### 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 1 Data and Probability

The learning for data and probability in grade 1 is essential for solidifying a concrete foundation upon which to build more abstract representations and analysis.  As with kindergarten, grade s are using familiar life events to explore how the language of probability is used and how data can be represented concretely.  This step of representing one thing, such as the weather, as another, such as coloured squares or tally marks is the basis for understanding the representation of amounts and relationships in bar (grades 3-5)  and line (grade 6+)  graphs.  It is essential that grade 1s are given many opportunities to examine and concretely represent data in a variety of contexts, not only in math, but also in other subjects.  For example, if students are tracking weather data, it is important that students spend time, at various intervals, discussing the meaning of that data and making predictions and decisions based on the trends they find.  In this sense, the predictive nature of patterns can be highlighted during a patterning activity, how many times they can jump rope in a row in gym and deciding how many of each flavour of freezies are needed for Sports Day all become part of the learning.  It is therefore valuable to see data and probability less as a unit of study and more as tools that we use to practice investigating and reasoning throughout the year.

The grade 1 science curriculum lends itself well to studying patterns in data.  A year long study of the trends in local weather patterns, for example, can be connected to the seasonal rounds of the local Indigenous peoples, class gardens, clothing choices and other activities and decisions.  Allowing students the space to use what they are learning about data and probability in authentic contexts will help them see math, not as a school subject, but as a tool that can be used in their lives.  Grade 1s need to grasp the idea that these are not separate things, but that the ability to determine probability is a function of their ability to identify patterns in data.  Likewise, data can be used to make increasingly specific predictions.  This understanding paves the way for the use of more comparative probability language, as well as pictorial representations of concrete data in grade 2.

### Key Concepts

#### Data

Students create, describe and compare concrete graphs using one to one correspondence.

#### Probability

Students use an expanding vocabulary of probability language (always, sometimes, never, more/less likely) to predict the likelihood of familiar events and cycles.

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

##### Overview

In Grade 1, students are still working on a one-to-one correlation with data points.  This makes sense, as they are also still developing their number sense and counting abilities.  Routines such as Counting Collections, where students are expected to both count and record their counts accurately are good ways to practice both their number sense and their ability to track data.  At this age students are generally enthusiastic about opportunities to both collect and talk about different kinds of data.  They should be encouraged to gather data whenever possible, especially using tallies.  Snap cubes and grid paper are also important tools to introduce for keeping track of things, so that children begin to expand their toolkit of ways to manage and organize data. Questions such as, “What do you notice? What do you wonder? Are there any patterns? What does this data tell us?” help students make connections between their data and their decisions and actions.

##### Math Foundations:
• Sorting
• Counting
• Tracking: One to one correspondence

Strategies for organizing and tracking: categories, charts, tallies, pictographs (1 to 1)

##### Progression:
• Tracking using one to one correspondence: adding one tally or cube each time an event occurs
• Tracking multiple outcomes using 1 to one correspondence: types of weather
• Noticing trends: patterns in data they collect themselves or that has been collected by someone else
• Conduct surveys using tallies
• Data as evidence: Connecting data patterns to predictions
• Create and conduct survey questions for specific purposes
• Analyze their own data and data from other sources to solve problems and answer questions

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

##### Overview

It is important to recognize that probability in grade 1 is confined to the language used to predict familiar events.  The concept of experimental probability, that is working with simulations, is not introduced until grade 3 in the BC curriculum.  The key idea is the connection between prediction and evidence and the use of the familiar allows children to ground their understanding of this connection in repeated experience.  Allowing students the time to build a strong concrete understanding of the relationship between evidence and their predictions will set them up for success in later grades as they begin to deal with more abstract predictions and data.  While the difference between a guess and a prediction may seem obvious to us as adults, this understanding needs to be developed in young children.  Understanding the nature of probability when connected to data (evidence) also helps students understand the nature of estimation, if these concepts are explored concurrently. Students can be encouraged to define their level of faith in their estimation and predictions using the language of probability and appropriate evidence (data).  Repeated experiences with making predictions based on evidence and then examining the outcomes also helps students understand that even strong data, can lead to incorrect predictions.  For example, students may use the weather trends in their tracking, as well as knowledge of the seasons to predict if they will need umbrellas and rain boots on a field trip, but also realize that their prediction is not a guarantee.  As such, a deep understanding of probability, even at this early concrete level, is quite complex.

