Module 26 of 45Lesson 1 of 20 · Unit S01 · Data All Around
Statistics & Probability · Unit S01 · Foundations
Data All Around
Every number in every chart you will ever meet started life as somebody writing something down. This unit is about that moment: what data is, where it comes from, which kind you are holding, and how to get a messy list of twenty answers into a table that can be read at a glance.
The first unit of the Statistics & Probability strand. The only arithmetic in it is counting and adding whole numbers. What it asks for instead is precision with words — because in statistics the name you give a variable decides the table you may build and the chart you may draw.
By the end of this unit you can
- Describe what data is, and say whether a set of data is primary (you collected it) or secondary (somebody else did)
- Classify a variable as qualitative (words) or quantitative (numbers)
- Classify a quantitative variable as discrete (counted) or continuous (measured), and give the reason
- Turn a raw list into a tally chart, bundling in fives, and then into a frequency table
- Complete a frequency table and check it against its total
- Build and read a two-way table, and tell a cell from a row total
- Read a completed table back: how many of one kind, how many altogether, how many more than another
- Draw a first bar chart from a frequency table, with the frequency axis starting at 0
The one idea behind all of it
Data is just answers, written down. Everything else in statistics — every average, every chart, every probability — is something you do to answers that somebody wrote down. So the first skill is not calculating. It is classifying: knowing what kind of answer you are holding, because the kind decides the table you build and the chart you may draw.
collect → classify → count → chart
and the frequencies must always add back to the number you started with
Three questions do all the work in this unit. Who collected it? — primary or secondary. Words or numbers? — qualitative or quantitative. Counted or measured? — discrete or continuous. Answer those three about any dataset and you already know what you are allowed to do with it.
Where this unit sits
This is the first unit of the Statistics & Probability strand. It needs no new number work at all — counting and adding whole numbers is the whole of the arithmetic. Number Unit N01 Know Your Numbers covers that, and nothing from it is retaught here.
What comes next. S02 asks how to word the question you collect the data with, S03 asks whether the data can be trusted, and S04 draws the full family of charts — pie, pictogram, stem-and-leaf — from tables exactly like the ones you build here.
Where the numbers in this unit come from. The traffic count is a list of twenty vehicles, printed in full; the two-way table is sixteen named-nothing learners; the sibling table is twenty-four. Every count in every chart on this page can be checked by hand against the list it came from, and that is deliberate.
1
What counts as data
primary and secondary
Data is a set of facts you have collected, written down so they can be counted. A variable is the thing you are collecting — way to school, shoe size, favourite colour. The individual answers are the data values. If you collected them yourself, the data is primary. If you took them from somebody else — a textbook, a website, last year’s class — it is secondary. Primary data you can vouch for; secondary data you have to trust.
EXAMPLEYou and your class count vehicles at the gate for twenty minutes. That is primary data. You then look up how many combis are licensed in Gaborone in a reference book. That is secondary data. Both are useful; only the first one is yours.
Practice · What counts as data · 10 questions
P1State the word for data you collected yourself.Fluency
Answerprimary
P2State the word for data you took out of somebody else’s report.Fluency
Answersecondary
P3Your class counts the vehicles at the school gate for twenty minutes. Classify the data as primary or secondary.Fluency
Answerprimary
P4You copy last year’s rainfall figures out of a textbook. Classify the data as primary or secondary.Fluency
Answersecondary
P5You measure every learner’s hand span in your class. Classify the data and describe why.Applied
AnswerprimaryYou did the measuring, so the data is yours: it is primary. You can say exactly how it was collected, with what ruler, and on what day, which is the whole advantage of primary data.
P6You use the results of a survey another class ran last term. Classify the data and describe one risk.Applied
AnswersecondarySomebody else collected it, so it is secondary. The risk is that you cannot check how they asked: who they asked, how many, and whether the question was fair. You are trusting their method as well as their numbers.
P7“Way to school” is collected from 16 learners. Describe the difference between the variable and a data value here.Applied
Answerthe variable is “way to school”; a data value is one learner’s answer, such as “combi”The variable is the thing being collected — it has one name and one meaning for the whole class. A data value is one single answer from one single learner. Sixteen learners give one variable and sixteen data values.
