Tales by Dots and Lines · Lesson 13 of 13
Chapter Summary and Practice
“Connect the chapter’s ideas through mixed problems and finish with Game of Hex.”
• Choose an appropriate method for a mean, median, or frequency problem. • Interpret tables, line graphs, indices, and activity strips accurately. • Support conclusions with calculations or counterexamples. • Play Game of Hex using its placement and connection rules.
You have used numbers, dots, lines, colours, and time intervals to understand data. These representations answer different questions. A mean gives a fair share or balance point, a median gives a middle, and a graph helps you see how values vary or change.
This final lesson brings those ideas together. Read the recap, work through the mixed examples, and attempt the practice before opening the solutions. The chapter ends with Game of Hex, where careful observation and planning are used in a different kind of mathematical activity.
Chapter Summary
Begin every data problem by identifying what one observation represents and what unit is used. Then choose the method that answers the question. A request for total births needs addition; a request for births per month needs a mean; a question about middle family size needs ordered positions.
Keep exact values and estimates distinct. A decimal table entry can support a precise calculation with those recorded values, while a point read from a graph usually supports an approximate answer. The wording of your conclusion should reflect that difference.
| Idea | Method | Check |
|---|---|---|
| Mean | Add all observations and divide by their count | Repeated values and zero values still count |
| Balance point | Compare total distances on the two sides | Equal distances do not require equal numbers of dots |
| New observation | Change both total and count | Compare the inserted value with the old mean |
| Uniform change | Apply the same shift or scaling to the mean | Every observation must undergo the same operation |
| Missing value | Mean × count − known subtotal | Use the count of the complete dataset |
| Median | Sort, then locate the middle position or pair | Insertion, removal, and ties can change the outcome |
| Frequency data | Use value × frequency and total frequency | Running counts locate median positions |
| Line graph | Read axes, units, legend, and time scope | A large value is different from a large increase |
| Infographic | Decode the defined colour or size scale | An index is not automatically a percentage |
| Time strip | Convert cell counts using the interval length | The whole day must total 24 hours |
Choosing a Method
A useful first sentence in your working is “I need the total”, “I need the middle positions”, or “I need to compare the same times”. This tells you what information matters. It also prevents the common habit of taking an average just because several numbers appear in the question.
Problem
A group has mean mass 65.3 kg. One person loses 2 kg and two people gain 1 kg each. What can be said about the new mean and median?
- 1.The net change in total mass is −2 + 1 + 1 = 0 kg.
- 2.The number of people is unchanged.
- 3.The mean is therefore unchanged at 65.3 kg.
- 4.The median cannot be determined from these changes alone. It depends on which observations changed and how the ordered middle positions are affected.
Problem
Find the mean of the first 50 positive integers, the first 50 positive odd integers, and the first 50 positive multiples of 4.
- 1.For 1 through 50, pair 1 with 50, 2 with 49, and so on. Each of 25 pairs totals 51. The mean is (25 × 51) ÷ 50 = 25.5.
- 2.The first 50 odd numbers run from 1 to 99. Opposite-end pairs each total 100, so their mean is 50.
- 3.The first 50 positive multiples of 4 are 4 times the numbers 1 through 50. Their mean is 4 × 25.5 = 102.
- 4.These shortcuts work because of the regular structure; do not apply the endpoint midpoint to an arbitrary dataset.
Problem
Values 2, 4, and 7 occur with frequencies 3, 2, and 3. Find the mean and median.
- 1.The count is 3 + 2 + 3 = 8.
- 2.The weighted total is 2 × 3 + 4 × 2 + 7 × 3 = 35, so the mean is 35 ÷ 8 = 4.375.
- 3.Positions 1–3 contain 2, positions 4–5 contain 4, and positions 6–8 contain 7.
- 4.The fourth and fifth values are both 4, so the median is 4.
Problem
Fill three blanks in 5, 21, 14, __, __, __ so that the median is 13. Are there finitely many possibilities if positive whole numbers are allowed?
- 1.For six ordered values, the third and fourth must average to 13. One suitable middle pair is 12 and 14.
- 2.Choose the blanks as 1, 12, and 22. The sorted list is 1, 5, 12, 14, 21, 22.
- 3.The median is (12 + 14) ÷ 2 = 13.
- 4.Replace the final blank 22 with any positive integer at least 21. The middle pair stays 12 and 14.
- 5.There are infinitely many possibilities because the large value has no upper bound.
Problem
A region has a strongly positive rice-side index, and a group there has mean hobby time 1 hour. Explain two claims we cannot make.
- 1.The positive index describes a mapped consumption comparison; it does not prove that nobody eats wheat.
- 2.A mean hobby time of 1 hour does not prove that every person spends exactly 1 hour.
- 3.For example, hobby times 0, 1, and 2 hours have mean 1 hour.
- 4.Both measures summarise groups, so individual claims require more detailed evidence.
Reasoning with Always and Sometimes
A statement that is true for one dataset may fail for another. To disprove “always”, one valid counterexample is enough. To justify a rule for all datasets, explain why it follows from the definitions or a general calculation.
The mean of two even numbers is always a whole number, but it is not always even: 2 and 4 have mean 3. Similarly, the mean of two multiples of 5 need not be a multiple of 5; 5 and 10 have mean 7.5. Five multiples of 5 have an integer mean, but it need not be a multiple of 5 either.
Quiz
Five values have mean 8. Their total is what?
