### 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.

## Data and Probability

In Grade 6, students continue to explore how graphs display data. In earlier grades, students create and interpret concrete graphs, picture graphs, and bar graphs. In Grade 6, students transition to line graphs. They analyze these graphs in order to identify patterns and relationships between two variables.

Students formalize their use of the terms theoretical and experimental probability in grade 6.  After exploring a variety of probability experiments, mostly through play, in grade 5, students are now expected to learn how to calculate and list all of the theoretical outcomes of a given experiment and compare that to the outcome (experimental probability).  Students will explore two independent events experimentally in Grade 7.

### Key Concepts

#### Line graphs

Students interpret and create line graphs to display, as connected points, data that involves two variables. They look for and describe patterns and relationships between two variables that are communicated by graphs.

#### Single-outcome probability, both theoretical and experimental

Students determine all possible outcomes of a situation to determine the theoretical probability and then compare those projections to the experimental outcome.

#### Key Data and Probability Concept 1: Line Graphs

##### Overview

Students differentiate between situations that involve categorical data (e.g., favourite book, number of siblings) and situations that involve two variables (e.g., distance and time, length and width). Students learn that it is appropriate to use line graphs for the latter. They learn to plot data as connected points. As students analyze and create line graphs, they attend to characteristics of graphs such as labels and scales. Scale is directly related to students’ study of many-to-one correspondence within bar graphs in Grades 4 and 5. These experiences help students answer the essential question, “What story does this graph tell?”

##### Math Foundations:
• interpret and create bar graphs and double bar graphs (categorical data)
• understand many-to-one correspondence (e.g., one square may represent several items)
##### Progression:
• interpret and plot individual data points on the coordinate plane
• analyze and interpret line graphs
• decide when it is appropriate to display data using line graphs
• create line graphs, attending to important characteristics such as labels and scales
##### Sample Week at a Glance:

This sequence of lessons shows the formal introduction to this concept within Grade 7. Students may have revisited picture graphs and bar graphs prior to this week. This is illustrated briefly in Monday’s warm-up.

Warm-Up. Display the following graph:

https://slowrevealgraphs.com/2022/04/22/how-loud-is-too-loud-english-espanol/

Ask “What do you notice? What do you wonder?” Possible responses include:

• each vertical ‘stack’ shows the noise level of different sounds
• the categories of sounds are arranged in increasing order of noise
• each rectangle within a bar represents 10 decibels
• one-half of a rectangle represents 5 decibels
• a pink region shows sounds that can damage hearing
• three of seven sounds can damage hearing

Note that the data in this picture graph is categorical. Later in the week, call back to this particular graph. Discuss that it is not appropriate to use a line graph to represent this type of data since the horizontal categories of raindrops, conversation, traffic, etc. are not numerical.

Lesson. In pairs, have students complete the Desmos activity Robots: What a Point in a Scatterplot Means. Please refer to the Teacher Moves and Sample Responses tabs for Desmos’ three-part lesson plan.

Before. Activate prior knowledge developed through yesterday’s activity by displaying the following scatterplot:

Ask “What do you notice? What do you wonder?” Possible responses include:

• data for four brands of cereal is shown
• each point tells the serving size and amount of sugar
• Green has the most sugar; Red has the least
• Red has the greatest serving size; Black has the least
• the horizontal scale: one space represents 10 g of cereal
• the vertical scale: one space represents 2 g of sugar

During. Display the slides for the line graph Materials Recycled in Pounds per Person per Day (US) from Slow Reveal Graphs.

See the Instructional Routines page to learn more about Slow Reveal Graphs, including teacher moves.

After. Have students record their individual responses to the following questions:

• What is being compared in the graph? What does the horizontal axis represent? What does the vertical axis represent?
• What observations did you make? Are there sudden spikes?
• What trend do you see? What predictions can you make?
• What’s going on in this graph? What story does this graph tell?

Discuss student responses. If time permits, repeat the Slow Reveal Graphs routine using a different line graph. The graph The Arctic Ice Cover is Receding maintains the theme of environmentalism.

