### 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

The focus of Data for grade 5 is on double bar graphs. This builds on students’ experience with single bar graphs in grades 3 & 4. By the end of grade 4, students should be fluent with making single bar graphs that use either one-to-one correspondence (i.e., vertical scale goes up by 1) or many-to-one correspondence (vertical scale goes up by a number greater than 1). Grade 5 introduces the concept of comparing data, such as multiple grades, multiple ages, multiple teams, etc. using the double bar graph. Care should be taken when using binary genders such as boys vs girls, as this does not cover the full range of genders that may be represented in your classroom and can reinforce dated gender norms. Next year, students will learn about line graphs, which are useful continuous data (e.g., time, distance). Bar graphs use discrete data.

The focus of Probability for grade 5 is on probability experiments. It is very play-based. A variety of experiments with single events are to be considered, such as using spinners, rolling dice, picking marbles out of a bag, etc. Students make predictions about probabilities and may draw on language from previous years, such as: likely, unlikely, certain, uncertain, more/less/equally likely. They are also expected to use fractions to describe probabilities in grade 5, though the terms theoretical probability and experimental probability are not formalized until grade 6. For example, students may have drawn green three times out of 10 trials from a bag of green and red marbles and say that the probability is 3/10. If there were actually 4 green marbles and 6 red marbles in the bag, then the expected (i.e., theoretical) probability for drawing green is 4/10.

### Key Concepts

#### Double Bar Graphs

Students use one-to-one correspondence and many-to-one correspondence with double bar graphs.

#### Probability Experiments

Students predict the outcome of single events (e.g., spinners, dice) and represent probabilities as fractions.

#### Key Data and Probability Concept 1: Double Bar Graphs

##### Overview

Students have been working up to the bar graph since primary. Concrete graphs with linking blocks, and colouring in squares in grids to make vertical bars are the beginnings of the bar graph. In grade 4, students use legitimate bar graphs to represent discrete data (i.e., data with certain values or items). Examples of discrete data include number of students, tickets, items, etc. Since data is discrete, the bars should not be touching as in a histogram, which is a common error. Double bar graphs take the bar graph further by comparing discrete data. For example, comparing two different groups of people, companies, products, etc. All graphs should have a concise yet descriptive title, labelled axes, and scale with units if appropriate. The frequency is most often the vertical axis by convention.

Double bar graphs are a visual that makes it easier to compare similar data, whereas a single bar graph would not highlight these differences. For example, in a grade 4/5 combined class, you might want to compare the grade 4 data to the grade 5 data rather than the class as a whole. The graph makes this easier than viewing raw data, and a single bar graph would not show differences between grades. Students can draw conclusions or make inferences directly from the graphs, which connects Data and Probability very well. For example, if comparing data on two grades, you might be able to predict which grade is more or less likely to have a certain outcome. For this reason, it is useful to combine Data and Probability when teaching, which is one reason for having an integrated week plan for the two strands.

Students can pose their own questions and collect data from classmates in an organized way, such as using a tally chart. The data is used to create double bar graphs. This can be done by hand or using technology, such as Microsoft Excel. There are also many online apps that make this easy. With the “heavy lifting” of drawing the graphs out of the way, students can concentrate their efforts on drawing conclusions and making predictions or inferences based on the data in the graph. This is one argument for using technology. It is a good idea to give students both experiences: creating double bar graphs by hand and using technology.

##### Math Foundations:

The following concepts and competencies are foundational in supporting understanding of double bar graphs in grade 5:

• Skip counting (for many-to-one correspondence in graphs)
• Creating single bar graphs by hand and with technology
• Explaining and justifying ideas drawn from graphs
• Connecting math concepts from Data and Probability (e.g., thinking about probability when analyzing a graph)
##### Progression:
• Review of single bar graphs (from grade 4)
• Looking at many examples
• Noticing the different components (title, axes)
• Interpreting graphs (drawing conclusions, linking to probability)
• When do we want to compare data? – creating the need for the double bar graph
• Asking good questions and collecting data efficiently
• Creating double bar graphs by hand and analyzing them
• Creating double bar graphs using technology and analyzing them

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

##### Overview

Students have been playing with probability experiments in previous years with single outcomes, such as flipping a coin, rolling a die, spinning a spinner, drawing marbles from a bag, or drawing a card from a deck of playing cards. They can describe probability using the likelihood of an event, such as: certain vs uncertain, likely vs unlikely, possible vs impossible, never vs always; as well as comparing the likelihood of an event, such as: more/less/equally likely. In grade 5, students continue to play with probability experiments but now connect their knowledge of fractions to probability by using fractions to represent the likeliness of an outcome. Though the curriculum does not formalize the concept of theoretical vs experimental probabilities until grade 6, students are still playing around with these ideas in grade 5. For example, students can use a fraction to describe the probability of rolling an odd number with a regular 6-sided die. 7/10 could describe rolling an odd number 7 times out of 10 separate rolls. Students might say that half the numbers on a die are odd so that they would expect to roll an odd number half the time or 5 out of 10 times, which could be described as 5/10 or 1/2.

