Two-way tables: a week of ready-to-teach maths

Five lessons for Year 10 Statistics. Print this pack and the week is prepared: each day has a one-page plan and a student worksheet with practice and a stretch section, plus 24 survey slips to sort into a table, twelve question and calculation cards to match, a mini-check and every answer with its working.

Start here: five minutes to Monday

1. Skim the week at a glance on the next page.
2. Print the five days. Each day is two A4 sheets: a plan and a worksheet.
3. Print the two cut-out sheets, one copy per pair. Cut the 24 survey slips before Day 1 and the 12 question cards before Day 4, and keep each set in an envelope.
4. Calculators are used from Day 3 on. Coloured pencils help with the segmented bar chart on Day 5.
5. Open the free interactive unit on the board. Every plan says which picture to show and which buttons to press.

The idea in plain words

A two-way table sorts the same people by two categorical variables at once, such as year level and how they get to school, and counts each combination. The totals around the edge, the margins, count each category on its own and meet at the grand total. A cell can be compared with everyone (a joint percentage) or with its own row or column (a conditional percentage), and the question decides which. When the conditional percentages differ clearly from group to group, the two variables are related; when they are about the same, they are not. A relationship in a table, like any association, does not show that one variable causes the other.

One day, one lesson

The five lessons fill a week of maths, one lesson of about 60 minutes a day. Run them in order: each day stands on the one before. Every plan marks a split point for two shorter sessions, and every worksheet ends with a stretch section for students who finish early.

How we write these

Percentages are rounded to one decimal place, so a row of rounded percentages can add to 99.9 or 100.1. We say “of all the students” for a joint percentage and name the group, as in “of the Year 12 students”, for a conditional one, because the words decide the calculation. Tables put one variable down the side and one across the top; either way round is correct.

The week at a glance

One lesson a day for a week. Each day stands on the day before, so run them in order.

How the week builds

Day 1 builds the table from raw answers; Day 2 adds and uses its totals; Day 3 compares cells with everyone; Day 4 compares cells within a group; and Day 5 uses those comparisons to decide whether two variables are related. It builds on the two-way tables and Venn diagrams students met in probability, and it prepares the analysis of categorical data in senior mathematics.

Materials for the week (one trip)

• From the classroom: calculators, rulers, coloured pencils, scissors, envelopes for the slips and cards, this pack printed.
• Cut-out sheet 1, one per pair: 24 survey slips (Day 1).
• Cut-out sheet 2, one per pair: 12 question and calculation cards (Day 4).
• Nothing else to buy. If you like, collect your own class’s answers to two questions on Day 1 and build a second table.

Words for the word wall

categorical variable, category, two-way table, cell, row, column, row total, column total, margin, grand total, joint percentage, marginal percentage, conditional percentage, given, row percentage, column percentage, segmented bar chart, relationship, association

Dear families

This week in maths, Year 10 is learning to sort people by two questions at once in a two-way table, such as year level and how they get to school, and to use percentages to decide whether the answers to the two questions are related.

Try this at home

• Ask two questions of everyone at home (for example: tea or coffee? morning person or night owl?) and put the answers in a small table.
• Ask: “80% of dog owners walk every day” and “80% of daily walkers own a dog”. Do they say the same thing? (No: each is out of a different group.)
• Look for a table in the news or on a food label. What are the rows and the columns? What do the totals count?

My two questions week

Each day, ask one person two yes-or-no questions of your choice and write the answers. On Friday, put the five people into a two-way table.

Building a two-way table

A two-way table counts the same people by two categorical variables at once. Students sort 24 survey slips into a table of year level by transport, tally a list of answers into a second table, and name the variables and their categories.

We are learning to

• tell a categorical variable from a numerical one, and list a variable’s categories,
• sort raw answers into a two-way table, one count for each combination,
• check that the cells account for everyone surveyed.

Success criteria

• I can build the table from the slips and check that its cells add to 24.
• I can name the two variables in a table and the categories of each.

You need

The survey slips (cut-out sheet 1), one set per pair; the worksheet.

Lesson flow (about 60 minutes)

Two shorter sessions? Stop after Part A; start the next with Part B.

Watch for these ideas

• Counting a slip once for each variable instead of once in its cell.
• Mixing up a variable (transport) with one of its categories (bus).
• Expecting every cell to hold a similar count.

Answers

• Part A: Year 9: walk 4, bike 1, bus 5, car 3; Year 10: walk 3, bike 4, bus 2, car 2. The cells add to 24.
• Part B: breakfast and tired 3, breakfast and not tired 9, no breakfast and tired 5, no breakfast and not tired 3; total 20.
• Part C: Part A has year level (Year 9, Year 10) and transport (walk, bike, bus, car); Part B has breakfast today (yes, no) and tired in period 1 (yes, no). A height in cm is numerical: it would need groups, such as 150 to under 160 cm, before it could go in a table.
• Part D: the categories overlap. Ask how students get to school on most days, or add a category such as “more than one way”.

