NCERT Solutions for Class 7 Maths Chapter 5 Connecting the Dots

This worksheet provides complete and accurate NCERT Solutions for Class 7 Maths Chapter 5 Connecting the Dots from the Ganita Prakash II textbook. This chapter introduces students to the world of data handling and statistics in a meaningful and engaging way. Students learn how to collect data, represent it visually, and analyse it using measures of central tendency such as mean and median. The chapter is important because it builds the foundation for logical thinking, data interpretation, and real-life problem-solving skills that students will need in higher classes and everyday life.

Chapter summary: themes and activities

Chapter 5 Connecting the Dots is a data-focused chapter that uses real-life contexts to help students understand statistics. The chapter does not contain stories or poems but is built around engaging activity-based and discussion-based learning. Students explore batting scores of cricketers, onion prices across two towns, heights of family members, class heights, sudoku-solving times, and more. Each context is used to teach a key statistical concept, making the learning feel relevant and grounded. The chapter also features a Puzzle Time activity called Connect the Dots, where students use logical clues to crack a 3-digit code. Several questions are open-ended and student-generated, encouraging independent thinking and observation.

What this NCERT chapter covers?

This chapter covers a wide range of important data handling concepts, including identifying statistical questions versus non-statistical questions, understanding the concept of representative values, calculating the arithmetic mean and median of a data set, understanding the effect of outliers on mean and median, interpreting and constructing dot plots and double-bar graphs, visualising data through different graphical representations such as line graphs and infographics, comparing data sets using measures of central tendency, and understanding the difference between a score of zero and a missing value in data.

How to use these NCERT solutions?

Students should first attempt each question on their own before referring to the solutions in this worksheet. This helps build independent thinking and problem-solving confidence. Parents and teachers can use these solutions to verify answers, understand the step-by-step method expected by NCERT, and identify areas where a student needs more practice. All solutions in this worksheet follow the exact order and structure of the NCERT Ganita Prakash II textbook, making it easy to navigate section by section. This worksheet is also a helpful revision tool before tests and examinations, as it covers every section of Chapter 5 in one place.

Student tips and learning tricks

Always read the question carefully to check whether it is asking for the mean, the median, or both. When calculating the mean, add all values first and then divide by the total count — do not skip any values. When finding the median, always arrange the data in ascending order first. If the number of values is even, the median is the average of the two middle values. Remember that a score of zero is different from a missing or absent value — only count the values that are actually present when calculating the mean. When a data set has an outlier (a value that is much higher or lower than the rest), the median is usually a better representative than the mean. In dot plots, look at the spread of dots to understand variability before drawing conclusions.

Why NCERT solutions are important?

NCERT solutions are the most reliable reference for Class 7 Maths because they are aligned directly with the official curriculum. Chapter 5 Connecting the Dots builds critical thinking by connecting mathematics to real-world data. Strong understanding of this chapter helps students perform confidently in school assessments, as questions on mean, median, and data representation are regularly tested. These solutions also help students understand where and why they made errors, which is more valuable than simply memorising answers. By following NCERT-aligned solutions, students develop a structured approach to answering questions that matches the expectations of their teachers and examiners.

Complete answer key – NCERT solutions

Statistical questions

Statistical questions are those where answers vary and data must be collected to understand the pattern.

Answer: The statistical questions are:
(b) How old are the dogs that live on this street?
(c) What fraction of the students in your class like walking up a hill?
(g) What was the rainfall pattern in Barmer last year?

Representative values

What do you think of Vaishnavi's statement?
Vaishnavi's statement is not appropriate because Yashasvi played only 4 matches while Shubman played 5. The total alone is not a fair comparison when the number of matches is different. We should use the average (mean) to compare.

Can a single number represent a group of numbers?
Yes. A single number like the average (arithmetic mean) can act as a representative of a group of numbers. For example, Shubman's batting in the second series can be represented by his average of 21 runs per match, and Yashasvi's by 24 runs per match.

Figure it out (Mean – Section 1)

1. Shreyas's bounces: 6, 2, 9, 5, 4, 6, 3, 5
Average = (40 ÷ 8) = 5

2. Student-generated activity. Students should perform the bat-and-ball activity themselves for 7 or more attempts, record the number of bounces each time, find the total, and divide by the number of attempts to find their own average.

