that are adherent to each other, while 2 is to consider the two orientations of the same pattern.
As the conclusion, the teachers can briefly discuss with the students how this STEM exemplar can integrate the theoretical calculation and real-life application of Smartphone security effectively, and let them see how the advanced Counting skills can be used in reality.
An extended after-lesson activity may be arranged that the teachers can write down the number of Android Mobile Lock patterns which can be constructed using 4 and 5 digits. In the meanwhile, all students are asked to think whether they can develop any mathematical pattern if the number of patterns is unlimited for an increasing number of dots available on the mobile phone.
Third Lesson Plan – Suitable for Students having Average
Like our Second Lesson Plan, the teachers are suggested to begin from less number of dots like one or two only, to enhance the chance of success and the confidence of students in forthcoming challenges when the number of dots is increased to 3 or more.
This can also enhance the opportunity for building up more proactive and constructive student participation.
The teachers can start the lesson by emphasising the security importance in online data transfer for the protection of personal data. Besides, a brief introduction of the Enigma machine, an encryption device invented by Alan Turing in the mid-20th century to protect the commercial, diplomatic and military communication, can be used to discuss with students on the selection of best security method (PIN, Pattern or Fingerprint) to protect the valuable data in their mobile phones. This introduction can help to relate this essential real-life problem to the upcoming activities for students’ attention.
In the first part of the lesson, the teachers introduce the principles of setting Secret Screen Lock in Android Smartphone so the students understand what kinds of codes are valid or invalid. Each student has to find out the number of security patterns starting from 1 and 2 dots. A few students are
then asked to present his/her findings of each case on the board in details for class discussion.
When the students know only some simple counting principles, the most foreseeable approach for 2-dot case should be 9 8 – 8 2 = 56. But from what I observed in the tryout lessons some students could give other innovative and creative approaches.
One example is as shown below:
I. For digit 1, there are 5 valid connections: 12, 14, 15, 16 and 18. Applying the concept of rotational symmetry, digit 3, 7 and 9 should construct 5 valid connections.
II. For digit 2, there are 7 valid connections: 21, 23, 24, 25, 26, 27 and 29. Applying the concept of rotational symmetry, digit 4, 6 and 8 should construct 7 valid connections.
III. For digit 5, it can be joined to all other 8 remaining digits as valid connections.
IV. Correct Number of ways = 4 5 + 4 7 + 8 = 56
The students can relate an S.1 concept: Rotational Symmetry to facilitate their simple Counting!
With carefulness and effort, it is believed that many students could understand how to find the right answer 56 patterns with the help of ideas brainstormed from class discussion.
After the students get familiar with the locking patterns, the teachers can classify the students of
3 to 4 students and ask each group to find the number of Smartphone
security patterns using 3 dots. The teachers can help the students by dividing the 9 dots of the pattern into 3 groups: 4 pieces of corner C ( i.e. 1, 3, 7 and 9 ), 4 pieces of side S ( i.e.
2, 4, 6, 8 ) and 1 central dot O ( 5 ) and each group is required to think about the corresponding number of combination case by case.
The teachers can then assist the students using the easiest 3 corners ( 3C ) to explain its invalidity. The other cases like 2 corners and 1 side, ( 2C 1S ), 1 corner and 2 sides ( 1C 2S ), etc., are also introduced and the students are brought to see various possibilities in each case and its number of patterns available. The teachers can use a few diagrams drawn on the board to illustrate that
Case Connection Patterns Number of Patterns Formed
3C None 0
2C 1S CSC 4 4 3 = 48 SCC 4 2 1 = 8 2C 1O COC 4 1 3 = 12
OCC 1 4 1 = 4
1C 2S
CSS 4 4 2 = 32 SCS 4 4 3 = 48 SSC 4 4 2 = 32
1C 1S 1O
CSO 4 4 1 = 16 COS 4 1 4 = 16 SCO 4 4 1 = 16 SOC 4 1 4 = 16 OCS 1 4 4 = 16 OSC 1 4 4 = 16 3S SSS 4 2 1 = 8
2S 1O
SSO 4 2 1 = 8 SOS 4 1 3 = 12 OSS 1 41 3 = 12 Total Number of Ways 320
The students are guided to identify the number of patterns formed in different cases above and see if their work can be combined to generate the final answer 320 patterns correctly.
If time permits, the teachers can ask each group to design the Smartphone pattern using 4 dots and kept it secret. In each series, when a group is selected at random, the other groups could ask their members “YES” or “NO” questions to clarify the unknowns and to guess the pattern. The number of questions is unlimited. The group which can offer the right answer wins the series. The students should be active in discussion and can guess some patterns correctly with hinted questions raised.
The teachers can ask them to use advanced approaches in finding the above answers should they have learnt Permutation and Combination later.
Using the “Android Mobile Secret Screen Lock” exemplar above, as an illustration, we find that an exemplar can be used on the students of various levels if we could slightly adjust the teaching approach to cater for student diversity. Its introduction serves as an objective of “throwing a sprat minnow to catch a whale”, putting a sample here for colleagues’ reference so they
can also prepare their exemplars for different topics of various levels themselves confidently. Through sharing hands-on experiences by our frontline teachers, hopefully, we can turn out not only the curriculum leaders but also life-long learners to strive for academic excellence in Hong Kong.