Data and Probability
Though smaller in scope in the curriculum, data and probability are prevalent in daily life and developing these concepts is an important part of becoming a numerate citizen.
Probability experiences usually involve the collection of data. Curricular content standards for data and probability can be developed simultaneously by interpreting and creating graphs that represent results from probability experiences.
Across K-7, the learning standards for data describe how data is represented, building from concrete and pictorial graphs up to bar, line, and circle graphs. Students learn to appreciate that how data is represented tells a story of the data, and by analyzing the data they can look for patterns, and make predictions, comparisons, and decisions. For data to have more meaning for students, it is important that they experience deciding what data they will collect, collecting the data, representing it, and analyzing it. Students will be engaged with data because it connects with their daily lives. Care should be taken when using binary genders such as boys vs girls when collecting or representing data, as this does not cover the full range of genders that may be represented in your classroom and can reinforce dated gender norms. Also be mindful of the type of data you might collect or represent about students’ lives that may signal or position students around socio-economic status or cultural values and beliefs.
Students encounter chance and uncertainty in their daily lives, and these underlie their learning journey through probability. In Primary, students develop the language of how likely events are to happen using comparative language. In Intermediate, students explore chance events more formally through experiments, the analysis of which helps them to describe the likelihood of different events, including using fractions. Students also learn about sample space which leads into determining theoretical probability. A big idea about probability is that the more data we have, the more we are able to describe trends and make predictions. In other words, the more data that is collected, the closer the experimental probability will approach the theoretical probability.
As students explore data and probability, there are many opportunities to connect to students’ lives, community, culture, and place. Data can help students understand themselves, their community and issues and events in the world around them. With these experiences we are honouring the following First Peoples Principle of Learning: Learning is holistic, reflexive, reflective, experiential, and relational (focused on connectedness, on reciprocal relationships, and a sense of place).
As we learn about key concepts in data and probability, we will also be developing many curricular competencies. Two that we have chosen to focus on in our designing of lesson ideas are:
- Explain and justify mathematical ideas and decisions
- Connect mathematical concepts to each other, other areas of learning and personal interests
Although these two curricular competencies have been highlighted, there will be many opportunities to develop many curricular competencies during the investigation of data and probability.
Learning Story for Grade 7
Data and Probability
Having represented and analyzed data using pictographs, bar graphs, and line graphs in earlier grades, in Grade 7 students will explore circle graphs. Circle graphs tell a story about data that the earlier representations do not do as well, namely how each category or part of the data compares to the whole. As in other grades, students will analyze given circle graphs, as well as create their own, including gathering their own data. Now that students have explored several different representations over the years, an important idea to develop is comparing the different representations and why one would choose to use a circle graph instead of other representations.
In the Intermediate grades, students have been exploring single event probabilities, both experimentally and theoretically (using sample space). In Grade 7, students explore two event probabilities involving events that are independent. Independence is an important concept that means that an event that is repeated is not affected by earlier results. Students will explore two independent events experimentally in Grade 7, then theoretically in Grade 8.
Key Concepts
Circle Graphs
Students apply their understanding of percents, angles, and circles to data to create and analyze circle graphs.
Experimental Probability with Two Independent Events
Students develop a sense of how likely different outcomes are using experiments to simulate two independent events.
Key Data and Probability Concept 1: Circle Graphs
Overview
Expanding their repertoire of different representations of data, in Grade 7 students explore circle graphs, which are also known as pie charts. Unlike the other representations they have learned, circle graphs describe well how each category relates to the whole. For example, in the circle graph below we can see that the blue category is about half, and the green is about a third.
As circle graphs focus on the relationship of each category to the whole, it is not focused on the specific numbers for each category. Though one can convert from the percents to the numbers, this is a less important aspect of this concept.
One can also make comparisons between categories though not as well as they can using a bar graph. Because of these differences, students will learn not just about circle graphs but also when it is a good choice of representation.
Students will analyze given circle graphs, but they will also create them. Circle graphs also give students an opportunity to apply their understanding of percents and angles.
