Operations with Integers

A Magic Grid of Integers

A grid containing some numbers is given below. Follow the steps as shown until no number is left.

When there are no more unstruck numbers, stop. Multiply the circled numbers. An example is shown below.

Try again, and choose different numbers this time. What product did you get? Was it different from the first time? Try a few more times with different numbers!

Play the same game with the grid below. What answer do you get?

What is so special about these grids? Is the magic in the numbers or the way they are arranged or both? Can you make more such grids?

Let us try to find a few more pairs of numbers from their sums and differences:

(a) Sum = 27, Difference = 9

(b) Sum = 4, Difference = 12

(c) Sum = 0, Difference = 10

(d) Sum = 0, Difference = -10

(e) Sum = -7, Difference = -1

(f) Sum = -7, Difference = -13

Using the token interpretation, find the values of:

(a) $3 \times (-2)$

(b) $(-5) \times (-2)$

(c) $(-4) \times (-1)$

(d) $(-7) \times 3$

If $123 \times 456 = 56088$, without calculating, find the value of:

(a) $(-123) \times 456$

(b) $(-123) \times (-456)$

(c) $(123) \times (-456)$

Try to frame a simple rule to multiply two integers.

Consider the numbers represented by the following tokens in the diagram:

Find the following products.

(a) $4 \times (-3)$

(b) $(-6) \times (-3)$

(c) $(-5) \times (-1)$

(d) $(-8) \times 4$

(e) $(-9) \times 10$

(f) $10 \times (-17)$

Find the values of:

(a) $14 \times (-15)$

(b) $-16 \times (-5)$

(c) $36 \div (-18)$

(d) $(-46) \div (-23)$

A freezing process requires that the room temperature be lowered from $32^\circ\text{C}$ at the rate of $5^\circ\text{C}$ every hour. What will be the room temperature 10 hours after the process begins?

A cement company earns a profit of ₹8 per bag of white cement sold and a loss of ₹5 per bag of grey cement sold. [Represent the profit/loss as integers.]

(a) The company sells 3,000 bags of white cement and 5,000 bags of grey cement in a month. What is its profit or loss? (b) If the number of bags of grey cement sold is 6,400 bags, what is the number of bags of white cement the company must sell to have neither profit nor loss?

Replace the blank with an integer to make a true statement.

(a) $(-3) \times \underline{\quad\quad} = 27$

(b) $5 \times \underline{\quad\quad} = (-35)$

(c) $\underline{\quad\quad} \times (-8) = (-56)$

(d) $\underline{\quad\quad} \times (-12) = 132$

(e) $\underline{\quad\quad} \div (-8) = 7$

(f) $\underline{\quad\quad} \div 12 = -11$

Find the values of the following expressions: (a) $(-5) \times (18 + (-3))$ (b) $(-7) \times 4 \times (-1)$ (c) $(-2) \times (-1) \times (-5) \times (-3)$

Find the values of the following expressions:

(a) $(-27) \div 9$

(b) $84 \div (-4)$

(c) $(-56) \div (-2)$

Find the integer whose product with $(-1)$ is:

(a) 27

(b) -31

(c) -1

(d) 1

(e) 0

If $47 - 56 + 14 - 8 + 2 - 8 + 5 = -4$, then find the value of $-47 + 56 - 14 + 8 - 2 + 8 - 5$ without calculating the full expression.

Do you remember the Collatz Conjecture from last year? Try a modified version with integers. The rule is — start with any number; if the number is even, take half of it; if the number is odd, multiply it by $-3$ and add 1; repeat. An example sequence is shown below.

Try this with different starting numbers: $(-21)$, $(-6)$, and so on. Describe the patterns you observe.

In a test, (+4) marks are given for every correct answer and (-2) marks are given for every incorrect answer.

(a) Anita answered all the questions in the test. She scored 40 marks even though 15 of her answers were correct. How many of her answers were incorrect? How many questions are in the test?

(b) Anil scored (-10) marks even though he had 5 correct answers. How many of his answers were incorrect? Did he leave any questions unanswered?

Pick the pattern — find the operations done by the machine shown below.

Imagine you're in a place where the temperature drops by 5°C each hour. If the temperature is currently at 8°C, write an expression which denotes the temperature after 4 hours.