3. Try this – flowering plant activity: Student-generated activity. Students should identify a flowering plant in their neighbourhood, track the number of flowers blooming each day over a week, find the total, and divide by 7 to get the average number of flowers per day.

4. Nikhil: (17+18+17+16+19+17+18) ÷ 7 = 17.43
Sunil: (20+18+18+17+16+16+17) ÷ 7 = 17.43
Answer: Both Nikhil and Sunil have the same average time of approximately 17.43 seconds. Neither ran quicker on average; their average speeds are equal.

5. Mean enrolment during six consecutive years:
Data: 1555, 1670, 1750, 2013, 2040, 2126
Total = 11154
Number of years = 6
Mean = 11154 ÷ 6 = 1859
Answer: The mean enrolment in the school during this period is 1859 students.

Math talk – Know your onions!

Find the average price of onions at Yahapur and Wahapur:
Yahapur prices: 25, 24, 26, 28, 30, 35, 39, 43, 49, 56, 59, 44
Total = 458; Number of months = 12
Mean (Yahapur) = 458 ÷ 12 = 38.17 (approx.) ₹ per kg

Wahapur prices: 19, 17, 23, 30, 38, 35, 42, 39, 53, 60, 52, 42
Total = 450; Number of months = 12
Mean (Wahapur) = 450 ÷ 12 = 37.5 ₹ per kg

Answer: The average price of onions in Yahapur is approximately ₹38.17 per kg and in Wahapur is ₹37.5 per kg. Yahapur has a slightly higher average price.

Math talk – Dot plot discussion

Does this visualisation capture all the data presented in the tables?
The dot plot captures all the data values but loses the month-wise (sequential) order of the data. We can see how many times each price appears, but we cannot identify which price belongs to which month.

Can we tell the price of onions in Yahapur in January from the dot plot?
No. The dot plot shows only the values and their frequency, not which month each value belongs to. We cannot tell that 25 corresponds to January.

Can you think of any other ways to compare the data?
Yes. Other ways include comparing the range (difference between maximum and minimum), comparing month-by-month prices, comparing how many months each place had higher prices, and using the median.

What else do you wonder about?
Student-generated activity. Students may discuss: Why do onion prices rise in certain months? Are prices affected by rainfall or harvest seasons? How do price changes affect farmers and consumers? Students should discuss with peers, teachers, or family.

Math talk – Height of a family

Find the average height of each family. Can we say Yaangba's family is taller?
Yaangba's family: 169, 173, 155, 165, 160, 164
Total = 986; Number of members = 6
Mean = 986 ÷ 6 = 164.33 cm (approx.)

Poovizhi's family: 170, 173, 165, 118, 175
Total = 801; Number of members = 5
Mean = 801 ÷ 5 = 160.2 cm

Answer: Yaangba's family has a higher average height (164.33 cm) compared to Poovizhi's family (160.2 cm). However, we cannot confidently say Yaangba's family is taller because in Poovizhi's family, 4 out of 5 members are taller than 164.33 cm. The low average for Poovizhi's family is due to one young child (118 cm), who is an outlier. The median would be a better representative here.

Can you think of a number that can represent the data better?
Yes. The median is a better representative when there are outliers. The median of Poovizhi's family is 170 cm, which better reflects the heights of most family members.

Find mean and median of Poovizhi's data without the outlier 118:
Data without outlier: 165, 170, 173, 175
Total = 683
Mean = 683 ÷ 4 = 170.75 cm
Median: Sorted data = 165, 170, 173, 175 → two middle values = 170 and 173
Median = (170 + 173) ÷ 2 = 171.5 cm

Answer: Without the outlier, the mean rises from 160.2 cm to 170.75 cm and the median rises from 170 cm to 171.5 cm. Both values now more accurately represent the family's heights. The change in mean is more significant than the change in median, showing that the mean is more affected by outliers.

Are you a bookworm?

Data: 6, 8, 5, 15, 3, 0, 2, 7, 12, 40, 0, 8, 5, 1, 10
Sorted data: 0, 0, 1, 2, 3, 5, 5, 6, 7, 8, 8, 10, 12, 15, 40
Number of values = 15; Total = 122
Mean = 122 ÷ 15 = 8.13 (approx.)
Median = 8th value in sorted data = 6

Answer: Mean = approximately 8.13 stories; Median = 6 stories
Since 40 is a very high outlier at the upper end, the mean will be greater than the median. Answer confirmed: Mean (8.13) > Median (6).
The median value of 6 means that half of the class members have read 6 or more stories.