##### Math Foundations:
• Sorting: Events can be likely or unlikely
• Patterns: Help us predict
##### Progression:
• Sorts familiar events into likely and unlikely
• Uses personal experiences as evidence
• Organizes events by whether they are more or less likely
• Uses data they or others have gathered as evidence
• Makes a plan based on predictions
• Makes an evidence-based plan that includes contingencies for the uncertainties inherent in predictions. For example, decides not to wear a rain jacket and boots, but packs an umbrella.
##### Sample Week at a Glance

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

As described previously, Data and probability activities are best integrated throughout the year.  With this in mind, teachers may wish to do a short unit in September or October, reviewing/introducing concepts shared in Kindergarten and grade 1.  These concepts include an understanding of the terms likely and unlikely, the use of tally marks for keeping track of data and discussions of the features of concrete graphs.  Students should also have experience with counting and sorting, although these skills will continue to develop concurrently with the data and probability activities.  An important concept to develop early on is the difference between a guess and a prediction or estimation.

The activities in this week are in no particular order and can be repeated throughout the year, either with different topics or as part of routines, projects or extended studies.

Routine: Always, Sometimes, Never

• Present students with a statement
• Students categorize the statement as always, sometimes or never true
• Search for examples and non-examples

Note: The discussion component here is very important.  Students should be encouraged to listen for evidence from others that confirms their answer or causes them to shift their thinking.

Always, Sometimes, Never Resources : Tracy Zager; NRich

Learning Focus: Use evidence to justify categorizations involving  probability language (always, sometimes, never)

Warm Up (whole class):

Display a chart with the categories always, sometimes and never.

• Ask, “Where would we put the statement, ‘birds fly.’ (Choose a statement that students have sufficient background knowledge.)
• Once a student places the statement, ask them to justify their answer.
• Ask for agreement or disagreement and respective justifications until students agree which category to use.
• If students are in agreement that ‘birds fly’ belongs in the always category, ask if they can think of any birds that cannot fly (non-example). Explain that non-examples are important in checking our thinking. We might be able to think of lots of examples, but even one non-example can change our thinking.
• Students will likely think of penguins or ostriches and realize that the appropriate category is ‘sometimes,’ if the original student did not already do so.
• Ask students for an example of a ‘never’ statement and repeat the process.

Exploration (groups of 3):

*Note: The first few times this routine is used, it is important to use statements that are fairly simple for students to categorize.  When assessing for understanding of the terminology (probability grade 1) it is important that the teacher fully understand WHY a student is placing a statement in a particular category. This routine can also be used with mathematical statements to help students more deeply explore their understanding of the math itself. For examples of these types of statements, see the resources in the Always, Sometimes, Never section of the Routines page.

• Give students statements or pictures to sort into always, sometimes, never charts. For example:
• Animals have 4 legs
• Humans need food to live
• Remind them to discuss with their groups and think hard for non-examples.
• Circulate during this time to assess progress. Note specific groups using strategies or having conversations that would be helpful to share during the consolidation discussion.

Consolidate (whole class):

• Sequence the groups/students whose ideas you noted that highlight the learning focus. For example:
• A group clarified the meaning of one of the terms for one of their members
• A student changed their opinion after listening to a new piece of evidence.

A student explains how their group made a particularly difficult or contentious decision.

Learning Focus: Collecting, representing and interpreting data over time

This “day” indicates the time that the booklets, data station or project is launched.  The time period can be anywhere from a week to the year, depending on the project for which the data will be used.

The activity can be set up in 2 ways:

1. Independent Data Booklets (See example): Students complete daily as a transitional routine (morning or after recess/lunch)
2. Classroom Data Station: Create a space in the classroom for various data collection charts, graphs, etc (see the examples in the data booklet for ideas). Assign as classroom jobs to students to track the data.

*Data tracking activities have traditionally been included in primary classrooms as ‘calendar’ activities.  This is a great time for Data Talks about the data being collected.  However, avoid having the routine be the class watching one student colour in the data for the day, as this is something better done as an independent activity.

• Data Collection & Concrete graphs: gathering thematic data (water use, rainfall, weather, exercise, garbage collected from the playground, etc.) This data should be meaningful to students in that it will be used for data talks, projects and other classroom activities.

• Data discussions (What do you notice? What do you wonder?): patterns and relationships (weather+seasons, clothing, activities; Where could we conserve water?). These discussions should arise naturally from projects, student questions and teachable moments identified by the teacher.  Students should be encouraged to notice patterns in the data, draw conclusions based on the evidence in the data and take action in their projects and communities based on what they notice in the data.  Some examples:
• Students track weather data throughout the year. Once per month they analyze the data and look at the change of clothing and footwear that they keep at school to see if it is appropriate.
• Students collect litter on the playground once a week for a month. They track what they collect and, at the end of the month, discuss what they notice.  The class plans a campaign based on what they notice to reduce litter on the playground.
• Students learn about the seasonal rounds (see next lesson) and moon calendars of a local Indigenous Nation. They collect survey data on or track throughout the year activities that children in their school do during different months or seasons. Using their data and knowledge of how moon calendars work, students create their own calendars or rounds.

