D. Data · Data and Probability

Data Detectives: Analysing Data and Making Decisions

D1.6 D1.5 A1About 50 minutes

Learning goals

I can ask and answer different kinds of questions about data in multiple-bar graphs and stem-and-leaf plots.
I can draw conclusions that the data supports.
I can make a convincing argument with a claim, evidence and reasoning.
I can use data to make an informed decision.

Words to know

analyse
Look closely at data to find out what it tells us.

conclusion
Something you decide is true after studying the data.

evidence
Numbers or facts from the data that support what you say.

convincing argument
A claim backed up by evidence from the data and clear reasons.

informed decision
A choice made by using data and facts, not just guessing.

Remember from Grade 3

You analysed pictographs and bar graphs, compared graphs with different scales, and used data to make decisions. This year you will analyse multiple-bar graphs and stem-and-leaf plots, and use the median and mode as evidence.

The carnival committee needs your help

The school held a two-day winter carnival. Next year the committee wants to order the right amount of each snack. They have the sales records. Can the data detectives tell them what to do?

Day 1 bars are green and Day 2 bars are blue. Popcorn 30 then 45, hot cocoa 50 then 40, apples 15 then 20, muffins 25 then 25. The scale counts by 10s.

Three kinds of questions

Good data detectives ask three kinds of questions:

Read the data: find one fact on the graph. "How many muffins were sold on Day 1?"
Read between the data: compare or combine. "How many cups of hot cocoa were sold altogether?"
Read beyond the data: predict or decide. "What should we order more of next year?"

Worked example

Example 1: Ask and answer

Answer one question of each kind about the snack graph.

Read the data: Muffins on Day 1: the bar reaches 25 (halfway between 20 and 30).
Read between the data: Hot cocoa altogether: 50 + 40 = 90 cups. Popcorn altogether: 30 + 45 = 75 bags.
Read beyond the data: Popcorn sales went up from 30 to 45, and hot cocoa went down from 50 to 40. Maybe Day 2 was warmer, or the popcorn smell drew people in! We can't be sure why, but we can plan for both to be popular.

Muffins Day 1: 25. Hot cocoa total: 90. Popcorn is growing in popularity.

Worked example

Example 2: A convincing argument

Make a convincing argument about which snack the committee should order the most of.

Claim: Order the most hot cocoa.
Evidence: Hot cocoa sold 90 cups in two days, more than any other snack (popcorn 75, muffins 50, apples 35).
Reasoning: The snack that sold the most will probably be wanted the most again. But popcorn rose from 30 to 45, so order plenty of popcorn too.

Claim + evidence (numbers) + reasoning = a convincing argument.

Comparing two stem-and-leaf plots

Two Grade 4 classes timed how long each student could hold a plank, in seconds.

Stem (tens)	Leaf (ones)
1	5 8
2	0 4 4 6
3	0 2 5
4	1

Class A plank times in seconds. Key: 2 | 4 means 24 seconds.

Stem (tens)	Leaf (ones)
2	2 5 8
3	0 4 4 4 6
4	0 5

Class B plank times in seconds. Key: 3 | 4 means 34 seconds.

Worked example

Example 3: Which class held longer?

Use the two plots. Which class generally held a plank longer? Use the median and mode as evidence.

Each class has 10 students.
Class A in order: 15, 18, 20, 24, 24, 26, 30, 32, 35, 41. Middle values (5th and 6th): 24 and 26. Median 25. Mode 24.
Class B in order: 22, 25, 28, 30, 34, 34, 34, 36, 40, 45. Middle values: 34 and 34. Median 34. Mode 34.
Class B's median is 9 seconds higher. Most of Class B's leaves are in the 30s and 40s, but most of Class A's are in the 10s and 20s.

Class B generally held a plank longer: median 34 seconds compared with 25 seconds.

Each dot is one student. Class B's dots sit further to the right, and its median (34 s) is 9 seconds higher than Class A's (25 s).

Worked example

Example 4: An informed decision

Grade 4 and Grade 5 (28 students each) voted on a day for a family skating party. Which day should they choose?

