##
* *Reflection: Real World Applications
Cat Island (Day 2 of 2): Cats can’t add but they do multiply! - Section 4: Improving Solutions

WARNING: This was a very time consuming assignment to 'grade.' I would definitely recommend solving the problem for yourself before trying to attempt the read students' solutions.

The first thing that I learned from analyzing my students’ work is that my students need more practice on CLEARLY demonstrating their thought process, justifying their work, and stating assumptions. It was so difficult to follow a majority of my students’ thinking and nearly all of my students did not clearly state their assumptions as they were asked to do.

These were my **top 4 favorite student approaches **for clearly demonstrating their thought process:

**Error Analysis of a few Samples of Student Work**

Many of my students (and myself) missed including one or more of the given constraints.

- Constraint not included:
*Kittens can get pregnant at 4 months of age.*

- Constraint not included:
*Cats on average only have 3 litters a year. (or maybe up to 6 in the 18 months)*

- Constraint not included:
*Time constraints on future generations being able to reproduce in 18 months.*

Many students also didn’t account for males’ line of descents, but I understand that was difficult to track.

- Constraint not included:
*Only factoring in that female cats can create decedents of first cat.*

**My Attempts**

To help me really get a good grasp of the problem, I made a few attempts at solving before I read through my students’ work and before I read the solutions from MARS. After two attempts on my part, I somewhat sympathized with my students. This problem gets complex fast and it really was a challenge to keep my thoughts clear and organized to myself, much less another reader. Interestingly, I missed a major given assumption in both of my attempts: female cats can only carry 3 litters of kittens a year. (Note: The MARS solutions really are much more efficient and clear than either of my methods. It’s fun when I teach a lesson and get to learn something new myself!)

My first attempt:Cat Island, my first attempt

Assumptions:

- first cat got pregnant at the start of time, so first litter was born at 2 months
- high estimation, assumed 6 cats per litter with 3 boys and 3 girls
- Only counting decedents from the female cats, as male reproduction is hard to track
- Giving the females cats one month to get pregnant after giving birth

My second attempt: Cat Island, my second attempt

I failed to include the fact that cats can only have about 3 litters a year.

Assumptions:

- Mom was already pregnancy, so first litter was born at 0 months
- Cats have 6 kittens every litter
- Pregnant back-to-back (oops… missed that cats can only have 3 litters a year)
- Assuming now that males will reproduce at the same rate as a female (which is probably way too low of a rate)

*Real World Applications: Analysis of Students' Final Responses*

# Cat Island (Day 2 of 2): Cats can’t add but they do multiply!

Lesson 2 of 14

## Objective: SWBAT investigate an exponentially increasing sequence and make sensible estimates and assumptions based on this sequence.

#### Revisit Cat Island

*2 min*

Refresh students' memory about Cat Island and the problem they explored in the last class period by reprsenting pg. 2-9 of the flipchart (it’s just a repeat of the story from yesterday).

This lesson follows the MARS Mathematical assessment Project *Modeling: Having Kittens* lesson that can be found here. However, I am cutting a few pieces and speeding things up a bit so that we can wrap it up in today’s 50 minute class period (see below for details on that!).

*expand content*

#### Group Solving/Make Poster

*20 min*

Pass back students’ individual papers from Day 1 with comments on them. See page 4 of the PDF *Modeling: Having Kittens* lessons from the Mathematics Assessment Project for some great starting points for questions to prompt students to think about their current misunderstandings or pieces of the problem they are missing. (If you don’t have time to leave individual feedback for students I recommend looking over the papers and getting a general idea of which prompting questions would best suit your class.) In my resources, have left pg.10 of the flipchart blank so that I can add common issue questions to discuss with the whole class after reviewing their work on Day 1.

Once papers are returned to students, group students into new teams. I am going to break apart the teams students typically work with on this problem. My goal is to enable new approaches to the problem for each group. I hope richer mathematical conversations will result from mixing up the teams at this point. I may even number student’s papers as I review them and have all student with the same number group up. If I do this, I will consider different approaches to the problem as well as who will work well together. If the teams work well together, I may make these teams their new quarter 2 teams.

Teams will work together to produce a final answer, showing their work to explain their strategy. Students will need to be able to explain their solution and should make these visuals to help explain it. I am not sure exactly yet what form I want these visuals to take. Posters? Butcher paper? Just notebook paper? Maybe large whiteboards? I am leaning more toward whiteboards (with a huge emphasis on not erasing until the end up class!) since I am not planning on collecting it.

#### Resources

*expand content*

#### Stay and Stray

*8 min*

Once students have had some time to formalize and agree on a solution in their teams, I am going to ask the teams to complete a **Stay and Stray**. One student will stay with their poster and be the expert. While the other two team members go visit different posters. The idea here is that students will check in with other teams and maybe find some ways they can improve their solutions. I am going to allow students about 2 minutes per rotation. I only plan on doing two rotations. So each ‘straying’ student will only visit two other posters, but collectively each team will gather information from 4 different posters. Then that should give students about 3 minutes to report back to their teams and discuss their findings.

#### Resources

*expand content*

#### Improving Solutions

*20 min*

To help get students refocus after the stay and stray I present the prompting questions I prepared on page 11 of the flipchart and lead a whole class discussion about the student work on pages 12-14. The goal of this part of the lesson is to address the misconceptions that were identified in students' individual solutions. I know there are many great conversations that could occur in analyzing the work and we could probably use a whole class period doing this, but I plan to take about 5 minutes discussing some of the constraints of and assumptions in students’ work. After this brief discussion, the remaining class time will be given to students to improve their solutions and write their final conclusions.

For me, the primary goal of the Cat Island Investigation was for students to see what it means for a population to increase exponentially. I will be refering back to this problem throughout the Exponential Functions unit as a real life example of how quickly growth can occur when the underlying process is exponential. So I will not view the lesson as unsuccessful if all students do not come to a correct mathematical conclusion. The focus was more on developing their ability to understand the situation and to construct a model that enables further exploration.

*expand content*

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- UNIT 1: Basic Functions and Equations
- UNIT 2: Polynomial Functions and Equations
- UNIT 3: Rational Functions and Equations
- UNIT 4: Exponential Functions and Equations
- UNIT 5: Logarithmic Functions and Equations
- UNIT 6: Conic Sections
- UNIT 7: Rotations and Cyclical Functions
- UNIT 8: Cyclical Patterns and Periodic Functions
- UNIT 9: Trigonometric Equations
- UNIT 10: Matrices
- UNIT 11: Review
- UNIT 12: Fundamentals of Trigonometry

- LESSON 1: Cat Island (Day 1 of 2): Cats can’t add but they do multiply!
- LESSON 2: Cat Island (Day 2 of 2): Cats can’t add but they do multiply!
- LESSON 3: Graphing Exponential Functions
- LESSON 4: Shifting Exponential Functions
- LESSON 5: Counting the Change: Linear, quadratic, or exponential? (Day 1 of 2)
- LESSON 6: Counting the Change: Linear, quadratic, or exponential? (Day 2 of 2)
- LESSON 7: Sorting out the Change
- LESSON 8: Stretching Exponential Functions (and your mind)
- LESSON 9: Credit Card Investigation: What is interest? (Day 1 of 4)
- LESSON 10: Credit Card Investigation: What is interest? (Day 2 of 4)
- LESSON 11: Credit Cards Investigation: How is interest really calculated? (Day 3 of 4)
- LESSON 12: Credit Card Investigation: A Dastardly Scheme (Day 4 of 4)
- LESSON 13: Exponential Functions Test Review
- LESSON 14: Exponentials Functions Unit Test