Which value is an outlier?
40 is the outlier (it is far higher than all other values in the data).

Find the mean and median without the outlier 40:
Data without 40: 0, 0, 1, 2, 3, 5, 5, 6, 7, 8, 8, 10, 12, 15
Total = 82; Number of values = 14
Mean = 82 ÷ 14 = 5.86 (approx.)
Median = average of 7th and 8th values = (5 + 6) ÷ 2 = 5.5

Answer: Without the outlier, the mean drops from 8.13 to approximately 5.86. The median changes from 6 to 5.5. The mean is more affected by removing the outlier than the median, confirming that the median is a more stable measure in the presence of outliers.

Are we on the same page?

Data (pages Mon–Sun): 16, 18, 20, 22, 26, 16, 10
Total = 128; Number of values = 7
Mean = 128 ÷ 7 = 18.29 (approx.)
Sorted data: 10, 16, 16, 18, 20, 22, 26
Median = 4th value = 18

Answer: Mean ≈ 18.29 pages; Median = 18 pages
Note: The outlier 10 (Sunday, fewer pages) is at the lower end, pulling the mean slightly below the median.

Math talk – Observe variability in the three examples

(a) Mean and median close to each other: Yaangba's family heights — no extreme outlier, data is balanced.
(b) Mean < Median: Poovizhi's family heights — outlier at the lower end (118 cm) pulls the mean downward.
(c) Mean > Median: Short stories read — outlier at the higher end (40 stories) pulls the mean upward.

Math talk – Discuss the effect on mean and median when outliers are present on both sides

When outliers are present on both sides (one very high, one very low), their effects on the mean may cancel each other out, keeping the mean close to the median. However, if one outlier is more extreme than the other, the mean will shift in that direction. The median remains largely unaffected because it depends only on the middle value(s), not the extreme values.

How tall is your class?

Math talk – How many students are taller than the class average height (Mean = 144.4)?
Boys: 147, 135, 130, 154, 128, 135, 134, 158, 155, 146, 146, 142, 140, 141, 144, 145, 150
Girls: 143, 136, 150, 144, 154, 140, 145, 148, 156, 150, 150
Combined class = 28 students

Boys above 144.4: 147, 154, 158, 155, 146, 146, 145, 150 → 8 boys
Girls above 144.4: 150, 154, 145, 148, 156, 150, 150 → 7 girls
Total = 15 students

Answer: 15 students are taller than the class average height of 144.4 cm.
8 boys are taller than 144.4 cm.

Math talk – How long is a minute? (Group A and Group B dot plot discussion)

Both groups performed well at estimating 1 minute.
Group A: Mean = 58.21 seconds, Median = 60 seconds. Since Mean < Median, the data has some lower outliers pulling the mean down. A few students opened their eyes quite early (around 40–45 seconds), making the mean lower than the median.
Group B: Mean = 59.28 seconds, Median = 59.5 seconds. Mean and median are very close to each other, suggesting Group B's data is more balanced with fewer extreme outliers. Group B's estimates are clustered more tightly around 60 seconds.
Overall, Group B performed slightly better as their estimates are closer to 60 seconds and more consistent.

Zero vs no value

A score of 0 means the player played the match but scored zero runs. A dash (—) or "Did not play" means the player was absent and that match should not be counted in the total number of matches when calculating the average. Only matches actually played are counted in the denominator.

Figure it out (Median – Section 2)

1. Yahapur prices sorted: 24, 25, 26, 28, 30, 35, 39, 43, 44, 49, 56, 59
Number of values = 12 (even); Two middle values = 6th and 7th = 35 and 39
Median (Yahapur) = (35 + 39) ÷ 2 = 37 ₹ per kg

Wahapur prices sorted: 17, 19, 23, 30, 35, 38, 39, 42, 42, 52, 53, 60
Number of values = 12 (even); Two middle values = 6th and 7th = 38 and 39
Median (Wahapur) = (38 + 39) ÷ 2 = 38.5 ₹ per kg

Answer: Median price in Yahapur = ₹37 per kg; Median price in Wahapur = ₹38.5 per kg.

