D1. Data Literacy
Specific Expectations
Data Collection and Organization
D1.1
sort sets of data about people or things according to two and three attributes, using tables and logic diagrams, including Venn, Carroll, and tree diagrams, as appropriate
- two-way table (two attributes) showing data for choice of material and tools to make a greeting card:
- Venn diagram:
- The data in the Venn diagram is drawn from the two-way table example above.
- The Venn circles indicate the number of students whose favourite material is art paper and the number of students whose favourite tool is markers. The intersection of the circle shows the number of students who prefer both art paper and markers.
- Fifteen students do not fall into these categories, so the number 15 is placed outside the circles but inside the rectangle.
- Carroll diagram:
- The data in the Carroll diagram is drawn from the two-way table example.
- The "type of material" category is grouped in two complementary categories – Art Paper and Not Art Paper.
- The "type of tool" category is grouped in two complementary categories – Marker and Not Marker.
- tree diagram for three attributes:
- Students at School B were also asked “Would you like to use an envelope for your card?”. The responses for students who selected “Markers” or “Crayons” are listed in the tree diagram below:
- Data can be sorted in more than one way.
- Two-way tables are used to sort data into all of the possible combinations for the characteristics of two attributes.
- A three-circle Venn diagram can be used to sort data based on three characteristics (e.g., red, large, stripes) for three attributes (e.g., colour, size, markings).
- A Carroll diagram can be used to sort data into complementary sets for two characteristics (e.g., red – not red, stripes – no stripes) for two attributes (e.g., colour, markings).
- A tree diagram can be used to sort data into all the possible combinations of characteristics for two or more attributes (e.g., red stripes, red dots, blue stripes, blue dots, green stripes, green dots).
Note
- A variable is any attribute, number, or quantity that can be measured or counted.
- The number of possible combinations of categories can be determined by multiplying together the number of possibilities for each attribute (variable) under consideration. For example, there are 24 possible combinations for four shapes – circle, rectangle, triangle, hexagon – combined with three colours – red, blue, green – and combined with two sizes – large, not large (i.e., 4 × 3 × 2 = 24).
Provide students with objects (e.g., beads, buttons, feathers, ribbon) of many different shapes, colours, and sizes. Have them create a tree diagram to show all the possible combinations. Next, have them select one shape (e.g., circle), one colour (e.g., green), and one size (e.g., small), and create a Venn diagram to sort their objects using these three attributes.
D1.2
collect data through observations, experiments, and interviews to answer questions of interest that focus on qualitative and quantitative data, and organize the data using frequency tables
- questions of interest and type of collection:
Questions of Interest | Data Collection Methods |
|
Observation:
|
|
Experiment:
|
|
Interview:
|
- frequency table:
- Data compiled after asking primary students in School A “How many minutes do you spend doing your favourite activity?”:
- The type and amount of data to be collected is based on the question of interest.
- Data can be either qualitative (e.g., colour, type of pet) or quantitative (e.g., number of pets, height).
- Data can be collected through observations, experiments, interviews, or written questionnaires over a period of time.
- Frequency tables are an extension of tally tables, in which the tallies are counted and represented as numerical values for each category.
Note
- In the primary grades, students are collecting data from a small population (e.g., objects in a container, the days in a month, students in Grade 3).
- When students are dealing with a lot of categories for data involving two attributes, one strategy is to reorganize the categories into their complements and use a Carroll diagram to organize the data.
Have students collect data that answers the questions “Do you prefer a vegetarian sandwich? Do you prefer your sandwich to be toasted or not?”. Have students organize the data in a Carroll diagram showing the frequencies.
Have student collect data for a question of interest, and then ask them to organize the data in a frequency table.
Data Visualization
D1.3
display sets of data, using many-to-one correspondence, in pictographs and bar graphs with proper sources, titles, and labels, and appropriate scales
- pictograph with many-to-one correspondence:
- bar graph with a scale of an interval of 2:
- displaying the same data as the pictograph example:
- The order of the categories in graphs does not matter for qualitative data (i.e., the categories can be arranged in any order).
- The categories for pictographs and bar graphs can be drawn either horizontally or vertically.
- Graphing data using many-to-one correspondence provides a way to show large amounts of data within a reasonable view and is indicated by the scale of the frequency on the bar graph and the key for a pictograph.
- The source, titles, and labels provide important information about data in a graph or table:
- The source indicates where the data was collected.
- The title introduces the data contained in the graph or table.
- Labels provide additional information, such as the labels on the axes of a graph that describe what is being measured (the variable).
Note
- Have students use scales of 2, 5, and 10 to apply their understanding of multiplication facts for 2, 5, and 10.
Have students recreate a pictograph that uses one-to-one correspondence using two-to-one correspondence. Support students in making connections between the pictographs, so that they can identify how the pictographs change according to the correspondence used, and what stays the same.
Have students create a pictograph and a bar graph for the same data set using the ratios 2:1, 5:1, and 10:1. Make sure that there is at least one situation that requires them to either use half a picture or estimate halfway between two values to determine the height of a bar. These types of experiences help students understand the role of the fraction one half ($$\frac{1}{2}$$) in context.
