Tales by Dots and Lines - Chapter 5 Class 8 (Ganita Prakash II)

Find the mean of the following data and share your observations: (i) The first 50 natural numbers.Natural numbers are numbers starting from 1 like
1, 2, 3, 4, 5, ….

The dot plot below shows a collection of data and its average; but one dot is missing. Mark the missing value so that the mean is 9 (as shown below).We can use the "seesaw" balancing method, which is
Total distance from Mean from Left side
= Total distance from Mean on Right side

Sudhakar, the class teacher, asks Shreyas to measure the heights of all 24 students in his class and calculate the average height. Shreyas informs the teacher that the average height is 150.2 cm. Sudhakar discovers that the students were wearing uniform shoes when the measurements were taken and the shoes add 1 cm to the height. (i) Should the teacher get all the heights measured again without the shoes to find the correct average height? Or is there a simpler way?Yes, there is a simpler way

The three dot plots below show the lengths, in minutes, of songs of different albums. Which of these has a mean of 5.57 minutes? Explain how you arrived at the answer.A mean of 5.57 indicates the balance point of the data is between 5 and 6.
Plot B's data only goes up to 5, so its mean must be lower than 5.
Plot C's data is clustered entirely between 3.5 and 4.5, so its mean falls in that range.
Only Plot A contains data exclusively clustered around the 5 to 6.5 range, making it the only possible option with a mean of 5.57.

Find the median of 8, 10, 19, 23, 26, 34, 40, 41, 41, 48, 51, 55, 70, 84, 91, 92. (i) If we include one value to the data (in the given list) without affecting the median, what could that value be? (ii) If we include two values to the data without affecting the median what could the two values be? (iii) If we remove one value from the data without affecting the median what could the value be?Let’s find Median First

Examine the statements below and justify if the statement is always true, sometimes true, or never true. (i) Removing a value less than the median will decrease the median.Removing a smaller value shifts the middle position to the right, which will either increase the median or keep it the same.
Thus, this statement is Never True

The mean of the numbers 8, 13, 10, 4, 5, 20, y, 10 is 10.375. Find the value of y.Finding Mean of these numbers
8, 13, 10, 4, 5, 20, y, 10

The mean of a set of data with 15 values is 134. Find the sum of the data.Now,
Mean = (𝑆𝑢𝑚 𝑜𝑓 𝑎𝑙𝑙 𝑣𝑎𝑙𝑢𝑒𝑠)/(𝑁𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑣𝑎𝑙𝑢𝑒𝑠)
134 = (𝑺𝒖𝒎 𝒐𝒇 𝒂𝒍𝒍 𝒗𝒂𝒍𝒖𝒆𝒔)/𝟏𝟓
134 × 15 = Sum of all values
2010 = Sum of all values
Sum of all values = 2010

Consider the data: 12, 47, 8, 73, 18, 35, 39, 8, 29, 25, p. Which of the following number(s) could be p if the median of this data is 29? (i) 10 (ii) 25 (iii) 40 (iv) 100 (v) 29 (vi) 47 (vii) 30There are 10 known values:
8, 8, 12, 18, 25, 29, 35, 39, 47, 73

The number of times students rode their cycles in a week is shown in the dot plot below. Four students rode their cycles twice in that week. (i) Find the average number of times students rode their cycles.Here,
Total number of students = 3 + 1 + 4 + 7 + 7 + 5 + 4 + 6 + 3 + 0 + 2
= 42
And
Total rides = 0 × 3 + 1 + 2 × 4 + 3 × 7 + 4 × 7 + 5 × 5 + 6 × 4 + 7 × 6 + 8 × 3 + 10 × 2
= 193

A dart-throwing competition was organised in a school. The number of throws participants took to hit the bull’s eye (the centre circle) is given in the table below. Describe the data using its minimum, maximum, mean and median.Here,
Total number of students = 1 + 0 + 0 + 1 + 4 + 9 + 12 + 15 + 10 + 10
= 62
And,
Number of trials
= 1 × 1 + 2 × 0 + 3 × 0 + 4 × 1 + 5 × 4 + 6 × 9 + 7 × 12 + 8 × 15 + 9 × 10 + 10 × 10
= 473
= 473

