Class 6 › Mathematics › Ganita Prakash Chapter 10: The Other Side of ZeroClass 6 Mathematics — summary, notes, extra questions & NCERT solutions
The numbers \(\dots, -3, -2, -1, 0, 1, 2, 3, \dots\) are together called
A factors B fractions C integers D multiples
This final chapter extends the number line to the left of zero, introducing integers. Beginning with Bela\u2019s multi-storey building of fun, with floors above the ground (positive) and below the ground (negative), students see that numbers less than 0 are needed to describe situations like underground floors, debts and temperatures below zero. The complete number line now runs \(\dots, -3, -2, -1, 0, 1, 2, 3, \dots\); positive numbers lie to the right of 0 and negative numbers to the left. Together with zero, the positive and negative whole numbers form the integers. Every integer has an opposite, such as \(+3\) and \(-3\), which are the same distance from 0. The chapter teaches comparing and ordering integers (a number to the right is greater, so \(-1 > -5\)) and then adding and subtracting them using movements on the number line and the building\u2019s lift: moving up adds and moving down subtracts. Adding a number and its opposite gives 0. Real contexts \u2014 tokens, floors, gains and losses \u2014 make the rules for signed numbers meaningful, completing the journey from counting numbers to the full set of integers.
Integer
Any of the numbers \(\dots, -3, -2, -1, 0, 1, 2, 3, \dots\): the positive and negative whole numbers together with zero.
Negative number
A number less than 0, lying to the left of zero on the number line.
Positive number
A number greater than 0, lying to the right of zero on the number line.
Number line
A line on which integers are marked at equal spacing, with 0 in the middle.
Opposite (additive inverse)
For an integer such as \(+3\), its opposite is \(-3\); they are equally far from 0 and add to give 0.
Zero
The integer separating positive and negative numbers; adding 0 to any number leaves it unchanged.
Q1 What are integers? Answer Q2 Where do negative numbers lie on the number line? Answer Q3 Give a real-life situation needing negative numbers. Answer Q4 Which is greater, \(-1\) or \(-5\)? Answer Q5 What is the opposite of \(+7\)? Answer Q6 What do you get when you add a number and its opposite? Answer Q7 How is adding a positive integer shown on the number line? Answer Q8 How is subtracting shown using the building of fun? Answer Q9 Is zero a positive or a negative number? Answer Q10 Why was the number ray extended into a full number line? Answer Q1 1. You start from Floor +2 and press –3 in the lift. Where will you reach? Write an expression for this movement. Answer Q2 2. Evaluate these expressions (you may think of them as Starting Floor + Movement by referring to the Building of Fun). a. (+1)+(+4) = _______ Answer Q3 2. Evaluate these expressions (you may think of them as Starting Floor + Movement by referring to the Building of Fun). b. (+4)+(+1) = _______ Answer Q4 2. Evaluate these expressions (you may think of them as Starting Floor + Movement by referring to the Building of Fun). c. (+4)+(– 3) = _______ Answer Q5 2. Evaluate these expressions (you may think of them as Starting Floor + Movement by referring to the Building of Fun). d. (–1)+(+2) = _______ Answer Q6 2. Evaluate these expressions (you may think of them as Starting Floor + Movement by referring to the Building of Fun). e. (–1)+(+1) = _______ Answer Q7 2. Evaluate these expressions (you may think of them as Starting Floor + Movement by referring to the Building of Fun). f. 0+(+2) = _______ Answer Q8 2. Evaluate these expressions (you may think of them as Starting Floor + Movement by referring to the Building of Fun). g. 0+(–2) = _______ Answer Q9 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 Q1 a. (+1)+(+4) = _______ Answer Q2 b. (+4)+(+1) = _______ Answer Q3 c. (+4)+(– 3)+(–2) = _______ Answer Q4 d. (–1)+(+2)+(–3) = _______ Answer Q1 1. Compare the following numbers using the Building of Fun and fill in the boxes with < or >. a. –2 +5 Answer Q2 1. Compare the following numbers using the Building of Fun and fill in the boxes with < or >. b. –5 +4 Answer Q3 1. Compare the following numbers using the Building of Fun and fill in the boxes with < or >. c. –5 –3 Answer Q4 1. Compare the following numbers using the Building of Fun and fill in the boxes with < or >. d. +6 –6 Answer Q5 1. Compare the following numbers using the Building of Fun and fill in