### 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 3

#### Data and Probability

In grade 3, students build on the work they did in grade 2 by continuing to develop pictorial and symbolic data representation and working with simulated events in work on probability.

Students leverage previous work with concrete and pictorial data representation and begin to represent data in charts, tables, pictographs, and bar graphs. They have experience creating (individually and with others) concrete representations of data, such as using loose parts or math materials to characterize data. Additionally, in grade 2, they began to represent data pictorially with symbols, pictures, colours, etc. In grade 3, students extend this pictorial representation to more formal pictographs to organize data. They are introduced to interpreting and creating charts, tables, and bar-graphs. At this level, students work with one-to-one correspondence. In grade 4, they will continue working with charts, tables, pictographs, and single-bar graphs and will explore many-to-one correspondence in data representation.

For probability, grade 3 students begin to explore simulated events and apply language developed in previous grades to these events. In grade 2, students used language such as always, sometimes, never, likely, certain and uncertain as well as comparative terms (e.g., more likely, less likely, equally likely) in relation to familiar life events. In grade 3, students are provided opportunities to apply this language to simulated events through play and inquiry. They may work with coins, dice, cards, or spinners among other materials. In grade 4, they will engage with probability experiments that they track using tally charts.

When working with simulated events, such as tossing coins or rolling dice, students may collect and record their findings as data. Content standards for data and probability can be worked on simultaneously by interpreting and creating charts, tables, or graphs that represent results from play and inquiry involving simulated events. The sample week plan provided is an example of how this can be achieved.

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Data and probability work involves connections to other areas of math, especially number concepts. Students will draw on their understanding of addition or skip-counting, for instance, to calculate totals from a tally chart of dice rolls. There will be opportunities to perform operations on data to represent it in graphs or to interpret graphs.

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Connections between math and other subject areas can be made by using graphs from content areas such as science and social studies. Data represented in graphs in local news stories connects to place. Students may connect graphing and probability to their own interests and culture by collecting data or exploring probability of outcomes in specific activities (e.g., investigating probability of certain outcomes in preferred games). Making meaningful connections honours the First Peoples Principle of Learning “Learning is holistic, reflexive, reflective, experiential, and relational (focused on connectedness, on reciprocal relationships, and a sense of place).”

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Be mindful of the type of data you might collect or represent about students’ lives that may signal or position students around gender, socio-economic status, or cultural values and beliefs. 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, for example.

### Key Concepts

#### Data Collection and Graphing (pictographs and bar graphs)

Grade 3 students collect data and represent it in charts, tables, bar graphs, and pictographs. They interpret and create graphs with one-to-one correspondence.

#### Probability

Grade 3 students express the likelihood of simulated events (e.g., coin toss, dice roll) with descriptive language.

#### Key Data and Probability Concept 1: Data Collection and Graphing (pictographs and bar graphs)

##### Overview

Learning how data can be organized in tables, charts, and graphs supports the recognition of patterns and trends. Grade 3 students continue to explore how to interpret and represent data in graphs. Students develop critical thinking skills by describing, comparing, and discussing what they notice and wonder about charts, tables, and graphs. Looking at different graphs for the same data set can help students understand what stories can be told with the data.

Grade 3 students come with experience in previous grades working with concrete and pictorial graphs with one-to-one correspondence (one picture or square in a graph represents one instance/item/data piece). They have contributed to co-created pictographs and possibly bar graphs for things like tracking weather, favourite sports or pastimes, etc. They have some experience creating simple, informal pictographs on their own using concrete materials (e.g., small toys, loose parts, etc.) and visually on paper (e.g., with stamps, hand-drawn images, or gluing small pictures). They have seen data recorded in tally charts and T-charts, though they may not have been required to create these themselves.

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Grade 3 students collect and analyze data in charts and tables, such as tally charts and T-charts. They build upon their work with pictorial representation of data in grade 2 to create (co-create and individual work) formal pictographs and single-bar graphs using one-to-one correspondence. It is appropriate to continue providing experience with concrete and pictorial representations of data as a starting point; then students are taught to move these representations into pictographs and bar-graphs. Data might be collected about student-generated questions of personal interest. It also can be collected in relation to investigations in other subjects, such as science, or with respect to probability experiments in mathematics, as examples.