Have students graph previously collected quantitative data using many-to-one correspondence. Support students in understanding the difference between numbers used as categories and numbers used as frequencies.
Data Analysis
- mode
- using data in a bar graph:
- using data in a line plot:
- using data in an unorganized list:
- mean:
- When students are first learning about “mean” as it relates to data, it might be helpful to support them in understanding the idea by talking about “levelling out” or redistributing the data. For example, Student A has 6 markers, Student B has 4 markers, and Student C has 5 markers. To find the mean for the number of markers, the quantities must be “levelled out” or redistributed so that each student has 5 markers, which is the mean.
- Modes can be identified for qualitative and quantitative data. A variable can have zero, one, or multiple modes.
- The mean can only be determined for quantitative data. The mean of a variable can be determined by dividing the sum of the data values by the total number of values in the data set.
- Depending on the data set, the mean and the mode may be the same value.
Note
- The mean and the mode are two of the three measures of central tendency. The median is the third measure and is introduced in Grade 4.
- The mean is often referred to as the average. Support students in conceptually understanding the mean by decomposing and recomposing the values in the data set so that all values are the same.
Have students identify the mode(s), if any, for a variety of data sets. For example, give students a set that has one number appearing more than any other, a set that has two or more numbers appearing more than any other, a set in which all the numbers appear the same number of times, and a set where all the numbers are different.
In this early work with mean, it is important that the data sets students work with be represented with concrete materials only and contain only small numbers that can easily be manipulated. This enables students to learn concretely – to manipulate the numbers using concrete tools until the numbers in all the categories are the same.
Start with a scenario where three students have three different numbers of cubes: 3, 2, and 1. Ask “What do we need to do so that everyone has the same number of cubes?”
Repeat with 4, 3, and 2 cubes; then with 5, 3, and 1 cubes; and so on.
Have students determine the mean for a data set of 6, 3, 3, 2, and 1 using a concrete graph. To groups of 5 students, distribute rods of 6, 3, 3, 2, and 1 interlocking cubes. Ask students to imagine that this is the number of pieces of fruit in all their lunches. Ask them to organize the rods in ascending order and to find the mode. Now ask them to redistribute the cubes so that everyone has the same number. This represents the mean or average number of pieces of fruit that each student had in their lunch. Ask whether every student had three pieces of fruit (in reality, some may have had more, some may have had fewer, and some may have had three). Through these kinds of activities, students will begin to develop a conceptual understanding of the idea of central distribution.
D1.5
analyse different sets of data presented in various ways, including in frequency tables and in graphs with different scales, by asking and answering questions about the data and drawing conclusions, then make convincing arguments and informed decisions
- various ways of presenting data:
- Different representations are used for different purposes to convey different types of information.
- Frequency tables show numerically how often an item or value occurs in a set of data. They are quicker and easier to read than tallies.
- Graphs of quantitative data show the distribution and the shape of the data. For example, the data on a vertical bar graph may be skewed to the left, skewed to the right, centred, or equally distributed among all of the categories.
- It is important to pay attention to the scale on pictographs and other graphs. If the scale on a pictograph is that one picture represents two students, then the frequency of a category is double what is shown.
- Data that is presented in tables and graphs can be used to ask and answer questions, draw conclusions, and make convincing arguments and informed decisions.
- Questions of interest are intended to be answered through the analysis of the representations. Sometimes the analysis raises more questions that require further collection, representation, and analysis of data.
Note
- There are three levels of graph comprehension that students should learn about and practise:
- Level 1: information is read directly from the graph and no interpretation is required.
- Level 2: information is read and used to compare (e.g., greatest, least) or perform operations (e.g., addition, subtraction).
- Level 3: information is read and used to make inferences about the data using background knowledge of the topic.
- Analysing data can be complex, so it is important to provide students with strategies that will support them in building these skills.
One way for students to analyse graphs is to create a different representation of the same data. Provide them with a pictograph in which each picture represents a count of 2, 5, or 10. Have them create a concrete graph that represents the same data using one-to-one correspondence, trading each picture for 2, 5, or 10 blocks. Ask questions about the pictograph, and encourage students to verify their responses using the one-to-one concrete graph.
Provide students with a frequency table, a pictograph, or a bar graph, and model some questions about the data to support them in thinking about and forming their own questions. For example:
- question that requires reading and interpreting data from a graph or table:
- How many people visited the school on Thursday?
- question that requires finding data from a graph or table and using it in a calculation:
- How many more people visited the school on Friday than on Monday?
- question that requires using data from a graph to make an inference or prediction:
- Why do you think more people visited the school on Friday than on any other day?
Ask students what questions they have about the data, and encourage them to answer each other’s questions. Model asking questions using the three types of questions outlined in Task 2, and have students pose and answer their own questions, requiring them to think critically about the data.
Throughout the year, have students collect representations of real-life data about topics that are of interest to them. As students share, pose questions to support them in thinking critically about the data.