the boxes with < or >. e. 0 –4 Answer Q6 1. Compare the following numbers using the Building of Fun and fill in the boxes with < or >. f. 0 +4 Answer Q7 2. Imagine the Building of Fun with more floors. Compare the numbers and fill in the boxes with < or >: a. –10 –12 Answer Q8 2. Imagine the Building of Fun with more floors. Compare the numbers and fill in the boxes with < or >: b. +17 –10 Answer Q9 2. Imagine the Building of Fun with more floors. Compare the numbers and fill in the boxes with < or >: c. 0 –20 Answer Q10 2. Imagine the Building of Fun with more floors. Compare the numbers and fill in the boxes with < or >: d. +9 –9 Answer Q11 2. Imagine the Building of Fun with more floors. Compare the numbers and fill in the boxes with < or >: e. –25 –7 Answer Q12 2. Imagine the Building of Fun with more floors. Compare the numbers and fill in the boxes with < or >: f. +15 –17 Answer Q13 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 Q14 4. Mark the following floors of the building shown on the right. a. –7 Answer Q15 4. Mark the following floors of the building shown on the right. b. – 4 Answer Q16 4. Mark the following floors of the building shown on the right. c. +3 Answer Q17 4. Mark the following floors of the building shown on the right. d. – 10 Answer Q1 a. (+1) –(+4) = _______ Answer Q2 b. (0)–(+2) = _______ Answer Q3 c. (+4)–(+1) = _______ Answer Q4 d. (0)– (–2) = _______ Answer Q5 e. (+4) – (–3) = _______ Answer Q6 f. (–4) –(–3) = _______ Answer Q7 g. (–1)–(+2) = _______ Answer Q8 h. (–2) –(–2) = _______ Answer Q9 i. (–1) – (+1) = _______ Answer Q10 j. (+3)– (–3) = _______ Answer Q1 a. (+40)+ ______ = +200 Answer Q2 b. (+40)+_______=–200 Answer Q3 c. (–50)+ ______ = +200 Answer Q4 d. (–50)+_______=–200 Answer Q5 e. (–200) – (–40) = _______ Answer Q6 f. (+200) – (+40) = _______ Answer Q7 g. (–200) –(+40) = _______ Answer Q1 1. Mark 3 positive numbers and 3 negative numbers on the number line above. Answer Q2 2. Write down the above 3 marked negative numbers in the following boxes: Answer Q3 3. Is 2 > – 3? Why? Is –2 < 3? Why? Answer Q4 4. What are a. –5+0 Answer Q5 4. What are b. 7+(–7) Answer Q6 4. What are c. –10+20 Answer Q7 4. What are d. 10 –20 Answer Q8 4. What are e. 7–(–7) Answer Q9 4. What are f. –8–(–10) Answer Q1 a. –125 + (–30) = _______ Answer Q2 b. +105 – (–55) = _______ Answer Q3 c. +80 – (–150) = _______ Answer Q4 d. –99 – (–200) = _______ Answer Q1 1. Complete the additions using tokens. a. (+6)+(+4) Answer Q2 1. Complete the additions using tokens. b. (–3) + (–2) Answer Q3 1. Complete the additions using tokens. c. (+5)+(–7) Answer Q4 1. Complete the additions using tokens. d. (–2) + (+6) Answer Q5 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. Answer Q6 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. Answer Q1 1. Evaluate the following differences using tokens. Check that you get the same result as with other methods you now know: a. (+10) –(+7) Answer Q2 1. Evaluate the following differences using tokens. Check that you get the same result as with other methods you now know: b. (–8)– (–4) Answer Q3 1. Evaluate the following differences using tokens. Check that you get the same result as with other methods you now know: c. (–9) –(–4) Answer Q4 1. Evaluate the following differences using tokens. Check that you get the same result as with other methods you now know: d. (+9) –(+12) Answer Q5 1. Evaluate the following differences using tokens. Check that you get the same result as with other methods you now know: e. (–5)– (–7) Answer Q6 1. Evaluate the following differences using tokens. Check that you get the same result as with other methods you now know: f. (–2)– (–6) Answer Q7 2. Complete the subtractions: a. (–5) –(–7) Answer Q8 2. Complete the subtractions: b. (+10) –(+13) Answer Q9 2. Complete the subtractions: c. (–7)–(–9) Answer Q10 2. Complete the subtractions: d. (+3) – (+8) Answer Q11 2. Complete the subtractions: e. (–2) –(–7) Answer Q12 2. Complete the subtractions: f. (+3)–(+15) Answer Q1 1. Try to subtract: –3– (+5). How many zero pairs will you have to put in? What is the result? Answer Q2 2. Evaluate the following using tokens. a. (–3) – (+10) Answer Q3 2. Evaluate the following using tokens. b. (+8)– (–7) Answer Q4 2. Evaluate the following using tokens. c. (–5)–(+9) Answer Q5 2. Evaluate the following using tokens. d. (–9) –(+10) Answer Q6 2. Evaluate the following using tokens. e. (+6) –(–4) Answer Q7 2. Evaluate the following using