### 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 4

#### Data and Probability

Grade 4 students use knowledge, language, and skills developed in primary years to extend their understanding of how data can be represented with graphs and to conduct and describe probability experiments.

In primary years, students have experience working with concrete and pictorial graphs with one-to-one correspondence. In grade 4, students leverage this knowledge to interpret and create single bar graphs and pictographs. They apply and reinforce their understanding of one-to-one correspondence and extend it to represent data using many-to-one correspondence in both pictographs and bar graphs. In grade 4, students work with single-bar graphs. They will investigate double-bar graphs in grade 5.

For probability, grade 4 students engage with single-outcome probability experiments. They have been introduced to language to describe likelihood of outcomes in grades 2 and 3 and continue to use this language in grade 4. Grade 4 students record data from probability experiments (with cards, spinners, dice, etc.) using tallies. In grade 5, students will reinforce ideas and concepts from grade 4.

Probability experiments involve the collection of data. Content standards for data and probability can be worked on simultaneously by interpreting and creating graphs that represent results from probability experiments. The sample week plan provided is an example of how this can be achieved.

Data and probability work involves connections to other areas of math, especially number concepts. Students will draw on their understanding of multiplication and division, for instance, to work with many-to-one correspondence. There will be opportunities to perform operations on data to represent it in graphs or to interpret graphs.

Connections between math and other subject areas can be made by using graphs from and creating graphs in 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 designing probability experiments based on unique ideas (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).”

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 (bar graphs and pictographs)

Grade 4 students collect data and represent it in bar graphs and pictographs using one-to-one and many-to-one correspondence.

#### Probability

Grade 4 students work with single-outcome probability experiments and record data with tally charts.

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

##### Overview

Learning how to interpret and represent data in graphs contributes to students’ critical thinking development through describing, comparing, and discussing data. Looking at different graphs for the same data set can help students critically examine how data can be manipulated in various ways to tell different stories.

Students come into grade 4 with experience 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 used concrete materials such as tiles or linking cubes to create bar graphs. They have created graphs on paper by colouring squares on grids or using stamps or other pictures for pictographs. It is likely they have been introduced to titles for graphs and labeling each axis, at least for graphs co-created as a whole class with the teacher.

In grade 4, students leverage their knowledge from primary years to interpret and create pictographs and single-bar graphs. They apply and reinforce their understanding of one-to-one correspondence and extend it to represent data using many-to-one correspondence in both pictographs and bar graphs.

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 4, students are working on many-to-one correspondence so each symbol represents two or more instances of the data. Because of this, it can be important to choose symbols that can be partially represented based on the data.  For example, using a circle to represent four instances means that half a circle would represent two instances and a quarter circle would represent one instance. Labels should be included and a legend should be provided when working with many-to-one correspondence. 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. In grade 4, it may be appropriate to introduce digital tools for creating graphs. Bars should be the same width, be equally spaced apart, and not touch each other, and each bar should have a category label. Axis headings and labels should be provided and the graph should have a clear title related to the context of the graph. In grade 4, students work with both one-to-one correspondence (each square represents one instance of the data) and many-to-one correspondence (each square represents two or more instances of the data). A scale should be indicated when working with many-to-one correspondence. In grade 4, students work with single-bar graphs and will apply their understanding to double-bar graphs in grade 5.

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 4 students collect simple data in scientific inquiries 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).”

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 4:

• Collecting simple data (e.g., with categories and tallies)
• Interpreting (describing and comparing) and creating pictographs with one-to-one correspondence with concrete materials and pictorially (on paper)
• Interpreting (describing and comparing) and creating simple single-bar graphs with concrete materials (e.g., linking cubes or tiles) and pictorially (on paper), including with grid paper
• Skip counting by 2s, 5s, and 10s (to support many-to-one correspondence)
• Multiplication concepts of 2, 5, and 10 (to support many-to-one correspondence)
##### Progression:

Interpreting (describing and comparing) pictographs and single-bar graphs with one-to-one correspondence (such as those encountered in grade 3 as review)

• Pictographs
• Interpreting (describing and comparing) horizontal and vertical pictographs with many-to-one correspondence
• Noting inclusion of a scale or legend, labels, and title
• Creating horizontal and vertical pictographs with one-to-one and many-to-one correspondence using paper/pencil and/or technology
• Inclusion of title, labels, and scale or legend as necessary
• Making appropriate choices for scale and between one-to-one and many-to-one correspondence
• Single-bar graphs
• Interpreting (describing and comparing) horizontal and vertical single-bar graphs with many-to-one correspondence
• Noting inclusion of title, axis labels, and scale
• Discussion of what data is best represented with bar graphs
• Discussion of instances when one-to-one correspondence or many-to-one correspondence is more useful
• Creating horizontal and vertical single-bar graphs with one-to-one and many-to-one correspondence using paper/pencil and/or technology
• Noting inclusion of title, axis labels, and scale
• Making appropriate choices for scale and between one-to-one and many-to-one correspondence

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

##### Overview

Grade 4 students engage with single-outcome probability experiments. They may be using dice, spinners, coins, two-sided counters, or cards, as some examples. They track outcomes using tally charts. Probability experiments can be connected to data by representing outcomes in single-bar graphs.