Find 3 consecutive numbers with a product of (a) -6, (b) 120.

An alien society uses a peculiar currency called 'pibs' with just two denominations of coins — a+13 pibs coin and a -9 pibs coin. You have several of these coins. Is it possible to purchase an item that costs +85 pibs?

Find the values of:

(a) $(32 \times (-18)) \div ((-36))$

(b) $(32) \div ((-36) \times (-18))$

(c) $(25 \times (-12)) \div ((45) \times (-27))$

(d) $(280 \times (-7)) \div ((-8) \times (-35))$

Arrange the expressions given below in increasing order:

(a) $(-348) + (-1064)$

(b) $(-348) - (-1064)$

(c) $348 - (-1064)$

(d) $(-348) \times (-1064)$

(e) $348 \times (-1064)$

(f) $348 \times 964$

Given that $(-548) \times 972 = -532656$, write the values of:

(a) $(-547) \times 972$

(b) $(-548) \times 971$

(c) $(-547) \times 971$

Given that $207 \times (-33 + 7) = -5382$, write the value of $-207 \times (33 - 7) =\_\_\_?$

Use the numbers $3, -2, 5, -6$ exactly once and the operations '+', '-', and 'x' exactly once and brackets as necessary to write an expression such that —

(a) the result is the maximum possible

(b) the result is the minimum possible

Fill in the blanks in at least 5 different ways with integers:

(a) $\square + \square \times \square = -36$

(b) $(\square - \square) \times \square = 12$

(c) $(\square - (\square - \square)) = -1$

Rakesh gives you a challenge.

“I have thought of two numbers”, he says. “Their sum is 25, and their difference is 11.”

Can you tell me the two numbers?

You don’t need to use any formulas. Just try different pairs of numbers and then check:

1. Do the two numbers add up to 25?
2. Is the difference between them 11? (Remember: the difference means first number – second number.)

Write your guesses like this:

Context: Rakesh gives you a challenge. "I have thought of two numbers", he says. "Their sum is $25$, and their difference is $11$."

Can you tell me the two numbers? You don't need to use any formulas. Just try different pairs of numbers and then check:

1. Do the two numbers add up to $25$?
2. Is the difference between them $11$? (Remember: the difference means first number $-$ second number.)

Write your guesses like this:

Q. Did you find the right pair?

Context: Rakesh's Puzzle: A Number Game Rakesh gives you a challenge. “I have thought of two numbers”, he says. “Their sum is $25$, and their difference is $11$.”

Can you tell me the two numbers? You don't need to use any formulas. Just try different pairs of numbers and then check:

1. Do the two numbers add up to $25$?
2. Is the difference between them $11$? (Remember: the difference means first number $-$ second number.)

Write your guesses like this:

Q. Now that you’ve found the correct pair, Rakesh gives you a second challenge:

“Think of two numbers whose sum is $25$, but their difference is $-11$.”

Use the same method. Try different pairs of numbers and fill in the table again. You will notice that if you swap the numbers from the first puzzle, you get the answer to Rakesh’s second puzzle. That is, the first number is $7$ and the second is $18$!

The coin is at 0. If it is struck twice (the direction of the two strikes may be the same or different) can you give a formula for the final position of the coin?

Context: Suppose the first strike moves the coin rightward by 5 units from 0, and the second strike leftward by 7 units, then we take the First Movement = 5 units and Second Movement = -7 units.

Q. What is the final position of the coin?

Based on this new model, answer the following questions:

1. If the first movement is -4 and the final position is 5, what is the second movement?
2. If there are multiple strikes causing movements in the order 1, -2, 3, -4, ..., -10, what is the final position of the coin?

If there are multiple strikes causing movements in the order $1, -2, 3, -4, \dots, -10$, what is the final position of the coin?

From the figures below, what can you conclude about the magnitudes of $a$ and $b$ compared to each other, and what are their directions? Remember to start from 0.

Using tokens, argue out the following statements.

(a) $7 - 18 = 7 + (-18)$ (additive inverse of $18$ is $-18$)

(b) $4 - (-12) = 4 + 12$ (additive inverse of $-12$ is $12$)

Similarly find the values of $4 \times (-6)$ and $9 \times (-7)$? How can we interpret $(-4) \times 2$?