Building a two-way table

A. Sort the slips

Each slip is one student’s answers to two questions: what year are you in, and how do you mostly get to school? Sort the 24 slips into the table. One count is done for you.

Check: the cells should add to the number of slips, and each slip should be counted once.

B. From a list

Twenty students answered: did you eat breakfast today, and did you feel tired in period 1? Each box shows one student (breakfast, tired).

C. Variables and categories

D. Stretch: a better question

Totals and gaps

The totals around a two-way table, its margins, count each category on its own and meet at the grand total. Students add margins, check the grand total two ways, and use totals to fill the gaps in tables, one of them described only in words.

We are learning to

• add row and column totals and check the grand total both ways,
• use a row or column total to find a missing cell,
• build a two-way table from a description in words.

Success criteria

• I can fill every gap in the lunch table and say which total I used each time.
• I can turn the sentences about pets into a complete table.

You need

The worksheet; the Day 1 tables.

Lesson flow (about 60 minutes)

Two shorter sessions? Stop after Part A; start the next with Part B.

Watch for these ideas

• Adding a total into the next total, so people are counted twice.
• Subtracting from the wrong total: check the row or column the gap is in.
• Writing a “both” count in two cells.

Answers

• Part A: Year 9 13, Year 10 11; walk 7, bike 5, bus 7, car 5; grand total 24 both ways.
• Part B (rows in order, totals last): library 20, 15, 35; 30, 25, 55; 50, 40, 90. Lunch 40, 18, 6, 64; 30, 19, 7, 56; 70, 37, 13, 120.
• Part C: dog and cat 28, dog and no cat 17, no dog and cat 13, neither 22; cat total 41.
• Part D: two. Once two cells of the top row are known, its total gives the third, and each column total then gives the cell below. In general (rows − 1) × (columns − 1).

Totals and gaps

A. Add the margins

Your table from Day 1. Fill in the totals, then find the grand total two ways.

B. Fill the gaps

Use the totals to fill every empty box. Write which total you used beside each answer if it helps.

C. From words

In a survey of 80 people, 45 own a dog. 28 people own both a dog and a cat, and 22 own neither.

D. Stretch: how few?

Joint and marginal percentages

Dividing a cell by the grand total gives a joint percentage, the share of everyone in that combination; dividing a margin by the grand total gives a marginal percentage. Students turn a table of counts into percentages, move between a Venn diagram and a table, and test claims.

We are learning to

• find a joint percentage: one cell out of everyone,
• find a marginal percentage: one row or column total out of everyone,
• move between a Venn diagram and a two-way table of the same people.

Success criteria

• I can turn the phone table into percentages that add to 100%.
• I can say whether a claim is about one cell or a whole row or column.

You need

Calculators; the worksheet.

Lesson flow (about 60 minutes)

Two shorter sessions? Stop after Part A; start the next with Part B.

Watch for these ideas

• Dividing by a row total when the question is about everyone.
• Percentages that do not add to 100% across the whole table.
• Counting the overlap of a Venn diagram twice when filling the table.

Answers

• Part A (rows in order, totals last): 12.5%, 47.5%, 60%; 17.5%, 22.5%, 40%; 30%, 70%, 100%.
• Part B: sport only 15, both 7, music only 11, neither 17; in the table, sport and music 7, sport and no music 15, no sport and music 11, neither 17.
• Part C: 1 true, 2 false, 3 true (60% have a case, 17.5% are cracked with no case, 30% are cracked).
• Part D: 12.5% is one eighth, so the total must be a multiple of 8: 8, 16, 24, 32, 40, 48, 56, 64, 72, 80, 88, 96.

Joint and marginal percentages

A. Counts to percentages

A survey of 120 students: does your phone have a case, and is its screen cracked?

Write each count as a percentage of all 120 students. Two boxes are done for you.

B. A Venn diagram and a table

Of 50 students, 22 play a sport, 18 play music and 7 do both. Fill the Venn diagram, then the table.

C. True or false?

D. Stretch: which totals?

Conditional percentages

A conditional percentage is out of one group, a row or a column, not out of everyone. Students find row and column percentages for part-time jobs by year, say which group each percentage is out of, and match twelve question and calculation cards.

We are learning to

• find a row percentage and a column percentage and say which group each is out of,
• read “of the Year 12 students” or “given Year 12” as: divide by that group’s total,
• tell “Year 12 students who have a job” from “job holders who are in Year 12”.

Success criteria

• I can match each question card to its calculation and explain the number I divide by.
• I can explain why 50.8% and 42.3% both come from the same 33 students.

You need

The question cards (cut-out sheet 2), one set per pair; calculators; the worksheet.

Lesson flow (about 60 minutes)

Two shorter sessions? Stop after Part A; start the next with Part B.

Watch for these ideas

• Dividing by the grand total when the question names a group.
• Turning the condition round: of Year 12, or of the job holders?
• Adding row percentages down a column.