Add the votes for each day. Friday: 14 + 9 = 23. Saturday: 6 + 12 = 18. Sunday: 8 + 7 = 15.
Friday has the most votes altogether.
Notice: Grade 5's favourite was Saturday (12). A fair committee might mention this, and maybe plan a Saturday event another time.

Friday, with 23 of the 56 votes. That is an informed decision because it is based on the data.

Day	Grade 4	Grade 5	Total
Friday	14	9	23
Saturday	6	12	18
Sunday	8	7	15

Votes for the skating party day. Source: class votes.

Careful conclusions

Use words like most common or more, not everyone or always, unless the data really shows it.
Small groups can surprise you. Data from 5 people may not match the whole school.
If two groups are different sizes, be careful comparing frequencies.
Check the scale before you judge by bar heights.
The data tells you what happened, not always why.

Working together

Data detectives don't always agree. That's okay! When you disagree, point to the evidence: "I think popcorn, because it went up by 15." Listen to other reasons. A good decision often uses ideas from more than one person.

Just for fun

Why did more popcorn sell on Day 2? The data can't tell us why, only what happened. (But we suspect the smell of fresh popcorn makes a very convincing argument.)

Hands-on activity

Family decision with data

You need: paper, a pencil and a ruler.

Choose a real family decision, like what to have for Sunday dinner, which game to play on family games night, or where to go for a walk.
List 3 or 4 choices. Collect votes from the kids and from the grown-ups separately.
Make a frequency table and a multiple-bar graph.
Write one question of each kind (read the data, read between, read beyond) and answer them.
Present a convincing argument: claim, evidence and reasoning. Then decide together!

Math talk: "Did kids and grown-ups agree? Is the choice with the most votes fair to everyone? What could we do for the people who voted differently?"

Your turn

Snack graph: how many apples were sold on Day 2?

How many muffins were sold altogether?

Which snack sold the same amount on both days?

How many more bags of popcorn were sold on Day 2 than on Day 1?

How many snacks were sold altogether on Day 1?

Which kind of question is this: "Should we order more apples next year?"

Plank plots: what is the longest time in Class A? In Class B?

More practice: warm-up

Snack graph: how many cups of hot cocoa were sold on Day 1?

What does the scale count by?

What is the shortest plank time in Class A?

What are the three parts of a convincing argument?

Skating party: which day had the fewest votes in all?

More practice: on your own

How many snacks were sold altogether on Day 2? Was it more or less than Day 1?

How many students in Class B held a plank for 30 seconds or more?

Jamal says, "Everyone in Class B can hold a plank longer than everyone in Class A." Is he right?

Write a convincing argument for ordering more popcorn next year.

Skating party: why might a Grade 5 student feel the decision is unfair? What could the committee do?

Challenge

The committee has room for only 150 snacks per day next year. Using Day 2 data, is that enough? Explain.

Class A got one new student who held a plank for 60 seconds. Would Class A's median change much? Would its mean? Explain.

A survey of 5 Grade 4 students found that 4 like hot cocoa better than popcorn. Can you conclude that most Grade 4 students in Ontario like hot cocoa better? Explain.

Check your understanding

Quiz

"How many muffins were sold on Day 1?" is which kind of question?

Which is the best evidence for "Hot cocoa was the top seller"?

Class A's median plank time is 25 seconds. Class B's is 34 seconds. What can you conclude?

What is an informed decision?

Skating party votes: Friday 23, Saturday 18, Sunday 15. Which day has the most votes?

Which sentence is a careful conclusion?

For parents and teachers

What this lesson teaches: analysing data in multiple-bar graphs and stem-and-leaf plots by asking three kinds of questions (read the data, read between the data, read beyond the data), drawing careful conclusions, and making convincing arguments (claim, evidence, reasoning) and informed decisions (D1.6), using the median and mode as evidence (D1.5). Respectful disagreement and teamwork (A1) are built in.

How to help at home:

When your child makes a claim ("Everyone at school has that game!"), ask for evidence: "How many? How do you know?"
Use the family-decision activity for something real, and follow through on the decision.

Common difficulties:

Over-general conclusions ("everyone", "always") from data that shows "most" or "more".
Arguments with no numbers. Ask "Which number proves it?"
Confusing what the data shows with why it happened.
Forgetting that small groups may not represent everyone.