pricing.

3. Write the correct algebraic expression to calculate the total cost of the ride as a function of the minutes $m$.

A courier company charges its customers based on the weight of the parcels and the distance of the delivery. The base fee is $B$ euros. To this amount is added:

• $2.5$ euros for each kilogram of weight ($w$).

• $0.15$ euros for each kilometre of distance ($d$).

If the customer is "Premium", a discount of $15\%$ is applied to the total delivery price. Write the algebraic expression that determines the total cost of sending a parcel for a "Premium" customer in terms of $B$, $w$ and $d$.

Translate the following properties regarding integers into algebraic language:

1. The square of the sum of any two numbers, $a$ and $b$, is equal to the sum of their squares plus double their product.

2. The sum of the squares of three consecutive numbers, $n$ being the smallest of them.

3. The difference between the cube of a number $x$ and the square of its half.

4. The quotient of the sum of a number and its reciprocal, and three times that number.

Two 2nd Year students, Cian and Aoife, want to prove in a general way that "the sum of three consecutive integers is always divisible by $3$". Each proposes a different algebraic approach:

• Cian's method: he calls the first number $n$, the second $n + 1$, and the third $n + 2$.

• Aoife's method: she calls the central number $n$, the previous one $n - 1$, and the following one $n + 1$.

• Expand and simplify the sum proposed by Cian. Justify why the expression obtained proves that the result is always a multiple of $3$, regardless of the value of the integer $n$.

• Expand and simplify the sum proposed by Aoife. Justify why it also proves the same property.

• Compare both methods: which of the two is more direct and elegant when simplifying the numerical terms? Justify your answer.

• A classmate says: "Why complicate things with letters? Just by checking that $4+5+6=15$ and that $10+11+12=33$, you can already see that the rule works". Refute their argument by explaining what the essential function of algebra is compared to simply testing with numerical examples.

In the local school futsal tournament, $n$ teams participate. In the preliminary phase, all teams must play against each other exactly once (single round-robin system). The organisation uses the algebraic expression $P = \frac{n \times (n - 1)}{2}$ to calculate the total number of matches that must be scheduled.

1. Interpret the logical structure of the formula: explain why the number of teams $n$ is multiplied by $(n - 1)$, and why it is mandatory to divide that product by $2$.

2. Since the number of matches must necessarily be an integer for any number of teams $n$ ($n \ge 2$), prove algebraically that $P$ always yields an exact integer result by studying the two possible cases separately:

3. Case A: The number of teams is even ($n = 2k$, where $k$ is a natural number).

4. Case B: The number of teams is odd ($n = 2k + 1$, where $k$ is a natural number).

(Substitute each form into the formula for $P$ and simplify the fraction).

1. If the municipal sports centre only has available hours to play a maximum of $20$ matches over the weekend, determine using the algebraic formula whether it is possible to admit a tournament of $7$ teams.

Copy and complete the following table by identifying the components of these monomials:

Calculate the numerical value of the following expressions for the given values:

1. a) $3x + 10$ for $x = 4$
2. b) $x^2 - 5$ for $x = 3$
3. c) $2(a + b)$ for $a = 5$ and $b = 2$
4. d) $\frac{12}{x}$ for $x = 6$

Analyse the following algebraic expressions and complete the required information in each case:

1. In the expression $7x^2y - 5x + 3$: How many terms does it have? What is the constant term?

2. Given the monomial $-\frac{2}{3}ab^4$: State its coefficient, its literal part and its total degree.

Find the algebraic expression that represents the area of the following rectangle, knowing that its base is $5$ units longer than its height ($x$):

Find the algebraic expression for the perimetre of the following figure, which represents the floor plan of a small rectangular play area.

On a square steel sheet with side length $x$ centimetres, a worker must cut out four identical circles, one at each corner, to manufacture an industrial part. The radius of each circle is $r$ centimetres.

1. Write the algebraic expression for the remaining area of the sheet after making the four cut-outs (use $\pi$ as a symbol).

2. Knowing that the radius of the circles is a quarter of the side length of the sheet, that is $r = \frac{x}{4}$, simplify the previous expression so that it depends only on the variable $x$.

Observe the following sequence of patterns made with dots and determine the general term $a_n$ that allows you to calculate the number of dots needed for any pattern $n$.

A local council wants to install a wooden fence around a new playground. The fence is built by joining rectangular panels using vertical posts. Each panel requires $3$ horizontal slats. To support the structure, a post is placed at the beginning of the fence and another at the end of each panel.

1. Write an algebraic expression to calculate the total number of horizontal slats needed for a fence with $n$ panels.

2. Write an algebraic expression for the total number of posts needed for a fence with $n$ panels.

3. If each post costs $p$ euros and each horizontal slat costs $l$ euros, write the expression for the total cost of materials for a fence with $n$ panels.

Observe the following sequence of patterns made of small hexagons. The first pattern has a central hexagon surrounded by six others. In each step, a new layer of hexagons is added around the previous pattern.

1. Determine how many hexagons layer number $n$ has (without counting those in the inner layers).

2. Find the algebraic expression that represents the total number of hexagons that form the complete pattern when it has $n$ layers around the central hexagon.

(Hint: The number of hexagons added in each layer follows a sequence).