P8You time your own walk to school every day for a week. Classify the data and state how many data values you have.Applied
Answerprimary, and 7 data valuesYou collected it yourself, so it is primary. One timing per day for seven days is seven data values of the one variable “time taken to walk to school”.
P9A learner says “anything from the internet is secondary data”. Describe a case where data from a website would be primary.Challenge
Answerif your own class collected it and put it on the website, it is still primaryPrimary and secondary are about who collected the numbers, not about where they are stored. If your class ran the traffic count and then typed the results onto a class web page, the data is primary for you and secondary for anyone else who downloads it. The same numbers can be primary for one person and secondary for another.
P10Describe one advantage of primary data and one advantage of secondary data.Challenge
Answerprimary: you know exactly how it was collected; secondary: it is much quicker and can cover far more peoplePrimary data is trustworthy because you controlled the method — you know who was asked and how. Secondary data is cheap and fast, and can reach a scale you could never collect yourself, such as a whole country. Neither is better in general; they answer different sized questions.
2
Words or numbers?
qualitative and quantitative
A variable is qualitative when its answers are words or categories — combi, Setswana, blue. It is quantitative when its answers are numbers you can do arithmetic with — 7 vehicles, 152 cm, 4 siblings. The test is not whether digits appear. A jersey number is written with digits, but adding two jersey numbers gives nothing, so it behaves as a category.
Worked example 1 · Sort the school bag
1
Shoe size — counted, measured or neither?
quantitative, discrete
2
Favourite colour — counted, measured or neither?
qualitative
3
Height — counted, measured or neither?
quantitative, continuous
4
Number of brothers and sisters — counted, measured or neither?
quantitative, discrete
5
Home language — counted, measured or neither?
qualitative
6
Mass of the school bag — counted, measured or neither?
quantitative, continuous
7
Number of pencils in the bag — counted, measured or neither?
quantitative, discrete
8
Time taken to get to school — counted, measured or neither?
quantitative, continuous
Eight variables, two questions each time: words or numbers, and if numbers, counted or measured. Nothing else is needed, and nothing else counts as an answer.
EXAMPLE“How did you get to school?” gives combi, walk, car — qualitative. “How many minutes did it take?” gives 12, 25, 8 — quantitative. Same learners, same morning, two different kinds of data.
Practice · Words or numbers? · 10 questions
P11Classify the variable “favourite colour” as qualitative or quantitative.Fluency
Answerqualitative
P12Classify the variable “number of brothers and sisters” as qualitative or quantitative.Fluency
Answerquantitative
P13Classify the variable “home language” as qualitative or quantitative.Fluency
Answerqualitative
P14Classify the variable “shoe size” as qualitative or quantitative.Fluency
Answerquantitative
P15Classify “eye colour” and describe the reason.Applied
AnswerqualitativeThe answers are words — brown, black, hazel. There is no arithmetic you could do with them: brown plus black is not a colour, and there is no order from smallest to largest. That is exactly what qualitative means.
P16Classify “temperature at noon” and describe the reason.Applied
AnswerquantitativeThe answers are numbers on a scale — 31°C, 28°C — and arithmetic on them means something: one day really can be 3 degrees warmer than another.
P17A netball team records each player’s jersey number. Describe why this is qualitative data even though it is written with digits.Applied
Answerthe digits are labels, not amountsJersey 7 is not “one more” than jersey 6, and the average jersey number of a team means nothing at all. The digits name a player the way a word would, so the variable behaves as a category. The test is never whether digits appear; it is whether arithmetic on them means anything.
P18Classify “day of the week” and describe the reason.Applied
AnswerqualitativeThe answers are the names of days. They do have an order, but they are still categories: Tuesday plus Thursday is not a day, and there is no sensible average day.
P19Four variables were collected: home language, number of pencils in the bag, favourite colour, height. Classify each one as qualitative or quantitative.Challenge
Answerqualitative: home language, favourite colour; quantitative: number of pencils in the bag, heightTwo of them give words — a language and a colour — and two give numbers you can add and compare. Sorting first, before any chart is chosen, is what stops a pie chart of heights or a mean favourite colour.