tokens. f. (–2)–(+7) Answer Q1 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? Answer Q2 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? Answer Q3 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? Answer Q1 1. Looking at the geographical cross section, fill in the respective heights: a. Answer Q2 1. Looking at the geographical cross section, fill in the respective heights: b. Answer Q3 1. Looking at the geographical cross section, fill in the respective heights: c. Answer Q4 1. Looking at the geographical cross section, fill in the respective heights: d. Answer Q5 1. Looking at the geographical cross section, fill in the respective heights: e. Answer Q6 1. Looking at the geographical cross section, fill in the respective heights: f. Answer Q7 1. Looking at the geographical cross section, fill in the respective heights: g. Answer Q8 2. Which is the highest point in this geographical cross section? Which is the lowest point? Answer Q9 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 Q10 4. What is the highest point above sea level on Earth? What is its height? Answer Q11 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 Q1 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? Answer Q2 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. Answer Q1 1. Do the calculations for the second grid above and find the border sum. Answer Q2 2. Complete the grids to make the required border sum: Border sum is +4 Answer Q3 2. Complete the grids to make the required border sum: Border sum is – 2 Answer Q4 2. Complete the grids to make the required border sum: Border sum is – 4 Answer Q5 3. For the last grid above, find more than one way of filling the numbers to get border sum –4. Answer Q6 4. Which other grids can be filled in multiple ways? What could be the reason? Answer Q7 5. Make a border integer square puzzle and challenge your classmates. Answer Q1 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! Answer Q2 2. Play the same game with the grids below. What answer did you get? First grid Answer Q3 2. Play the same game with the grids below. What answer did you get? Second grid Answer Q4 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? Answer Q1 1. Write all the integers between the given pairs, in increasing order. a. 0 and –7 Answer Q2 1. Write all the integers between the given pairs, in increasing order. b. –4 and 4 Answer Q3 1. Write all the integers between the given pairs, in increasing order. c. –8 and –15 Answer Q4 1. Write all the integers between the given pairs, in increasing order. d. –30 and –23 Answer Q5 2. Give three numbers such that their sum is –8. Answer Q6 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 Q7 4. Solve these: 8 – 13 Answer Q8 4. Solve these: (– 8) – (13) Answer Q9 4. Solve these: (– 13) – (– 8) Answer Q10 4. Solve these: (– 13) + (– 8) Answer Q11 4. Solve these: 8 + (– 13) Answer Q12 4. Solve these: (– 8) – (– 13) Answer Q13 4. Solve these: (13) – 8 Answer Q14 4. Solve these: 13 – (– 8) Answer Q15 5. Find the years below. a. From the present year, which year was it 150 years ago? Answer Q16 5. Find the years below. b. From the present year, which year was it 2200 years ago? Answer Q17 5. Find the years below. c. What will be the year 320 years after 680 BCE? Answer Q18 6. Complete the following sequences: a. (–40), (–34), (–28), (–22), _____, ______, ______ Answer Q19 6. Complete the following sequences: b. 3, 4, 2, 5, 1, 6, 0, 7, _____, _____, _____ Answer Q20 6. Complete the following sequences: c. _____, ______, 12, 6, 1, (–3), (–6), _____, ______, ______ Answer Q21 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). Answer Q22 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) Answer Q23 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) Answer Q24 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) Answer Q25 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) Answer Q26 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) Answer Q27 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) Answer Q28 9. This string has a total of 100 tokens arranged in a particular pattern. What is the value of the string? Answer Q1 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 Q2 2. Give your own examples of each rule. 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Any of the numbers \(\dots, -3, -2, -1, 0, 1, 2, 3, \dots\): the positive and negative whole numbers together with zero.
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