Note differences. In this second graph, the vertical axis is truncated (i.e., starts at 5.0, not 0.0); individual data points are shown, not implied and “filled in” within decades; the overall trend is downward, not upward; etc.

Repeat the Slow Reveal Graphs routine. This time, students will explore both a bar graph and a line graph. Display the slides for the bar graph Boxed Mac & Cheese Sales in the US.

When you consolidate, highlight that the horizontal axis shows categorical data (i.e., mac ‘n’ cheese brands). Each bar represents mac ‘n’ cheese sales for a single brand for a single year.

Display the Slow Reveal Graphs slides for the line graph Domino’s Pizza Has Quietly Become the Biggest Pizza Chain in the World.

Discuss similarities (e.g., theme of food; popularity, whether measured in sales or number of restaurants) and differences. Highlight that the horizontal axis shows not categorical data but a second value, namely year. Categories (i.e., pizza chain) are shown not using bars but three line graphs (i.e., Papa John’s [green], Domino’s Pizza [blue], and Pizza Hut [red]). Explain that it would be inappropriate to use a line graph to display the mac ‘n’ cheese data; a line connecting “Kraft Macaroni and Cheese” and “Velveeta” would not make sense. It would be possible to use a bar graph to display the pizza data but only for one particular year.

Place students in groups of three (“home groups”). Within each group, assign students the number 1, 2, or 3 (“expert groups”). Each number will correspond with a different graph. Have students form new groups of three with others who were assigned the same number.

Browse line graphs on Slow Reveal Graphs. Select three for your students to analyze and interpret. Assign your chosen graphs the number 1, 2, or 3. Provide expert groups of 1s with graph 1, expert groups of 2s with graph 2, and expert groups of 3s with graph 3. This time, provide students with “fully revealed” graphs (i.e., the last graph within each slide deck).

Turner’s Graph of the Week is another source to find graphs for your students. For example, Smartphone Ownership Among Youth, College Graduation Stats, and Youth Sports Injuries all include line graphs.

On Thursday, have students, in their “expert groups,” analyze and interpret their assigned graph. Invite students to answer “What’s going on in this graph?” and “What story does this graph tell?” Encourage them to apply what they learned from analyzing and interpreting graphs through the Slow Reveal Graph routine. Point them toward important characteristics such as labels and scales, if needed. Ask other prompting questions (e.g., “What observations did you make?” “What trends do you see?”). Tell students that they will return to their “home groups” on Friday; they will be expected to be “experts” on their assigned graph.

On Friday, have students, in their “home groups,” take turns telling the story of and answering questions about their assigned graph. If time permits, display a fourth line graph for students to analyze and interpret together.

Throughout the school year, look for interdisciplinary opportunities for students to revisit line graphs in other areas of learning.

Next, students will decide which type of graph–picture graph, bar graph, or line graph–is appropriate to display data. Students will discuss situations in which line graphs can be used (e.g., hours spent listening to music/playing a sport over time; number of followers on social media over time) and those in which they cannot (e.g., favourite genre of music/favourite sport; hours spent on social media by day of the week).

Finally, students will create their own line graphs, first from data that is provided to them and then from data that they collect. They use graphing conventions in order to communicate effectively.

##### Suggestions for Assessment

By the end of Grade 6, students should be able to interpret line graphs. This includes explaining the meaning, in context, of individual points; noticing trends; and making predictions. Students should also be able to accurately create line graphs, where appropriate, to display data. This includes making decisions about variables, labels, scales, and titles.

#### Key Data and Probability Concept 2: Single-Outcome Probability

##### Overview

Students formalize their use of the terms theoretical and experimental probability in grade 6.  After exploring a variety of probability experiments, mostly through play in grade 5, students are now expected to learn how to calculate and list all of the theoretical outcomes of a given experiment and compare that to the outcome (experimental probability).