Pictographs use a symbol (or symbols) to represent data. This symbol can be designed as a picture related to the data category. For example, if exploring the preferred types of ball sports in the class, the symbol could be a circle to represent balls or a stick figure or emoji to represent students who prefer that sport. It could also be a square or other geometric figure distinct from the context of the pictograph. Symbols can be arranged vertically or horizontally and symbols in each category start at the same place and are equally spaced so that comparisons can be easily made between categories. In grade 3, students are working on one-to-one correspondence so each symbol represents one instance of the data. Labels should be included and a legend can be provided. A title helps convey the meaning of the pictograph.

Bar graphs can be horizontal or vertical and use bars of different lengths or heights to represent instances of data. Bars are created using a sequence of squares that are the same size and represent the same amount; usually, students colour squares on grid paper to create bar graphs. Bars should be the same width, be equally spaced apart, and not touch each other, and each bar should have a category label. Axes should be labeled and the graph should have a clear title related to the context of the graph. A scale can be included. In grade 3, students work with one-to-one correspondence (squares represent one instance of the data. In grade 4, they will build graphs with many-to-one correspondence (one square represents two or more instances of the data).

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There are many opportunities to give students experience interpreting graphs and creating their own graphs in other subject areas, especially social studies and science. Using data tables and bar graphs from text books, reference materials, or online sources (such as **Our World in Data**) In science, grade 3 students collect simple data during scientific inquiries with drawings or provided tables and analyze and communicate it through bar graphs (and other representations). Making meaningful connections across subjects and to students’ lives honours the First Peoples Principle of Learning “Learning is holistic, reflexive, reflective, experiential, and relational (focused on connectedness, on reciprocal relationships, and a sense of place).”

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Be mindful of the type of data you might collect or represent about students’ lives that may signal or position students around gender, socio-economic status, or cultural values and beliefs. 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, for example.

##### Math Foundations:

The following concepts and competencies are foundational in supporting understanding of data in grade 3:

- Counting with one-to-one correspondence
- Collecting simple data in provided tables, individually or with others (e.g., partner, class)
- Interpreting (describing and comparing) pictorial representations of data (may not be formal pictographs)
- Creating pictorial representations of data (though not formal pictographs) with one-to-one correspondence, with stamps,
- Co-creation of pictorial representations of data and simple bar-graphs (e.g., class weather charts)

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##### Progression:

- Class exploration of how information/data can be represented – through picture books (e.g., Me the World: An Infographic Exploration by Mereia Trius and Uma Wimple Charts Her House by Reif Larsen and Ben Gibson) and websites (Our World in Data, Slow Reveal Graphs)
- Pictographs
- Interpreting (describing and comparing) horizontal and vertical pictographs with one-to-one correspondence
- Noting inclusion labels and title (optional: legend)

- Creating horizontal and vertical pictographs with one-to-one correspondence using paper/pencil (symbolically)
- Inclusion of title, labels, and scale or legend as necessary
- Making appropriate choices for pictorial symbol

- Single-bar graphs
- Interpreting (describing and comparing) horizontal and vertical single-bar graphs with one-to-one correspondence
- Noting inclusion of title, axis labels, and, optionally, scale
- Discussion of what data is best represented with bar graphs

- Creating horizontal and vertical single-bar graphs with one-to-one correspondence using paper/pencil (symbolically)
- Noting inclusion of title, axis labels, and, optionally, scale

- Interpreting (describing and comparing) horizontal and vertical single-bar graphs with one-to-one correspondence

- Interpreting (describing and comparing) horizontal and vertical pictographs with one-to-one correspondence

#### Key Data and Probability Concept 2: Probability

##### Overview

Grade 3 students continue to develop comparative language, reinforcing and building on language learned in grade 2. Additionally, they begin to build a formal understanding of chance. This can be done by discussing familiar events and by engaging with classroom materials such as dice, spinners, coins, two-sided counters, or cards, as some examples. Although tracking of outcomes is not prescribed in grade 3 for probability, documenting outcomes with pictures or symbols connects to learning standards for data.