2. Sanskruti's class – domestic animals and pets data:
Data: 0, 1, 0, 4, 8, 0, 0, 2, 1, 1, 5, 3, 4, 0, 0, —, 10, 25, 2, —, 2, 4
(Two students were absent, so — values are excluded)
Valid data: 0, 1, 0, 4, 8, 0, 0, 2, 1, 1, 5, 3, 4, 0, 0, 10, 25, 2, 2, 4
Number of values = 20; Total = 72
Mean = 72 ÷ 20 = 3.6
Sorted data: 0, 0, 0, 0, 0, 0, 1, 1, 1, 2, 2, 2, 3, 4, 4, 4, 5, 8, 10, 25
Two middle values = 10th and 11th = 2 and 2
Median = (2 + 2) ÷ 2 = 2

Answer: Mean = 3.6 animals; Median = 2 animals
The data has an outlier at the higher end (25 animals), which pulls the mean (3.6) above the median (2). Most students have 0 to 4 animals. The median of 2 is a better representative of the typical number of animals per household. Since mean > median, the data is skewed to the right due to the high outlier.

3. Date palm trees:
Sorted heights: 43, 44, 45, 46, 49, 50, 51, 52, 52, 54, 54, 55, 55, 56, 56, 57, 58, 59, 60, 60, 60, 60, 61, 61, 62, 63, 65, 66, 67
Total number of trees = 29
For odd number 29, median = 15th term = 56
Sum of heights = 1621
Mean = 1621 ÷ 29 = 55.89

Answer: Median height = 56 feet; Mean height ≈ 55.89 feet; 13 trees are shorter than the average height.

4. Water usage:
(a) No, mean/median cannot be 25–30.
(b) Mean/median always lies between minimum and maximum.

5. The weight of boys is between 2.6 kg and 4.1 kg. The weight of girls lies between 2.5 kg and 4 kg. The heaviest baby is a boy, and the lightest is a girl.

6. Grade 5 (other section):
Whole class mean = 141.21, median = 142.5
Boys mean = 142.05, median = 143
Girls mean = 140.14, median = 140
Answer: Boys are taller on average in this section.

7. Sumo vs Ballet:
Average sumo ≈ 240 kg, ballet ≈ 42 kg
Ratio ≈ 6 times heavier.

Section 5.3 – Visualising data

What is the scale used in the double column graph of onion prices?
The scale used is 1 unit = 10 rupees (each grid line represents 10 rupees on the vertical axis, going from 0 to 60).

Is it now easier to compare month-wise prices in both places?
Yes. The double column (clustered) graph makes it easy to directly compare Yahapur and Wahapur prices for each month side by side. The relative heights of the bars immediately show which place had higher prices in any given month.

All it takes is a minute

1. No, we cannot clearly tell which team batted first. We also cannot confidently determine who won the match from this graph alone.
2. The blue team scored 10 runs in over 12.
3. The red team scored the least runs in over 8.
4. No, it is not easy to tell the target directly from this graph because it shows runs per over but not the total score clearly.

Figure it out (Visualising data)

1. a. The scale used is 10 km/h per division.
b. Sample Answer: It is interesting to see how differently animals move in air, land, and water. For example, birds like the peregrine falcon are extremely fast in the air, while animals like the cheetah are the fastest on land.
c. Sample Answer: The cheetah (~100 km/h) is about twice as fast as a lion (~50 km/h).
d. Yes, a sailfish is roughly four times faster than a humpback whale. However, we cannot say that the sailfish is the fastest aquatic animal in the world, because the infographic only shows a few animals and does not include all aquatic animals.

2. Grade 5 — Aquatic: 6, Aerial: 13, Spaceborne: 2, None: 4
Grade 9 — Aquatic: 5, Aerial: 8, Spaceborne: 9, None: 2
Scale: 1 unit = 2 students (appropriate scale)
Explanation: Draw a double-bar graph with 4 groups on the horizontal axis (Aquatic, Aerial, Spaceborne, None). For each group, draw two bars side by side — one for Grade 5 (e.g., blue) and one for Grade 9 (e.g., red) — using the scale 1 unit = 2 students. Label axes and add a legend.
Observations:
Aerial is the most popular choice in Grade 5 (13 students). Spaceborne is the most popular in Grade 9 (9 students). Fewer Grade 9 students chose None compared to Grade 5. Grade 5 students prefer Aerial over Spaceborne, while Grade 9 students show the opposite trend.