Math Foundations:
Foundational, supporting concepts and related competencies that are needed to develop this grade level concept:
- Data collection: determining question(s) to survey and how to track the data (e.g., tally marks)
- Angles: measuring and constructing angles up to 360° using a protractor
- Percents of a number, including mental math strategies
- Circles: centre, radii
Progression:
- Understand what a circle graph represents
- Analyze and interpret a given circle graph. This may also include translating percent values to the actual data values. For example, if the whole represents 80 people and one section is 20%, then that section represents 20% of 80 which is 16.
- Create a circle graph from given data
- Given percent data, then given raw data to be converted to percents
- Using a percent wheel
- Using angles
- Gather data and represent it in a circle graph
- They need to think about what kind of data lends itself well to a circle graph
- Given percent data, then given raw data to be converted to percents
Sample Week at a Glance:
It would not take a week of lessons to fulfill what students need to understand about circle graphs, so below is a suggested plan for a sequence of learning experiences. Prior to this concept, students should have already learned grade-level material about percents.
Focus: Analyzing and interpreting given circle graphs
Before: Compare a bar graph to a circle graph
- Do a Same But Different routine with two graphs:
- Show the class a bar graph and a circle graph of the same data. For example, 20 students were asked which season was their favourite.
- Discuss how they are the same, and how they are different. For example:
- They each use the same colours for each season.
- They each show the relative size for each season, but the bar graph makes the comparison easier to see.
- The bar graph shows the actual number of who selected each season.
- The circle graph shows how each season compares to the whole. It’s easier to see that almost half the students chose “summer” from the circle graph.
During: Analyze and interpret a circle graph
- Show students a circle graph and have them work in pairs or small groups to determine as many things as they can about the data that the circle graph represents.
- For example, the circle graph below represents water use in the Okanagan Basin (source).
After:
- Have a class discussion about what they learned about water use from the circle graph. Students may give specific percents for each category, but there is more to analyze. For example,
- A little more than half of the water is used for agriculture.
- Residences use 31% of the water (combining indoor and outdoor), which is almost 1/3.
- Golf courses use almost as much as residences do indoors.
- Ask the class what they think the “other” category represents. For example: schools, industries, businesses/commercial (stores, restaurants, etc), and parks.
- Tell the class that the Okanagan Basin uses about 220 billion litres of water each year. Have them work in pairs to choose one or more categories and to estimate how many billion litres of water are used for that each year.
Provide other circle graphs and/or have students search for circle graphs in the media. Ask them to do as much analysis of they can – what story about the situation does the circle graph tell?
Focus: Creating circle graphs given percents
Note: The data in this lesson uses a lot of categories. You may wish to use a simpler set of data instead using a context of your choice. For this lesson, provide the data as percents rather than the raw numbers.
Before:
- To make sense of how percents are represented in a circle graph, present examples in order (i.e., one at a time) in increasing levels of precision. For each, ask them to determine the % of each colour. For example:
- The final one uses what is called a percent wheel, which allows one to colour in sections to the nearest percent. You can find a master of these here.
- Before being given a set of data, it is helpful to think about the context first so that the data has more meaning
- Show a list of movie genre’s: Adventure, Action, Drama, Comedy, Thriller/Suspense, Horror, Romantic Comedy
- Ask the class to think about the box office results in 2023 for each genre. [Note: If you do this lesson in a later year, you may wish to get data for that year (source)]
- Have a class discussion about which genre they think will have the most box office revenue in 2023, and why. Then discuss which they think will have the least.
During:
- Present the data:
- Have the class work in pairs to create a circle graph of the data using a percent wheel. As you circulate, support as needed to make sure students understand how the percent value of each piece of data is represented on the percent wheel.
- Tell the class to add details to their circle graphs, specifically a title, legend, and percent values.
After:
- Have a group share their completed graph, or re-create one as a class.
- Ask the class what they notice and wonder about the data. You may also ask your own questions to prompt discussion. For example:
- Which two categories combine for over half of the box office? Why do they think that is?
- Why do they think Romantic Comedies are so low?
- How do these box office results compare with their own movie preferences?
- Would these results change year by year?
- What genres could be in the “Other” category? Why have an “Other” category instead of graphing every genre?
