The Other Side of Zero

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.

Complete these expressions.

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. $+1500 - (-1500)$

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

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

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

What are: a. $-5 + 0$ b. $7 + (-7)$ c. $-10 + 20$ d. $10 - 20$ e. $7 - (-7)$ f. $-8 - (-10)$?

Complete the additions using tokens.

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

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

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

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

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?

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

Complete the subtractions:

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

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

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

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

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

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

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

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

Evaluate the following using tokens.

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

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

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

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

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

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

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?

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?

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?

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

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

Can you write the points A, B, ..., G in a sequence of decreasing order of heights? Can you write the points in a sequence of increasing order of heights?

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

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).

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

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.

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

Complete the grids to make the required border sum:

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

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

Make a border integer square puzzle and challenge your classmates.

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

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

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?

Write all the integers between the given pairs, in increasing order. a. 0 and -7 b. -4 and 4 c. -8 and -15 d. -30 and -23

Give three numbers such that their sum is -8.

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.

Solve these:

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.

Complete the following sequences:

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 $(-30)$.

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)

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

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

Give your own examples of each rule.

Can there be a number less than 0? Can you think of any ways to have less than 0 of something?

Observe that some of the floors in the 'Building of Fun' are below the ground. What are the shops that you find on these floors? What is there on the ground floor?

In Bela's Building of Fun, a lift is used to go up and down between the floors. It has two buttons: '+' to go up and '-' to go down. Pressing '+' once goes up one floor (+1), and pressing '-' once goes down one floor (-1).

Q. What do you press to go four floors up? What do you press to go three floors down?

Number all the floors in the Building of Fun.

Start from the Food Court and press +2 in the lift. Where will you reach?

If Basant now presses +4 and then presses -4 in the lift, where will he reach?

Write the inverses of these numbers: +4, -4, -3, 0, +2, -1.

Connect the inverses by drawing lines.

Comparing numbers using floors

Who is on the lowest floor?

Should we write -3 < -4 or -4 < -3?

Evaluate 15 – 5, 100 – 10 and 74 – 34 from this perspective.

In the other exercises that you did above, did you notice that subtracting a negative number was the same as adding the corresponding positive number?

Take a look at the 'infinite lift' above. Does it remind you of a number line? In what ways?

Use unmarked number lines to evaluate these expressions:

Suppose you open a bank account at your local bank with the ₹100 that you had been saving over the last month. Your bank balance therefore, starts at ₹100. Then you make ₹60 at your job the next day and you deposit it in your account. This is shown in your bank passbook as a 'credit'.

Q. Your new bank balance is _______.

Suppose you open a bank account with a starting balance of ₹100 and deposit ₹60 (making the balance ₹160). The next day you pay your electric bill of ₹30 using your bank account. This is shown in your bank passbook as a 'debit'.

Q. Your bank balance is now _______.

Suppose you open a bank account with a starting balance of ₹100, deposit ₹60 (making the balance ₹160), and pay an electric bill of ₹30 (making the balance ₹130). The next day you make a major purchase for your business of ₹150. Again this is shown as a debit.

Q. What is your bank balance now? _______

Is this possible?

Your bank balance was -₹20. You make ₹200 at your business the next day.

Q. What is your balance now?