In any case, students need to have a firm understanding on how to represent probability using a fraction. What do the top and bottom numbers in the fraction represent with respect to probability? It depends on whether you are thinking about experimental or theoretical probability. For experimental probability, the numerator is the number of trials where the desired outcome is achieved, while the denominator is the total number of trials in the experiment. For theoretical probability, the numerator represents the number of desired equally likely outcomes, while the denominator is the total number of equally likely outcomes. The words “equally likely” are used because not all outcomes may be equally represented. For example, if a spinner is 75% red and 25% green, then you cannot say there are simply two desired outcomes. They are weighted differently. Students do not use the language of theoretical and experimental probability, but we can get around this by saying things like “what do you expect the probability to be” for theoretical probability and “what did the results of your experiment show” for experiment probability.

##### Math Foundations:

The following concepts and competencies are foundational in supporting understanding of probability experiments in grade 5:

• Describing the likelihood of an event using comparative language
• Understanding the meanings of the top and bottom numbers in a fraction, among other fraction concepts
• Representing fractions
• Comparing and ordering fractions
• Equivalent fractions
• Connecting math concepts from Data and Probability (e.g., thinking about probability when analyzing a graph)
##### Progression:
• Probability games/experiments… getting an intuitive feel of probability
• Which results happen more often?
• How does understanding probability help with this game?
• Probability games/experiments… describing likelihood of an event
• Using comparative language
• Using fractions
• Using probability to interpret data from double-bar graphs (and other graphs)
##### This sample week integrates both data and probability key concepts for this grade level.

Prior to this week, students will have reviewed the foundational algebraic ideas outlined in the learning progression above. They will have already studied pattern rules for increasing and decreasing patterns.

Lesson Topic: Introduction to Double Bar Graphs

Before: Data Talks. Use a double bar graph that is relevant to the students in your class. Here is an example. What does the graph tell us? How is this similar/different to a single bar graph? Why would you want to use a double bar graph vs a single bar graph for this data?

During: Ask students to brainstorm some questions they might want to explore where comparing two things is useful. You can do this in small groups while you give support to those who need it, and then have students share as a class. Record these ideas for all students to see. As a class, vote on one to explore that is doable in class. Then do it!

By a show of hands, have students answer the question while you record the data in a tally chart which clearly shows two columns that can be put into a double bar graph. Individually or in pairs, ask students to graph the data on grid paper or a whiteboard with a grid.

Adapt by leading a small group of students through this process or doing it as a whole class if needed. Extend by asking students who are done early to check in on their peers for support and feedback.

After: Reconvene as a class. Share how you did it or use one or more student samples as teaching points. Did students transfer the data correctly? Is there a title? Are axes labelled appropriately? Then ask students to interpret the data. What does the data tell them? How is this useful?

Ask students to hand in their graphs for formative feedback. Based on this, you may need to walk students through another day of co-creating a double bar graph and interpreting it.

Lesson Topic: Creating and Analyzing Double Bar Graphs

Before: Slow Reveal Graphs. Select a double bar graph that your students might find interesting. Here are some examples. Get students to share what they notice and wonder at each step of the reveal. What does the graph tell us? What does the double bar graph tell us that a single bar graph would not?

During: Math Workshop. Some ideas include:

• Meet with teacher to go over the key points in creating a double bar graph and how to interpret the data
• Some examples of double bar graphs that are incomplete or have errors. Students have to find these and describe them or fix them.
• Examples of double bar graphs. Students have to interpret them and write three ideas that the data shows.
• Online practice (e.g., Mathletics)
• Data is given. Students create a double bar graph by hand.
• Data is given. Students use technology (e.g., MS Excel) to create a double bar graph following instructions (written or video).

After: Class share on Math Workshop. Then ask students to write in their math journals. What do they understand or feel good about? What are they still needing help with? Collect for formative feedback.

Lesson Topic: Probability Interlude

Before: Number Talks. Do a number talk with a simple probability task such as rolling a die. Ask about the probability of…

• Rolling a 1
• Rolling a 1 or a 5
• Rolling an odd number
• Rolling a number greater than 2

How do students describe their probabilities? Students may use language like “likely” or “unlikely”. Some students may say things like “1 in 6” or use a fraction. Record these ideas together using a graphic like a number line.