- Provide students with other sets of data that are given in terms of percents and ask them to create a circle graph for each. Include at least one analysis-type of question for each.
- Though not as important, you may also wish to ask them to convert a circle graph to numerical data, which involves applying the percent of each category to the number of the whole.
Focus: Creating circle graphs from data
The data for a circle graph are related in three ways: the raw data (numbers, including the total), percents, and the central angles for each sector (i.e., each piece of the pie). When given a set of data, one needs to decide which conversion to do first, to percents or to angles. The size of the whole influences which of these two may be easier to do first, as will be shown in the During part of the lesson. The Before focuses on how to convert from percents to angles, and vice versa.
Before: Converting between percents and angles
- Sketch a circle graph with partially filled values that relate each percent to each central angle. Have students work in small groups to determine the missing percents and central angles. The answers are shown in red.
- Discuss the solutions as a class. Students may have a variety of strategies for figuring out the missing values.
- Because the circle is cut in half, the missing percents can be determined by the missing parts to make 50%. Similarly, the missing angles can be determined by the missing parts to make 180°.
- They may also notice relationships between the sectors. For example 10% is half of 20%, so its central angle should be half of 72°.
- It’s also important that they make sense of why each percent has that angle, and vice versa. 100% is 360°, so 10% is 36°, 5% is 18°, etc.
- With other data, the relationship between the percent and the angle may not be as direct to determine, in which case a calculator is appropriate. For example, if the data requires a 60° central angle, the percent is which is 16.7%. Conversely, if the data requires 24%, the central angle would be , in which case they could approximate the angle measurement.
- Ask the class to recreate this circle graph using a protractor. They may use a compass to construct a circle, or you may provide a circle template that includes the centre point.
During:
- Provide a set of data and have the class work in groups to create a circle graph. They’ll need to decide whether it’s better to convert the data to angles first, or to percents. Remind them to give their graph a title, legend, and to label each sector with its percent. For example, here’s a set of data for how a sample group of Grade 7 students spent a week day on average:
- Alternatively, the class could brainstorm their own list of activities and estimate a class average for each.
- You may decide to have a turn and talk, then a class discussion, about whether it’s easier to convert the data to angles or to percents, or leave the groups on their own to figure that out and then debrief the decision in the After.
- Note that the whole is 24 hours, which is not a friendly number for percents, but works well with 360°. Each hour can be measured by a 15° angle.
- Provide support through prompting questions as needed. Some may need reminding how to measure an angle. For those needing extra support, you could provide a percent wheel for them to use instead.
After:
- Have a group share their completed graph, or re-create one as a class.
- Have a discussion about the strategies they used.
- As noted above, each hour is worth 15°, so to graph for 8 hours, we need 120°
- Using a calculator, you can divide each number of hours by 24 and round to the nearest percent.
- Ask the class what activities could be part of the “Other” category?
- Have a discussion about whether a circle graph is an appropriate choice to represent this data.
- It is because 24 hours as a whole matters, and it shows how much of a whole day each activity used. We can also see enough of a comparison between the different activities.
- Provide students with one or two more sets of data and ask them to create a circle graph for each. Include at least one analysis-type of question for each.
Project: Gathering and Representing Data through a Survey
A project is more than can fit into one day, so it is described briefly here instead. Either individually or in groups, students will put together a mini-project to gather and represent data. The project would include:
- The survey question
- Why did they choose it?
- Why is it appropriate to represent this using a circle graph?
- Who is their target population? (e.g., classmates)
- How many did they survey?
- What are the survey results?
- Creating a circle graph of the data including a title, legend, and the percents for each category.
- An analysis of their data. For example, what did they learn? What surprised them? Why do they think <this> happened, etc.
At other times during the year it would be worth revisiting different circle graphs with a focus on analyzing and interpreting. A worthwhile routine to do is a slow-reveal graph. In particular this example explores how long animals sleep.
Suggestions for Assessment
By the end of this grade students will be able to analyze a given circle graph to answer questions and describe the story of the data. They will also be able to create a circle graph given a set of data, or gathering their own set of data. Throughout they will be mindful of when a circle graph is an appropriate representation.