P20A learner writes the rule “if it is written with digits it is quantitative”. Describe why the rule fails, with an example.Challenge
Answerit fails for qualitative labels written as digits, such as a jersey number or a house number, which are never quantitativeHouse number 42 is not twice house number 21, and the two cannot be added to give anything. Digits are being used as a name. The working rule is: if arithmetic on the values means something, the variable is quantitative; if the values only name a category, it is qualitative.
3
Counted or measured?
discrete and continuous
Quantitative data splits again. Discrete data is counted: it lands on separate, fixed values with nothing in between — 3 goals, 4 siblings, size 6 shoes. Continuous data is measured: between any two values there is always another one, and measuring more carefully just adds decimal places — height, mass, time. The question to ask is never “is it a number?” but “did I count it or measure it?”
Worked example 2 · Spot the impostor
1
Number of learners in a class — counted or measured?
counted, so discrete
2
Number of goals scored — counted or measured?
counted, so discrete
3
Length of a pencil — counted or measured?
measured, so continuous
4
Number of pages in a book — counted or measured?
counted, so discrete
5
So which of the four was wrongly called discrete?
length of a pencil
6
Give the reason.
a pencil can be any length; measuring more carefully just adds decimal places
Three of the four are counts of separate things. The impostor is the one you would reach for a ruler to find, and reaching for a ruler is the test.
EXAMPLEShoe size looks like a measurement, and it is not. Sizes go 5, 6, 7 — and half sizes if the shop stocks them — but never 6.31. It is discrete. The length of the foot inside the shoe is continuous.
Practice · Counted or measured? · 10 questions
P21Classify “number of goals scored” as discrete or continuous.Fluency
Answerdiscrete
P22Classify “height” as discrete or continuous.Fluency
Answercontinuous
P23Classify “number of pages in a book” as discrete or continuous.Fluency
Answerdiscrete
P24Classify “mass of a loaf of bread” as discrete or continuous.Fluency
Answercontinuous
P25Classify “shoe size” as discrete or continuous and describe the reason.Applied
AnswerdiscreteSizes come in fixed steps — 5, 6, 7, and half sizes where the shop stocks them. There is no size 6.31, so the values are separate and countable. The foot inside the shoe is continuous; the size printed on it is not.
P26Classify “time taken to get to school” and describe the reason.Applied
AnswercontinuousTime is measured, not counted. Between 12 minutes and 13 minutes every value is possible, and a stopwatch with more decimal places just finds more of them. Rounding the answer to the nearest minute does not make the variable discrete — it only makes the record rougher.
P27Classify “number of pencils in the bag” and describe the reason.Applied
AnswerdiscretePencils are counted one at a time, so the answers are whole numbers with nothing in between. A broken pencil is still one pencil, or none.
P28Classify “length of a pencil” and describe the reason.Applied
AnswercontinuousA pencil can be any length at all. Measuring it more carefully adds decimal places rather than landing on a new fixed value, and that is exactly what continuous means.
P29Three of these four were called discrete. Identify the one that is really continuous and describe why: number of learners in a class, mass of a loaf of bread, number of siblings, number of pages in a book.Challenge
Answermass of a loaf of breadLearners, siblings and pages are all counted in whole numbers, so all three are discrete. Mass is read off a scale: a loaf can be 748 g or 748.3 g or anything between, so it is continuous. The test that finds the impostor every time is “would I reach for a measuring instrument?”
P30Describe why the number of learners in a class can never be 27.5, and state what that tells you about the variable.Challenge
Answerlearners are counted in whole numbers, so the variable is discreteA class has 27 learners or 28 learners; there is no learner-and-a-half to be had. Values that can only land on separated whole numbers are exactly what discrete means. Note that an average class size of 27.5 is perfectly sensible — the average is a calculation, not a data value.
4
Tally it, table it
from a raw list to a frequency table
A raw list is unreadable. A tally chart makes it countable: one mark per value, and every fifth mark drawn across the previous four so the marks read in fives. Counting in fives is the whole point — it is why the bundles exist. A frequency table then records the totals, one row per category, with the frequency being how many times that category occurred. The frequencies must add up to the number of values you started with, and checking that is not optional.
EXAMPLEThe raw list, in full. Twenty vehicles went past the gate of a school, in this order:
Car; Car; Combi; Car; Bakkie; Car; Combi; Motorbike; Combi; Combi; Combi; Bakkie; Combi; Bakkie; Combi; Lorry; Motorbike; Car; Motorbike; Bakkie.
