The Other Side of Zero — Class 6 Mathematics

What are integers?

Integers are the numbers : the positive and negative whole numbers together with zero.

Where do negative numbers lie on the number line?

To the left of zero; positive numbers lie to the right of zero.

Give a real-life situation needing negative numbers.

Floors below the ground in a building, temperatures below zero, or money owed (a debt) all need negative numbers.

Which is greater, or ?

is greater, because it lies to the right of on the number line.

What is the opposite of ?

Its opposite is ; they are the same distance from 0 on opposite sides.

What do you get when you add a number and its opposite?

You get 0, for example .

How is adding a positive integer shown on the number line?

By moving that many steps to the right (or up in the building lift).

How is subtracting shown using the building of fun?

By moving down in the lift; going down from a floor subtracts, just as moving left on the number line.

Is zero a positive or a negative number?

Zero is neither positive nor negative; it separates the positive numbers from the negative numbers.

Why was the number ray extended into a full number line?

Because numbers less than 0 exist (to the left of 0), so the ray starting at 0 is completed into a true number line of integers.

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

You start at and move . . You reach Floor , the Toy Store.

2. Evaluate these expressions (you may think of them as Starting Floor + Movement by referring to the Building of Fun). a. (+1)+(+4) = _______

. Start at (Food Court) and go up 4 floors to (Sports Centre).

2. Evaluate these expressions (you may think of them as Starting Floor + Movement by referring to the Building of Fun). b. (+4)+(+1) = _______

. Start at and go up 1 floor to .

2. Evaluate these expressions (you may think of them as Starting Floor + Movement by referring to the Building of Fun). c. (+4)+(– 3) = _______

. Start at and go down 3 floors to .

2. Evaluate these expressions (you may think of them as Starting Floor + Movement by referring to the Building of Fun). d. (–1)+(+2) = _______

. Start at and go up 2 floors to .

2. Evaluate these expressions (you may think of them as Starting Floor + Movement by referring to the Building of Fun). e. (–1)+(+1) = _______

. Start at and go up 1 floor to the ground floor .

2. Evaluate these expressions (you may think of them as Starting Floor + Movement by referring to the Building of Fun). f. 0+(+2) = _______

. Start at and go up 2 floors to .

2. Evaluate these expressions (you may think of them as Starting Floor + Movement by referring to the Building of Fun). g. 0+(–2) = _______

. Start at and go down 2 floors to .

3. 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.

Any pair where . Examples: , , , , . All represent going from the starting floor to with a downward movement.

a. (+1)+(+4) = _______

b. (+4)+(+1) = _______

c. (+4)+(– 3)+(–2) = _______

. , .

d. (–1)+(+2)+(–3) = _______

. , .

1. Compare the following numbers using the Building of Fun and fill in the boxes with < or >. a. –2 +5

. Floor is below Floor .

1. Compare the following numbers using the Building of Fun and fill in the boxes with < or >. b. –5 +4

1. Compare the following numbers using the Building of Fun and fill in the boxes with < or >. c. –5 –3

. is lower than .

1. Compare the following numbers using the Building of Fun and fill in the boxes with < or >. d. +6 –6

1. Compare the following numbers using the Building of Fun and fill in the boxes with < or >. e. 0 –4

1. Compare the following numbers using the Building of Fun and fill in the boxes with < or >. f. 0 +4

2. Imagine the Building of Fun with more floors. Compare the numbers and fill in the boxes with < or >: a. –10 –12

2. Imagine the Building of Fun with more floors. Compare the numbers and fill in the boxes with < or >: b. +17 –10

2. Imagine the Building of Fun with more floors. Compare the numbers and fill in the boxes with < or >: c. 0 –20

2. Imagine the Building of Fun with more floors. Compare the numbers and fill in the boxes with < or >: d. +9 –9

2. Imagine the Building of Fun with more floors. Compare the numbers and fill in the boxes with < or >: e. –25 –7

2. Imagine the Building of Fun with more floors. Compare the numbers and fill in the boxes with < or >: f. +15 –17

3. 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.

With , , , the line increases upward by 1 per tick. So , , , , depending on scale; counting ticks between and shows intermediate floors at integer steps.

4. Mark the following floors of the building shown on the right. a. –7

is 5 floors below / 5 floors above . Mark one tick above C.

4. Mark the following floors of the building shown on the right. b. – 4

is 3 floors below . Mark three ticks above D.

4. Mark the following floors of the building shown on the right. c. +3

is 2 floors above . Mark two ticks above E.

