C1.3 Determine pattern rules and use them to extend patterns, make and justify predictions, and identify missing elements in growing and shrinking patterns involving rational numbers, and use algebraic representations of the pattern rules to solve for unknown values in linear growing and shrinking patterns.
Activity 1: Find Regularities (Growing and Shrinking Patterns)
Divide the class into teams. Give them a situation. From this, individually, they create a number pattern representing the situation (encourage students to use a table of values to organize their data). Then, among themselves, the students describe their pattern rule. Each situation should lead the students to question the type of pattern found and, subsequently, the different properties found.
Examples of scenarios
Sophia buys a membership at a gym. She has to pay a $35 registration fee. In addition, there is monthly fee of $45.50.
- Create a table of values that represents the relationship between the fees and the number of months in a year.
- What type of pattern is this? Justify your answer.
- What is the initial value and what is the constant rate in this pattern?
- Use an algebraic equation to answer the following question: With $490, how many months of membership can she afford?
Source: En avant, les maths!, 8e année, CM, Algèbre, p. 4.
- For the following repeating pattern, create a table of values to represent the relationship between the term number and the term value (number of squares).
- Use an algebraic equation to determine the term number that will have 99 squares.
Source: En avant, les maths!, 8e année, CM, Algèbre, p. 6.
A plumbing company offers its services to do work in the home at a cost of $80 per hour, plus a $100 service call fee.
- Create a table of values that represents the relationship between the total invoice amount and the number of hours required to complete the work.
- What type of pattern is this? Justify your answer.
- What is the initial value and what is the constant rate in this pattern?
- What will the bill be if it takes five hours to complete the work?
- Use an equation to determine the number of hours needed to complete the work if the cost of the invoice is $380.
Activity 2: Carousel of Patterns
Divide the class into teams. Present different cards to each team in the form of stations (about 10 minutes per station; rotate). Each card contains a situation, a table of values or a graphical representation. Members must answer a series of questions at each station during the allotted time. Circulate and record students' different strategies. Involve the teams in the feedback.
Examples of Stations in the Carousel
Station 1
The following table of values represents the number of necklaces made by Jalal during his shift:
| Number of Hours (h) | 2 | 4 | 6 | 8 |
|---|---|---|---|---|
| Number of Necklaces (n) | 1,5 | 3 | 4,5 |
- Determine how many necklaces he manages to make in three hours.
- Determine how many necklaces he manages to make in eight hours.
- If he worked 12 hours, how many necklaces could he make?
- Which equation represents the number of necklaces made by Jalal?
Station 2
The maximum temperature has been decreasing by 0.75 degrees every day for the past week. If it was 23 degrees a week ago,
- write an equation representing the change in temperature in the last week. Explain what the constant rate means in this situation.
- If this steady decrease continues, what will the temperature be after five days?
- Put the data into a table of values and find today's temperature.
- Explain how to get the term 8 in this situation.
Station 3
The following table represents the total cost to own and use a cell phone. Usage fees are billed monthly.
| Number of Months of Use (n) | 0 | 2 | 5 | 10 |
|---|---|---|---|---|
| Total Cost (c) | 500 | 575,40 | 688,50 | 877 |
- Why do you think there is a $500 upfront cost?
- Determine the total cost after 1 month of use.
- Determine the rule that represents this situation and write it as an equation.
- What will be the total cost after 1 year of use?