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Students have some familiarity with probability in their own lives. For example, they can talk about the probability of snow in the last week of October, the likelihood that a soccer game will be canceled due to poor weather, how likely they are to roll doubles when playing Snakes and Ladders, etc. These relevant contexts are considered concrete representations for probability. Students have some sense of chance/randomness and independence of events through their experiences with dice and coin flips, as two examples.

In kindergarten, grade 1, and grade 2, students have been developing the language of probability in relation to familiar life events. The shift in grade 3 is to connect probability language to simulated events. These may involve dice, coins, spinners, or cards, as some examples. They may talk about the likelihood of an event as *always, certain, uncertain, probable, impossible, *or *never *and they may compare events using the terms *more likely, less likely*, or *equally likely*. It may be useful to have a word-wall or other display with words related to describing or comparing the probability of outcomes. A display could include a probability line with certain/always at one end and impossible/never at the other (examples can be found in Marian Small’s Making Math Meaningful for Canadian Students K-8 on p. 625).

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One misconception that many students (and adults) have is that an outcome is influenced by a previous outcome(s). For example, after flipping a coin 3 times and having it land as heads, students may believe that tails is more likely on the next flip. However, tails and heads are still equally likely. It is important to challenge this misconception while students engage in simulated events.

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When students move into grade 4, they will use their understanding of chance from grade 3 to predict outcomes in probability experiments. They will formally record outcomes of experiments in tally charts.

##### Math Foundations:

The following concepts and competencies are foundational in supporting understanding of probability in grade 3:

- A developing understanding and use of the language of probability including describing using impossible, uncertain, never, likely, unlikely, always, possible and comparing using more likely, less likely, equally likely to describe familiar events
- Experiences with classroom materials such as dice, cards, coins or two-sided counters, and spinners (e.g., through playing games)
- Counting with one-to-one correspondence
- Addition of numbers to 50

##### Progression:

- Discussions about where students encounter chance and probability in their own lives to teach and reinforce vocabulary of probability (describing using
*impossible, uncertain, never, likely, unlikely, always, possible*and comparing using*more likely, less likely*,*equally likely*) - Engage in activities with classroom materials such as spinners, dice, coins/two-sided counters, and cards
- Using descriptive language of impossible, uncertain, never, always, certain to describe outcomes
- Using comparative language for two or more possible outcomes – more likely, less likely, equally likely

- (Optionally: connecting data and probability by representing documented results in a pictograph or single bar graph)

##### Sample Week at a Glance

**This sample week integrates both data and probability key concepts for this grade level. **

*Because graphing and probability concepts can be done with smaller numbers and language about probability has been introduced in grade 2, it is appropriate to do a week focused on graphing and probability earlier in the year.*

**Focus**: exploring how information/data can be represented

**Before**: Picture Book and discussion – Me the World: An Infographic Exploration by Mereia Trius. Choose several pages to look at and discuss as a class (consider projecting with a document camera/ipad-AppleTV combo). Guide a discussion with questions such as, “What is this about? How do you know?”, and inviting comparisons of categories of data (e.g., “Which is more…A or B?”). Discuss why people choose to display information in these different ways (what are the advantages?).

Alternatively, read Uma Wimple Charts Her House by Reif Larsen and Ben Gibson and lead a discussion similar to the one described above.

**During**: Workshop/Stations. Provide 3-5 stations for students to engage with. Students change stations every 10-15 minutes. For example:

- Provide some tables/charts of information from Social Studies or Science materials. Ask students to answer some questions about the information.
- Students choose 3 of their regular activities and create a representation of how many times a week they do those activities (leave the representation open for students to create).
- Provide a collection of materials, such as, different colours of interlocking cubes or other blocks. Ask students to create a pictorial representation of the number of each colour.
- Provide a pictograph about a topic of general interest with a few questions to answer (alternatively, “What do you notice? What do you wonder?”)
- Provide a bar-graph about a topic of general interest with a few questions to answer (alternatively, “What do you notice? What do you wonder?”)
- Provide books such as Animals by the Numbers by Steve Jenkins and Dinosaurs by the Numbers by Steve Jenkins for students to look through with a partner

These stations can be used as activities students can do throughout the week during choice times or when other work is finished.

