### 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 Kindergarten: Data and Probability

The concepts of data and probability are interwoven through children’s daily lives. In Kindergarten, children collect data and/or represent data with concrete materials or pictorially.

The following lesson idea shares examples of these Kindergarten concepts integrated together. For example, students might be asked to each take off one of their shoes. The shoes could be sorted by colour or type (pull on, velcros, laces, buckles, etc) and then lined up in rows and columns to essentially create a concrete bar graph. Students can then analyze the class shoe graph and make comparative statements such as “Most of us have shoes with velcro, “ “Four of us have laces,” and “Two more people have laces than shoes with buckles.” A probability (often referred to as chance and uncertainty in primary grades) connection could be made with an extension question like: “Is it likely or unlikely that our graph will be the same tomorrow?”

It is important to include data and probability concepts throughout the school year and connect these concepts to other math concepts, other areas of learning and to the children’s personal interests. There are always many opportunities to collect data and create graphs in areas of learning such as science and social studies.

### Key Concepts

#### Likelihood of events

Using familiar events in children’s lives, children think about they likelihood of these events happening using terms like likely or unlikely

#### Key Data and Probability Concept 1: Concrete or Pictorial Graphs

##### Overview

In Kindergarten, understanding of quantity is very connected to creating and analyzing data and graphs. Concrete graphs are created with real items such as coloured blocks, shoes, type of fruit, type of sports equipment, etc and the items are sorted by type and then compared. Students learn by placing the concrete item in an organized way, such as rows, makes it more clear to analyze and count the quantity of items. Creating and talking about graphs can help students get to know each other and feel part of a learning community such as heard in comments like “My favourite sport is soccer too. I play games on Saturdays.” Some items or events are not easily represented concretely such as the weather or types of animals and this is where pictorial representations can be used to create graphs with. Students can draw pictures or use rubber stamps to create pictures to represent data. Tally marks can also be considered a form of pictorial or visual representation of data.

##### Math Foundations:

Students will come into Kindergarten with informal experiences with data and graphs from home and pre-K learning context such as:

- sorting materials by attributes (colours, size, shape
- responding to survey type questions such as What is your favourite colour? Or “Do you like to play outdoors? (yes/no/sometimes etc).
- informal one-to-one counting experiences

##### Progression:

- Sorting concrete materials as data or to represent data (colour blocks, personal items, rocks with children’s names on them)
- Comparing data sets of concrete materials (more, less, about the same)
- Creating concrete graphs (rows, groups, etc) to represent the quantity of data
- Use pictures to represent concrete items or events and then create visual graphs with the pictures
- Counting data sets using one-to-one correspondence
- Compare data using language such as two more than, the same amount, the most popular, the least favourite, etc
- Analyze data, drawing upon personal experiences and knowledge – what story does this data tell?

#### Key Data and Probability Concept 2: Likelihood of Familiar Life Events

##### Overview

Children in Kindergarten are introduced to math concept of probability through discussion of daily life and events that are familiar to them. At this grade level, probability is more general and focuses on concepts of chance and uncertainty. What are the changes of something happening? Could this happen or not? In Kindergarten, we introduce students specifically to the terms of “likely” and “unlikely” as we discuss the likelihood of certain actions or events happening. To support the understanding of these terms, we can use visuals and gestures such as nodding yes/ shaking head now or thumbs up and thumbs down. Probability is an abstract concept that will develop over time with connections to games and experiences with dice and spinners and to number and computations as students progress into the intermediate grades. Students will connect their understanding of likelihood that they have developed in Kindergarten as they engage in more formalized probability experiences in later grades such as predicting the likelihood of rolling two sixes with dice.