Suggested Links and Resources
- How to use Polypad to create a circle graph (scroll down to Pie and Donut Charts)
Key Data and Probability Concept 2: Experimental Probability with Two Independent Events
Overview
Students have been exploring probability with independent events for years, but in Grade 7 the concept of independence gets formalized as students experiment to determine probabilities with two independent events. Events are independent when the results of one event do not influence what happens in the next event. For example, if you toss a fair coin, what you get (heads or tails) does not affect what you will get the next toss.
There are many different tools that students can use to experiment with independent events:
- Coins
- Dice
- Spinners
- Objects (colour tiles, cubes, marbles, etc) in a bag
- Playing cards
Polypad also provides a nice suite of virtual materials for probability.
An experiment with two independent events could be two consecutive events involving one tool (e.g., toss a coin twice) or a combination of tools (e.g., roll a die and spin a spinner). Even with one tool there are many possible outcomes that could be explored. For example, with two dice they could explore the sum of the dice, the products, whether you get a double, and so on.
It is not the expectation in Grade 7 to explore theoretical probability. In doing experiments, one develops a sense of the probability of an outcome by doing many trials. The more trials they do, the better picture of the probability they will get. Before carrying out an experiment, it is beneficial for students to predict or estimate the results.
By Grade 7, students should feel comfortable using the notation of P(event) to describe a probability. For example, when tossing one coin, P(heads) = 1/2 or 0.5.
Math Foundations:
Foundational, supporting concepts and related competencies that are needed to develop this grade level concept:
- Language of likelihood (e.g., likely/unlikely, always/sometimes/never, etc)
- Expressing probability using a fraction
- Experimental vs theoretical probability
Progression:
- Meaning of independence
- Experimental probability with two independent events
- Simple events with one tool (e.g., toss 2 coins)
- What is the probability that they are both heads?
- Simple events with two tools (e.g., toss a coin and roll a die)
- What is the probability of getting heads and a 3
- More complex events (e.g., sum of two dice)
- Simple events with one tool (e.g., toss 2 coins)
Sample Week at a Glance
It would not take a week of lessons to fulfill what students need to understand about probability, so below is a suggested plan for a sequence of learning experiences.
- Ask the class this problem, and do a think-pair-share:
- A fair coin was tossed 4 times, and every time it came up heads (HHHH).
- What is more likely to happen on the 5th toss?
- Hopefully the class will realize that because the coin is fair, it is just as likely to come up heads as it is to come up tails. The other answers reveal misconceptions about independence that are sometimes called gamblers’ fallacies:
- I’m on a roll, it’s more likely heads!
- The odds need to balance out, it’s more likely tails!
- Explain to the class that some probability events are independent, meaning that the probability of each event is not affected by what events happened before. For example, no matter whether one gets heads or tails on a toss of a fair coin, heads and tails are equally likely to be the result of the next toss.
- Have a brief discussion about which situations are not independent. A common example is when something is drawn (e.g., a colour tile from a bag or a card from a deck) that is not replaced before drawing a second one.
- Have students work in groups to rotate through up to 4 stations. For each station they will carry out many trials and keep track of the results in order to estimate the probability of a particular event. Some suggestions include:
- Two coins: repeatedly toss 2 coins. Use the results to estimate the probability of getting one heads and one tails.
- Two dice: repeatedly toss 2 six-sided dice. Use the results to estimate the probability of rolling two even numbers.
- Spinner with 4 equal colours: Spin twice. Use the results to estimate the probability of getting the same colour twice.
- Deck of 52 cards: Choose a card and put it back. Choose a second card. Use the results to estimate the probability that at least one of the cards is a face card (Jack, Queen, or King)
- Debrief each of the stations. Have students share their estimated probabilities and compare. A big idea about experimental probability is that the more trials that are made, the closer the experimental probability will approach the theoretical. So combining students results for one of the two of the stations would be worthwhile.
- Students are not expected to determine the theoretical probabilities, but it may be helpful to know what they are for comparison purposes. In the above examples, the theoretical probabilities are:
- Two coins: P(one heads and one tails) = 0.5
- Two dice: P(both dice even) = 0.25
- Spinner with 4 equal colours: P(same colour twice) = 1/16 = 0.0625
- Deck of 52 cards with replacement: P(at least one face card) = 69/169 which is about 0.41.