**After**: As a class, generate a list of “I wonder” questions about representing information/data. Keep this list posted and revisit over the next lessons to see what has been addressed.

Consider adding additional lessons/activities based on this list.

**Focus**: Interpreting and creating pictographs (day 1)**Before**: Slow reveal graphs routine – follow the instructions for Slow Reveal Graphs and use a pictograph from https://slowrevealgraphs.com, such as https://slowrevealgraphs.com/2020/02/12/dont-waste-water/ (ideally one with one-to-one correspondence). Include focus on the picture chosen and how this helps understand the information in the graph. Use this graph to teach about the components of a pictograph.

**During**: Introduce the 2-day project of students designing a survey, collecting data in a chart, and making a pictograph of the data.

Teacher models the design of a survey question (teacher will do the project alongside students as a demonstration) and creates a chart for recording data.

In pairs (or individually, if students prefer), students design a survey question that can be answered by other students in the class. Students design a system for collecting data.

If time allows, students may start collecting survey data (from 10-15 students). Teacher collects survey data.

**After**: Teacher displays data collected for the model survey. Teacher leads a “What do you notice? What do you wonder?” short discussion. Students help the teacher decide on a symbol to use for the corresponding pictograph and how many data instances the symbol could represent. Ask students to think about their symbols between this class and next.

(Before next class, teacher should create a pictograph from data she/he/they collected)

Revisit “I wonder” list from Monday and see what questions might be answered.

Focus: interpreting and creating pictographs (day 2)

**Before**: Display teacher-created pictograph from survey data collected in previous lesson. Lead a class discussion about the pictograph and the data displayed in it. Include questions about what category had the most/least data points, how many data points a particular category had, and comparisons between categories. Use this pictograph to teach about the components of a pictograph (title, labels, and, optionally, legend). Discuss how symbols/pictures are chosen. Leave this pictograph on display for the remainder of the lesson as a reference.

**During**: Students finish collecting survey data. Students create a pictograph based on collected data. Teacher circulates to check in and help as needed.

**After**: Assign individuals/pairs of students into groups so there will be 3 pictographs to share. Each individual or pair shares their pictograph and the other students offer TAG (Tell one thing you noticed, Ask a question, Give a suggestion) (one student per letter, chosen by the presenter[s]).

Revisit “I wonder” list from Monday and see what questions might be answered.

*Collect survey data and pictographs to use for formative assessment. Consider adding another lesson on interpreting and/or creating pictographs if the evidence of learning suggests this is needed.*

**Focus:** language describing likelihood of simulated events, including comparative language

**Before**: Have a brief discussion about the likelihood of certain events in the students’ lives to review/teach language related to probability (impossible, uncertain, never, likely, unlikely, always, possible and comparing using more likely, less likely, equally likely). For example, the likelihood that the bell will ring at 3pm, that it will rain tomorrow, that they will have cereal for breakfast tomorrow, that there will be school on Saturday, etc. Then, lead a discussions about a few probability situations, preferably with concrete materials. For example, flip a coin and discuss the likelihood of a heads/tails/heads or tails/not a heads or tails. Draw cards from a pile (once letting students know what is in the pile) – what is the likelihood of a red card/a black card/a red card compared to a black card/a certain number/etc.

Make a list of vocabulary for a word wall during the discussion. Consider introducing a probability line and placing the vocabulary in the appropriate place on the line (examples can be found in Marian Small’s Making Math Meaningful for Canadian Students K-8 on p. 625).

**During**: Making and describing spinners. Provide students with circles (with the centre marked) to make spinners. Students mark off and colour portions of the spinner. Allow them to play with their spinners using a pencil and a paper-clip (the pencil holds the paperclip by ‘standing’ on the circle’s centre and then the paperclip can be flicked to spin it). Have them write several statements about the likelihood of the spinner landing on different colours using language from the vocabulary list developed in the *Before *section of the lesson. Encourage comparative language (more likely, equally likely, less likely) as well as the likelihood of a single colour.

(spinner image created at https://www.nctm.org/adjustablespinner/ )

**After**: Students share their spinners with a partner. They ask the partner to describe the likelihood of the spinner landing on various colours and comparing the likelihood of two colours. End with a class discussion where 2-3 students share their spinners and other students describe the likelihood of spinning certain colours.