##### Math Foundations:
• Predicting single outcomes using spinners, dice, coloured blocks
• Expressing single probabilities as fractions
• Performs probability experiments
##### Progression:
• Understands that there are both theoretical and experimental probabilities.
• Lists all the theoretical probabilities of a single outcome event using spinners, dice, etc.
• Compares the theoretical probabilities to the experimental outcomes.
• Analyzes factors that affect the outcomes of probability.
##### Sample Week at a Glance:

The week outlined here provides a sequence of lessons that explicitly highlight the progression above.  It is not meant to suggest that it requires 5 lessons for students to learn about theoretical and experimental probability, nor that the exploration of such should be confined to one week out of the year.  Probability can be integrated throughout the year, particularly in science and ADST where the experimental process is used, as well as any time games are played that involve predictable outcomes.  Teachers may wish to introduce the concept of the 2 probability types early in the year and revisit their use and relationship throughout the year.

This activity both accesses background knowledge from Grade 5 probability and allows for the introduction of the new components in the grade 6 curriculum.

Part 1:

• Present the story of Kaia’s father’s ties from It’s a tie (NRich) to the class
• Ask: Why does Kaia’s father think it is impossible for him to have worn one tie more than once in a week? What does Kaia realize, that he does not? (Since he returns the tie to the collection each evening, the probability of drawing out that tie is the same the next day.)

Part 2:

• Have students work in groups of 3 to complete the task
• For early finishers, you might ask them to consider how Kaia’s father could develop a system so that he wears each tie equally and doesn’t repeat during the week.

Part 3:

• Introduce the terms experimental and theoretical probability.
• Ask: What is the theoretical probability that each tie will be drawn? How might Kaia explain to her father why he could potentially wear one tie multiple times per week? How do the theoretical probabilities for each tie compare to your experimental results? If Kaia’s father wanted to make sure that he didn’t repeat wearing a tie each week, what could he change about his system?

Note: NRich has many tasks that can be used to explore the relationship between theoretical and experimental probability (as well as other areas of the math curriculum), such as this one Winning the Lottery

Ace In the Hole: This lesson can be done with students using Mathigon’s Polypad if you have enough devices (1 device/2-3 students) or with physical playing cards.

Note: The provided lesson plan goes beyond the expectations for grade 6.  If time permits and students are engaged, they may enjoy the challenge of the later parts of the lesson. In this case, assess only the students ability to list the theoretical probabilities and compare them to their experimental outcomes.

Have students work in pairs or groups of 3.

Opening (Whole class Number Talk):

• Display the image of the 16 card decks in the Mathigon lesson plan
• Ask: What is the theoretical probability of drawing an ace?
• After some think time, accept any answers offered. It is likely that at least one student will say 1/16.  If this is the only answer offered or if students offer only incorrect answers, ask them to consider the meaning of the 1 and the 16.  If necessary prompt them to think about the number of aces available and where that is reflected in the fraction.
• If both 1/16 and 4/16 are offered, ask students to describe the strategies for achieving those answers. Through this facilitated discussion, students should come to agreement that the probability is actually 4/16 or ¼

Investigation (groups of 2-3):

• Give students time to conduct the experiment and compare their outcomes to the theoretical.
• If time permits: pair groups to discuss their results.
• Have students calculate the theoretical probability for the ace in the second experiment where card 1 is drawn before the probability is calculated. To stay within grade 6 parameters they may look at the card, so they only have to calculate the single known probability.
• Have them conduct the experiment and compare results
• Circulate as students work to assess their understandings. Make note of groups or individuals sharing strategies that would be helpful in the consolidation.

Consolidation (whole class):

• Ask: What did we learn in this investigation about calculating theoretical probabilities?
• Sequence the students/groups that you have chosen to share their thinking/strategies. Some you might choose include:
• A student who initially did not know how to determine probability and learned how from their group.
• A student who initially calculated the probability as 1/16 and now understands why it is actually 4/16 or 3/16
• A group who used a common strategy for listing the theoreticals
• A group who used a unique, efficient strategy for creating the list.

Learn to play Slahal: Although this activity is listed as one day, it will require more than one math block to do it justice.  If the teacher chooses not to include the cross curricular learning, then using a non-indigenous game, such as rock, paper, scissors is a more appropriate choice.  The game and questions can be revisited throughout the unit/year to provide opportunities for deeper learning.

