Page 243
Question: What do you press to go four floors up? What do you press to go three floors down?
Answer:
- To go four floors up, you press + + + + or +4.
- To go three floors down, you press – – – or -3.
Page 244
Question: Number all the floors in the Building of Fun.
Answer:
(Based on the complete building diagram shown on Page 246.)
- Floor +6: Space
- Floor +5: Sports
- Floor +4: Ice Cream
- Floor +3: Book Store
- Floor +2: Art Centre
- Floor +1: Food Court
- Floor 0: Welcome Hall
- Floor -1: Toy Store
- Floor -2: Video Games
- Floor -3: Cinema
- Floor -4: Ghost House
- Floor -5: Dinosaur
Page 245
Figure it Out
Question 1: You start from Floor +2 and press -3 in the lift. Where will you reach? Write an expression for this movement.
Answer:
You will reach the Toy Store (Floor -1).
Expression: (+2) + (-3) = -1.
Question 2: Evaluate these expressions (you may think of them as Starting Floor + Movement by referring to the Building of Fun).
a. (+1) + (+4) =
b. (+4) + (+1) =
c. (+4) + (-3) =
d. (-1) + (+2) =
e. (-1) + (+1) =
f. 0 + (+2) =
g. 0 + (-2) =
Answer:
a. (+1) + (+4) = +5
b. (+4) + (+1) = +5
c. (+4) + (-3) = +1
d. (-1) + (+2) = +1
e. (-1) + (+1) = 0
f. 0 + (+2) = +2
g. 0 + (-2) = -2
Question 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.
Answer:
- (+1) + (-6) = -5
- (-2) + (-3) = -5
- 0 + (-5) = -5
Page 246
Figure it Out
Question: Evaluate these expressions by thinking of them as the resulting movement of combining button presses:
a. (+1) + (+4) =
b. (+4) + (+1) =
c. (+4) + (-3) + (-2) =
d. (-1) + (+2) + (-3) =
Answer:
a. (+1) + (+4) = +5
b. (+4) + (+1) = +5
c. (+4) + (-3) + (-2) = -1
d. (-1) + (+2) + (-3) = -2
Question: Write the inverses of these numbers: +4, -4, -3, 0, +2, -1.
Answer:
- Inverse of +4 is -4
- Inverse of -4 is +4
- Inverse of -3 is +3
- Inverse of 0 is 0
- Inverse of +2 is -2
- Inverse of -1 is +1
Question: Connect the inverses by drawing lines.
Answer:
- (+5) connects to (-5)
- (-7) connects to (+7)
- (-8) connects to (+8)
- (+9) connects to (-9)
Question: Who is on the lowest floor?
- Jay is in the Art Centre. So, he is on Floor +2.
- Asin is in the Sports Centre. So, she is on Floor [ ].
- Binnu is in the Cinema Centre. So, she is on Floor [ ].
- Aman is in the Toys Store. So, he is on Floor [ ].
Answer:
- Asin is on Floor +5.
- Binnu is on Floor -3.
- Aman is on Floor -1.
- Binnu is on the lowest floor.
Page 247
Figure it Out
Question 1: Compare the following numbers using the Building of Fun and fill in the boxes with < or >.
a. -2 [ ] +5
b. -5 [ ] +4
c. -5 [ ] -3
d. +6 [ ] -6
e. 0 [ ] -4
f. 0 [ ] +4
Answer:
a. -2 < +5
b. -5 < +4
c. -5 < -3
d. +6 > -6
e. 0 > -4
f. 0 < +4
Question 2: Imagine the Building of Fun with more floors. Compare the numbers and fill in the boxes with < or >:
a. -10 [ ] -12
b. +17 [ ] -10
c. 0 [ ] -20
d. +9 [ ] -9
e. -25 [ ] -7
f. +15 [ ] -17
Answer:
a. -10 > -12
b. +17 > -10
c. 0 > -20
d. +9 > -9
e. -25 < -7
f. +15 > -17
Question 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.
Answer:
- B = -9
- C = -6
- F = +2
- G = +6
- H = +11
Question 4: Mark the following floors of the building shown on the right.
a. -7
b. -4
c. +3
d. -10
Answer:
- P is -7
- Q is -4
- R is +3
- S is -10
Page 248
Question: Evaluate 15-5, 100-10 and 74-34 from this perspective.
