Operations with integers

At a ski resort in Sierra Nevada, the thermometer read $-5^{\circ}C$ at seven in the morning. At noon, the temperature rose by $12^{\circ}C$ and, by nightfall, it dropped again by $9^{\circ}C$. What temperature did the thermometer read at the end of the day? Write the complete calculation using integers.

Calculate the absolute value of the following numbers and order them from smallest to largest:

$|-12|, |+5|, |0|, |-3|, |+1|$

Look at the following number line and the moves made by a grasshopper starting from point $0$. State the integer corresponding to each final position after making the jumps shown.

1. a) Which number does it reach after the first jump of $+4$?
2. b) Which number does it finally land on after the second move?

A delivery driver enters a building and parks his van on the second basement level (floor $-2$). He makes the following trips:

1. He goes up 5 floors to deliver a package.

2. He goes down 4 floors to pick up another box.

3. He goes up 7 floors for the last delivery.

Calculate which floor the delivery driver is on when he finishes his work. Use the following diagram, which shows the starting floor; you can mark each trip on it.

In your savings account for the end-of-year trip, you have an initial balance of $\text{€}\ 45$. During the week, you make the following transactions:

• You deposit $\text{€}\ 20$ given to you by your grandmother.

• You pay for a cinema ticket costing $\text{€}\ 9$.

• You buy a second-hand book for $\text{€}\ 6$.

• You are paid back $\text{€}\ 3$ that you had lent to a friend.

Calculate the final balance using an expression with positive and negative numbers.

In a young saver's bank account, the following transactions are recorded in one week: a deposit of $45$ €, a card payment of $-12$ €, a transfer received of $20$ €, and three automatic subscription charges of $-9$ € each.

If the balance at the start of the week was $-15$ €, what is the final balance? Write down the process using an addition of positive and negative numbers.

Fill in the empty boxes in this integer pyramid, knowing that the value in each box is equal to the sum of the two numbers directly below it. You must first calculate the boxes at the base to be able to complete the rest.

Solve the following multiplications and divisions, correctly applying the rule of signs:

1. a) $(+6) \times (-3) =$
2. b) $(-20) : (-5) =$
3. c) $(-8) \times (-4) =$
4. d) $(+35) : (-7) =$

Solve the following multiplication number pyramid. The value in each box is the product of the two integers located directly below it.

Expand the brackets and calculate the final result of the following expressions:

1. a) $+(+7) + (-3) - (+5) =$
2. b) $-(-4) - (+8) + (-2) =$
3. c) $+(-10) - (-6) - (-1) =$

Cian has solved the following combined operation with integers on the board:

His step-by-step working was as follows:

1) $12 - 4 = 8$

2) Inside the bracket: $(-3) - 5 = -8$

3) Multiplies inside the bracket: $(-8) \times (-2) = 16$

4) Final result: $8 \times 16 = 128$

Identify and explain in a reasoned manner the two mistakes made by Cian regarding the order of operations. Next, solve the operation correctly step by step yourself, carefully applying the rules of signs.

Calculate the result of these combined operations, respecting the order of operations:

1. a) $10 + 2 \times (-3) =$
2. b) $(-15) : 3 - 4 =$
3. c) $8 - (-10) : 2 =$
4. d) $(-5) \times (-2) + (-6) =$

In a survival video game, a player starts with $150$ energy points. For every hour spent exploring dark caves, they lose $25$ points, but for every "light crystal" they collect, they gain $40$ points.

If the player spends $4$ hours in the caves and manages to collect $3$ crystals, write an expression with integers that represents their final energy and calculate it.

Calculate the result of the following combined operations, respecting the order of operations and paying special attention to the signs:

1. a) $-5 \times [ -3 + 2 \times (4 - 7) ]$

1. b) $( -12 + 16 ) : ( 1 - 5 ) - 3 \times ( -2 )$

Fill in the blanks with the corresponding integer to make the equations true:

1. a) $-15 + ( \square ) = -32$

1. b) $4 \times ( \square ) = -36$

1. c) $( \square ) : (-3) = +12$

1. d) $-10 - ( \square ) = -4$

Solve the following combined operation, paying special attention to the order of operations and the use of brackets, square brackets, and braces:

$15 - \{ -3 \times [8 - (12 - 15) \times (-2)] + (-18) : (5 - 11) \}$

Find the value that the unknown $x$ must have for the following equation to hold true. Show all intermediate steps of your working:

$24 : \{ -2 \times [x - (12 - 17)] \} = -3$

Apply the distributive property to solve the following expressions in two different ways (first calculating inside the brackets and then distributing the factor):

1. a) $-3 \times ( 8 - 12 )$

1. b) $5 \times [ -4 + ( -6 ) ]$

1. c) $( -10 + 2 ) \times ( -4 )$

To calculate the numerical value of the expression:

two 2nd Year students propose two different strategies:

• Cian says: "The best way is to apply the distributive property: multiply $-7$ by each term inside the bracket and then add and subtract the results."

• Aoife says: "It is much faster to first solve the operations grouped inside the bracket and then multiply the result by $-7$."

1. a) Solve the expression following Cian's procedure in detail.
2. b) Solve the expression following Aoife's procedure in detail.
3. c) Compare both paths: which of the two methods was more advantageous in this case? Explain what numerical feature the terms inside the bracket had that justifies your choice.

Use the distributive property in reverse (factorising out a common factor) to calculate the result of the following expression as quickly and easily as possible:

$(-12) \times 35 + (-12) \times (-18) + (-12) \times 33$

Once the common factor is taken out, solve the expression inside the brackets and obtain the final result.