1. Draw a picture or a graph of what you think the growth of the Jactus might look like.

Transcription

1 Memo #1 Welcome to Plants R Us, the world s biggest provider of strange plants! Your first project is to investigate a new plant we just discovered, the Jactus plant. Under a normal grow light, the Jactus grows extremely fast, faster than any plant we ve ever seen! 1. Draw a picture or a graph of what you think the growth of the Jactus might look like. 2. What variables do you think might contribute to the Jactus growth?

2 Memo #2 We ve developed a computer model that shows people how the Jactus grows. 1. How can you keep track of how the plant grows? 2. How tall will the Jactus be after 10 weeks? How do you know?

3 Memo #3 We want to share some information with customers about how the Jactus plant grows. Please provide us with the following: 1. What is the relationship between how much time passes and how tall the plant is?

4 2. Here s a picture of the Jactus plant. Compared to this picture, draw another picture of what the Jactus will look like after 2 weeks, and then a picture of what it will look like after 4 weeks.

5 Memo #4 Yesterday we saw two tables on the board showing the growth of the Jactus. One had days, and one had weeks: Days Inches Weeks Inches Which do you think is easier to work with, days or weeks? 2. How tall will the Jactus be after 210 days (30 weeks)?

6 Memo #5 Here s a picture of the Jactus plant. Compared to this picture, draw another picture of what the Jactus will look like after 1 week, and then a picture of what it will look like after 3 weeks.

7 Memo #6 Describe a way to figure out how tall the plant is on any given week:

8 Memo #7 Say you have a Jactus growing in the yard, and you go on vacation for 1 week. How tall is it going to be when you return?

9 Memo #8 Here s a picture of the Jactus plant that is ½ tall at 0 weeks. Draw a picture and write how tall it will be at: 1 week: 2 weeks: 3 weeks: 4 weeks:

10 Memo #9 How much taller was the Jactus compared to how tall it was originally after: 1 week? 2 weeks? 3 weeks? 4 weeks?

11 Memo #10 Somebody at the plant factory measured the height of the Jactus after different weeks and made the following table. Can you fill in the missing parts? Week Height (inches)

12 Memo #11 You have 3 Jactus plants and each one grows in a different way. The flowering jactus doubles every week. The desert jactus triples every week. The tropical jactus quadruples every week. Which jactus grows the fastest? The blooming jactus starts out at 10 tall and it doubles every week. The evergreen jactus starts out at only 1 tall, but it triples every week. Which jactus do you think is going to be bigger in the long run?

13 Memo #12 Here s a table for the height of a Jactus in inches after a certain number of weeks. Can you figure out how to find how tall the Jactus will be after any number of weeks? Weeks Inches x?

14 Memo #13 Here s another table for the height of a Jactus in inches after a certain number of weeks. Can you figure out how to find how tall the Jactus will be after any number of weeks? Weeks Inches x?

15 Memo #14 Here s another table for the height of a Jactus in inches after a certain number of weeks. Can you figure out how to find how tall the Jactus will be after any number of weeks? Weeks Inches x?

16 Memo #15 Here s another table for the height of a Jactus in inches after a certain number of weeks. Can you figure out how to find how tall the Jactus will be after any number of weeks? Weeks Inches , x?

17 Memo #16 Last week each of you divided the inches for successive weeks to find out whether the plant was doubling, tripling, etc: Weeks Inches = It must be tripling! 9 3 = What if you couldn t do that, because you didn t have successive weeks in your table? How would you figure out if it was doubling, tripling, or doing something else? Weeks Inches ,719,476,736

18 Memo #17 Here s another table for the height of a Jactus in inches after a certain number of weeks. Can you figure out how to find how tall the Jactus will be after any number of weeks? Weeks Inches , ,374, x?

19 Memo #18 What if you didn t know the height of the plant at 0 weeks when it started growing? How would you figure out how tall it was initially? Week Inches x?

20 Memo #19 Here s a Jactus that starts out at 1 cm tall and doubles every week: Week CM Week CM *Have them draw picture of one that starts at 2 cm 0 2 *Have them write equations for both 1 4 *Ask them to explain why the 2 nd one has the x2 in equation

21 Memo #20 Here s another table for the height of a Jactus in inches after a certain number of weeks. Can you figure out how to find how tall the Jactus will be after any number of weeks? Weeks Inches , x?

22 Memo #21 Last week each of you divided the inches for successive weeks to find out whether the plant was doubling, tripling, etc: Weeks Inches = It must be tripling! 9 3 = What if you couldn t do that, because you didn t have successive weeks in your table? How would you figure out if it was doubling, tripling, or doing something else? Weeks Inches ,719,476,736

23 Memo #22 Here s another table for the height of a Jactus in inches after a certain number of weeks. Can you figure out how to find how tall the Jactus will be after any number of weeks? Weeks Inches , ,374, x?

24 Memo #23 What if you didn t know the height of the plant at 0 weeks when it started growing? How would you figure out how tall it was initially? Week Inches x?

