Exploring the function zoo A 60-80 minute exploratory lesson prior to an extensive functions sequence.
Some preliminary comments for teachers
Welcome to Advanced Mathematics!
THE CHALLENGE: Senior mathematics is a roller coaster ride.
In my jurisdiction we cover 95 key concepts and skills in the first 10 weeks. True much of it is revision from junior school – but it’s A LOT of content.
How it feels for many students after the first few months ….
How they likely organised the information in their mind …
What it looks like after two years of senior school ….
Where it ends up after their last exam……
Are there ways to manage so much information? YES! Try this : can you recognise most of the animals in here?
How do you do this? Most likely because you have organised related animals into collections of ‘families’ – in to a type of ‘zoo’.
Could we do this with all the mathematical creatures (the functions)? ABSOLUTELY!
The Function Zoo Lesson is a teaching sequence to help students recall and organise everything they know about the different functions.
Exploring the function zoo This lesson is an adaptation of an idea from Mary Barnes. It provides an excellent opportunity to observe and listen to student mathematical discussion and see how they (if) they can work collaboratively, and how they engage with high and low tech mathematical tools. I recommend using this lesson BEFORE intensive investigation of the different functions. The idea is to provide a framework for storing the information to come.
The Function Zoo: Lesson Sequence
1. Explain the idea
4. Class tour: whole class visits each group to discuss the function zoo drawings.
2. Provide the tools
5. Teacher led discussion on making connections – within and between function groups. Hint at future connection.
3. Give lots of time to students (at least 20 minutes. 30-40 is best). OBSERVE!
6. Student make a record of their learning.
What I saw the students doing • • •
High quality sustained peer tutoring and learning. Learning and practicing the meta language. Friendly competition between the groups.
How it helped me • • •
Listening to extended student conversations about maths. Differentiation: extension questions tailored to each group. Efficient : can work with 4 students at once.
High Tech – Low Tech • • •
Butcher paper forced working at a high levels of abstraction. Mathematical software allowed exploration of harder equations. Results recorded in traditional formats.
Highly targeted lesson • • •
Clear goals for the lesson. Class debrief at the end to ensure shared learning. Concrete student output reused throughout the topic.
Our subject is too special to be thrown out after the final exams
What does your zoo look like?
Can you help your students organise their subject knowledge so they will be able to use it for the rest of the lives?
Teacher annotation in these boxes. You will probably wish to delete them for your lesson. Or use the section below ready to go without annotations.
I like to open with the video “Animal Beatbox” (3 mins) http://www.youtube.com/watch?v=vxiSP_ch_oI It engages the students and it demonstrates to students they know A LOT about all the different animals.
Question: Can you make connections? Can you organise in a way that makes sense?
We learn to make our own groupings as young children
Later we learn more formal organising principles.
Which helps you make sense of this!
What if we could organise mathematical functions in this way?
𝒚 = 𝒙𝟐
𝒚 = 𝒙𝟐 + 𝟐 𝒚 = −(𝒙 − 𝟏)𝟐
Group work: 20 minutes • Can you sketch 8 (or more) different function families? • Organise the function families in a way that makes sense to you.
GeoGebra: A tool to explore ideas (or any other graphing tool will do. desmos.com is another excellent tool
Student output: on butcher paper. Insist the whole group work on the same sheet - not individual sheets
Your turn! Groups of 3 30 minutes
• Can you sketch and label
8 (or more) different function families?
I turn a blind eye if students ‘cheat’ by looking at other groups – so long as they have made a good initial start themselves.
Use this time to observe! Fascinating to watch and listen to the mathematical conversations.
10 minutes before the end, throw this up on your data projector – but don’t say anything. I don’t even tell the class I put it up – they will notice eventually.
How many of these do you have?
Lesson Closure
Take the whole class on a visit to each zoo. Have each group explain in turn what they drew. Ask them to evaluate their own work.
Connections within each family:
I then go into some more explicit teaching to ensure the see key idea of how variations on one type of function – via translation, reflection and scaling fit into one family. All these seemingly different functions are really just variations on one key idea. Just like the difference between a lion, a tiger and a jaguar.
𝒚 = 𝒙𝟐
𝒚 = 𝒍𝒏(𝒙)
𝒚 = 𝒙𝟐 + 𝟒
𝒚 = 𝒍𝒏(𝒙) + 𝟒
𝒚 = (𝒙 − 𝟓)𝟐
𝒚 = 𝒍𝒏(𝒙 − 𝟓)
At first these look different:
… later we see the connections.
At first these look different: And then show how even quite different looking functions are related. Your students have likely already built some connections between the different degree polynomials. This is a nice set up for later on when students will see connections via gradient functions.
… later we see the connections.
At first these look different:
… later we see the connections. At first these look different:
This is useful is you plan on teaching 𝒆𝒊𝜽 = 𝐜𝐨𝒔𝜽 + 𝒊 𝐬𝐢𝐧𝜽 later on in the course
… later we see the connections.
15 minutes
I ask the students to record their understanding of the function zoo in their notebooks.
Page 4 of 57. Week 3 & 4. Week 5 & 6. Week 7 & 8. Week 9 & 10. Week 1 & 2 95 concepts and skills. In my jurisdiction we cover 95 key concepts and skills in the first 10 weeks. True much of it is revision from junior school â but it's A LOT of content. Page 4 of 57. The-Function-Zoo-A-Group-Exploration-Lesson-Design-v7.pdf.