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 randomness and independence of events through their experiences with dice and coin flips, as two examples.

In primary grades, students have been introduced to probability experiences and the language used to describe them. 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. Students have been developing their understanding of chance when engaging in probability experiences using manipulatives such as coins, two-sided counters, and dice.

Grade 4 students will continue using the language developed in primary grades (impossible, uncertain, probably, never, always, more likely, less likely, 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).

Grade 4 students engage with probability experiments designed by the teacher and by themselves. They may contribute data to a whole class experiment or combine data with a partner in addition to collecting data for their own experiment.

Grade 4 students are expected to track events using tallies. This data can be translated to a visual representation using a single-bar graph, connecting concepts of data and probability.

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 probability experiments.

Understanding developed in grade 4 will be reinforced in grade 5. Additionally, in grade 5 students will represent probabilities for single-outcome events as fractions.

##### Math Foundations:

The following concepts and competencies are foundational in supporting understanding of probability in grade 4:

• Language of probability including describing using impossible, uncertain, never, likely, unlikely, always, possible and comparing using more likely, less likely, equally likely
• Understanding of chance (e.g., with coin toss or rolling dice)
• Engaging in probability experiences with a variety of concrete materials with different number of possible outcomes (e.g., coin tosses (2 outcomes), rolling dice (6 outcomes))
##### Progression:
• Discussions about where students encounter chance and probability in their own lives and review language of probability
• Engage in probability games and experiments and describe events and track outcomes in a tally chart
• Using descriptive language of impossible, uncertain, never, always, certain
• Using comparative language for two or more possible outcomes – more likely, less likely, equally likely
• Begin with fewer outcomes (e.g., coin flip, drawing a red or black card from a deck) and move to more outcomes (e.g., rolling a die or the sum of two dice, drawing a particular suit from a deck of cards)
• Design own probability experiment, track outcomes using tally chart
• (Optionally: connecting data and probability by representing results from a tally chart in a single bar graph)
##### Sample Week at a Glance

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

Because grade 4 students are generally building on experiences from grade 3, this week plan could represent a week in the first half of the school year once it is determined that the students have skills in skip-counting and with multiplication facts of 2, 5, and 10 (supports many-to-one correspondence).

Focus: review and interpretation of pictographs and day 1 of creating pictographs

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/2023/02/14/heartbeats-per-minute-and-body-weight-of-different-species/ (ideally one with many-to-one correspondence). Include focus on the legend and how that 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 create a chart for collecting data.

If time allows, students may start collecting survey data.  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.

Focus: interpreting pictographs and day 2 of creating pictographs

This class will need a longer session or can be extended over two days.

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/2019/11/25/dinosaur-fossil-finds-by-continent/ (ideally one with many-to-one correspondence). Include focus on the legend and how that helps understand the information in the graph. Use this graph to review the components of a pictograph.

During: Second day of creating pictographs. Teacher demonstrates setting up title and labels for the pictograph from his/her/their data collected in the previous class. Review the choice of symbol and what it represents from the “After” discussion last class. Begin to enter symbols for one or more categories.

Students continue data collection, as necessary. Once data collection is complete, students work on creating their pictographs. Teacher checks in with individuals/pairs regarding how many data points each symbol will represent.

After: Gallery Walk.  Display the students’ and teacher’s pictographs on vertical surfaces (walls, windows, boards, doors).  Ask students to visit each pictograph and think about what they notice and wonder about each graph and the graphs in comparison to each other.  Gather students together to discuss the “notices” and “wonders”. Ask students to think about the data in the chart and the data in the graphs. Ask, “How do pictographs help us read data?” and discuss.

Focus: 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/2022/03/02/colors-per-national-flag/ (ideally one with many-to-one correspondence). Include focus on the scale and how that helps understand the information in the graph. Use this graph to teach the components of a bar-graph.

During: Teacher uses the survey data they collected on Monday to demonstrate the construction of a bar graph. Provide students with grid paper. Students work in pairs (or individually if that is how they chose to work on Monday) to display their previously collected data (for the pictograph activity)

After: Math Journal or Exit Ticket. Students answer the questions: Which graph do you prefer for your survey, the pictograph or the bar-graph? Why? Alternatively, use these questions for a class discussion.