Answer:
- 15 – 5 = 10 (Because 5 + 10 = 15)
- 100 – 10 = 90 (Because 10 + 90 = 100)
- 74 – 34 = 40 (Because 34 + 40 = 74)
Page 249
Figure it Out
Question: Complete these expressions. You may think of them as finding the movement needed to reach the Target Floor from the Starting Floor.
a. (+1) – (+4) =
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) =
Answer:
a. (+1) – (+4) = -3
b. (0) – (+2) = -2
c. (+4) – (+1) = +3
d. (0) – (-2) = +2
e. (+4) – (-3) = +7
f. (-4) – (-3) = -1
g. (-1) – (+2) = -3
h. (-2) – (-2) = 0
i. (-1) – (+1) = -2
j. (+3) – (-3) = +6
Page 251
Figure it Out
Question: Complete these expressions.
a. (+40) + [ ] = +200
b. (+40) + [ ] = -200
c. (-50) + [ ] = +200
d. (-50) + [ ] = -200
e. (-200) – (-40) = [ ]
f. (+200) – (+40) = [ ]
g. (-200) – (+40) = [ ]
Answer:
a. (+40) + (+160) = +200
b. (+40) + (-240) = -200
c. (-50) + (+250) = +200
d. (-50) + (-150) = -200
e. (-200) – (-40) = -160
f. (+200) – (+40) = +160
g. (-200) – (+40) = -240
Page 253
Figure it Out
Question 1: Mark 3 positive numbers and 3 negative numbers on the number line above.
Answer:
Example markers:
- Positive: +2, +5, +8
- Negative: -1, -3, -7
Question 2: Write down the above 3 marked negative numbers in the following boxes.
Answer: -7, -3, -1
Question 3: Is 2 > -3? Why? Is -2 < 3? Why?
Answer:
- Yes, 2 > -3 because on the number line 2 is on the right side of -3.
- Yes, -2 < 3 because 3 is on the right of -2.
Question 4: What are:
a. -5 + 0
b. 7 + (-7)
c. -10 + 20
d. 10 – 20
e. 7 – (-7)
f. -8 – (-10)
Answer:
a. -5 + 0 = -5
b. 7 + (-7) = 0
c. -10 + 20 = 10
d. 10 – 20 = -10
e. 7 – (-7) = 14
f. -8 – (-10) = 2
Page 255
Question: Use unmarked number lines to evaluate these expressions:
a. -125 + (-30) =
b. +105 – (-55) =
c. +80 – (-150) =
d. -99 – (-200) =
Answer:
a. -125 + (-30) = -155
b. +105 – (-55) = +160
c. +80 – (-150) = +230
d. -99 – (-200) = +101
Page 257
Figure it Out
Question 1: Complete the additions using tokens.
a. (+6) + (+4)
b. (-3) + (-2)
c. (+5) + (-7)
d. (-2) + (+6)
Answer:
a. (+6) + (+4) = +10
b. (-3) + (-2) = -5
c. (+5) + (-7) = -2
d. (-2) + (+6) = +4
Question 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?
Answer:
- (3 positive, 5 negative)
- Floor: -2
- Statement: (+3) + (-5) = -2
- (6 positive, 3 negative)
- Floor: +3
- Statement: (+6) + (-3) = +3
Page 258
Figure it Out
Question 1: 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)
Answer:
a. (+10) – (+7) = +3
b. (-8) – (-4) = -4
c. (-9) – (-4) = -5
d. (+9) – (+12) = -3
e. (-5) – (-7) = +2
f. (-2) – (-6) = +4
Question 2: Complete the subtractions:
a. (-5) – (-7)
b. (+10) – (+13)
c. (-7) – (-9)
d. (+3) – (+8)
e. (-2) – (-7)
f. (+3) – (+15)
Answer:
a. (-5) – (-7) = +2
b. (+10) – (+13) = -3
c. (-7) – (-9) = +2
d. (+3) – (+8) = -5
e. (-2) – (-7) = +5
f. (+3) – (+15) = -12
Page 259
Figure it Out
Question 1: Try to subtract: -3 – (+5). How many zero pairs will you have to put in? What is the result?
Answer:
- You have to put in five zero pairs.
- The result is -8.
Question 2: Evaluate the following using tokens.
a. (-3) – (+10)
b. (+8) – (-7)
c. (-5) – (+9)
d. (-9) – (+10)
e. (+6) – (-4)
f. (-2) – (+7)
Answer:
a. (-3) – (+10) = -13
b. (+8) – (-7) = +15
c. (-5) – (+9) = -14
d. (-9) – (+10) = -19
e. (+6) – (-4) = +10
f. (-2) – (+7) = -9
Question: Your bank balance starts at ₹100. Then you make ₹60. Your new bank balance is?
Answer: ₹100 + ₹60 = ₹160
Question: The next day you pay your electric bill of ₹30. Your bank balance is now?
Answer: ₹160 – ₹30 = ₹130
Question: The next day you make a major purchase for your business of ₹150. What is your bank balance now?
Answer: ₹130 – ₹150 = -₹20
Question: The next day you make ₹200. What is your balance now?
Answer: -₹20 + ₹200 = ₹180
Page 260
Figure it Out
Question 1: Suppose you start with 0 rupees 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?
Answer:
(+30) + (+40) + (+50) – (40) – (50) – (60)
Balance = -30
Question 2: Suppose you start with 0 rupees 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?
