Question

mathematical thinking and reasoning
5 mtr use patterns and structure to help understand and
connect mathematical concepts
mathematicians who use patterns and structure to help understand and connect
mathematical concepts look for similarities among problems.
work with a partner. a researcher studies the populations of carnivore and
omnivore species in a region over time and records annual population estimates.

1. the table shows the numbers y of panthers in the region after x years.

time (years), x	1	2	3	4	5

a. write a model to represent the data. explain how you used a
pattern to determine which type of function to use for your model.

b. when would you expect there to be no panthers in the region when
the population continues to decrease at the same rate?

1. the table shows the numbers y of black bears in the region after x years.

time (years), x	1	2	3	4	5

a. write a model to represent the data. how does the model compare to
the model you created in exercise 1? explain your reasoning.

b. when would you expect there to be 650 black bears in the region?

Explanation:

1a.

Step1: Check the pattern

Calculate the differences between consecutive \( y \)-values: \( 56 - 48 = 8 \), \( 48 - 40 = 8 \), \( 40 - 32 = 8 \), \( 32 - 24 = 8 \). The common difference is \( -8 \) (since it's decreasing), so it's a linear function.

Step2: Find the slope and y-intercept

The slope \( m = -8 \). Using point-slope form with \( (x_1, y_1) = (1, 56) \): \( y - 56 = -8(x - 1) \). Simplify: \( y = -8x + 8 + 56 \), so \( y = -8x + 64 \).

Step1: Set \( y = 0 \)

We want to find \( x \) when \( y = 0 \) in \( y = -8x + 64 \).

Step2: Solve for \( x \)

\( 0 = -8x + 64 \) → \( 8x = 64 \) → \( x = 8 \).

Step1: Check the pattern

Calculate differences: \( 631 - 618 = 13 \), \( 644 - 631 = 13 \), \( 657 - 644 = 13 \), \( 670 - 657 = 13 \). Common difference is 13, so linear function.

Step2: Find the model

Slope \( m = 13 \). Using \( (1, 618) \): \( y - 618 = 13(x - 1) \) → \( y = 13x - 13 + 618 \) → \( y = 13x + 605 \). Compared to 1a's model (\( y = -8x + 64 \)), this has a positive slope (increasing) vs negative (decreasing), different slopes and y-intercepts.

Answer:

The model is \( y = -8x + 64 \) (linear function, common difference of -8).

1b.