### 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 2

#### Data and Probability

In grade 2 students continue to deepen their foundational understanding of the language and representations used to describe and record one to one data and probability.  In the previous grades, students learn to build, describe and compare concrete graphs and to describe probability using always, sometimes, never, more and less likely.  In grade 2, students add a level of abstraction to their representation ability, by learning how to translate their concrete graphs into pictorial representations.  They might colour in squares on grid paper, stamp dots or draw individual symbols/pictures to represent data they have collected.  They practice methods of recording data in organized ways, such as by using tallies and they add to their vocabulary of comparative probability language to use more nuanced terms like certain, uncertain and likelihood, including “equally likely.”  At this stage, students are building an important foundation for more abstract understandings of data representation, such as using bar graphs and tables, which will happen in grade 3.  In grade 2 students continue to work with familiar life events, so that they can focus on building connections and meaning before moving on to simulated events in subsequent grades.  It is important that they get the chance to collect and represent data for a variety of things involved in their daily routines and environments.  Common activities include surveys of favourite things (colours, foods, animals, etc.) and tracking events (daily weather, days in school, plant growth, etc.).

### Key Concepts

#### Data: pictorial representation of concrete graphs

Students gather data, build concrete graphs, using one-to-one correspondence, and create pictorial graphs using grids, drawings, stamps, etc.

#### Probability: likelihood of familiar life events

Students continue to expand their understanding and use of the comparative language of probability

#### Key Data and Probability Concept 1: Pictorial representation of concrete graphs, using one to one correspondence.

##### Overview

At this stage of development, students have had 2 years of experience with discussing methods and reasons for collecting and organizing data using tallies, concrete materials and pictographs with one to one correspondence.  In grade 2 they are continuing to practice these methods with increasing independence.  For example, rather than collecting data as a group within their classroom community, students may work with partners to survey the school community on various topics they are interested in and then construct their own representations of the data they gather.  They are reinforcing basic concepts of graphing, such as labeling and choosing appropriate tools (tally, grid, symbols) to represent “how many” are reflected in each of their topics or categories.  They continue to deepen their understanding of how to keep data organized and how the various forms of representation are connected.

It is important that students’ explorations of data are integrated into their daily experiences in order to create the meaning that will lead to strong foundational understanding.  Students at this age usually love to survey people within their school and home communities on topics of interest that they can generate individually or in groups.  They can make choices about how to organize their collection sheets and how their pictographs will be constructed.  Look for opportunities to integrate data collection and representation within other subjects like science, social studies and ADST.  Students can gather data over time using logs (minutes of exercise, growth of plants in school gardens, types of weather) and use representations to compare data (height in centimeters, sunny vs rainy days).  To deeply develop the meaning of why data is collected and the different ways in which it can be represented, it is important that students have ample opportunities to discuss their data and use what they discover to make decisions within their school and local communities.  Questions that begin with how can data help us understand… can be useful for launching inquiries that spark meaningful learning.

##### Data and Probability Foundations:

The following concepts and competencies are foundational in supporting understanding of data collection and graphing in Grade 2 and are the focus of learning in Kindergarten and Grade 1:

• Sorting using a single attribute
• Counting with 1 to 1 correspondence
• Co-creation of concrete/pictorial data
• Placing names into categories (present/absent, favourites)
• Tracking daily weather
• Parts of graphs (title, labels, 1 to 1 representations)
• Understanding the purpose of organizing and representing data
• Keep track of things
• Make predictions
• Compare
##### Progression:
• Sort concrete data/items into categories, using more than one attribute simultaneously, to organize and create concrete representations.
• Organize and keep track of one to one data using pictorial representations (pictographs, tallies).
• Develop clear survey questions in order to gather data to answer specific questions. At this stage students are refining their ability to ask questions that are not leading and that obtain the required information to answer their question.
• Connect various representations of the same data to each other and discuss how each is useful and supports understanding. Make choices about how their data should be represented.
• Analyze concrete and pictorial data graphs to make comparisons and answer questions.

#### Key Data and Probability Concept 2: Likelihood of familiar life events.

##### Overview

In grade 2 students continue to work on the development of the language of probability that they have been building since grade 1.  They continue to use previously learned terminology, like always, sometimes, never, to discuss the likelihood of familiar life events.  They expand their probability vocabulary to include certain and uncertain, as they develop more nuanced understandings of chance and factors that affect the likelihood of something happening.  Students at this stage of development should be demonstrating a strong understanding of the connection between data and the language of probability when they are drawing conclusions and using evidence to back up their use of the terminology.  They understand that different things affect the quality of their data and are becoming more clear and precise with both their creation of questions and the use of probability language to discuss their conclusions.