- For some of the stations, have a discussion or have them write an exit ticket on why the probability results make sense.
Focus: Using experimental probability to predict
Note: This is an interesting and worthwhile activity rather than a lesson, and as such is optional.
Tiles in a Bag
- Place an assortment of 10 colour tiles in a bag. Only you know which colours you used and how many of each colour are in the bag.
- Tell the class you put 10 tiles in the bag and ask them to make a prediction about what’s in the bag. Ask them how confident they feel about their decision.
- Do several trials as described below:
- Shake the bag.
- Randomly pull a tile from the bag and record its colour.
- Put the tile back in the bag.
- Tell the class they can change their prediction.
- Periodically ask how confident they feel about their prediction.
- As more and more trials are done, the fewer changes they will make with their predictions, and the more confident they will feel.
- Once most of the class agrees on what they think is in the bag, reveal the contents of the bag one tile at a time. Chances are, they will be correct, or at least very close.
- This activity highlights how experimental probability can lead to accurate results if enough trials are done. Sometimes this is the only way a probability can be determined (e.g., with weather).
Focus: Experimental probability with two independent events: Exploring outcomes with two tools (e.g., toss a coin and roll a die)
Before:
- Explain to the class that up to this point, they have only used one tool at a time. Today they will do experiments using two different tools.
- As an example, show the class a coin and a six-sided die. Ask for some ideas of events they could determine the experimental probability of. For example:
- P(heads and 3)
- P(tails and an even number)
- P(heads and a number lower than 5)
During:
- Have students work in groups.
- They can choose their own set of two tools to use, choosing from: a coin, a die, a variety of spinners, objects like colour tiles in a bag, a deck of cards, or other. They could even use two spinners if the spinners are different.
- Ask them to determine an experimental probability for an event that they choose. They should first make a prediction, and then perform and keep track of many trials.
After:
- Have a few groups share their experiment and the results. For each one ask the class to estimate what the probability could be.
- Ask groups to share if any of the results were different from what they expected, and to think of a reason why.
Tell the class to return to their groups. Ask them to create an event that they think will have about a 0.5 probability. This would make for a fair game. Ask them to play the game a few times and see if it turns out to seem fair.
Focus: Experimental probability with more complex events
You could explore more complex events with a variety of tools. This lesson is based on using six-sided dice.
Before:
- Give every student two dice.
- Ask them to roll their dice.
- Ask how many had a matching pair, and how many did not. Record the results.
- Do this another two or three times.
- The experimental probability of matching dice should be around 1/6. Ask the class why this makes sense [whatever one die has, the chances that the other die has the same number is 1 out of 6].
During:
- Tell the class they will be rolling two dice, and adding up the sum of the two dice. Ask them to think about which sums may be more likely, or all sums will have the same probability.
- Have students work in groups.
- Members of the group should roll two dice many times each and record the sums, keeping track of how many times each sum occurred.
After:
- Have a few groups share their results. You may also wish to add up the results of the groups to make a larger sample.
- Ask the class what they noticed about the sum of two dice. For example:
- 7 seems to be the most likely.
- Some pairs of numbers seem to be about the same probability. Like 6 and 8, 2 and 12, …
- Ask the class if any of the results surprised them. Some may be surprised that the sums have different probabilities. Discuss why this turns out to be the case.
- Here is a nice challenge to consider doing: Is this dice game fair?
- It costs $1 to play each game.
- 2 dice are rolled
- If the difference between the dice is 3, 4 or 5, you win $2.
- If the difference between the dice is 0, 1, or 2, you lose.
- Would you play?
- Students can experiment and will see that the game isn’t fair. Ask them to think about why.
- Ask them to explore if you won $3 instead of $2.
Suggestions for Assessment
By the end of Grade 7, students should understand what makes some probability events independent, and be able to describe some examples. They should be able to perform experiments involving two independent events, keep track of the results, and describe the experimental probability of the event. They should appreciate that they more trials they do, the closer their experimental probability will approach the theoretical.