Car; Car; Combi; Car; Bakkie; Car; Combi; Motorbike; Combi; Combi; Combi; Bakkie; Combi; Bakkie; Combi; Lorry; Motorbike; Car; Motorbike; Bakkie.
One mark per vehicle, bundled in fives. The tally column and the frequency column say the same thing twice, which is the check.
The finished frequency table. The five frequencies add to 20, the number of vehicles in the raw list above.
Two frequencies rubbed out. Count them again from the raw list — not from the total, which would only ever give you the pair.
Worked example 3 · Tally the traffic
1
Write the five vehicle types down the side of a tally chart.
five rows
2
Work along the list of 20, one mark per vehicle, bundling every fifth mark across the four before it.
20 marks in all
3
Count the Combi row in fives.
Combi: 7
4
Count the Car row.
Car: 5
5
Count the Bakkie, Lorry and Motorbike rows.
4, 1 and 3
6
Add the five frequencies and check against the list length.
7 + 5 + 4 + 1 + 3 = 20
The check at the end is the whole discipline. If the frequencies had come to 19 or 21, the tally would be wrong and the chart drawn from it would be wrong too.
EXAMPLETwenty vehicles go past. You do not write “combi” twenty times; you make one mark per vehicle in the right row and bundle every fifth mark across the other four. At the end you count in fives, and the five row totals must come to 20.
Practice · Tally it, table it · 10 questions
P31Twenty vehicles are tallied. State how many tally marks the chart has altogether.Fluency
Answer20
P32The traffic table is above. State the frequency of Combi.Fluency
Answer7
P33The traffic table is above. State the frequency of Bakkie.Fluency
Answer4
P34The traffic table is above. State the frequency of Lorry.Fluency
Answer1
P35Calculate the total of the five frequencies and describe what the total checks.Applied
Answer207 + 5 + 4 + 1 + 3 = 20, which is the number of vehicles in the list. The total is the check: if the frequencies had come to 19 or 21, a mark has been lost or double-counted and the table cannot be used.
P36Calculate how many of the twenty vehicles were not combis.Applied
Answer13Total minus that one frequency: 20 − 7 = 13. Adding the other four frequencies gives the same answer, which is a useful check.
P37Calculate how many more combis than bakkies went past.Applied
Answer3“How many more” is a subtraction of the two frequencies, never an addition: 7 − 4 = 3. Say the answer as a sentence — there were 3 more combis than bakkies — because that is what stops the subtraction being done the wrong way round.
P38Draw the tally marks for a frequency of 13, and describe how you would count them back.Applied
Answer|||| |||| |||Two full bundles of five and three singles. You count them back in fives: 5, 10, then the three singles, giving 13. Bundling is what makes the count in fives possible, and counting in fives is what stops the miscount.
P39A learner’s tally of the same twenty vehicles gives frequencies adding to 19. Describe what has gone wrong and what to do.Challenge
Answerone vehicle has been missed; go back to the raw list and tally againThe frequencies must add back to the number of values in the list. Nineteen against a list of twenty means exactly one mark was never made, or one row was miscounted. There is no way to repair it from the table alone — the raw list is the only evidence, so the tally is done again against it.
P40Calculate how many of the twenty vehicles had four wheels or more, using the frequency table.Challenge
Answer17Only the motorbikes have fewer than four wheels, so this is the total minus that frequency: 20 − 3 = 17. The same answer comes from adding combi, car, bakkie and lorry: 7 + 5 + 4 + 1 = 17.
5
Two ways at once
the two-way table
A two-way table classifies one set of data by two variables at once. Every learner is counted exactly once, in exactly one cell. The row totals tell you about the first variable, the column totals about the second, and the grand total sits in the corner where the two agree. If the row totals and the column totals do not give the same grand total, something has been counted twice or missed.
Every learner sits in exactly one cell, so the grand total can be reached down the rows and across the columns.
Worked example 4 · Two ways at once
1
Sixteen learners of Form 1C. Put gender down the side and way to school across the top.
a 2 by 3 grid
2
Count the girls who walk.
