Grade by Grade
D. Data · Two-Variable Data, Scatter Plots and Probability

Collecting Continuous Two-Variable Data

D1.2 A1About 50 minutes
Learning goals
  • I can plan how to collect continuous data for a two-variable question.
  • I can measure carefully and choose a sensible level of precision.
  • I can organize two-variable data in a table of values.
  • I can use repeated trials and the mean to make my data more reliable.

Words to know

continuous data
Data that is measured and can take any value in a range, including decimals, like 152.5 cm or 9.1 s.
table of values
A table with one column for each variable. Each row is one ordered pair from one person, object or trial.
trial
One run of an experiment, like one drop of a ball or one timing of a pendulum.
precision
How exact a measurement is, like to the nearest centimetre or the nearest tenth of a second.
fair test
An experiment where you change only the independent variable and keep everything else the same.

Remember from Grade 7

Discrete data is counted (number of pets: 0, 1, 2...). Continuous data is measured and can be any value in a range (a height of 158.4 cm). In Grade 7 you planned fair samples and unbiased questions. Those ideas still matter. Now you will collect continuous data for two variables at once.

A question worth measuring

Maya wondered: "Does the length of a pendulum's string affect how long it takes to swing?" A pendulum is just a weight hanging on a string, like a playground swing.

Both variables are continuous: string length (in cm) and time (in seconds). She can't just count them. She has to measure them, carefully, and keep each length together with its time.

A metal weight hangs on a string from a lab stand, with a metre stick and a stopwatch on the table beside it.
These are the tools for Maya's investigation: the metre stick measures the string length and the stopwatch measures the swing time. Each length is recorded together with its time as one ordered pair. AI-generated image

Planning a two-variable investigation

  1. Write the question. Name both variables.
  2. Decide which variable is independent (the one you change or choose) and which is dependent (the one you measure in response).
  3. Choose units and tools. A metre stick for length, a stopwatch for time.
  4. Choose your precision. Nearest centimetre? Nearest tenth of a second? Use a level that your tool can really measure.
  5. Make it a fair test. Change only the independent variable. Keep everything else the same (same weight, same starting angle, same person timing).
  6. Plan enough data. At least 5 to 10 ordered pairs, spread across a wide range of values.
  7. Repeat trials when you can, and use the mean of the trials. This evens out small timing mistakes.
  8. Make a table of values before you start, with units in the headings.
Worked example

Example 1: Maya's pendulum plan

Plan Maya's investigation and design her table of values.

  1. Independent variable: string length, in cm. Maya will choose lengths of 20, 40, 60, 80 and 100 cm.
  2. Dependent variable: time for 10 swings, in seconds. One swing is too quick to time well, so timing 10 swings makes the measurement more accurate.
  3. Tools and precision: measuring tape (nearest cm), phone stopwatch (nearest tenth of a second).
  4. Fair test: same washer on the end, pulled back to the same starting position each time, same timer.
  5. Repeat: 3 trials for each length, then find the mean.

Table headings: String length (cm) | Trial 1 (s) | Trial 2 (s) | Trial 3 (s) | Mean time for 10 swings (s). The independent variable goes in the first column.

top of doorframetapewasherstringlengthtape to middleof washer1 swing = over AND backtime 10 swings, then repeat 3 timesFair testCHANGE only:string length(20, 40, 60, 80, 100 cm)KEEP the same:the same washersame start (about 10 cm)same person timing
Maya measures the string from the tape to the middle of the washer. One swing is over and back. Only the length changes; everything else stays the same.
String length (cm)Trial 1 (s)Trial 2 (s)Trial 3 (s)Mean time for 10 swings (s)
209.09.39.09.1
4012.512.712.612.6
6015.615.415.815.6
8017.917.817.717.8
10020.120.420.120.2
Maya's results. Each mean is the sum of the three trials divided by 3, rounded to the nearest tenth. Example: (9.0 + 9.3 + 9.0) ÷ 3 = 27.3 ÷ 3 = 9.1.
Three trials for the 20 cm string (time for 10 swings)8.88.99.09.19.29.39.4secondsTrial 1Trial 3Trial 2mean = 9.1 s(9.0 + 9.3 + 9.0) ÷ 3 = 27.3 ÷ 3 = 9.1 s
The three trials for the 20 cm string are close together. The mean (9.1 s) sits between them, so one slightly slow click of the stopwatch (9.3 s) only nudges the result a little.
Worked example

Example 2: Organize messy notes

Jonah dropped a ball from different heights and wrote these notes: "75 cm drop, bounced 44. From 25 cm it bounced 15. 150 cm gave 88. 50 cm → 29. From 125 it went up 74. 100 cm drop, bounce 58." Organize his data in a table of values.

  1. The independent variable is drop height. The dependent variable is bounce height. Both are in cm.
  2. Put drop height in the first column and sort it from least to greatest. Sorting makes patterns easier to see.
  3. Keep each bounce height on the same row as its drop height.

Two columns, Drop height (cm) and Bounce height (cm), with rows 25 → 15, 50 → 29, 75 → 44, 100 → 58, 125 → 74, 150 → 88 (shown below). Each row is one ordered pair, for example (100, 58).

Drop height (cm)Bounce height (cm)
2515
5029
7544
10058
12574
15088
Jonah's ball-bounce data as a table of values, sorted by the independent variable.
Worked example

Example 3: Data about people

The Okafor family wants to know: "Can you predict a person's arm span from their height?" They measure 10 family members and friends. How should they measure, and what does the table look like?