Revisit “I wonder” list from Monday and see what questions might be answered.

Keep spinners in a safe place for Friday’s lesson.

Focus: collecting data in simulated events and creating single-bar graphs

**Before**: Slow reveal graphs routine – follow the instructions for Slow Reveal Graphs and use a pictograph from https://slowrevealgraphs.com, such as https://slowrevealgraphs.com/2022/10/10/consumption-of-kraft-dinners-mac-cheese-across-canada/ (ideally one with one-to-one correspondence). Include focus on the picture chosen and how this helps understand the information in the graph. Use this graph to teach about the components of a pictograph.

**During**: Gathering data and creating single-bar graphs.

Using a spinner with 4 sections created on https://www.nctm.org/adjustablespinner/ , spin the spinner 10-15 times and discuss the data in the chart. Demonstrate how to create a bar graph of the data.

Using spinners created in Thursday’s lesson, students gather data and create single-bar graphs. Allow students to work in partners (picking one spinner) or individually. Students spin the chosen spinner 10-15 times and record the results. Provide students with grid paper to construct a bar graph.

**After**: Class Discussion. Choose 2-3 spinners and the corresponding bar graphs to discuss. Ideally select at least one spinner-graph pair for which the collected data might not match the expected outcomes. Start with the spinner and ask for statements that describe the likelihood of spinning different colours. Record these statements. Show the corresponding bar graph. Ask students what they notice. Discuss how having more data will show results closer to expected outcomes (bar graphs will match the statements more closely with more data collected).

Revisit “I wonder” list from Monday and see what questions might be answered.

Use formative assessment strategies to determine if the pace in this week plan is suitable for your students. Extend the activities for more days as necessary.

If these content standards are introduced early in the year, find opportunities to continue examining, reading, and creating graphs in other subject areas and contexts.

##### Suggestions for Assessment

When students are engaged in data collection and graphing activities, gather for evidence through observations, conversations, and student work, that students can, by the end of grade 3:

- Develop a survey question with discrete choices
- Record survey data in an organized way that they can read and interpret (they can explain it)
- Can interpret pictographs; can identify the category with the most/least instances and can use the data to make comparisons
- Can interpret single-bar graphs; can identify the category with the most/least instances and can use the data to make comparisons
- Create a pictograph (one-to-one correspondence) with a meaningful title, category labels, and an appropriate picture/symbol to represent data; pictures/symbols are evenly spaced across categories in order to make comparisons; the data in the pictograph that matches collected data
- Create a single-bar graph (one-to-one correspondence) with a meaningful title, axis headings, and labels; the data in the graph matches collected data in charts or tables or other representations

There are opportunities to document student learning for data collection and graphing in a portfolio (digital or physical). For a physical portfolio, students may select a favourite graph they created. For a digital portfolio, students might take a picture of a graph. Students may be asked to describe why they chose that graph as an example of their understanding, what questions they have, and what their next steps might be to continue their learning.

When students are engaged in activities related to probability, gather for evidence through observations, conversations, and student work, that students can, by the end of grade 3:

- Accurately use language of probability to describe and compare different outcomes related to simulated events with classroom materials (dice, spinners, etc.) in class discussions and in individual work
- Understand chance in a variety of simulated events (e.g., flipping coins, flipping cups, rolling a die, etc.)

##### Suggested Links and Resources

**Websites and Digital Documents**

Slow Reveal Graphs website **https://slowrevealgraphs.com/**

Slow Reveal Graphs Instructional Routine – **https://coastmetro.ca/elementary-math-project/instructional-routines/**

Data Talks – Youcubed: **https://www.youcubed.org/resource/data-talks/**

Our World in Data **https://ourworldindata.org/**NCTM Adjustable Spinner

**https://www.nctm.org/adjustablespinner/**

**Books**

Me and the World: An Infographic Exploration by Mereia Trius

Uma Wimple Charts Her House by Reif Larsen and Ben Gibson

Animals by the Numbers by Steve Jenkins

Dinosaurs by the Numbers by Steve Jenkins

Making Math Meaningful for Canadian Students K-8 by Marian Small (Nelson)