3. (a) Marking bars for Gujarat and Delhi:
Gujarat: 2022 = 69,000; 2023 = 89,000; 2024 = 78,000
Delhi: 2022 = 62,000; 2023 = 74,000; 2024 = 81,000
(b) The graph compares vehicle registrations across different states and years.
Answer: The X-axis shows states. The Y-axis shows number of registrations (in thousands). The scale increases at equal intervals.
c. Most states show a steady increase in registrations from 2022 to 2024. Some states show a large rise, while others show only a small increase.
d. Assam received approximately 10,000 more registrations in 2023 compared to 2022.
e. The registrations increased to about 1.5 times the 2022 value.
f. No, the statement is not correct.

4. Guess of months:
Answer: Day 1 shows maximum temperature of 34°C and minimum of 16°C — this is likely a winter month (December or January in Jodhpur). Day 2 shows maximum of 43°C and minimum of 30°C — this is likely a summer month (May or June in Jodhpur).

Section 5.4 – Data detective

Math talk – Dot plots of School A and School B (Grades 6, 7, 8):

School A — Means:
Grade 6 Boys: 134.8, Grade 6 Girls: 137.78
Grade 7 Boys: 141.8, Grade 7 Girls: 141.83
Grade 8 Boys: 149.35, Grade 8 Girls: 147.81

School B — Means:
Grade 6 Boys: 149.84, Grade 6 Girls: 150.2
Grade 7 Boys: 156.14, Grade 7 Girls: 155.41
Grade 8 Boys: 156.14, Grade 8 Girls: 156.83

Observations: Students in School B are considerably taller than students in the same grades in School A. In School A, girls are slightly taller than boys in Grades 6 and 7, while in Grade 8 boys are slightly taller. In School B, the heights of boys and girls in the same grade are very similar. Heights increase with grade in both schools. The large difference between schools could be due to geographic location, nutrition, or socioeconomic factors.

Which of the following statements can be justified using the data?

1. The average heights of both boys and girls at every age increased from 1989 to 2019.
Answer: TRUE. For every age from 5 to 19, both boys' and girls' heights in 2019 are higher than in 1989.

2. The average height of 13-year-old girls in 1989 is more than the average height of 14-year-old girls in 2009.
Answer: FALSE. 13-year-old girls in 1989 = 143.2 cm; 14-year-old girls in 2009 = 148 cm. 143.2 cm < 148 cm. The statement cannot be justified.

3. The average height of 15-year-old boys in 2019 is more than the average height of 16-year-old boys in 1989.
Answer: TRUE. 15-year-old boys in 2019 = 159 cm; 16-year-old boys in 1989 = 158.9 cm. 159 cm > 158.9 cm. The statement can be justified (by a very small margin).

4. All girls aged 13 are taller than all girls aged 11.
Answer: FALSE. The table shows averages, not individual heights. There will be individual girls aged 13 who are shorter than some girls aged 11. We cannot make this claim about all individuals from average data.

5. Throughout the age period 5 to 19, the average boy's height is more than the average girl's height.
Answer: This is NOT always true. At ages 5–9, boys are slightly taller. At ages 10–12, girls are slightly taller (e.g., age 11: boys 137 cm, girls 138.6 cm; age 12: boys 142.2 cm, girls 143.8 cm). The statement cannot be fully justified.

6. Boys keep growing even beyond age 19.
Answer: CANNOT BE FULLY JUSTIFIED from this data. The data only extends to age 19 and shows near-plateau growth (age 18 = 166 cm, age 19 = 166.5 cm). The trend suggests growth slows significantly. We cannot fully justify this from the given data alone, though it is biologically possible.

Math talk – In 2019, between which two successive ages did boys grow the most? Girls?

Boys' heights in 2019: 107.1, 113.1, 118.6, 123.5, 128.1, 132.6, 137.0, 142.2, 148.4, 154.4, 159.0, 162.3, 164.6, 166.0, 166.5
Largest growth for boys: Between ages 12 and 13 (difference = 6.2 cm)

Girls' heights in 2019: 107.2, 112.9, 118.0, 122.7, 127.6, 132.8, 138.6, 143.8, 147.7, 150.4, 152.4, 153.8, 154.7, 155.2, 155.2
Largest growth for girls: Between ages 10 and 11 (difference = 5.8 cm)

Answer: In 2019, boys grew the most between ages 12 and 13. Girls grew the most between ages 10 and 11.

