The Other Side of Zero | FIO

Question 49

Can you explain each of Brahmagupta's rules in terms of Bela's Building of Fun, or in terms of a number line?

Solution

Understand the Question

Bela's Building of Fun & Number Line Models:
- Positive numbers ( $+$ ): Represent moving up floors in the building or moving right on a number line.
- Negative numbers ( $-$ ): Represent moving down floors in the building or moving left on a number line.
- Addition ( $+$ ): Performing consecutive movements starting from ground level ( $0$ ).
- Multiplication ( $\times$ ): Performing repeated movements in a given direction (a positive multiplier repeats the same direction; a negative multiplier reverses the direction).

Step 1 · Explain Addition Rules

Rule 1: $(+) + (+) = (+)$ (Bigger gain)

Building: Bela goes up $3$ floors, then goes up $2$ more floors $\rightarrow$ ends $5$ floors higher.
Number line: Start at $0$ , move right $3$ units, then move right $2$ units $\rightarrow$ land on $+5$ .

$(+3) + (+2) = +5$

Rule 2: $(-) + (-) = (-)$ (Bigger loss)

Building: Bela goes down $2$ floors, then goes down $4$ more floors $\rightarrow$ ends $6$ floors lower.
Number line: Start at $0$ , move left $2$ units, then move left $4$ units $\rightarrow$ land on $-6$ .

$(-2) + (-4) = -6$

Rule 3: $(+) + (-) = ?$ (Depends on which is greater)

Building: Bela goes up $7$ floors, then goes down $4$ floors $\rightarrow$ ends $3$ floors above ground.
Number line: Start at $0$ , move right $7$ units, then move left $4$ units $\rightarrow$ land on $+3$ .

$(+7) + (-4) = +3$

Rule 4: $(-) + (+) = ?$ (Depends on which is greater)

Building: Bela goes down $6$ floors, then goes up $2$ floors $\rightarrow$ ends $4$ floors below ground.
Number line: Start at $0$ , move left $6$ units, then move right $2$ units $\rightarrow$ land on $-4$ .

$(-6) + (+2) = -4$

Step 2 · Explain Multiplication Rules

Rule 5: $(+) \times (+) = (+)$

Building: Go up $3$ floors, $2$ times $\rightarrow$ $6$ floors higher.
Number line: Move right $3$ units, repeated $2$ times $\rightarrow$ land on $+6$ .

$(+3) \times (+2) = +6$

Rule 6: $(-) \times (-) = (+)$

Building: Moving down $3$ floors with a negative multiplier reverses the action into going up $3$ floors, $2$ times $\rightarrow$ $6$ floors higher.
Number line: Moving left $3$ units reversed $2$ times gives moving right $3$ units twice $\rightarrow$ land on $+6$ .

$(-3) \times (-2) = +6$

Rule 7: $(+) \times (-) = (-)$

Building: Going up $3$ floors reversed $2$ times $\rightarrow$ going down $6$ floors.
Number line: Moving right $3$ units reversed $2$ times $\rightarrow$ move left $6$ units to land on $-6$ .

$(+3) \times (-2) = -6$

Rule 8: $(-) \times (+) = (-)$

Building: Go down $2$ floors, $3$ times $\rightarrow$ $6$ floors lower.
Number line: Move left $2$ units, repeated $3$ times $\rightarrow$ land on $-6$ .

$(-2) \times (+3) = -6$

Step 3 · Provide Real-World Examples

Using financial transactions (earnings as positive, losses/debts as negative) and game scoring:

Rule 1: Find $\text{₹}5$ , then find $\text{₹}4$ more $\implies (+5) + (+4) = +9$
Rule 2: Lose $\text{₹}3$ , then lose $\text{₹}6$ more $\implies (-3) + (-6) = -9$
Rule 3: Earn $\text{₹}10$ , spend $\text{₹}7 \implies (+10) + (-7) = +3$
Rule 4: Owe $\text{₹}12$ , earn $\text{₹}5 \implies (-12) + (+5) = -7$
Rule 5: Gain $2$ points, $4$ times $\implies (+2) \times (+4) = +8$
Rule 6: Cancel a loss of $5$ points, $3$ times $\implies (-5) \times (-3) = +15$
Rule 7: Reverse an earn of $4$ points, $2$ times $\implies (+4) \times (-2) = -8$
Rule 8: Lose $3$ points, $5$ times $\implies (-3) \times (+5) = -15$

Answer

Brahmagupta's integer rules can be represented using upward/downward movement in a building, right/left steps on a number line, or real-life gain/loss situations.

Common Mistakes

Adding vs. Multiplying Negatives: Confusing $(-a) + (-b) = -(a+b)$ with $(-a) \times (-b) = +ab$ . Adding debts increases debt (negative), but multiplying two negatives results in a positive.
Sign in Addition of Opposite Signs: For $(+) + (-)$ , the sign of the result depends on the integer with the larger absolute value, not automatically positive or negative.

More questions in FIO

You start from Floor $+2$ and press $-3$ in the lift. Where will you reach? Write an expression for this movement.

Evaluate these expressions (you may think of them as Starting Floor + Movement by referring to the Building of Fun).

Starting from different floors, find the movements required to reach Floor $-5$ . For example, if I start at Floor $+2$ , I must press $-7$ to reach Floor $-5$ . The expression is $(+2) + (-7) = -5$ .

Find more such starting positions and the movements needed to reach Floor $-5$ and write the expressions.

Evaluate these expressions by thinking of them as the resulting movement of combining button presses:

Compare the following numbers using the Building of Fun and fill in the boxes with $<$ or $>$ .

Imagine the Building of Fun with more floors. Compare the numbers and fill in the boxes with $<$ or $>$ :

If Floor A $= -12$ , Floor D $= -1$ and Floor E $= +1$ in the building shown on the right as a line, find the numbers of Floors B, C, F, G, and H.

Mark the following floors of the building shown on the right.

a. $-7$ b. $-4$ c. $+3$ d. $-10$

Complete these expressions. You may think of them as finding the movement needed to reach the Target Floor from the Starting Floor.

Q10

Complete these expressions.

Q11

Try evaluating the following expressions by similarly drawing or imagining a suitable lift:

a. $-125 + (-30)$

b. $+105 - (-55)$

c. $+105 + (+55)$

d. $+80 - (-150)$

e. $+80 + (+150)$

f. $-99 - (-200)$

g. $-99 + (+200)$

h. $+1500 - (-1500)$

Q12

Mark 3 positive numbers and 3 negative numbers on the number line above.

Q13

Write down the above 3 marked negative numbers in the following boxes:

Q14

Is $2 > -3$ ? Why? Is $-2 < 3$ ? Why?

Q15

What are:

(a) $-5 + 0$

(b) $7 + (-7)$

(d) $10 - 20$

(e) $7 - (-7)$

(f) $-8 - (-10)$ ?

Q16

Complete the additions using tokens.

a. $(+6) + (+4)$

b. $(-3) + (-2)$

c. $(+5) + (-7)$

d. $(-2) + (+6)$

Q17

Cancel the zero pairs in the following two sets of tokens. On what floor is the lift attendant in each case? What is the corresponding addition statement in each case?

Q18

Evaluate the following differences using tokens. Check that you get the same result as with other methods you now know:

a. $(+10) - (+7)$

b. $(-8) - (-4)$

c. $(-9) - (-4)$

d. $(+9) - (+12)$

e. $(-5) - (-7)$

f. $(-2) - (-6)$

Q19

Complete the subtractions:

a. $(-5) - (-7)$

b. $(+10) - (+13)$

c. $(-7) - (-9)$

d. $(+3) - (+8)$

e. $(-2) - (-7)$

f. $(+3) - (+15)$

Q20

Try to subtract: $-3 - (+5)$ .

How many zero pairs will you have to put in? What is the result?