The average number of customers visiting a shop and the average number of customers actually purchasing items over different days of the week is shown in the table below. Visualise this data on a line graph.While drawing a line graph, we follow these steps
First we decide what to put in x-axis and y-axis
We always use time in x-axis – so Day of the week is in x-axis
So, Number of Customers are in y-axis
Then, we find out our Scale
Then, we find out our Scale
Since numbers go from 7 to 35
We choose Scale 1 unit length = 4 units
Lastly, we decide our legend – which color to denote
We choose Blue for Visiting, Red for Purchasing

The average number of days of rainfall in each month for a few cities is shown in the table below: (i) What could be the possible method to compile this data?Just like temperature, meteorologists collect daily rainfall data over several years.
They then add up all the rain that fell in a specific month (say, all the Junes over a 10-year period) and average it out to find the "typical" amount of rain for June.
Question 2 (ii) Mark the data for Mangaluru, Port Blair, and Rameswaram in the line graph shown below. You can round off the values to the nearest integer.Since the graph is already given, we just plot points and join them by straight lines.

The following line graph shows the number of births in every month in India over a time period: (i) What are your observations?The graph shows a distinct, repeating wave pattern.
Births dip low at the beginning of the year and hit a peak in the later months of the year, every single year.
Question 3 The following line graph shows the number of births in every month in India over a time period: (ii) What was the approximate number of births in July 2017?The dot is between 1.5M and 2M
Thus, we can say It is approximately 1.75 Million.
Question 3 The following line graph shows the number of births in every month in India over a time period: (iii) What time period does the graph capture?The graph starts slightly before July 2017 and ends slightly after January 2020.
So, it roughly covers mid 2017 to early-2020.
Question 3 The following line graph shows the number of births in every month in India over a time period: (iv) Compare the number of births in the month of January in the years 2018, 2019, and 2020.We notice that
Dot of Jan 2019 is higher than Dot of Jan 2018
Dot of Jan 2020 is same as than Dot of Jan 2019
Question 3 The following line graph shows the number of births in every month in India over a time period: (v) Estimate the number of births in the year 2019.We can calculate the approximate values for each month
For each month, value is
Jan 2019 – 1.75 M
Feb 2019 – 1.55 M
Mar 2019 – 1.8 M
Apr 2019 – 1.5 M
May 2019 – 1.6 M
Jun 2019 – 1.6 M
July 2019 – 1.75 M
Aug 2019 – 2 M
Sep 2019 – 1.95 M
Oct 2019 – 2 M
Nov 2019 – 1.9 M
Dec 2019 – 1.8 M Thus,
Number of births in 2019
= 1.75 + 1.55 + 1.8 + 1.5 + 1.6 + 1.6 + 1.75 + 2 + 1.95 + 2 + 1.9 + 1.8
= 21.2 M

Mean Grids: (i) Fill the grid with 9 distinct numbers such that the average along each row, column, and diagonal is 10.Here, the average of 3 numbers is 10,
So, their sum must be 30

Give two examples of data that satisfy each of the following conditions: (i) 3 numbers whose mean is 8.Here,
Sum of the 3 numbers = 8 × 3 = 24

Fill in the blanks such that the median of the collection is 13: 5, 21, 14, _____, ______, ______. How many possibilities exist if only counting numbers are allowed?Our data is
5 , 21, 14, _____, ______, ______

Fill in the blanks such that the mean of the collection is 6.5: 3, 11, ____, _____, 15, 6. How many possibilities exist if only counting numbers are allowed?Let the two numbers be x & y
So, our data is
3, 11, x, y, 15, 6

Check whether each of the statements below is true. Justify your reasoning. Use algebra, if necessary, to justify. (i) The average of two even numbers is even.We take an example
Let the two even numbers be 2 & 4
Average = (2 + 4)/2
= 6/2
= 3

There were 2 new admissions to Sudhakar’s class just a couple of days after the class average height was found to be 150.2 cm. (i) Which of the following statements are correct? Why? (a) The average height of the class will increase as there are 2 new values. (b) The average height of the class will remain the same. (c) The heights of the new students have to be measured to find out the new average height. (d) The heights of everyone in the class has to be measured again to calculate the new average height.To find the average, we need to Sum of all heights
Thus, we cannot find average without knowing the heights of the new students
So, option (c) is correct
Question 6 There were 2 new admissions to Sudhakar’s class just a couple of days after the class average height was found to be 150.2 cm. (ii) The heights of the two new joinees are 149 cm and 152 cm. Which of the following statements about the class’ average height are correct? Why? (a) The average will remain the same. (b) The average will increase. (c) The average will decrease. (d) The information is not sufficient to make a claim about the average.The two new kids have heights of 149 and 152.
Thus,
Average of two new kids = (149 + 152)/2
= 301/2
= 150.5 cm

