New Senior Secondary Mathematics Curriculum Sharing of SBA Task Exemplars (Draft) HKEAA September 2006

Resources for Setting Assessment Tasks (1) Exemplars from HKEAA & CDI 2008 15 tasks 2009 2011 10 tasks 2013

NSS Math Curriculum Sharing of SBA Task Exemplars First Exemplar: Structures Second Exemplar: Sweets in boxes Third Exemplar: Performance of Broadband Internet Service Providers

First Exemplar: Structures Part A John uses some square tiles to pave the walkway in his garden. He uses six white tiles to form a 2 x 3 rectangle.

He adds more and more tiles to the original six tiles, forming the following three patterns. Students are required to count the number of tiles in these patterns and investigate the number of tiles in the nth pattern.

Part B Peter uses some cubes to make 3-D models. He uses six cubes to form a 2 x 3 x 1 rectangular block.

He adds more and more cubes to the original six cubes, forming the following three models. Students are required to count the number of cubes in these models and investigate the number of cubes in the nth model.

Teacher Guidelines This task requires students to observe number patterns in some structures; make a conjecture through generalization; test / justify the conjectures. It is anticipated that in general, students can complete the task in 2 hours. However, teachers can exercise their professional judgment to adjust the time allowed to cater for the needs of their students.

If necessary, teachers may give a brief description of the task at the beginning, e.g. models made with multilink cubes may be useful for introducing Part B. Teachers may feel free to modify the question to cater for the needs of their students, e.g. using other shapes or solids as the starting pattern. Feedback should be provided to students after marking the task, e.g. different approaches in handling each part could be discussed.

Second Exemplar: Sweets in boxes Parts A and B Investigate the existence of integral solutions of some simple linear equations, such as 2 x + y = 7 2 x + 3 y = 13 x + 2 y = 12 2 x + 4 y = 26 3 x + 6 y = 5

Parts C and D Donald broke open all large and small packets and poured out all the sweets inside. Students are required to find the numbers of sweets originally contained in the packets.

Part E Students, being managers of logistic companies, are required to pack 108160 chocolates into 192 large size boxes and 80 small size boxes, subject to some extra requirements.

Part F Given that the linear equation a x + b y = c has infinitely many integral solutions. Students are required to investigate some relations among a, b and c .

Annex 1: Diophantine Equations Diophantine equations are equations with two or more variables whose values are restricted to integers. E.g. x + y = 4 is a linear Diophantine equation which has infinitely many integral solutions, such as (-2,6), (-1,5), (0,4), (1,3), (2,2), … If x and y are restricted to positive integers only, then the equation will only have three solutions, namely (1,3), (2,2) and (3,1).

Teacher Guidelines This task requires students to solve linear equations in two variables; understand some basic knowledge about Diophantine equations, choose a suitable strategy to handle daily life problems, carry out the plan and evaluate the solution obtained.

Annex 1 provides a simple introduction to Diophantine equations and some techniques in determining whether a Diophantine equation has infinitely many integral solutions or not. Teachers may present it to students to study before the assessment activity (either at home or in class), or describe it briefly at the beginning of the assessment activity.

Third Exemplar: Broadband ISPs Part A The downloading times of the same file through ISP X and ISP Y are given. Students are required to represent the given data by suitable statistical measures and comment which ISP performs better.

Part B A researcher records the downloading times of files of different sizes. Students are required to explain which ISPs performs better, state some limitations of the research and suggest some possible improvements.

Teacher Guidelines This task requires students to select and use appropriate statistics measures to analyse data; select and use appropriate charts to analyse data; interpret results; point out limitations and suggest improvements.

If necessary, teachers may give a brief description of the task at the beginning, e.g. teachers may consider to explain technical terms involving internet services such as ISP, file size (MB) and downloading speed (Mbps) etc. Teachers may feel free to modify the question to cater for the needs of their students. For example, students using different ISPs can be grouped in pairs. Instead of providing the data set, teachers may ask students to collect data for this task at home and tabulate them as in the tables given in Part A and Part B.

Thank you !!