4. Mark the following floors of the building shown on the right. d. – 10

is 2 floors above . Mark two ticks above A.

a. (+1) –(+4) = _______

. Movement needed to go from to is .

b. (0)–(+2) = _______

c. (+4)–(+1) = _______

d. (0)– (–2) = _______

e. (+4) – (–3) = _______

f. (–4) –(–3) = _______

g. (–1)–(+2) = _______

h. (–2) –(–2) = _______

i. (–1) – (+1) = _______

j. (+3)– (–3) = _______

a. (+40)+ ______ = +200

b. (+40)+_______=–200

c. (–50)+ ______ = +200

d. (–50)+_______=–200

e. (–200) – (–40) = _______

f. (+200) – (+40) = _______

g. (–200) –(+40) = _______

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

Choose any 3 positives to the right of 0 and 3 negatives to the left, for example positives and negatives marked on the line to .

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

For example (any three negatives you marked).

3. Is 2 > – 3? Why? Is –2 < 3? Why?

Yes, because lies to the right of on the number line and all positives are greater than negatives. Yes, because lies to the left of .

4. What are a. –5+0

4. What are b. 7+(–7)

4. What are c. –10+20

4. What are d. 10 –20

4. What are e. 7–(–7)

4. What are f. –8–(–10)

a. –125 + (–30) = _______

b. +105 – (–55) = _______

c. +80 – (–150) = _______

d. –99 – (–200) = _______

1. Complete the additions using tokens. a. (+6)+(+4)

. Six greens plus four greens = ten greens.

1. Complete the additions using tokens. b. (–3) + (–2)

. Three reds plus two reds = five reds.

1. Complete the additions using tokens. c. (+5)+(–7)

. Five zero pairs cancel leaving two reds.

1. Complete the additions using tokens. d. (–2) + (+6)

. Two zero pairs cancel leaving four greens.

2. 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? a.

Set a has 3 positives and 5 negatives. Three zero pairs cancel leaving . Statement . Attendant on Floor .

2. 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? b.

Set b has 6 positives and 3 negatives. Three zero pairs cancel leaving . Statement . Attendant on Floor .

1. Evaluate the following differences using tokens. Check that you get the same result as with other methods you now know: a. (+10) –(+7)

1. Evaluate the following differences using tokens. Check that you get the same result as with other methods you now know: b. (–8)– (–4)

1. Evaluate the following differences using tokens. Check that you get the same result as with other methods you now know: c. (–9) –(–4)

1. Evaluate the following differences using tokens. Check that you get the same result as with other methods you now know: d. (+9) –(+12)

. Add one zero pair to make 10 positives, remove 12 leaves 1 negative and 2 negatives? Result .

1. Evaluate the following differences using tokens. Check that you get the same result as with other methods you now know: e. (–5)– (–7)

1. Evaluate the following differences using tokens. Check that you get the same result as with other methods you now know: f. (–2)– (–6)

2. Complete the subtractions: a. (–5) –(–7)

2. Complete the subtractions: b. (+10) –(+13)

2. Complete the subtractions: c. (–7)–(–9)

2. Complete the subtractions: d. (+3) – (+8)

2. Complete the subtractions: e. (–2) –(–7)

2. Complete the subtractions: f. (+3)–(+15)

1. Try to subtract: –3– (+5). How many zero pairs will you have to put in? What is the result?

. Start with 3 negatives, need to remove 5 positives. Add 5 zero pairs, remove 5 positives, left with 8 negatives.

2. Evaluate the following using tokens. a. (–3) – (+10)

2. Evaluate the following using tokens. b. (+8)– (–7)

2. Evaluate the following using tokens. c. (–5)–(+9)

2. Evaluate the following using tokens. d. (–9) –(+10)

2. Evaluate the following using tokens. e. (+6) –(–4)

2. Evaluate the following using tokens. f. (–2)–(+7)

1. 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?

Credits , debits . Balance . So rupees, i.e. rupees overdrawn.

2. 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?

Sum debits . Debits , credit . Balance rupee.

3. 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?

Positive balance avoids fees and interest charged on overdrafts and shows financial stability. Temporarily negative may be worthwhile for an essential purchase, emergency, or investment that will soon generate income to repay, like buying stock for a business that quickly yields profit.

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

Point A approx m above sea level (highest peak).

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

Point B approx m (below sea level, valley).

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

Point C approx m

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

Point D approx m (deepest below sea level).

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

Point E approx m

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

Point F approx m

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

Point G approx m

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

Highest is A, lowest is D.

3. 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?

Decreasing (high to low): . Increasing (low to high): .

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

Mount Everest, about m (about m) above sea level.

