Cronkenstein’s Eucalyptus Tree
You are on your way to the local cemetery to plan for your school’s haunted house. In preparing for halloween, the cemetery is examining trees to makes sure they are stable. Cronkenstein, the cemetery ground keeper, focussed on a eucalyptus tree and explained that the height difference of the trees occurred because after the first tree was planted, additional trees were planted in 5 year intervals (that is every five years). The tree nearest you is 85 feet tall and the next one, which is five years older, is 90 feet tall. You are curious about their age, so you decide to graph the pattern to learn more about the age of the tree.
a) Copy the data table
b) Fill in the height of the tree for each year, assuming the tree continues to grow at the same rate.
c) How does the height of the tree change as the years go by? Study your table and describe in words the pattern you see.
d) Write a word equation that relates the height of the tree to the number of years.
e) How tall will the tree be in 30 years? In 50 years? in any future year? Answer in complete sentences.
f) How tall was the tree 10 years ago? 20 years ago? How many years ago was the tree planted?
g) Follow the steps below to graph the data from Cronkenstein and the eucalyptus tree at the cemetary.
Scale the horizontal (x) axis by fives. Each tick mark represents five years. As you number to the right of zero, you are looking at the future. AS you number to the left of zero, you are looking at the past so -5 means “five years go.” Scale the vertical (y axis) by tens. Each tick mark will represent the ten feet of height.
Plot the points from your table. Remember to go across to the year (x) and then move up to the height (y).
Connect the points to show the tree’s continuous growth
Does it make sense for this line to continue forever in both directions? Explain.
h) Use your graph to answer the following questions.
Find the point on your graph where your line crosses the horizontal (x) axis. What is the height of the tree at this point? Record this in your table.
What does this point represent?
Cronkenstein Practice
Use the table below to complete parts (a) through (c)
a) Fill in the missing numbers in the table
b) Use complete sentences to describe the pattern that relates input values to output values.
c) Write the equation that represents the pattern in y=form and add that to your table.
Garzilla’s Coyote Pack
Cronkenstein’s friend Garzilla decided to bring her study team to the cemetery to research the coyotes living there. She explained that there are 108 coyotes in the cemetery and their number is decreasing at a rate of 3 coyotes per year. She is concerned about the decline in the coyotes.
a) copy and complete the data table to study the coyote population population from 20 years ago to help study it in the future. Now is represented by 0 years.
b) Describe the pattern in the table using complete sentences
c) Write an equation to represent the pattern
d) Create a graph in which “Time in Years” is on the horizontal (x) axis. It should be scaled so that one tick mark on the graph represents ten years.
e) The vertical (y) axis will represent the number of coyotes and should be scaled so one tick mark represents ten coyotes.
f) Plot the points from the data table
g) Using a straightedge, draw a line to connect the points. Extend your line as far as possible on your graph.
h) The point where the line crosses the y-axis is called the y-intercept. What are the coordinates of this point?
i) The point where the line crosses the x axis is called the x-intercept. What are the coordinates of this point.
j) When will the population be fewer than 50 coyotes.
k) When would there have been 200 coyotes?
l) In how many years will the wolves be gone from this park?
Garzilla Practice
Here is another practice.
a) copy and complete the table on your paper
b) Make a graph of the points
c) Draw a line to connect the points. Be sure to extend your line as far as possible on the graph.
d) Name the coordinates of the x intercept.
e) Name the coordinates of the y intercept.
Garzilla Practice Part 2
Complete each table then explain the pattern between the input and output values using complete sentences. Last, write the rule that describes the pattern.
1)
Explanation:
Rule:
2)
Explanation:
Rule:
Copy and complete the tables
The Big Hike
Mr. Salamonster and Mr. Cartwolf are camping with iMiddle students near Cronkenstein’s cemetery and Garzilla’s coyotes. Mr. Salamonster, and Cartwolf each led a group hike. Salamonster’s group hiked to Grizzly Peak, which has an elevation of 182 feet. Mr. Cartwolf’s group hiked to Mt. Sawtooth, which has an elevation of 324 feet. The elevation of their campsite is 120 feet below sea level. If Mr. Cartwolf’s group hikes downhill at a rate of 3 feet per minute and Salamonster’s group hikes downhill at a rate of 2 feet per minute, predict which group will arrive at the campsite first.
Answer the following sub problems for Salamonster’s hiking group in complete sentences.
1) If they have not yet started hiking, what is the elevation of Salamonster’s group now?
2) If “x” represents minutes spent hiking and “y” represents elevation in feet, write an ordered pair to represent Salamonster’s group now. (Remember they have not started hiking yet.)
3) What is the elevation of Salamonster’s group after one minute of hiking? Two minutes? Three minutes? Write an ordered pair to represent each situation.
4) Write a variable expression to represent the rate of descent as feet per minute for Salamonster’s group. (How fast are they hiking downhill)
5. Copy the table below for Salamonster’s group and complete the table for the first five minutes of hiking.
6. Use your table and what you have learned from playing silent board games to find the rule for Salamonster’s Group.
7. Follow the example below and the rule for Salamonster’s group to calculate the remaining values in the table. Use these values to complete your table.
8. Graph the information from the table starting from their elevation now and using all the remaining values you calculated (multiples of five minutes).
9. Draw a continuous line through the points you graphed.
10. Find the range in elevation from Grizzly Peak to the Campsite.
11. At a rate of descent of two feet per minute, how long will it take Salamonster’s hiking group to reach camp?
12. According to your graph, how many minutes did it take Salamonster’s group to reach their campsite? Does this agree with your answer in part 11? Explain.
Now repeat all of the above problems for Mr. Cartwolf’s hiking group.
1) If they have not yet started hiking, what is the elevation of Mr. Cartwolf’s group now?
2) If “x” represents minutes spent hiking and “y” represents elevation in feet, write an ordered pair to represent Mr. Cartwolf’s group now. (Remember they have not started hiking yet.)
3) What is the elevation of Mr. Cartwolf’s group after one minute of hiking? Two minutes? Three minutes? Write an ordered pair to represent each situation.
4) Write an expression (slope) to represent the rate of descent as feet per minute for Mr. Cartwolf’s group. (How fast are they hiking downhill)
5. Copy the table below for Mr. Cartwolf’s group and complete the table for the first five minutes of hiking.
6. Use you table and what you have learned from playing silent board games to find the rule for Mr. Cartwolf’s Group.
7. Follow the example below and the rule for Mr. Cartwolf’s group to calculate the remaining values in the table. Use these values to complete your table.
8. On the same graph, graph the information from the table starting from their elevation now and using all the remaining values you calculated (multiples of five minutes). Label the two different graphs.
9. Draw a continuous line through the points you graphed.
10. Find the range in elevation from Mt. Sawtooth to the Campsite.
11. At a rate of descent of three feet per minute, how long will it take Mr. Cartwolf’s hiking group to reach camp?
12. According to your graph, how many minutes did it take Mr. Cartwolf’s group to reach their campsite? Does this agree with your answer in part 11? Explain.
Debrief:
1. Which group reached camp first? Does this agree with your prediction?