Girls, Walk: 3
3
Count the girls who come by combi, and the girls who come by car.
4 and 2
4
Add the girls’ row.
3 + 4 + 2 = 9
5
Add the boys’ row.
7 boys
6
Add the two row totals, then add the three column totals, and check they agree.
9 + 7 = 16, and 5 + 7 + 4 = 16
Every learner is in exactly one cell, so the grand total can be reached two ways. When the two ways disagree, the table is wrong — and that is the only warning you get.
EXAMPLESixteen learners, classified by gender and by way to school. The cell where Girls meets Combi holds 4. That is four girls who came by combi — not four girls altogether, and not four combi passengers altogether.
Practice · Two ways at once · 10 questions
P41The two-way table is above. State how many girls walk.Fluency
Answer3
P42The two-way table is above. State how many boys come by combi.Fluency
Answer3
P43The two-way table is above. State how many girls there are altogether.Fluency
Answer9
P44The two-way table is above. State how many learners come by car.Fluency
Answer4
P45Calculate how many boys there are altogether, and state which numbers you added.Applied
Answer7Add the whole Boys row: 2 + 3 + 2 = 7. A row total answers a question about the side variable only — here, gender.
P46Calculate how many learners are in Form 1C, two different ways.Applied
Answer16By rows: 9 + 7 = 16. By columns: 5 + 7 + 4 = 16. The two routes must agree, because every learner sits in exactly one cell and is therefore counted exactly once each way.
P47Calculate how many girls do not walk to school.Applied
Answer6The girls’ row total minus the walking cell: 9 − 3 = 6. Those are the girls who came by combi or by car.
P48Calculate how many more learners come by combi than by car.Applied
Answer3Two column totals, subtracted: 7 − 4 = 3. Note that both numbers are totals, not cells — the question is about the whole form, not about girls or boys separately.
P49A learner says “4 learners come by combi, because the table says 4 where Girls meets Combi”. Describe the error and state the correct number.Challenge
Answer7The cell 4 answers two questions at once: girls, and combi. “How many come by combi” asks about one variable only, so it is the column total: 4 + 3 = 7.
P50Describe the check that a two-way table is right, and show it for this table.Challenge
Answerthe row totals and the column totals must give the same grand total: 16Rows: 9 + 7 = 16. Columns: 5 + 7 + 4 = 16. They agree, so every learner has been counted exactly once. If they disagreed, at least one learner has been placed in two cells or in none, and the table has to be rebuilt from the raw list.
6
Read it back
answering questions from a table
Reading a table back is a skill in itself, and the questions come in four shapes: how many of one kind (read one cell), how many altogether (add the frequency column), how many more (subtract two frequencies), and how many are not (total minus one frequency). A bar chart is the same table drawn: one bar per category, the frequency axis starting at 0, and equal gaps between the bars because the categories are separate things.
A completed frequency table, ready to be read back.
The same table drawn. One bar per category, equal gaps, and the frequency axis starting at 0.
Worked example 5 · Read it back
1
How many learners have exactly 2 brothers and sisters? Read the row.
6 learners
2
How many learners answered altogether? Add the frequency column.
24 learners
3
How many have more than 3? Add the rows for 4 and 5.
2 + 1 = 3
4
How many do NOT have exactly 1? Total minus that frequency.
24 − 7 = 17
Four question shapes, four different moves: read a row, add the column, add two rows, subtract from the total. Deciding which move the question wants is most of the mark.
EXAMPLEFrom the sibling table: 6 learners have exactly 2 brothers and sisters; 24 learners answered altogether; 1 more learner has 1 than has 2; and 18 learners do not have exactly 2, because 24 − 6 = 18. Every one of those four answers is a different move on the same table.
Practice · Read it back · 10 questions
P51The sibling table is above. State how many learners have exactly 3 brothers and sisters.Fluency
Answer5
P52The sibling table is above. State how many learners answered altogether.Fluency
Answer24
P53The sibling table is above. State the most common number of brothers and sisters.Fluency
Answer1
P54The sibling table is above. State how many learners have no brothers or sisters at all.Fluency
Answer3
P55Calculate how many learners have more than 3 brothers and sisters.Applied
Answer3“More than 3” means 4 or 5, so add those two rows: 2 + 1 = 3. It does not include the row for 3 itself.