5. 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).

Challenger Deep in the Mariana Trench, about to m below sea level. On land, the Dead Sea shore is about m.

1. 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?

Places like Leh, Kargil, Siachen, Drass in Ladakh, Srinagar, Gulmarg, Shimla, Manali, Darjeeling. Common: high altitude in Himalayas, northern latitude, snow-covered. Cold due to altitude, distance from sea, and Himalayan winter winds.

2. 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. Temperature 14°C 8°C –2°C –4°C Time 02:00 a.m. 11:00 p.m. –2°C 02:00 p.m. –4°C 11:00 a.m.

Warmest at afternoon: at 02:00 p.m., at 11:00 a.m., at 11:00 p.m., coldest at 02:00 a.m. (night/early morning coldest).

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

Top row , bottom row , left column , right column . Border sum .

2. Complete the grids to make the required border sum: Border sum is +4

One solution for first grid to give border sum : needs filling missing numbers to make each row/column sum . Example: top row , bottom etc. One valid completion is top middle , bottom middle with appropriate corners as shown in guide.

2. Complete the grids to make the required border sum: Border sum is – 2

Second grid target : One completion is top right , bottom left , bottom right giving each border line .

2. Complete the grids to make the required border sum: Border sum is – 4

Third grid target : Example top middle , middle right , bottom middle , corners give each side .

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

Because only border cells matter and middle of each side is shared, there are degrees of freedom. E.g., increase top middle by and decrease bottom middle by while adjusting sides keeps sum . Many integer triples work.

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

All such hollow grids with 4 border cells per side overlapping corners have infinite solutions when enough empty cells, because you have 4 equations for more unknowns, leaving free variables.

5. Make a border integer square puzzle and challenge your classmates.

Create your own 3x3 hollow square, choose a target sum like or , fill some border numbers and leave others blank for classmates to solve.

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

Whatever four numbers you circle (one per row and column), the sum is always the same, for the given example grid. Repeating gives each time due to the special construction.

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

For the first grid, every transversal (broken-diagonal) sum is the same constant . For example and .

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

For the second grid, every transversal sum is the constant . For example .

3. 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?

Grid is formed by adding a row vector and column vector: each cell . Then any transversal sum constant independent of permutation. Both numbers and arrangement matter. You can make more by choosing any .

1. Write all the integers between the given pairs, in increasing order. a. 0 and –7

Integers between and :

1. Write all the integers between the given pairs, in increasing order. b. –4 and 4

1. Write all the integers between the given pairs, in increasing order. c. –8 and –15

1. Write all the integers between the given pairs, in increasing order. d. –30 and –23

2. Give three numbers such that their sum is –8.

Example . Many solutions, e.g., .

3. 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.

The dice faces are . Adding one face from each die, the sums that can occur are (note IS possible). So between and , the sums that cannot be obtained are .

4. Solve these: 8 – 13

4. Solve these: (– 8) – (13)

4. Solve these: (– 13) – (– 8)

4. Solve these: (– 13) + (– 8)

4. Solve these: 8 + (– 13)

4. Solve these: (– 8) – (– 13)

4. Solve these: (13) – 8

4. Solve these: 13 – (– 8)

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

If present is 2026 CE, 150 years ago = CE.

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

, but no year 0, so 175 BCE (or 174 BCE depending on inclusive count). Approximately 175 BCE.

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

i.e. 360 BCE.

6. Complete the following sequences: a. (–40), (–34), (–28), (–22), _____, ______, ______

Add :

6. Complete the following sequences: b. 3, 4, 2, 5, 1, 6, 0, 7, _____, _____, _____

Pattern alternating: (odd positions decreasing by 1, even increasing by 1).

6. Complete the following sequences: c. _____, ______, 12, 6, 1, (–3), (–6), _____, ______, ______

Differences then : preceding terms , following (as per pattern).

7. 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).

Example to get near : One exact: . Any expression evaluating close to is acceptable.

8. 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)

Always positive, because subtracting a negative is adding a positive: .

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

Can be positive, negative or zero depending on magnitudes.

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

Always negative.

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

Can be positive, negative or zero.

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

Always negative.

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

Can be positive, negative or zero.

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

Pattern of triangles: counts give value . (Each pyramid net value decreases, overall sum 20).

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

Yes. e.g., sum of two positives = go up then up, still up; sum of two negatives = go down then down, still down; mixed = go opposite directions, net is difference towards larger magnitude; number and its inverse return to 0; adding 0 causes no movement.

2. Give your own examples of each rule.

Examples: , , and , , . Subtraction: , , , , , .