Math talk – Estimate height of newborn (50 cm) and ages 1–4

Answer: If the average height of a newborn is 50 cm, estimated heights are approximately: Age 1: 72–75 cm; Age 2: 83–86 cm; Age 3: 92–95 cm; Age 4: 98–102 cm.

Math talk – Estimate heights for 2029 based on trend

The data shows a consistent increase in average heights from 1989 to 2019 (roughly 0.3–0.5 cm increase per age per decade). Based on this trend, heights in 2029 would be approximately 1–2 cm taller than 2019 values at each age for both boys and girls. Students should add approximately 1 cm to the 2019 values as an estimate for 2029.

How is the graph organised? What information is presented?

The horizontal axis lists various countries from left to right (roughly from shorter to taller average heights). The vertical axis shows height in cm, starting from 145 cm. For each country, four data points are shown: B-1989, B-2019, G-1989, and G-2019. This allows comparison of how heights have changed over 30 years across different countries.

What do you find interesting?
Student-generated activity (open-ended — students' personal responses accepted). Sample observations: Heights have increased in almost all countries from 1989 to 2019. Countries like the Netherlands have among the tallest averages, while countries like Timor-Leste have the shortest. Boys are consistently taller than girls at age 19 in all countries shown. India's heights have also increased over this period.

Figure it out (Section 5.4)

1. a. True — The boys' dot plot spreads across a wider range of pocket numbers.
b. True — The middle value for boys lies further to the right compared to girls.
c. False — Most girls have fewer pockets, so the mean is smaller.
d. True — The highest dot for boys is farther to the right.

2. a. Mean = (Sum of points) / (Number of games)
Suppose A scored: 10, 8, 6, 4; Total = 28; Mean = 28 ÷ 4 = 7
b. Divide by 3. C played only 3 games (one game marked "Did not play"). So mean must be calculated using only the games played.
c. The player with the highest average score. From comparison, Player A performs best.

3. Group 1: Number of students = 10
Sum of marks = 85 + 76 + 90 + 85 + 39 + 48 + 56 + 95 + 81 + 75 = 730
Mean = 730 ÷ 10 = 73
Sorted: 39, 48, 56, 75, 76, 81, 85, 85, 90, 95
Median = (76 + 81) ÷ 2 = 78.5

Group 2: Number of students = 9
Sum of marks = 68 + 59 + 73 + 86 + 47 + 79 + 90 + 93 + 86 = 681
Mean = 681 ÷ 9 = 75.67
Sorted: 47, 59, 68, 73, 79, 86, 86, 90, 93
Median = 5th score = 79

4. Observations:
Cricket is the most watched sport (1240) but participation is relatively low (320). Basketball and Swimming have balanced watching and participation numbers. Athletics has the least participation (105) compared to watching (510). Hockey shows moderate watching (320) and participation (250). Overall, watching is far more popular than participating across all sports.

5. 17 is an odd number that cannot be divided into two perfectly equal groups.
Sorted heights: 101, 102, 106, 109, 110, 110, 112, 115, 115, 115, 115, 115, 117, 120, 120, 123, 125
Median = 9th term = 115 cm
The height used to divide students into two groups is 112 < height < 115.
If we take 114 cm: Group 1 (Less than 114 cm): 101, 102, 106, 109, 110, 112 → 7 students; Group 2 (More than 114 cm): 115, 115, 115, 115, 115, 117, 120, 120, 123, 125 → 10 students.

6. To analyse the heights of students in a class, we can use measures of central tendency such as the mean and median.

7. (d) The mean height of students in the other section cannot be determined. Each section has the same number of students (30 students: 15 boys + 15 girls). However, knowing the mean height of students in only one section does not give any information about the heights of students in the other section.

8(a). Estimated values for the number of skyscrapers: New York – 38; Tokyo – 160; London – 305

8(b). (i) Only 12 cities have more skyscrapers than Mumbai (86).
Cities with more than 86 skyscrapers: Hong Kong (553), Shenzhen (367), New York (~300), Dubai (251), Guangzhou (188), Shanghai (183), Tokyo (~165), Kuala Lumpur (154), Chongqing (144), Jakarta (112), Bangkok (110), Singapore (95) → 12 cities. Statement is TRUE.