Q21

Evaluate the following using tokens.

a. $(-3) - (+10)$

b. $(+8) - (-7)$

c. $(-5) - (+9)$

d. $(-9) - (+10)$

e. $(+6) - (-4)$

f. $(-2) - (+7)$

Q22

Suppose you start with ₹0 in your bank account, and then you have credits of ₹30, ₹40, and ₹50, and debits of ₹40, ₹50, and ₹60. What is your bank account balance now?

Q23

Suppose you start with ₹0 in your bank account, and then you have debits of ₹1, 2, 4, 8, 16, 32, 64, and 128, and then a single credit of ₹256. What is your bank account balance now?

Q24

Why is it generally better to try and maintain a positive balance in your bank account? What are circumstances under which it may be worthwhile to temporarily have a negative balance?

Q25

Looking at the geographical cross section, fill in the respective heights:

Q26

Which is the highest point in this geographical cross section? Which is the lowest point?

Q27

Can you write the points $\text{A}, \text{B}, \dots, \text{G}$ in a sequence of decreasing order of heights? Can you write the points in a sequence of increasing order of heights?

Q28

What is the highest point above sea level on Earth? What is its height?

Q29

What is the lowest point with respect to sea level on land or on the ocean floor? What is its height? (This height should be negative).

Q30

Do you know that there are some places in India where temperatures can go below $0^\circ\text{C}$ ? Find out the places in India where temperatures sometimes go below $0^\circ\text{C}$ . What is common among these places? Why does it become colder there and not in other places?

Q31

Leh in Ladakh gets very cold during the winter. The following is a table of temperature readings taken during different times of the day and night in Leh on a day in November. Match the temperature with the appropriate time of the day and night.

Q32

Do the calculations for the second grid above and find the border sum.

Q33

Complete the grids to make the required border sum:

Q34

For the last grid above, find more than one way of filling the numbers to get border sum $-4$ .

Q35

Which other grids can be filled in multiple ways? What could be the reason?

Q36

Make a border integer square puzzle and challenge your classmates.

Q37

Try afresh, choose different numbers this time. What sum did you get? Was it different from the first time? Try a few more times!

Q38

Play the same game with the grids below. What answer did you get?

Q39

What could be 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?

Q40

Write all the integers between the given pairs, in increasing order.

(a) $0$ and $-7$

(b) $-4$ and $4$

(d) $-30$ and $-23$

Q41

Give three numbers such that their sum is $-8$ .

Q42

There are two dice whose faces have these numbers: $-1, 2, -3, 4, -5, 6$ . The smallest possible sum upon rolling these dice is $-10 = (-5) + (-5)$ and the largest possible sum is $12 = (6) + (6)$ . Some numbers between $-10$ and $+12$ are not possible to get by adding numbers on these two dice. Find those numbers.

Q43

Solve these:

Q44

Find the years below. a. From the present year, which year was it 150 years ago?

b. From the present year, which year was it 2200 years ago?

c. What will be the year 320 years after 680 BCE?

Hint: Recall that there was no year 0.

Q45

Complete the following sequences:

Q46

Here are six integer cards: $(+1), (+7), (+18), (-5), (-2), (-9)$ .

You can pick any of these and make an expression using addition(s) and subtraction(s).

Here is an expression: $(+18) + (+1) - (+7) - (-2)$ which gives a value $(+14)$ . Now, pick cards and make an expression such that its value is closer to $(-30)$ .

Q47

The sum of two positive integers is always positive but a (positive integer) - (positive integer) can be positive or negative. What about

a. (positive) - (negative)

b. (positive) + (negative)

c. (negative) + (negative)

d. (negative) - (negative)

e. (negative) - (positive)

f. (negative) + (positive)

Q48

This string has a total of 100 tokens arranged in a particular pattern. What is the value of the string?

Q49

Can you explain each of Brahmagupta's rules in terms of Bela's Building of Fun, or in terms of a number line?

Q50

Give your own examples of each rule.