Minds On

Difference rules

Record the first 6 terms of each pattern digitally, orally, or in print.

  1. Start at 3 add 2.5 each time.
  2. Start at 9 and increase by the next multiple of 2 each time, starting by adding 2 × 1.
  3. Start at $80 subtract $4.75 each time.

Describe each of the following geometric patterns. Which part changes? Circle, underline, or use another method of your choice to indicate the part that is always the same.

Pattern 1: Term one has 2 cubes side by side, Term 2 has 2 cubes side by side and 1 cube on top, Term 3 has 2 cubes side by side, and 2 cubes stacked on top, Term 4 has 2 cubes side by side and 3 cubes stacked on top.

Pattern 2: Term one has a central block and one cube against each side of the central block, Term 2 has one central block and 2 squares against each side of the central block, Term 3 has one central block and 3 squares against each side of the central block.

Pattern 3: Term 1 has 5 squares arranged in 2 columns (3 and 2), Term 2 has 8 squares which are arranged in 2 columns of 3 and 1 column of 2, and Term 3 has 11 squares arranged in 3 columns of 3 and 1 column of 2.

Action

Constant rate and initial values

The initial value (constant) of a linear growing pattern is the value of the term when the term number is zero. An example of a real-life application of an initial value is a membership fee.

A constant rate is a rate of change that remains the same and doesn’t go up or down.

Examine the following graph. To find the initial value, we look at when the term number is zero. In a table of values, the term number is on the left and the term value is on the right.

A line graph titled Local Fair Prices. The x-axis is labeled Number of Rides and Games and its scale starts at 0, going up by twos to 20. The y-axis is labeled Cost ($) and its scale starts at 0, going up by ones to 27. A straight line is plotted from coordinate (0,15) to (20, 25).

What is the term value on the graph when the term number is 0?

Press ‘Answer’ to reveal the term value.

The initial value is $15.

If this graph is called Local Fair Prices, and the number of rides is along the x-axis and the cost ($) is on the y-axis, what does it mean when a point is plotted at (0,15)? What does the initial value represent?

Press ‘Answer’ to reveal the what the initial value represents.

When there are 0 rides, the cost is $15. It must cost $15 to enter the park. The initial value represents the cost to get in the park.

Use the graph to complete the following table of values. You can complete this table in your notebook or using another method of your choice.

Term number Term value
0 15
1 15.50
2
3
4
5

Press ‘Answer’ to reveal the filled table of values.

Term number Term value
0 15
1 15.50
2 16
3 16.5
4 17
5 17.5

Using the information from the graph and the table of values, what is the cost of each ride/game? Remember: the cost to get in the park is $15.

Press ‘Answer’ to reveal the cost.

The cost per ride/game must be $0.50 because for 1 ride/game the price went from $15 to $15.50. 2 rides/games cost $16 which is $1 more than the cost of entry, and $1 divided by two is 50 cents.

This is the constant rate- the rate of change stays the same, it’s always $.50 no matter how many rides you go on or games you play. For every ride or game, there is the cost of entry (the initial value) plus the constant rate ($0.50)

Gym membership and monthly fees

A group of runners want to get gym memberships. If they sign up for the year, the gym charges a one-time fee of $40 and then a monthly fee of $34.50.

  1. Create a table, or a detailed description to indicate the total spending each month towards the membership. $40 is paid up front.

Complete the Gym Membership Table in your notebook or use the following fillable and printable document.

Press the ‘Activity’ button to access the Gym Membership Table.

Activity (Open PDF in a new window)
  1. How much would they spend on the membership after 12 months?
  2. Create an equation that can be used to calculate the total gym membership cost with y equal to total and x number of months.
  3. Graph, or provide a detailed written or audio recording describing the runner’s money spent for the gym. Use your T-table chart, or previous description for your coordinates. Do the values start at (0, 0)? Explain.
  4. What is the constant rate and initial value?

Consolidation

Gym membership paid in full

This is a decorative image and doesn’t require alt text

The gym offers the runners a deal if they pay the full year membership upfront. The total cost is $415 for the year including the one-time fee of $40.

What is the constant rate? What is the initial value?

How do these compare to the constant rate and initial value when the gym membership was paid monthly?

Use your gym fee equation to calculate how much the runners would save if they take the deal.

Show your work using a method of your choice.

Reflection

As you read the following descriptions, select the one that best describes your current understanding of the learning in this activity. Press the corresponding button once you have made your choice.

I feel...

Now, expand on your ideas by recording your thoughts using a voice recorder, speech-to-text, or writing tool.

When you review your notes on this learning activity later, reflect on whether you would select a different description based on your further review of the material in this learning activity.