(ii) Only 7 cities have fewer skyscrapers than Mumbai (86).
Cities with fewer than 86 skyscrapers: Seoul (82), Toronto (81), Melbourne (69), Miami (58), Istanbul (48), Moscow (46), London (~20) → 7 cities. Statement is TRUE.

(iii) The tallest building in the world is in Hong Kong.
Answer: FALSE. Hong Kong has the most skyscrapers (buildings taller than 150m), but this does not mean it has the tallest building in the world. The tallest building in the world is the Burj Khalifa in Dubai (828 m). The infographic shows the number of buildings taller than 150m, not which city has the tallest individual building.

9. Sum of differences = 0.5 + (-0.6) + 1.5 + (-2.4) + 0.5 = -0.5
Average difference = -0.5 ÷ 5 = -0.1 cm
Estimated values are 0.1 cm less than actual values on average.

10(a). Table of estimates and measurements:
Length of a pen: Estimate = 15 cm, Measure = 14.5 cm, Difference = 0.5
Length of an eraser: Estimate = 3 cm, Measure = 3.6 cm, Difference = (−0.6)
Length of palm: Estimate = 12 cm, Measure = 10.5 cm, Difference = 1.5
Length of geometry box: Estimate = 20 cm, Measure = 22.4 cm, Difference = (−2.4)
Length of math notebook: Estimate = 30 cm, Measure = 29.5 cm, Difference = 0.5

10(b). Aditi's Sudoku solving times (all 17 values combined):
Mean = 5470 ÷ 17 = 321.76 ≈ 321 seconds
Sorted: 220, 230, 240, 270, 280, 290, 310, 320, 320, 330, 340, 360, 370, 380, 400, 400, 410
Median = 9th value = 320 seconds

Observations: Week 2 times are consistently lower, showing Aditi's practice is paying off. The median dropped from 360 seconds (Week 1) to 240 seconds (Week 2), highlighting faster average performance.

Individual project – open-ended activities

(a) How long is a sentence? Student-generated activity. Students should pick two textbooks from different subjects, choose one page with a lot of text from each, count the number of words in every sentence on each page, record the data, make a dot plot for each page, and calculate the mean and median number of words per sentence for each page. Compare the two pages and describe differences in sentence length variability and central tendency.

(b) What is in a name? Student-generated activity. Students should list all classmates' names, count the number of letters in each name, then: (i) calculate mean and median name length; (ii) draw a dot plot of name lengths and describe variability and central tendency; (iii) identify which starting letters are most and least common; (iv) find the median starting letter; (v) draw a double-bar graph categorising boys' and girls' names by vowel/consonant start and end patterns.

Individual project (long term) – In and out

Student-generated activity. Over one month, students should track how many times they step out of the house each day and record this daily. After 30 days: (i) make a dot plot and calculate mean and median, and describe variability; (ii) look for interesting patterns (e.g., more outings on weekends, fewer on school days); (iii) optionally, collect data from family members and compare patterns.

Small-group project

(a) Our heights vs. family's heights: Student-generated activity. (i) Each student collects heights of all family members, makes a dot plot, calculates mean and median, and describes variability and central tendency. (ii) Draw a double-bar graph showing each student's own height next to their family's mean height. (iii) Share observations with the group — e.g., which students are taller than their family mean, which are shorter.

(b) Estimating time: Student-generated activity. (i) Each student (and family members) closes their eyes and opens after what they think is 1 minute or 3 minutes (no counting). The actual time elapsed is recorded. Make two dot plots — one for 1-minute estimates, one for 3-minute estimates. (ii) Mark the estimates on the dot plot. Calculate mean and median. Describe variability. (iii) Draw a double-bar graph showing each family's mean 1-minute and mean 3-minute estimates. (iv) Compare estimates across students — who is better at estimating 1 minute vs 3 minutes? Are the estimates for 3 minutes roughly 3 times the 1-minute estimates?

Puzzle time – Connect the dots

The final 3-digit code is 415.
Clue 3 told us digits 0, 3, and 6 are not in the code. Clue 1 showed that 5 is the last digit. Clue 4 proved that 4 is in the code but not at the end. Clue 2 showed that 1 is in the code but not in the last spot. Clue 5 confirmed that 4 and 5 are correct but in different positions.
So the only arrangement that fits all clues is 415.

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