  1. Height: stand with your back against a wall, heels together, no shoes. Mark the top of the head with a flat book and measure to the nearest centimetre.
  2. Arm span: stand with your back to the wall, arms stretched out at shoulder height. Measure fingertip to fingertip to the nearest centimetre.
  3. Measure each person the same way. Ask permission before measuring anyone, and give each person a letter instead of a name to keep the data private.
  4. Height is the independent variable (we predict from it), so it goes first.

The table of values is below. Each row is one person's ordered pair, like Person D: (162, 165).

PersonHeight (cm)Arm span (cm)
A150148
B155157
C160158
D162165
E165163
F168170
G170172
H172169
I175176
J180181
Height and arm span of 10 people. You will graph this data in the next lesson.

Good data habits

  • Write units in every column heading, not after every number.
  • Record every trial, even one that looks strange. You can decide later what to do with it.
  • Use the same precision for every value in a column (all to the nearest tenth, for example).
  • Spread your independent values across a wide range (20 cm to 100 cm, not 50 cm to 55 cm), so a pattern has room to show up.
Just for fun

According to a story told by his student, a young Galileo timed a swinging lamp in Pisa Cathedral using his own pulse as a stopwatch. Your phone's stopwatch is a big upgrade, and you don't have to count your heartbeat for 10 swings.

Hands-on activity

Kitchen-doorway pendulum

You need: string (about 1.2 m), a heavy washer, nut or key, tape, a measuring tape or metre stick, a phone or stopwatch, a partner if possible.

  1. Tie the washer to the string. Tape the other end to the top of a doorframe or the edge of a table so it can swing freely.
  2. Adjust the string so the length from the tape to the middle of the washer is 20 cm. Measure carefully.
  3. Pull the washer to one side (always about the same distance, around 10 cm) and let go. Time 10 complete swings (over and back counts as one swing).
  4. Do 3 trials and record them. Find the mean to the nearest tenth of a second.
  5. Repeat for 40, 60, 80 and 100 cm.
  6. Make a table of values like Maya's. Keep it: you will graph it next lesson.

What to look for: longer strings take more time, and the three trials for each length should be close (within a few tenths of a second). If one trial is far off, think about why (did you count 9 swings?).

Your turn

1

Is the time it takes to run 100 m discrete or continuous data?

2

For the question "Does the height of a ramp affect how far a toy car rolls?", which variable goes in the first column of the table of values?

3

Three trials gave 12.4 s, 12.8 s and 12.6 s. Find the mean.

4

Why did Maya time 10 swings instead of 1?

5

Sam plans to test ramp heights of 10, 11 and 12 cm. What would you suggest?

6

Write the ordered pair for Person H in the Okafor table.

7

In the ball experiment, name two things Jonah should keep the same for a fair test.

More practice: warm-up

1

Discrete or continuous? The mass of a pumpkin.

2

Discrete or continuous? The number of seeds in a pumpkin.

3

What goes in each row of a table of values?

4

Find the mean of 58 cm, 59 cm and 57 cm.

5

Where do you write the units in a table of values?

More practice: on your own

1

Plan an investigation for: "Does the amount of water in a glass bottle affect the pitch (frequency, in hertz) when you tap it?" Name the independent and dependent variables and one thing to keep the same.

2

A stopwatch shows hundredths of a second, but people react about 0.2 s late. Is it sensible to record times to the nearest hundredth? Explain.

3

Trials of 15.6 s, 15.4 s and 15.8 s were recorded for the 60 cm pendulum. Show that the mean is 15.6 s.

4

Priya records height in metres for some people (1.62) and centimetres for others (158). What problem will this cause, and how should she fix it?

5

Why should each person in the arm-span survey be given a letter instead of a name?

Challenge

1

Jonah drops the ball from 100 cm three times and gets 58, 59 and 27 cm. The ball hit the edge of a rug on the third drop. What should he record, and why?

2

A table of values has 8 rows and 2 variable columns. How many separate measurements were made? How many ordered pairs are there?

3

Design a table of values for: "Is there a relationship between the outdoor temperature at 7 a.m. and the time it takes to scrape the frost off the family car?" Include headings with units and explain how many days you would collect data.

Check your understanding

Quiz

1.

Which is continuous data?

2.

In a table of values, the independent variable usually goes...

3.

Three trials were 20.1 s, 20.4 s and 20.1 s. What is the mean?

4.

Why do scientists repeat trials?

5.

In a fair test, you should...

6.

Which set of drop heights would give the most useful data?

For parents and teachers

Big idea: The Ontario curriculum (D1.2) asks Grade 8 students to collect continuous data to answer questions of interest involving two variables, and to organize the data in a table of values. This is a hands-on lesson: the pendulum or ball-bounce experiment works well at a kitchen table.

Common difficulties:

  • Recording values without units, or mixing units (cm and m).
  • Choosing values too close together (50, 51, 52 cm), so no pattern can appear.
  • Ignoring or erasing strange trials. Encourage your child to record everything and write a note about anything that went wrong.
  • Over-precise numbers (hundredths of a second from a hand-held stopwatch).

How to help: Be the timer or the measurer. Ask, "Did we keep everything else the same?" and "Why do we need three trials?" Keep the table: the next lesson turns it into a scatter plot. The lesson also builds A1 skills: planning, persevering through repeated trials, and working together.