Answers

• Part A (job, no job): Year 10 25.7%, 74.3%; Year 11 41.5%, 58.5%; Year 12 50.8%, 49.2%; all students 39%, 61%.
• Part B: of the 78 with a job, Year 10 23.1%, Year 11 34.6%, Year 12 42.3%; together 100%.
• Part C: the card key is with the mini-check answers.
• Part D: the first holds (50.8% of Year 12 students have a job); the second does not (only 42.3% of job holders are in Year 12). Each is out of a different group.

Conditional percentages

A survey of 200 senior students: do you have a part-time job?

A. Row percentages

For each year, what percentage have a job and what percentage do not? Each row is out of that year’s total. The first row is done for you.

B. Column percentages

Of the 78 students with a job, what percentage are in each year?

C. Match the cards

Match each question card to the calculation that answers it. Write the letters.

D. Stretch: same words, different groups

Is there a relationship?

Two categorical variables are related when the conditional percentages differ clearly from group to group. Students draw a segmented bar chart of jobs by year, decide which tables show a relationship, and play the self-check quiz.

We are learning to

• compare conditional percentages across groups to decide whether two variables are related,
• draw and read a segmented bar chart,
• describe a possible relationship in context without claiming a cause.

Success criteria

• I can draw the segmented bar chart of jobs by year and say what it shows.
• I can say which tables show a relationship and quote the percentages that show it.

You need

Rulers; coloured pencils; calculators; the worksheet; the self-check quiz on the board.

Lesson flow (about 60 minutes)

Two shorter sessions? Stop after Part B; start the next with Part C.

Watch for these ideas

• Comparing counts instead of percentages when the groups differ in size.
• Calling any small difference a relationship.
• Saying that one variable causes the other.

Answers

• Part A: job shares of 25.7%, 41.5%, 50.8%: the share with a job grows with each year, so year level and having a job are related.
• Part B: phones: 20.8% of phones with a case are cracked against 43.8% without, so the variables are related. Hands: 10%, 9.2%, 10.8% left-handed, about the same in every year, so no relationship.
• Part C: 25% of breakfast eaters were tired against 62.5% of the others: a possible relationship, but only 20 students, and breakfast need not be the cause: students who sleep well may both eat breakfast and feel alert.
• Part D: two questions with yes-or-no answers, an empty 2 by 2 table with totals, and the two rates to compare, such as the share who say yes to the second question in each group of the first.

Is there a relationship?

A. A segmented bar chart

Use your row percentages from Day 4. In each bar, shade the part with a job from the bottom up; leave the part with no job white.

B. Related or not?

C. Breakfast and tiredness

The breakfast answers from Day 1.

D. Stretch: design a survey

Survey slips

For Day 1, one set per pair. Each slip is one student’s answers: year level, and how the student mostly gets to school. Sort the slips into a two-way table of year level by transport.

Teacher note: the finished table is with the mini-check answers.

Questions and calculations

For Day 4, one set per pair, with the job table from the worksheet. Six cards ask a question; six give a calculation. Match each question to the calculation that answers it.

Teacher note: the key is with the mini-check answers.

What we know: Two-way tables

Work on your own. A calculator is allowed. Show your working. Questions 1 to 7 use this survey of where people live and the holiday they prefer.

1. Complete the totals of the table.
2. What percentage of all the people live inland and prefer the bush? ______________________
3. What percentage of all the people prefer the beach? ______________________
4. What percentage of the people on the coast prefer the beach? ______________________
5. What percentage of the people who prefer the beach live on the coast? ______________________
6. To decide whether where people live and the holiday they prefer are related, compare, circle one:   the percentage who prefer the beach in each group  ·  the two largest cells  ·  the grand totals
7. Is there a relationship? Does living on the coast make people prefer the beach? Explain. ______________________
8. Another table has row totals 20 and 30, column totals 35 and 15, and 12 in its top-left cell. Find its other three cells. ______________________

Answers and marking guide

Answers with working

1. Row totals 80 and 80; column totals 90 and 70; grand total 160 both ways.
2. 50 ÷ 160 = 31.25% (a joint percentage).
3. 90 ÷ 160 = 56.25% (a marginal percentage).
4. 60 ÷ 80 = 75% (out of the coast group only).
5. 60 ÷ 90 ≈ 66.7% (out of the beach group: a different question from 4).
6. The first: the percentage who prefer the beach in each group.
7. Yes: 75% of coast people prefer the beach against 37.5% inland. It does not show cause: people who love the beach may choose to live on the coast.
8. Top-right 8, bottom-left 23, bottom-right 7.

A quick three-level guide

Eight questions, four ideas. A student at standard reads the wording to choose what to divide by, and compares groups with percentages, never with counts.

Card keys

Survey slips (Day 1): Year 9: walk 4, bike 1, bus 5, car 3; Year 10: walk 3, bike 4, bus 2, car 2. Question cards (Day 4): G with A; D with J; E with L; B with H; K with F; I with C.

Weekly class record

Jot a tick as you move around the room; the mini-check fills any gaps. A tick a day is plenty.

The five columns are the five days: building a table, totals, joint percentages, conditional percentages, and relationships.