As an extended activity, teachers can ask students to use the mathematical patterns they inquired from above to induce how many possible locks can be built using 5 to 9 dots and discuss in the next lesson should they have the interest to pursue advanced knowledge of Combination and Permutation.
Second Lesson Plan – Suitable for Students having Average
warm-up activity, each growarm-up is asked to design the Smartphone pattern using one digit only and kept it secret. They are then requested to calculate the number of possibilities, which is 9C1
= 9. It is believed they can work out the answer easily for some senses of success.
The teacher can then choose one digit by himself/herself as own secret code and let the students guess it. The students are allowed to raise creative “YES or NO” questions as hints for guessing the randomly set pattern, like “Did your selected dot lie above the middle horizontal line?”, “Did your selected dot lie on L.H.S. of the middle vertical line?”, etc. It is expected this simple activity can develop an engaging atmosphere that the students would maintain high-level participation and develop their generic skills by raising some meaningful questions throughout the process.
After the relatively simple warm-up activity, each group is told to design its Smartphone pattern using 2 digits this time and keep it secret. Teachers can ask the students to evaluate the number of patterns formed by selecting 2 out of 9 dots at random without restrictions, which is 9C2 2! or 9P2 = 72. The students are allowed again to raise some creative “YES or NO”
questions to guess the pattern like “Did your pattern contains
an axis of symmetry?”, “Did your pattern passes through the middle dot?”, etc. The answer should be guessed correctly after about 5 to 6 trials.
After successful identification of the pattern in this second activity, the teachers need to illustrate that the actual number of possible patterns is less than 72 due to some straight lines connecting 3 points. The teachers can give their students 3 to 4 minutes to analyze and to identify there are 8 straight lines (3 horizontal, 3 vertical and 2 diagonals) which contribute 8 2 = 16 illegal patterns. The real number of patterns that could be formed using 2 digits is thus 72 – 16 = 56.
After the student gets familiar with the rules of setting the lock pattern, the teachers can introduce the third activity in which each group will be asked to design the Smartphone pattern using 3 digits and again the “YES or NO” questions could be raised to guess the pattern. After two activities that the students have warm–up already, it is believed the discussion will be fiercer among students but no group may be able to guess the right answer due to too many possible choices.
The teachers can draw accordingly different figures on the board to teach the students how to find the correct number of
patterns that could be formed by using 3 digits. The teachers need to state that 9Pn for n > 1 ( i.e. 9P3 = 504 in this activity ), is not the correct answer. It is because there are illegal patterns generated, like 128, 537, 482, etc. which passes through either more than 3 points or some previously connected points that violated the principles. Teachers can apply a simple application of the Inclusion-Exclusion Theorem in Combinatorics to teach the students patiently in a step-by-step that
Correct Number of ways = 9P3 – 8 ( 2 2 6 + 2 ) + 12 2
= 504 – 8 26 + 24
= 504 – 208 + 24
= 320
The three components of the first step above can be carefully explained to students as follows:
I. The first term 9P3 is the number of ways selecting 3 digits among nine possible choices at random without restriction, which is the same as that appeared in the First Lesson Plan.
II. For the second term 8 ( 2 2 6 + 2 ), we have
(a) The digit 8 refers to 8 invalid straight lines (i.e. 13, 17, 39, 79, 19, 37, 28 and 46) with 2 points just counted but they have an intermediate uncounted point.
(b) The first 2 refers to the two orientations of the same invalid line, for instance, 136 and 631 are counted as 2 different ways even their appearance is the same.
(c) The second 2 refers to the flowing positions of an invalid line. For example, if it has three points 1, 5 and 7 chosen, it can be drawn either by 1475 or 5147.
(d) The digit 6 refers to the remaining 6 dots on the grid that can be chosen to draw the line.
(e) The last 2 refers to the number of invalid straight lines formed by 3 collinear points. For example, we consider the invalid line formed by three digits 1, 5 and 9. There are 3! = 6 different arrangements in which four ( 159, 951, 519 and 591 ) are valid while the other two ( 195 and 591 ) are invalid.
III. For 12 2, we have to aware that in II we have some invalid patterns double–counted that we need to deal with and take them out. There are 12 such patterns concerned ( 139, 317, 179, 397, 137, 319, 197, 379, 173, 391, 719, 739 ), all involve any 2 of the 8 invalid straight lines in II. (a) above
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.