P56Calculate how many learners do not have exactly 1 brother or sister.Applied
Answer17Total minus that one frequency: 24 − 7 = 17. Adding the other five rows gives the same number, which is the check.
P57Calculate how many more learners have 1 brother or sister than have 4.Applied
Answer5A “how many more” question subtracts the two frequencies: 7 − 2 = 5. Both numbers are frequencies — how many learners — and not numbers of brothers and sisters, and saying the answer as a sentence keeps the two apart.
P58The bar chart of the traffic count is above. State which bar is the tallest and describe what its height means.Applied
AnswerCombi, at a height of 7The tallest bar is the one with the largest frequency. Its height is read off the frequency axis as 7, and that means 7 combis went past the gate in the twenty minutes — not 7 minutes, and not 7 anything else. A bar’s height is always a frequency.
P59Describe why the frequency axis of a bar chart must start at 0.Challenge
Answerso that each bar’s height stays in proportion to its frequency, which only happens when the axis starts at 0If the axis started at, say, 4, the bar for a frequency of 5 would be one unit tall and the bar for 7 would be three units tall — the picture would say “three times as many” when the truth is “a bit more”. Starting at 0 is what makes the comparison you see the comparison that is true. Unit S03 is about charts that break this rule on purpose.
P60Calculate how many learners have fewer than 2 brothers and sisters, and state which rows you added.Challenge
Answer10“Fewer than 2” means 0 or 1, so the rows for 0 and 1 are added: 3 + 7 = 10. The row for 2 is not included — “fewer than” excludes the value itself, which is the commonest slip in this kind of question.
The classifying routine
Four moves, whatever the dataset turns out to be.
1
Who collected it?You, or somebody else? That is primary or secondary, and it is the first thing to say about any dataset.
2
Words or numbers?Qualitative or quantitative. If the answers are categories, no average will ever mean anything.
3
Counted or measured?Only for quantitative data. Counted is discrete, measured is continuous, and the reason matters as much as the word.
4
Tally, table, total.One mark per value, bundled in fives; one row per category; and the frequencies must add back to the number you started with.
The mistakes that cost the most marks
1Calling shoe size continuous because it is “a measurement”
Wrong“Shoe size is measured off your foot, so it is continuous.”
RightShoe size is discrete. Sizes come in fixed steps — 5, 6, 7 and perhaps half sizes — and there is no size 6.31.
The foot inside the shoe is continuous; the size printed on the shoe is not. Ask what is actually written down, not what it came from. The same trap catches dress sizes and shoe widths.
2Forgetting the bundle of five, and miscounting
WrongEighteen separate upright marks, counted one by one, and read off as 17.
RightBundle every fifth mark across the four before it, then count the bundles in fives: three bundles and three singles is 5 + 5 + 5 + 3 = 18.
The bundles are not decoration. Counting a long row of identical uprights is exactly where miscounts happen, and the whole table is wrong afterwards. The frequencies must add back to the number of values you started with — that is the check that catches it.
3Reading a two-way table cell as a row total
Wrong“4 girls are in Form 1C, because the table says 4 under Girls.”
RightThe cell says 4 girls came by combi. The number of girls altogether is the row total, which is 9.
A cell answers two questions at once and a total answers one. Say the cell out loud as a full sentence — “girls, and combi” — and the mistake becomes impossible to make.
Tasks to submit
Criterion B · Investigating patterns
Twenty questions
A sorting game. One player thinks of an object from a set laid out on the desk. The other player may only ask questions with a yes or no answer — “is it made of metal?”, “is it blue?” The question is how many objects a fixed number of questions can always tell apart.
Part 1 · the structured investigation
- Predict how many different objects 5 yes/no questions could always tell apart. Write the prediction down before you work anything out.
- Play the game with 1 question, then 2, then 3, and tabulate the largest number of objects that can always be told apart.
- Identify the pattern in your table. Say what happens to the number each time one more question is added.
- Determine the general rule for n questions, and write it as a formula.
- Test and verify the rule: build the full question tree for 4 questions, count the branches at the end, and compare the count with your formula.