A mean is 10. Insert 4 and 16. What happens?
Which dataset disproves “the mean of two even numbers is always even”?
Insert 9 into 2, 5, 5. What is the new median?
A graph has monthly totals. Which operation gives the yearly total?
How many hours do 19 half-hour cells represent?
A graph omits a country. What does that establish?
A value is recorded 12 too high among 6 observations. How much should the mean decrease?
The total is count × mean = 5 × 8 = 40.
Practice Problems
- Find the mean of 2, 3, and 10 and verify it using balanced distances.
- Seven values have mean 12. Insert a value of 20, then find the new mean.
- Six values have mean 9. Remove an existing value of 14. Find the remaining mean.
- A dataset has mean 18. Every value is halved and then increased by 4. Find the new mean.
- Ten measurements have mean 24.6. Nine of them total 219. Find the missing one.
- A mean of 15 is based on 8 values. One entry 26 should have been 18. Correct the mean.
- Find the median of 18, 5, 11, 7, 14, 9. Then insert 20 and find the new median.
- Values 1, 3, 6 have frequencies 2, 5, 3. Find count, mean, and median.
- Two groups have counts 4 and 6 and means 12 and 17. Find their combined mean.
- The numbers 3, 11, x, y, 15, 6 have mean 6.5. Find all ordered positive whole-number pairs (x, y).
- Give six numbers with mean equal to median, and six with mean greater than median.
- Must the mean of five multiples of 5 be a multiple of 5? Give a justified answer.
- A spreadsheet places six activity values in B3:G3. Write formulas for their total and mean.
- A line graph gives counts 120, 150, 140, 200 over four equally spaced days. Find the largest increase and test “increased every day”.
- A price rises from ₹9.47 to ₹23.99. Another rises from ₹20 to ₹29.80. Which has the greater absolute increase?
- An activity strip has 48 half-hour cells. Sleep uses 18 and travel uses 3. Find both durations and the remaining time.
- An observation begins at 23:25 and ends at 00:40 the next day. Find the duration.
- Design a short investigation comparing school-day durations. State one limitation.
Step 1: The total is 15 and the count is 3, so the mean is 5. Step 2: Left distances are 3 + 2 = 5 and the right distance is 5.
Key Takeaways
• Choose a summary that matches the question: total, mean, median, difference, or trend. • Use all observations and correct counts; sort before finding a median. • Check units, legends, time windows, and rounding before interpreting displays. • Use counterexamples when a rule might be only sometimes true. • A good conclusion states what the data supports and recognises what remains unknown.
Check Your Understanding
Before finishing, explain three ideas aloud: why the mean balances distances, why a high inserted value may leave the median unchanged, and why an average does not describe every individual. If one explanation is difficult, revisit the corresponding worked example and make your own small dataset.
Then choose one graph or strip you created. Ask someone to read it using only its title, labels, scale, and legend. Their questions will show whether your display communicates the information clearly.
Game of Hex
Finish the chapter with a two-player connection game. Hex uses a rhombus-shaped board made of hexagonal cells. The standard board below has 11 rows and 11 columns. Each player tries to connect a different pair of opposite sides with a continuous chain of their own pieces.
Print the board or copy its cell pattern onto paper. Use two colours of counters, or draw two distinct symbols such as a circle and a cross. Blue connects the top and bottom blue sides; Red connects the left and right red sides. The colours belong to the board’s sides, not to the edge of your sheet.
- Assign the two colours or symbols and decide who starts. Each player has a pair of opposite target sides.
- Take turns placing one piece on any empty cell. A piece stays where it is placed for the rest of that game.
- Join cells of your own colour that share a full edge. Neighbouring cells in a chain must touch along an edge; separated cells do not connect.
- Try to build an unbroken chain from one assigned side to the opposite assigned side. You may also place pieces to block the other player.
- The first player to complete their connection wins immediately. Clear the board only when starting a new round.
A chain may bend and wander. It does not have to be straight, and a single piece does not need to connect to your existing pieces as soon as it is placed. What matters is that, by the winning move, all the pieces along one complete route share edges and reach both target sides.
For example, cells F1, F2, F3, and so on down to F11 form an edge-sharing route from the top to the bottom. If all those cells belonged to Blue, Blue would have a winning connection. This is a way to check the rule, not a sequence the other player must allow.
Problem
Blue has pieces in F1, F2, F4, F5, F6, F7, F8, F9, F10, and F11. Is that listed route complete?
- 1.F1 touches the top side and F11 touches the bottom side.
- 2.The listed pieces have a gap at F3. F2 and F4 are not adjacent because F3 lies between them.
- 3.That listed route is incomplete. If F3 is empty, Blue could complete this route by placing there on a later turn.
- 4.A different connected route elsewhere on the board could still win; inspect the whole position before judging the game.
Play, Explain, and Play Again
Use the board for two rounds, changing which player moves first in the second round.
- After a few turns, point to your two target sides and identify which of your pieces already connect.
- Before each move, consider whether it extends your route, links two parts, or blocks an immediate opponent connection.
- At the end, trace the winning chain one cell at a time and check each shared edge.
- Discuss one move you would change, then clear the board and start a fresh round.
Keep a simple record of who started, who won, and the number of turns. After several rounds, you can use the chapter’s data skills to summarise your own games. A few rounds describe your play; they do not prove a general rule about every Hex game.
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Data Stories Across Time
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