25 Memo #24 Here s a Jactus that starts out at 1 cm tall and doubles every week: Week CM Week CM

26 Memo #25 Last time I gave you two values for the Jactus to see if you could figure out how it s growing and how tall it was when it started growing at week 0: Week Height See if you can figure out how it s growing. What happens as it grows from week 14 to week 16?

27 Memo #26 Can you figure out how this Jactus is growing? Week Height 8 19, ,122, ,615,843,020.8 How do you figure out how the Jactus is growing if you are given any two points in a table?

28 Memo #27 Explain why you multiply your equation for how the Jactus grows by the height of the plant at week 0:

29 Memo #28 Can you figure out a) How the Jactus is growing, and b) How tall it was at week 0 when it started growing? Week Height 24 3,355, ,374,182.4

30 Memo #29 Here s a Jactus that starts out at 1 cm tall and doubles every week: Week CM Week CM

31 Memo #30 Say you have a plant that is 1 tall when it starts growing at week 0, and it doubles. So a table of how the plant grows by weeks looks like this: Week Height How does the plant grow between Week 0 and Week 1? How tall is the plant after 1 day? Week Height Day 2 Days 3 Days 4 Days 5 Days 6 Days 7 Days (1 Week) 2

32 Memo #31 Explain what happens to the Jactus doubling plant when the initial height is something other than 1 inch. Why does it do this?

33 Memo #32 This is a table for the Jactus that doubles every week. The entries are approximate because the height values are actually decimals that keep going forever. How tall is it when it has grown for 0.25 weeks? Week Height ???

34 Memo #33 This is another table for the Jactus that doubles every week. How tall will the plant be at 2.5 weeks? Week Height ???

35 Memo #34 How is this Jactus plant growing? Week Height

36 Memo #35 We are going to take another look at the doubling Jactus that starts out at 1 at Week 0. These tables below are all for this plant: Week Height How does the plant grow from one week to the next? Week Height How does the plant grow every 3 weeks? Week Height , ,777,216 How does the plant grow every 10 weeks? Week Height ½ How does the plant grow every half week?

37 Memo #36 Can you used what you figured out before to see if this plant is doubling, tripling, or something else? Week Height ,719,476,736 How does this plant grow every week? How does this plant grow every 3 weeks? How does this plant grow every ½ week?

38 Memo #37 Can you used what you figured out before to see if this plant is doubling, tripling, or something else? Week Height , , How is this plant growing every 0.1 weeks? How is this plant growing every week? How is this plant growing every 10 weeks?

39 Memo #38 Yesterday you figured out something pretty cool. For any two weeks, you were coordinating the heights and the weeks like this: Week Height ,194, ,073,741,824 You wrote an equation: 11 = 4,194,304 1) What does this equation mean? 2) What does it have to do with how the plant is growing every week? 3) Will it always get 4,194,304 times as big each 11 weeks no matter what? Why or why not?

40 Memo #39 How does this plant grow every 0.1 weeks? How does it grow every week? Week Height

41 Memo #40 Can you used what you figured out before to see if this plant is doubling, tripling, or something else? Week Height ,719,476,736 How does this plant grow every week? How does this plant grow every 3 weeks? How does this plant grow every ½ week?

42 Memo #41 Can you used what you figured out before to see if this plant is doubling, tripling, or something else? Week Height , , How is this plant growing every 0.1 weeks? How is this plant growing every week? How is this plant growing every 10 weeks?

43 Memo #42 Here is a Jactus plant that s doubling every half a week: Week Height 0 1 ½ How is it growing every week?

44 Memo #43 If a Jactus plant doubles every DAY, how much does it grow every week? Make a table to show the Jactus growth.

45 Memo #44 The Flowering Jactus (FJ) doubles every half a week. The Evergreen Jactus (EJ) doubles every week. Both of them start at 1 tall at 0 weeks. Draw a graph of both the FJ and the EJ on the same set of axes.

46 Memo #45 Do you think that this Jactus is growing exponentially the same way throughout the table? Or is its growth speeding up or slowing down in some weird way? Week Height , ½ ,188,

47 Memo #46 a) If the Jactus grows 8 times every 3 weeks, how does it grow every week? b) How does it grow every half a week? c) How does it grow every day?

48 Memo #47 Say you looked at a table for the doubling jactus (you know that it doubles every week) and you found that from week 2 to week 5, the jactus got 8 times as big. And then you looked at some more points and saw that from week 23 to week 26, it got 8 times as big. And then you looked at some more points and saw that from week 50 to week 53, it got 8 times as big. Do you think it will ALWAYS get 8 times as big for any 3-week jump? Why?

49 Memo #48 Here is a table for a Jactus that is tripling every week: Week Height a) How much bigger would the plant grow from week 200 to week 202? b) How much bigger do you think the plant would grow from week 155 to week 160? c) How much bigger would the plant get in 10 weeks? d) How much bigger would the plant get in 1 day?

50 Memo #49 What does it mean for a plant to grow exponentially?

51 Memo #50 When you have a plant that doubles and has an initial height of 1, you have written equations like H = 1 2 w. a) If the initial height is, say, 5 instead of 1, what does your equation look like instead? b) Why does it make sense to write it the way you did?