Slahal (Lahal/Stick Game/Bone Game) is a game played by Indigenous peoples throughout Turtle Island (North America) for thousands of years. How the game is played varies slightly by region.  As with all Indigenous cultural connections it is important to include some learning around the game before playing it.  This can be done as part of the Social Studies and Language Arts curriculums using the following resources:

Once students have learned about the game and had an opportunity to play, students can be encouraged to explore and discuss the probability inherent in Slahal.  The questions below can guide discussions and activities for many games where chance is a factor.  If students are interested in this topic, teachers might deepen this learning through a project on probability and games.  There has been some interesting research conducted using games like Rock, Paper, Scissors that shows that, although we perceive these as games of chance, there are strategies that make experienced players better than new players.  Exploring these factors in conjunction with learning to calculate theoretical probabilities can deepen students’ comparisons of theoretical and experimental outcomes.

Questions to ask (These can be used to guide discussions over time as students become more proficient with the game and/or with a variety of games:

• What Math lives here? (probability, computation, various mathematical competencies)
• In what ways might an understanding of math affect game play? (knowing the theoretical probabilities can influence the choices we make)
• Are there strategies that increase your possibility of winning? (answers will vary based on student’s experience with playing the game)
• How might you calculate the theoretical probability of winning? (Rather than giving students step-by-step instructions or a formula, allow them first to think about, discuss and explore ways of finding the theoretical probability. Only after students have explored their own ways of doing this, should they be introduced to the formal procedure.)
• What factors, other than mathematical probability, affect the actual outcome of the game? (answers will vary based on student’s experience with playing the game)
• What is the significance of Slahal in the Indigenous cultures that play it? Are there comparable games in your culture? (This information is available in the links above)
• How might the skills developed playing Slahal and/or other games be applicable to other areas of life? (answers will vary based on student’s experience with playing the game)
• What happens when we adopt game play as a method of conflict resolution in our classroom/school community? (This is an interesting thought experiment that could be tested if the class wishes.)

3-Act Tasks are a great way of encouraging students to build their problem-solving strategies, as they have to think about the information that they require to solve the problem, as well as the strategy for solving.

Option 1: This Tim Horton’s Roll Up The Rim To Win task is fairly straightforward in process and students will be calculating probabilities using numbers in the hundreds of thousands.  Choose this option if students are still developing their understanding of how to calculate probabilities.

Option 2: This Darius Washington – Free Throws For The Win task is more complex and requires some modification to keep the probability calculations within the capabilities of grade 6 students.  Choose this task if the class is already fairly proficient at calculating probabilities and you have some avid sports/basketball fans in the group.  Some modifications:

• Help students convert the 72% probability to a fraction if percents have not yet been introduced.

Do not expect students to calculate the double probability of Darius’s 72% free throw rate and his ⅔ shot average.  Instead, discuss these 2 probabilities in Act 1 along with other factors that might affect the probability (stress, end of game/season fatigue,etc.) and have groups settle on a probability that makes sense to them.  They should then use this percentage/fraction to calculate the theoretical probabilities at each stage.

Revisit Slahal (or Rock, Paper, Scissors), do the 3-Act task that you did not choose for Thursday’s lesson or set up some probability stations using games and tasks from NRich and/or Mathigon.

While students are engaged in the activities, teachers can observe for evidence of learning or pull small groups for extra support.

Teachers might decide to begin with the Monday/Tuesday lessons early in the year and integrate the others at intervals or integrate other thematic activities that include comparisons of theoretical and experimental probabilities.

##### Suggestions for Assessment

Success Criteria (can be used throughout the unit/year by both the teacher and the students for self evaluation)

• Describes both theoretical and experimental probabilities.
• Lists all the theoretical probabilities of a single outcome event
• Compares the theoretical probabilities to the experimental outcomes.
• Analyzes factors that affect the outcomes of probability.

Observe students as they work in groups.  Note which areas of the success criteria they are able to do and which parts they are still working on.  Look for correlations of what students can do in your observations and discussions with students as they work, as well as any products (math journals, exit slips, calculations/rough work, formal answer write-ups) for activities where elements of probability are being discussed and implemented.  These opportunities may also occur in lessons involving probability outside of what are considered math lessons.