- Justify why one more question can at most double the number of groups — not treble it, and not add two.
| Number of questions, n | Objects that can always be told apart |
|---|---|
| 1 | 2 |
| 2 | 4 |
| 3 | 8 |
| 4 | |
| 5 | |
| n |
The first three rows are done for you as far as n = 3. The last three rows are yours.
Part 2 · your own investigation
Choose one of these, or an equally good question of your own, and investigate it in the same way — predict, tabulate, find the pattern, state a general rule, test it and justify it. (a) What happens when each question has three possible answers instead of two? (b) What is the smallest number of yes/no questions needed for a set of 30 objects, and how do you know it is the smallest? You choose the method, the table and the way you present the result.
Criterion B · Investigating patterns. Levels 1–2: applies the given procedure and records some correct values. Levels 3–4: completes the table correctly and describes the pattern in words. Levels 5–6: states a correct general rule for n questions and verifies it with the tree for 4. Levels 7–8: justifies the doubling from the structure of the tree — each existing group is split into a yes group and a no group and no group can split further than that — and carries Part 2 through to a tested general statement of its own.
Remember, you should communicate in an organized and coherent manner.
Criterion D · Applying mathematics in real-life contexts
The tuck-shop record sheet
A tuck shop at a school in Gaborone keeps one line of a record sheet for every sale. The manager wants to know what sells most at break time, so that she orders the right amounts. Break runs from 10:15 to 10:45. The sheet carries more columns than the question needs.
Record sheet · every sale of one day
| Item | Price (P) | Time | Weather | Seller |
|---|---|---|---|---|
| Fat cake | 5.00 | 07:50 | sunny | MK |
| Juice | 6.50 | 10:18 | sunny | TB |
| Fat cake | 5.00 | 10:20 | sunny | TB |
| Chips | 8.00 | 10:22 | sunny | MK |
| Fat cake | 5.00 | 10:24 | sunny | MK |
| Apple | 3.00 | 10:27 | sunny | TB |
| Juice | 6.50 | 10:30 | sunny | MK |
| Fat cake | 5.00 | 10:33 | sunny | TB |
| Chips | 8.00 | 10:36 | sunny | MK |
| Sandwich | 12.00 | 10:39 | sunny | TB |
| Fat cake | 5.00 | 10:42 | sunny | MK |
| Juice | 6.50 | 13:10 | cloudy | TB |
| Chips | 8.00 | 13:15 | cloudy | MK |
| Fat cake | 5.00 | 15:05 | cloudy | TB |
This record sheet is an invented illustrative example written for the task. It is not taken from any business, price list or publication, and no organisation, person or source is named or implied.
- Interpreting the situation. Describe which columns of the sheet are needed to answer “what sells most at break time”, and which are surplus to that question. Say what each surplus column would be useful for instead.
- Selecting the mathematics. Classify each of the five columns as qualitative or quantitative, and where it is quantitative, as discrete or continuous. Give the reason in each case.
- Applying the mathematics. Build the tally chart and then the frequency table of the items sold at break time only. State which item sold most.
- Accuracy. Check that your frequencies add to the number of rows you used, and state that number. Say how you decided which rows belonged to break and which did not.
- Sense in context. Describe what the manager should order more of, and describe one thing this sheet cannot tell her — something she would need to collect differently to find out.
Criterion D · Applying mathematics in real-life contexts. Levels 1–2: identifies some of the needed columns and makes a start on the tally. Levels 3–4: selects the right columns, classifies most of them correctly and completes the frequency table for break time. Levels 5–6: all five classifications correct with reasons, the frequency table complete and checked against the row count, and a described recommendation. Levels 7–8: in addition, describes clearly what the sheet cannot answer — for example how many learners wanted something the shop had run out of, which no record of sales can ever show — and says what would have to be collected instead.
When you are ready
The mastery quiz is 10 questions drawn at random from a bank of 25. You need 70% to pass and you have three attempts. Type a frequency as digits only, and type a term in lower case — discrete, not “Discrete data”. A multi-gap answer separates its gaps with the bar character, in the order the gaps appear. Work through the practice first, especially Concept 3, where the reason matters as much as the word.
Printable version
Download Unit S01 as a PDF — the same lesson, laid out for printing and for working offline.
Download Unit S01 as a PDF — the same lesson, laid out for printing and for working offline.