##### Math Foundations:

Students may come into Kindergarten with informal experiences with probability from home and pre-K learning context such as:

- playing card or board games
- hearing adults talk about things like weather forecasts (it’s probably going to rain later today)

##### Progression:

- using daily life contexts such as games, weather or a daily schedule, students predict what could happen
- students then begin to consider the chance of something happening and then using language to describe that chance with terms such as likely or unlikely

##### Sample Week at a Glance

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

** **Prior to this week of lessons, students will be familiar with their attendance routine of moving their name stone from one basket to the “I’m here” basket each morning as they come into the classroom. The class will also have been adding to a weather calendar for each day of school where on a piece of grid paper for the month, they draw a picture of that day’s weather. Please note that these lessons are shorter lessons (10-20 minutes) that could be considered the “before” or “during” part of the lesson and a closing and connecting discussion that could be considered the “after” part of the lesson. These tasks may be followed by Math Workshop or centres where students practice or review other math concepts.

**Attendance Graph**: Have students sit in a circle on the carpet and bring the two baskets of name stones to the carpet. Prompt with questions such as “Which basket has more?” “Is this attendance the same or different from Friday?” Remove the stones from each basket, inviting the students to choral count with you as you count each stone into a pile. Ask students to think about how they could organize the stones to make it more clear how many there are? For example, line up the stones in two rows, one for students who are at school and a second row for students who are absent. Practicing counting with one-to-one correspondence, with either you or a student touching each stone as you count. Ask students to think about how putting the stones in rows makes the “data” clearer for them to visualize.

If there are no students absent, you could sort and graph the name stones by the initial letter of each name or by the number of syllables.

Create a tally graph to record students’ responses and ask students to share their reasoning. (And please note that for this specific example, pediatricians do recommend that young children only eat grapes that are cut into smaller pieces). |

**Math discussion**: Have students consider the likelihood of familiar events with prompts such as: “Ms. N. will read us a story today,” “We will play outside at recess,” “We will have centre time this afternoon, or “The principal will visit our class.” Choose some very unlikely events to inject some humour into the discussion such as, “Is it likely or unlikely that an elephant will walk into our classroom?”

As you provide a prompt, invite students to show you and the class whether they think it is likely to happen (thumbs up) or unlikely to happen (thumbs down). If there are some questions where there is a clear mixture of responses, invite students to share their thinking. You could also record their responses in a tally chart to make their responses more visible to the whole class.

**Opening circle**: Have a large basket of Unifix cubes and pass it around the circle asking students to choose one cube that is their favourite colour. After everyone has chosen a colour, have students create piles of the different colours they have chosen in the middle of the circle. Ask students: What do you notice? What do you wonder? Make a connection to the name stones task on Monday and ask: How could we organize the cubes to make the data easier to visualize? Invite one or two students to snap each colour of cubes together and line them up in rows. Count each colour focusing on one-to-one correspondence. Invite students to share what they notice and pose their own questions for each other to think about such as “What colour has one more than red?”

**Weather data**: Look at the pictorial weather data that you have recorded for this week or month. Ask students questions like: *How many sunny days were there? How many cloudy days? Did we have more sunny or rainy days? *Make a photocopy of the grid of students’ weather drawings for the week or month. During centre time or Math Workshop, have one table set up for students to cut up the weather grid and sort the pictures into piles such as sunny, rainy, windy, cloudy, etc. At the end of centre time, come back onto the carpet and create a picture graph with the pictures of each type of weather in rows. Have students compare the amount/frequency of the different types of weather. Return back to your weather grid and ask students to notice if there is a pattern or “trend” for this week and have them predict what the weather will be tomorrow. Review the language of likelihood by asking questions such as: “Is it likely or unlikely that it will rain tomorrow? Be sunny? Snow?”

These examples of routines and lessons can be used throughout the school year, connecting to new contexts and areas of learning. The concepts provided good contexts for meaningful practice of counting.

##### Suggestions for Assessment

Most assessment of these concepts will be through observing and listening. Create a chart of related learning standards where you can make notes of your observations and how students are discussing these ideas.

By the end of Kindergarten, students will be able to:

- Create and describe a concrete or pictorial graph. (“There are five fruits. Three bananas and two apples. There are more bananas.”)
- Use language like “likely or unlikely” to describe an event such as “It Is likely that it will rain again tomorrow.”

##### Suggested Links and Resources

**Would You Rather Math? **(K-2 examples)

Unifix cubes or other snap cubes