Answer:
(-1) – 2 – 4 – 8 – 16 – 32 – 64 – 128 + 256
Balance = +1 (Rupee 1)
Page 261
Figure it Out
Question 1: Looking at the geographical cross section, fill in the respective heights:
Answer:
- A: +1500 m
- B: -500 m
- C: +300 m
- D: -1200 m
- E: +1200 m
- F: -200 m
- G: +100 m
Question 2: Which is the highest point in this geographical cross section? Which is the lowest point?
Answer:
- Highest point: A
- Lowest point: D
Question 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?
Answer:
- Decreasing: A, E, C, G, F, B, D
- Increasing: D, B, F, G, C, E, A
Question 4: What is the highest point above sea level on Earth? What is its height?
Answer:
The highest point is Mount Everest. Its height is 8848 m.
Question 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).
Answer:
The lowest known point on Earth is the Challenger Deep (in the Mariana Trench). Its height is approximately -10,994 m.
Page 262
Figure it Out
Question 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.
Answer:
- 14°C: 02:00 p.m.
- 8°C: 11:00 a.m.
- -2°C: 11:00 p.m.
- -4°C: 02:00 a.m.
Page 263
Figure it Out
Activity: A hollow integer grid.
Question 1: Do the calculations for the second grid above and find the border sum.
Answer:
The border sum is -3.
Question 2: Complete the grids to make the required border sum:
Answer:
Grid 1 (Border Sum +4):
- Top Row: -10, 10, 4
- Bottom Row: 9, -10, 5
- Left Column: -10, 5, 9
- Right Column: 4, -5, 5
Grid 2 (Border Sum -2):
- Top Row: 6, 8, -16
- Bottom Row: -2, -19, 19
- Left Column: 6, 11, -2
- Right Column: -16, -5, 19
Grid 3 (Border Sum -4):
- Top Row: 7, -2, -9
- Bottom Row: -8, -6, 10
- Left Column: 7, -3, -8
- Right Column: -9, -5, 10
Page 264
Figure it Out
Question 2: Play the same game with the grids below. What answer did you get?
Answer:
- Grid A (starts with 7): The sum is -14.
- Grid B (starts with -11): The sum is -14.
Page 265
Figure it Out
Question 1: Write all the integers between the given pairs, in increasing order.
Answer:
- a. 0 and -7: -6, -5, -4, -3, -2, -1
- b. -4 and 4: -3, -2, -1, 0, 1, 2, 3
- c. -8 and -15: -14, -13, -12, -11, -10, -9
- d. -30 and -23: -29, -28, -27, -26, -25, -24
Question 2: Give three numbers such that their sum is -8.
Answer:
One combination is -5, 7, -10.
Question 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.
Answer:
The missing numbers are: -9, -7, -5, 0, 2, 7, 9, 11.
Question 4: Solve these:
Answer:
(Completing the table)
- Row 1: -5, -21, -5, -21
- Row 2: -5, 5, 5, 21
Question 5: 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?
Answer:
(Note: Assuming current year 2025)
a. 1875 CE
b. 176 BCE
c. 360 BCE
Question 6: Complete the following sequences:
a. (-40), (-34), (-28), (-22), …
b. 3, 4, 2, 5, 1, 6, 0, 7, …
c. 12, 6, 1, (-3), (-6), …
Answer:
a. -16, -10, -4
b. -1, 8, -2
c. -8, -9, -9
(Reasoning for c: Subtract 6, then 5, then 4, then 3, then 2, then 1, then 0.)
Question 7: Here are six integer cards: (+1), (+7), (+18), (-5), (-2), (-9). Pick cards and make an expression such that its value is closer to (-30).
Answer:
One way is: (-2) + (-9) – (+18) – (+1) = -30.
Question 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)
b. (positive) + (negative)
c. (negative) + (negative)
d. (negative) – (negative)
e. (negative) – (positive)
f. (negative) + (positive)
Answer:
a. Positive
b. Can be positive or negative
c. Negative
d. Can be positive or negative
e. Negative
f. Can be positive or negative
Question 9: This string has a total of 100 tokens arranged in a particular pattern. What is the value of the string?
Answer:
The value is 20.
Page 268
Figure it Out
Question 1: Can you explain each of Brahmagupta’s rules in terms of Bela’s Building of Fun, or in terms of a number line?
Answer:
- Rule 1: Moving up from a high floor goes higher.
- Rule 2: Moving down from a basement floor goes deeper.
- Rule 3: Moving up then down lands you in between.
- Rule 4: Going up 5 floors then down 5 floors brings you back to the start.
- Rule 5: Pressing 0 means you stay on the same floor.
Question 2: Give your own examples of each rule.
Answer:
- 3 + 7 = 10
- (-2) + (-5) = -7
- 10 + (-4) = 6
- 8 + (-8) = 0
- (-5) + 0 = -5