It is important to note that vocabulary development is best achieved through inclusion in the learning, rather than through spelling tests or other rote memorization practices.  Teachers should encourage vocabulary use by using the appropriate terms both during math lessons and in other subjects.  This is a great time to look at culturally inclusive ways of exploring the language of probability, such as learning probability words in local Indigenous languages and exploring ways that other cultures determine probability.  For example, many BC Indigenous cultures use moon calendars and other natural cycles to predict and order events throughout the year.  This ties in to other parts of the grade 2 curriculum, such as patterns (math), life cycles (science) and goal-setting (career/health).

##### Data and Probability Foundations:
• Uses the terminology of always, sometimes, never, likely and unlikely to discuss concrete data and the probability of familiar events.
• Discusses (compares) the likelihood of familiar events
##### Progression:
• Discusses and compares the likelihood of familiar events using the language of probability learned in previous grades.
• Includes the language of certainty/uncertainty in probability discussions, as well as identifying that some events are equally likely/certain
• Increasingly uses more nuanced and specific vocabulary, supported by data, to describe the likelihood that something will happen.
##### Sample Week at a Glance

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

These are sample lessons that are adaptable and repeatable with different topics throughout the year.  Students should have lots of practice collecting, representing and discussing concrete/pictorial data in various authentic contexts throughout the year.  The topics can be determined by student/class interest or cross-curricular connection.  While each of these activities is presented as a single lesson plan, more than one block may be required depending on the class and the amount of time dedicated to the topic per day.

Science Connection: water conservation

Math Focus: Pictographs (creating and interpreting); data collection

Resource:

1. Follow the discussion questions in the slide deck as the graph is revealed.
2. Discuss: What other categories of water use could we add?
3. Co-create a template for tracking their water use at home and/or at school. How will we organize our data (categories)? How will we keep track of the data (tallies, log-book)? How long will we keep track of the data?

Students will begin keeping track of their water usage.  Further conversations about water use and conservation can take place during science learning times.

Consolidation/Journal: What are some important things to think about (focus on) when we are using graphs? How do I create an organized way to collect data over time?

Math Focus: Organizing and representing data

Read Aloud: The Great Graph Contest by Loreen Leedy

Note: before reading choose 3-4 pause points in the book to stop and ask questions about the choices the animals make about the type of graph they use to represent their data.  For example:

• What information do we know from looking at this graph? What doesn’t it show?  Are there other ways to show the same information? If I wanted to know ____, would this graph be useful?

Groupwork: Put students into groups of 2-3.  Tell them they are going to have their own Great Graph Contest.  They will pick a topic and ask a question about it.  They should then collect data and decide how to represent it pictorially in a way that will allow them to answer the question. Give them time to survey people and create their graphs.

Present: Groups will give a short presentation on the decisions they made and why those choices make sense.  They should include use of probability language and any changes they would make in their presentation. They can also include some likelihood and certainty questions  (2-3) that they can pose about their data that can be answered from their graph.

Math Connection: Patterns

Math Focus: Probability language: likelihood/certainty

Routine (Choral Counting):

• Skip count by 5 starting at 0 (2 columns). Stop after 30 and ask: What patterns do you notice? What comes next? What is the likelihood that 100 will appear in this count?  Are there numbers that you know for certain will appear? How do you know?
• Repeat the count by 5 starting at 1. Ask the same questions and add: How do patterns in data help us make predictions?

Encourage the use of specific vocabulary by rephasing, asking other students to explain what they heard another student say or if there are specific “math words” to say the same thing that a student said.  Avoid stopping students in the middle of their explanations to correct terminology.

Groupwork: In groups of 3 ask students to explore ways they might represent the patterns in skip counting.  Look for logical representations: 0,5,0,5 or building each number in steps 5, 10, 15…. Encourage students to represent the data 2-3 different ways (number line, pictograph, visual pattern).  Circulate as students work to ask questions and support probability language use. Select 2-3 groups to share what they did in consolidation.

Consolidation: Gather students around the examples.  Ask questions such as: What can you tell is certain/uncertain? How do you know?