Over time, students should be encouraged to make suggestions for the kinds of data they are interested in tracking based on their inquiries in science, social studies, health or personal interests.

Learning Focus: Incorporate First Peoples worldviews and perspectives to make connections to mathematical concepts

“Seasonal rounds refers to a pattern of movement from one resource-gathering area to another in a cycle that is followed each year”

• BC Curriculum, Grade 1 Science

Seasonal Rounds Resources: My Seasonal Round; The 13 Moons of the Wsanec

Learning Focus: Interpret data and create 1 to 1 graphs

Data Talks are a short activity that can be used as a warm-up or sponge activity to help students become familiar with the many different forms that data can take in a low risk, non-evaluative environment.  See the instructions for a data talk at the top of the Youcubed Data Talks page.

Youcubed: water usage (Data Talks)

This water usage talk is interesting in that it is a pictograph in a different form than students might have been exposed to previously.  Images for data talks can be gathered from other sources and/or created based on student interest, class projects, etc. Reminder: For grade 1, the data needs to be represented 1 to 1 and in concrete or pictorial ways. To extend a Data Talk to a full lesson the following format might be useful:

Warm Up: Whole class Data Talk

• Discuss the visual data
• What do you notice?
• What do you wonder?
• When might this representation be useful?

Explore:

Learning Focus: Develop, demonstrate, and apply mathematical understanding through play, inquiry, and problem solving

Data and Probability Playground (stations):

Note: The stations listed here are examples.  Teachers are encouraged to set up stations featuring books and tools that fit the current project or theme that students are exploring. Students may be provided templates at each station or not, depending on their experience with data tracking.  All station activities should be introduced and practiced as a class before being set up for independent or partner use.  Once students are used to the station playground routine, this time is an excellent opportunity for observational assessment, student interviews, or small group support.

Animals By The Numbers by Steve Jenkins

• What do you notice about the information in the book? What do you wonder?

• Domino Sort Graph (Inspired by Dice Sums)
• Domino Sets: remove any dominoes that add to more than 20
• How might we sort the dominoes by how many dots they have?
• Create a graph using the dominoes.
• Which number has the most dominoes? Which number has the most equations represented? Which has the least?

• Block Tower (data collection)
• How might we keep our data organized?

• Pigeon Math by Asia Citro Story Graph
• Before this station is used, students should be familiar with the story through read aloud and discussion.
• How can we keep track of how many pigeons are on the line after each page?
• Create a graph to show the story of how the pigeons move.

Probability Clothesline:

• Have students place pictures and/or statements on a clothesline based on the likelihood that they will happen (today/this month/ever). In this case one end of the clothesline is likely and the other is unlikely.

This activity can also be done as a category sort using always, sometimes, never

As explained previously, this learning is not strictly linear and should be embedded throughout the year.  As student understandings and skills deepen and progress, students should be given increasing control over the creation of their questions, organizational and collection systems, and the graphs they create.  Ultimately, this learning is most powerful when students are able to draw conclusions and take action within their communities based on their data.  Teachers should model probability language in discussions and activities in all subject areas and encourage students to do the same.  Routines like Always, Sometimes, Never, Data Talks and Playground stations can be used throughout the year to encourage the use of probability language and data explorations.

##### Suggestions for Assessment

Probability Playground days are excellent opportunities for both observational assessment and small group work.  The explore section of lessons also lends itself to gathering evidence of learning.  Success criteria can be used for this purpose as well as with students (with modified language) for self reflection throughout the unit.  Math journals can be used for students to reflect, using pictures, numbers and words, following the consolidation of learning or during writing times.

The key things to look for in grade 1 are:

• The development of the connection between evidence and predictions
• Accuracy and organization in the gathering and organization of data
• Development of understanding of the differences between what is known and what is predicted.
• The ability to create and interpret concrete graphs

Success Criteria:

• Answers, predictions and estimations are logical/evidence based
• Gather accurate data using tallies or another form of tracking
• Create concrete graphs based on data
• Use data to make logical plans and predictions
• Use the terms more/less likely, always, sometimes, never accurately and with increasing refinement of evidence