You may need to prompt these ideas. Then formalize how to write probability as a fraction:

Probability of event = Number of desired outcomes / Total number of outcomes

During: provide each student with a die. Students work in pairs to complete a die experiment similar to the one outlined here but without using the terms theoretical and experimental probability. They roll a die 60 times. They graph their results. You can make a double bar graph by plotting each student’s result in the pair side by side. What is the probability of rolling each number 1 through 6? Then connect this activity to the number talk with the follow up questions such as:

• What do you expect the probability to be for…
• Rolling a 6
• Rolling a 2 or 3
• Rolling an 8 (probability is 0)
• Rolling an even number
• Rolling a number less than or equal to 4
• Rolling a number greater than 0 (probability is 1)
• How did your experimental results compare?

Adapt by having pairs work together to do a single bar graph instead or simply record results in a chart. Extend by having students use equivalent fractions to compare their results more accurately.

After: Meet as a class to discuss the activity. Why are the expected results (often) different from the actual results?

Exit ticket: Give students an example with a spinner with equal regions (e.g., 1 green, 2 reds, 1 blue) to see if they are able to write probabilities as fractions for the questions like:

• Spinning green
• Spinning red
• Spinning red or blue
• Not spinning blue

Use this as formative feedback for the next lesson on probability.

Alternatively: If students need more practice mastering double bar graphs, you could give students additional practice and save the Probability Interlude lesson until next week.

Lesson Topic: Double Bar Graph Project (Day 1)

Before: Display a variety of double bar graphs, ideally with different ways to compare data (i.e., not just “boys” vs “girls”). Use a search engine like Google to help you. Have students work in pairs or small groups to ascertain what the survey question was for each graph and why they think so. Share some of these as a whole class. Discuss what makes a good survey question, why we need to be sensitive about the questions we ask (e.g., boys vs girls for gender non-conforming students), and how the question will compare data (i.e., create a double bar graph).

During: Students are asked to create their own survey questions that would be suitably graphed with a double bar graph. Have students work in pairs to brainstorm a variety of questions they could ask. A few students can share some of their examples to the class for inspiration.

Then students select ONE survey question that they will make the basis of their project. Again, they can do this in pairs. What answer options will they provide (if any)? How will they be comparing data? Is their question relevant and sensitive to the students they will be asking? The teacher could provide a template for this if necessary.

After: Have a class discussion on how the question process went. Ask pairs to swap their questions with another group (perhaps more than once) to get some feedback. You may want to collect them to give your own feedback.

Ask students how they will be collecting the data. Discuss some ways to do this efficiently and effectively. For example, how will they know they have surveyed everyone? Not double counted someone? Will they make a tally chart? This will take place tomorrow. Some students may want some time now to think this through.

Lesson Topic: Double Bar Graph Project (Day 2)

Before: If you provided feedback for survey questions, give students some time to make any necessary changes. Then ask students to collect data from their peers and to record it in an organized manner (e.g., tally chart).

During: Once data is collected, you might consider having students individually display the data and draw conclusions for assessment purposes. Alternatively, students work together but you can conference with them to assess each student’s understanding.

In any case, students will display their data using a double bar graph which includes a title and labelled axes. They are asked to make at least three inferences/conclusions based on the data.

After: Have students meet together to discuss how the project is going. Ask some reflection questions, which you may decide to have students answer as part of their projects. Here are some examples to choose from:

• What did you find most challenging?
• What did you learn from this activity?
• What would you do differently next time?

Students will likely need more time to polish their displays and to flesh out their conclusions or reflections. Consider providing some additional feedback before the final submission.

In the week to follow, provide students with some additional time to finish their projects. Once all students have finished, ask students to share their project findings. Here are some ways to do this efficiently:

• Gallery Walk. Students tour the class looking at other students’ displays and analysis of their data. As a class, a few students could share what they learned or found interesting.
• Small Group Share. Put students in small groups of around 4 students to take turns sharing their projects with one another. As a class, a few students could share what they learned or found interesting.
• Speed Pair Share. Students find another student to share their project for a few minutes. Then the teacher says “next” and students find a new partner. This can be repeated a few times. As a class, a few students could share what they learned or found interesting.

In addition to the above, the week continues with more probability experiments that give further practice of the probability concepts learned from the previous week. Alternatively, if you did not get to the Probability Interlude lesson, you could start with this lesson and then do more probability experiments with dice, spinners, cards, marbles in a bag, etc.

##### Suggestions for Assessment

By the end of grade 5, students should be able to ask a survey question to compare data and display the data using a double bar graph. They should be able to make inferences based on the data and draw conclusions. Students should be able to reflect on this process as well.

By the end of grade 5, students should be able to predict single outcomes in a variety of probability experiments (e.g., spinner, rolling a die) and represent the probability of a single event using a fraction.

Data Talks

Slow Reveal Graphs

Number Talks

mathisfun.com

Math Workshop (Jennifer Lempp)

Making Math Meaningful to Canadian Students, K-8 (Marian Small)