Math Focus: First Peoples worldviews and likelihood of familiar events

Note: Like many of the activities in this sample week, this activity can be expanded to more than one block of time or expanded into a year-long project. Calendar projects can be connected to many areas other of the mathematics curriculum, such as patterns and measurement, as well as other curricular areas, such as life cycles (science). As it is important to make our best efforts to connect to the Nations in whose territory we teach and avoid a pan-indigenous lens, effort should be made to learn about the moon calendars and activities associated with your local nations.  While many Indigenous peoples throughout the world use moon calendars, there is variation in the organization, language and activities associated with those calendars.  If possible, connect with and invite to your classroom a knowledge-keeper from your local Nation to speak about the traditions and protocols surrounding the calendar. If this option is not available, using a moon calendar from another part of BC is acceptable, as long as we are clear with students that not all Indigenous cultures are the same.

Moon Calendar Resources:

Notice/Wonder: Show students a picture of a moon calendar. Ask them what they notice and record their ideas on the board.  Ask students what they wonder about the calendar and record their thinking. Highlight any probability language in a different colour that students use when sharing their notice/wonders.

Explore:

• Explain to students that many Indigenous cultures used the moon and environmental factors, such as the salmon run to indicate seasons and predict when it was the best time to do certain activities. This would be an excellent time to invite a guest from one of your local Nations to speak to the class (remember to offer payment and to find out any protocols around making this request before doing so).
• Allow students to work in groups of 3 to compare and contrast our commonly used Gregorian calendar with the moon calendar. You may wish to provide a Venn Diagram template or some other graphic organizer appropriate to this task.

Consolidate: Ask students to think about how probability is connected to the use of calendars.  Consider the following questions:

• When is it most likely that it will be a good time to go fishing?
• What can you tell for certain from looking at the Moon calendar?
• In which months will we probably not be able to harvest food from plants?
• How might the probability of having different foods available have changed in modern times?

Extensions:

• Use the moon calendar to predict what you might see on a nature walk?
• What is the probability that we will see blossoms?
• What is the probability that we will need to dress for mud?

Have students create their own calendars using activities that they do at certain points in the year.

Math Focus:  Creating pictographs from collected data

Note: Depending on the length of time that was decided upon for tracking water consumption, the timing of this lesson can be adjusted.

Resource: Data that students have collected about their water consumption using their logs.

Discuss (accessing background knowledge): What are some things that we need to include when we are creating our pictograph(title, repeated symbols/pictures, straight lines, numbers, etc?

• You may wish to quickly revisit the slow reveal graph from Monday

Create: Put students into groups of 2 or 3.  Explain that they will use their groups to talk about the creation of their graphs, but that they will graph their own data. Alternatively, groups could choose one set of data to graph.  Have water themed stamps/stickers available for students to use to create their pictographs.  Circulate to ask questions about the choices and strategies students are making as they create their graphs.  Look for misconceptions in their graphs: 1 to 1 correspondence of numbers and symbols, unclear organization, etc. as you circulate select 2-4 graphs to share in the interpret section (discuss with the creators if they would like their work shared anonymously). Ensure that you choose a selection that highlights important things you wish to surface for consolidation.

Interpret: Gather students around (or project) the samples you have chosen one at a time.  Ask questions such as:

1. What can you tell about water usage from this graph?
2. What questions do you still have after looking at the graph?
3. What did the creator(s) of this graph do differently from what your group did?

Encourage students to use specific data and probability language and examples in their explanations.

Experiences such as the ones in this sample plan should be repeated using different contexts throughout the year in order to give students ample practice and time to deeply develop their understanding of the concrete/pictorial connection in various representations with one to one correspondence.  There is no shortage of opportunities to incorporate discussions of probability into daily activities.  Ask students about the probability of winning when playing games and the probability of rolling a certain number or landing on a certain place.  Discuss the probability of needing particular clothing or footwear based on weather reports and before field trips.  Highlight the language of probability when exploring patterns and cycles in all subjects.  Make predictions at various points when reading stories. Always ask students to provide evidence to support their predictions.  A solid foundation with this concept will allow students to more easily build connections to the increasingly abstract representations, language and analysis required in later grades.

##### Suggestions for Assessment

What to look for:

• Increasing refinement of probability language in discussions both related and unrelated to mathematics lessons.
• Ability to use estimation and other strategies to make accurate predictions
• Connection of evidence to predictions
• Using one object (blocks) to represent another (number of children who like different flavours of ice cream) to create concrete graphs
• Gathering data to answer their questions
• Accurate reflection of data using tallies, charts and pictographs

By the end of this grade students will be able to gather their own data related to questions that they are interested in, graph that data using pictographs and use the data to answer questions and make comparisons. They will be able to explain how they use data to make predictions and back up their reasoning and can use their graphic representations, as well as descriptive vocabulary.