Context: Consider the numbers represented by the following tokens:

We can see that all of them represent the number $(-2)$. Now, take 4 times each of these token sets. That is, place each set into the empty bag 4 times.

Q. What integer do we get as the final answer in each case? Do we get different answers because the sets look different, or the same answer because they all represent $-2$?

Context: We can see that all of them represent the number $-2$. Now, take $4$ times each of these token sets. That is, place each set into the empty bag $4$ times.

What integer do we get as the final answer in each case? Do we get different answers because the sets look different, or the same answer because they all represent $-2$?

Q. Check this for $5 \times 4$, by taking different token sets corresponding to $4$.

We have seen that $-4 \times 2$ is the number obtained by removing 2 positive tokens from the empty bag 4 times.

We know that removing or subtracting a number is the same as adding its inverse.

Using this, can $-4 \times 2$ be defined through a process of addition of tokens instead of removal of tokens?

What do you notice in this pattern? Can you describe it?

Will this pattern continue when the multiplier goes below zero and becomes a negative number?

What is the pattern when the multiplicand is a negative integer?

Context: Consider the expression $1 \times a$. We know that the value of this expression is 'a' for all positive integers.

Q. Is this true for all negative integers too?

What is the value of the expression $-1 \times a$?

In the case of integers, is the product the same when we swap the multiplier and the multiplicand? Try this for some numbers.

Observe the following pairs of multiplications (fill in the blanks where needed):

What do you notice in these pairs of multiplication statements?

Context: The product is the same when we 'swap' the multiplier and multiplicand. Earlier, we have seen a similar property with addition.

Q. Will this always happen?

Does the sign of the product change if we swap the multiplier and multiplicand?

Context: An exam has 50 multiple choice questions. 5 marks are given for every correct answer and 2 negative marks for every wrong answer.

Q. What are the maximum possible marks in the exam? What are the minimum possible marks?

Context: An elevator in a mining shaft moves above and below the ground. The elevator's positions above the ground are represented as positive integers and positions below the ground are represented as negative integers. In part (b), the elevator begins to descend from 15 m above the ground at the speed of 3 metres per minute. Method 1 models this using subtraction.

Q. Find the solution to part (b) using Method 1 described above.

Can you summarise the rules for integer division looking at the above pattern?

Take a few more examples of multiplication of 3 integers and check this property. What do you observe?

Are there orders in which $5 \times -3 \times 4$ can be evaluated? Will the product be the same in all these cases?

Multiply the expression $25 \times -6 \times 12$ in all the different orders and check if the product is the same in all cases.

Look at the following series of multiplications:

When $-1$ is multiplied $2$ or $4$ times the product is positive. When it is multiplied $3$ or $5$ times the product is negative. Can you generalise these statements further?

Using this understanding of multiplication of many integers, can you give a simple rule to find the sign of the product of many integers?

Now, consider the expression $5 \times (4 + (-2))$. As in the case of positive integers, is this expression equal to $5 \times 4 + 5 \times (-2)$?

Check if the distributive property holds for $(-2) \times (4 + (-3))$ (that is, if this expression equals $(-2) \times 4 + (-2) \times (-3)$), and for a few other such expressions of your choice.

What do you observe? We see that the distributive property seems to hold for integers, as well. Will this always happen?

Can you visually show the distributive property for an expression like $-4 \times (2 + (-3))$? [Hint: Use the fact that multiplying a number by $-4$ is adding the inverse of the number 4 times.]

Pick the Pattern

Two pattern machines are given below. Each machine takes 3 numbers, does some operations and gives out the result.

Find the operations being done by Machine 1.

Context: The operation done by Machine 1 is (first number) + (second number) - (third number). Written as an expression, this will be $a + b - c$, where $a$ is the first number, $b$ is the second number, and $c$ is the third number. For example, $5 + 8 - 3 = 10$, and $(-4) + (-1) - (-6) = 1$.

Q. So, the result of the last group will be, $(-10) + (-12) - (-9) = \text{\_\_\_\_\_\_\_}.$

Find the operations being done by Machine 2 and fill in the blank.

Make your own machine and challenge your peers in finding its operations.