Is 17 the average of the data shown in the dot plot below? Share the method you used to answer this question.We can use the "seesaw" balancing method,
Which is
Total distance from Mean from Left side
= Total distance from Mean on Right side

The weights of people in a group were measured every month. The average weight for the previous month was 65.3 kg and the median weight was 67 kg. The data for this month showed that one person has lost 2 kg and two have gained 1 kg. What can we say about the change in mean weight and median weight this month?For Mean
One person lost 2kg, and two gained 1kg.
Thus,
Total change in weight = –2 + 1 + 1
= 0

The following table shows the retail price (in ₹) of iodised salt in the month of January in a few states over 10 years. For your calculations and plotting you may round off values to the nearest counting number. (i) Choose data from any 3 states you find interesting and present it through a line graph using an appropriate scale.Let’s choose Andaman & Nicobar, Assam and Gujrat

Referring to the graph below, which of the following statements are valid? Why? (i) In 1983, the majority in rural areas used kerosene as a primary lighting source while the majority in urban areas used electricity.The graph clearly shows
Kerosene > Electricity for rural and
Electricity > Kerosene for urban in 1983

Answer the following questions based on the line graph. (i) How long do children aged 10 in urban areas spend each day on hobbies and games?Looking where the blue line hits the 10-year mark
Urban children aged 10 spend little more than 2 hours
Question 11 (ii) At what age is the average time spent daily on hobbies and games by rural kids 1.5 hours? (a) 8 years (b) 10 years (c) 12 years (d) 14 years (e) 18 yearsThe rural (red) line hits the 1.5-hour mark at (d) 14 years.
Question 11 (iii) Are the following statements correct? (a) The average time spent daily on hobbies and games by kids aged 15 is twice that of kids aged 10. (b) All rural kids aged 15 spend at least 1 hour on hobbies and games everyday.Statement (a)
At age 15, the average is roughly 1 hour. At age 10, it is 2 hours. 1 is half of 2, not twice!
Thus, given statement is incorrect

Individual project: Make your own activity strip for different days of the week. (i) Do you eat and sleep at regular times every day? Typically how long do you spend outdoors? (ii) Calculate the average time spent per activity. Represent this average day using a strip. (iii) Similarly, track the activities of any adult at home. Compare your data with theirs.Alright, we do something like this
And, then we answer these questions

Small group project: Make a group of 3 – 4 members. Do at least one of the following: (i) Track daily sleep time of all your family members for a week. Daily sleep time includes night sleep, naps, and any sleep during the day. (a) Represent this on strips. (b) Put together the data of all your group members. Calculate the average and median sleep time of children, adults, elderly. (c) Share your findings and observations.Let’s do this
And, then we answer these questions
Question 13 Small group project: Make a group of 3 – 4 members. Do at least one of the following: (ii) When do schools start and end? On a weekday, Manoj’s school starts at 9:30 am and ends at 4:30 pm, i.e., 7 hours which include class time and breaks. Collect information on the daily timings of different schools for Grade 8, including class time and break time (the schools can be anywhere in the country. You can ask your neighbours, relatives, parents and friends to find out). Analyse and present the data collected.Let’s do this
And, then we answer these questions

The following graphs show the sunrise and sunset times across the year at 4 locations in India. Observe how the graphs are organised. Are you able to identify which lines indicate the sunrise and which indicate the sunset? Answer the following questions based on the graphs: (i) At which place does the sun rise the earliest in January? What is the approximate day length at this place in January?Think about a normal day.
The sun rises in the early morning and sets in the evening. Therefore, the bottom lines (which hover between 04:00 and 08:00) represent sunrise, and the top lines (which hover between 16:00 and 20:00) represent sunset.

We all know the typical sunrise and sunset timings. Do you know when the moon rises and sets? Does it follow a regular pattern like the sun? Let’s find out. The following graph shows the moonrise and moonset time over a month: (i) Find out on what dates amavasya (new moon) and purnima (full moon) were in this month.The Rule of the Moon
A full moon (Purnima) sits exactly opposite the sun, meaning it rises right as the sun is setting (around 18:00). A new moon (Amavasya) sits right next to the sun, meaning it rises and sets at the exact same time the sun does (rising near 06:00 and setting near 18:00).