C1.2 Create and translate growing and shrinking patterns using various representations, including tables of values and graphs.
Activity 1: My friends' friends (represent a suite)
Present the following situation to the students.
Allow students to represent the situation using the strategy of their choice.
Example
Represent a situation using a table of values
| Day | Number of calls |
|---|---|
| 1 | 4 |
| 2 | 16 |
| 3 | 64 |
Represent a situation using a graphic representation
Source: Adapted from En avant, les maths, Grade 5, ML, Algebra, p. 20.
Represent a situation using semi-concrete materials
Source: Adapted from En avant, les maths, Grade 5, ML, Algebra, p. 20.
Activity 2: Does it take longer to get out of a room than to get in?
This activity integrates concepts from the Algebra and Data stands. It gives students the opportunity to see two different representations (table of values and graphical representation) of similar situations.
Present the following situation:
Teachers distribute a copy of the Entry and Exit Appendix to students and ask them to complete the two representations, observing the patterns in each. The teacher then invites students to examine each of the representations and make some observations.
Here are some possible observations:
- The two types of representations differ, the first is a table of values, the other is a graphical representation.
- On entry, the number of students in the gym increases, while on exit, it decreases.
- Twenty-five students per minute enter the gym.
- The number of students in the gym prior to the start of the field trip is 300.
- Thirty students a minute walk out of the gym.
- The time required to get all the students out of the gym is shorter than the time required to get them in.
Teachers then ask students to represent the student input with a graphical representation, and the output, with a table of values.
Note: By exploring both representations of the same situation simultaneously, students can more easily see the connection between the data in the table of values and the data in the graph. Graphical representations help visualize a set of numerical data; therefore, they help to see if it is a growing or shrinking situation and if there is a regularity.
As a class, the teacher goes deeper into the situation, pointing out some important elements and relationships.
Here are some suggested questions to stimulate this exchange, which vary depending on the students' progress in algebra:
- Which of the two representations do you find clearer visually? Why? (The graphical representation allows you to quickly see if it is a situation of growth or decrease. We can also see that there is a regularity)
- If there were more students, how many students would have entered the gym after 15 minutes? Can you determine this using the graph? (Based on the increase, we see that after 15 minutes there would be 375 students in the gym)
- Can you explain, in your own words, the rule that represents the relationship between the number of students in the gym and the amount of time that has passed since they entered the gym? (To determine the number of students in the gym at a certain time, multiply the elapsed time by 25)
- Can the same rule be used to represent the relationship between the number of students in the gym and the amount of time since the students began to leave? Explain your reasoning. (No, because the time required to enter the gym is not the same as the time required to exit)
- What rule represents the relationship between the number of students in the gym and the time elapsed since the start of the field trip? (To determine the number of students in the gym after a certain number of minutes from the start of the field trip, multiply the elapsed time by 30, and then subtract this product from the number of students in the gym at the start, which is 300)
Source: A Guide to Effective Instruction in Mathematics, Grades 4 to 6, pp. 107-108.
Appendix: Entry of students into the gymnasium
| Elapsed time (minutes) | 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Number of students in the gym | 0 | 25 | 50 | 75 |
Appendix: Gymnasium Student Exit
Source: A Guide to Effective Instruction in Mathematics, Grades 4 to 6, p. 116.
Activity 3: Long live recycled paper!
Teachers lead a class discussion on the environmental issues surrounding paper production and sound forestry. This will be an opportunity to discuss the relationship between making recycled paper and protecting forests. Teachers present the following situation to raise awareness of the benefits of using recycled paper:
It is estimated that, on average, the production of 10,000 sheets of recycled paper saves one tree. For this reason, one school chose to use only recycled paper and kept a cumulative count of the number of sheets of paper used at the end of each of the first three months.
After the first month, 10,000 sheets were used. After the second month, 15,000 sheets were used, and after the third month, 20,000 sheets.
As a class, teachers ask a variety of questions to draw out the different relationships that the situation presents:
- How many leaves will have been used after the fourth and fifth month, if consumption remains constant (i.e., if regularity is maintained)?
- If consistency is maintained, how many sheets will have been used after 10 months?
Students can represent the relationship between the number of months and the number of sheets used using a table of values.
| Number of months (n) | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 |
|---|---|---|---|---|---|---|---|---|---|---|
| Total number of sheets used (f) | 10 000 | 15 000 | 20 000 | 25 000 | 30 000 | 35 000 | 40 000 | 45 000 | 50 000 | 55 000 |
Continue the discussion with the following questions:
- How many trees were saved after one month by using recycled paper instead of non-recycled paper? (One tree was saved.)
- How many trees were saved after one school year (10 months) if recycled paper was used? (Five and a half trees were saved.)
- What rule could describe the relationship between the number of trees saved and the number of leaves used? (The number of trees saved is the number of leaves used divided by 10,000.) The table of values below could also represent this relationship.
| Total number of sheets used (f) | 10 000 | 15 000 | 20 000 | 25 000 | 30 000 | 35 000 | 40 000 | 45 000 | 50 000 | 55 000 |
|---|---|---|---|---|---|---|---|---|---|---|
| Number of trees saved(a) | 1 | 1.5 | 2 | 2.5 | 3 | 3.5 | 4 | 4.5 | 5 | 5.5 |
Note: Students may think that the number of trees saved is small (in one year, this school would have saved less than six trees). However, by putting the action in a global context (i.e., if each school uses only 100% recycled paper), students will better understand the benefit of this responsible practice.
To get students to grasp what such quantities of sheets represent, teachers show them a box of paper and present the relationship between the number of bundles in the box and the number of sheets in the box (10 bundles of 500 sheets, or 5000 sheets). The teacher asks questions to prompt them to create a table of values that represents the relationship between any number of packets of sheets and the corresponding total number of sheets, and another that represents the relationship between any number of packets of sheets and the corresponding number of boxes.
Here are some suggested questions to ask:
- How many packages would 2,000 sheets of paper represent? 3,000 sheets? 10,000 sheets?
- How can we determine how many bundles a given number of sheets represents?
- How many packages of leaves will there be in two boxes? three boxes? 10 boxes?
- How can we determine the number of leaf packs in a given number of boxes?
- How many boxes of recycled paper did the school in question use? (The school used 11 boxes of recycled paper.)
Source: Guide d’enseignement efficace des mathématiques de la 4e à la 6e année, pp. 110-112.
Note: Facilitate a discussion on the importance of making informed choices to protect and conserve the environment. Have students consider ways to reduce the use of recycled paper