Focus: language of probability, collecting data, (co-created) single-bar graph

Before: Lead a discussion about a probability situation, preferably with concrete materials.  For example, have a bag with several counters of different colours. Let the students know how many of each colour are in the bag (e.g., 3 red, 4 blue, 8 green) and ask a variety of questions to elicit responses with language of probability (e.g., “How likely is it that I pull a green?” [likely] “How likely is it that I pull a purple?” [impossible] “How likely is it that I pull a green, blue, or red?” [certain]). Include questions that invite comparison (e.g., “How likely is it to pull a green than a red?” [more likely])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: Using the Bone Game sticks (Blackfoot traditional game – p. 7 in https://intranet.csf.bc.ca/wp-content/uploads/sites/2/2019/12/ressources/EA_indigenous-games-for-children-en.pdf ) students work in pairs and  track the points thrown (0 – 11) over 10-20 throws (if students have not played this game before, you may want to have a session where students make the ‘bones’ and play the game). They document the data in a tally chart. Combine data into a class data chart. Have students work in pairs to make a single-bar graph of the data or create a bar-graph as a class. Use the graph to anchor a discussion about the likelihood of point values (e.g., “How likely is it to throw 7 points?” “How likely is it to throw 0 points?” “How likely is it to throw 12 points?” “How likely is it to throw a 0 points compared to 4 points?”). Discuss how the bar-graph helps illustrate the likelihood of different point totals.

After: Have students generate 3-5 statements about the likelihood of points, asking for at least one statement to be comparative. Collect these statements for formative assessment. Add another lesson on the language if formative assessment indicates this need.

Focus: gathering data in probability experiments and graphing it in single-bar graph

Before: Class discussion.  Before class, create a spinner using the NCTM adjustable spinner https://www.nctm.org/adjustablespinner/ – make some colours a larger piece than others. Choose the pie-chart option (near the bottom). Discuss which colours are more likely and which are less likely to be spun.  Spin 10 times and discuss results.  Spin 10 more times and discuss results, including how more spins (more data) the more confident we can be about the probability.

During:  Workshop/Stations. Create 4-6 probability experiments. For example, predicting and exploring the probability of different sums when rolling 2 dice, pulling different colour counters out of a bag, spinning numbers or colours on a given spinner (may have more than one spinner option), tossing a cup (landing on its bottom, top, side), flipping a coin, rolling a single die (6-sided or 10-sided). Do a quick rotation through the stations (2 minutes per station) to give students an understanding of each experiment. Students then choose a station to revisit. Students make predictions about the outcomes of the experiment using language of probability (e.g., make 3 statements about their predictions). They collect data in a tally chart related to the outcomes of the experiment. Finally, they represent the data in a single-bar graph.

After: Gallery Walk and discussion Post the single-bar graphs around the room. Students visit each graph. Lead a class discussion about what they noticed about the predictions made and the data collected. Reinforce that if the predictions seem reasonable and the data don’t support the predictions, collecting more data might give results closer to predictions.

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. Similarly, continue to provide opportunities to use language related to probability throughout the year.

##### 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 4:

• Develop a survey question with discrete choices
• Record survey data using charts or tables (e.g., tally chart)
• Can interpret pictographs with many-to-one correspondence; can identify the category with the most/least instances and can use the data to make comparisons
• Can interpret single-bar graphs with many-to-one correspondence; can identify the category with the most/least instances and can use the data to make comparisons
• Create a pictograph (many-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 (many-to-one correspondence) with a meaningful title, axis headings, and labels; the data in the graph matches collected data in charts or tables

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 4:

• Accurately use language of probability to describe and compare different outcomes related to simulated events with classroom materials (dice, spinners, etc.) in mathematical discussions and individual work
• Can make and explain/defend predictions related to probability experiments
• Can accurately collect data in tally charts during probability experiments

Websites and Digital Documents

Our World in Data https://ourworldindata.org/

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/

Creating a Probability Game – BC Curriculum resource  https://curriculum.gov.bc.ca/sites/curriculum.gov.bc.ca/files/contributed-resources/Creating%20a%20Probability%20Game.pdf

Indigenous Games for Children (HIGH FIVE Parks and Recreation Ontario) https://intranet.csf.bc.ca/wp-content/uploads/sites/2/2019/12/ressources/EA_indigenous-games-for-children-en.pdf

Books

Animals by the Numbers by Steve Jenkins

Dinosaurs by the Numbers by Steve Jenkins

Me and the World: An Infographic Exploration by Mereia Trius

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