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Week 1
Main points:
Seemingly repetitive patterns may not repeat indefinitely.
Inductive reasoning leads to a conjecture.
A conjecture may be disproved by a counterexample or be proven by mathematical induction (MI). See another example of MI.
In an arithmetic sequence, the difference of consecutive terms is always the same.
For an arithmetic sequence, the n th term is given by a + (n - 1)(d), where a is the first term and d is the common difference.
For a geometric sequence, the n th term is given by ar^(n - 1), where a is the first term and r is the common ratio. Note to TG2: Do not worry that we did not do this topic in class. Please refer to this commentary. (Note: The correct answers to Exercise 1.6 are 2^9 after 8 hours and 2^(n + 1) after n hours. Sorry!)
Week 2
Main points:
For enrichment: sum of n terms of an arithmetic sequence = 1/2 n [2a + (n - 1)d]. This result is useful for us to prove the conjecture in tutorial 1 question 4(c) directly.
The algebra method makes a good choice when the model method is cumbersome to be used for fractions that are not nice.
Week 3
Main points:
The balance analogy is useful for teaching students about addition and multiplication properties of algebra.
When we encounter a quadratic expression, it is convenient for us to complete the square. See this short discussion for a better understanding.
For next week, we'll use Geometer SketchPad. Here is a set of materials on GSP from DSM101 for your reference.
Week 4
Main points:
Knowledge of algebra is necessary in mathematics. Here is a resource site on various algebraic skills.
Hint to the hour- and minute-hand problem: Suppose the time after 12 is x.y, where x is the hour part of time and y is the minute part of time. For example, if the time is 1.15, then x = 1 and y = 15 and we note that x has to be a positive integer and y is any real number between 0 and 60 (inclusive of 0 and not including 60). Let's examine the time of 1.15 closely and check out the hour- and minute-hands by means of proportional reasoning. For this time, the minute-hand is at 90 degrees, obtained by 15/60 of 360 degrees. At 1 o'clock exactly, the hour-hand is at 1/12 of 360 degrees, i.e. 30 degrees. However, the hour-hand will be more than 30 degrees for the time of 1.15, and it is between '1' and '2', depending on the minute part of the time. Since the minute part is 15, we need to add another 15/60 of 30 degrees. Thus, the hour-hand makes an angle of 37.5 degrees, and we note that both hands make different angles. Now the problem requires us to find x, y such that the hour- and minute-hands coincide. Since the hands coincide, we will be able to establish a relationship between x and y. This link also discusses the same problem.
1.Redo the questions from Tutorial 3 but using Geogebra, which is freely downloadable from the Web, rather than GSP.
2. Note any differences between the two software packages which you think have an impact on teaching which might include: ...... technical aspects such as how angles are measured ..... or presentational aspects such as the way the labelling is handled.
3. Decide which of the two packages you think is the better teaching tool and write one paragraph explaining your decision.
4. Submit files with your work for each of points 1, 2 and 3 above to your tutor during e-learning week (deadline negotiable if you came to know this late).
Week 6
Main points:
Chapter 4 notes with comments, including last year's quiz questions on the last page. Answers to quiz questions: Q4 (a) False. Refer to the definitions of polyhedron and cone. (b) True. Which regular polyhedron satisfies the properties of a pyramid? (c) False. We have gone through this in class. (d) True. Are prisms cylinders? (e) False. Take a close look at the five regular polyhedra. (f) False. Is a quadrilateral a square?
Q5 (a) n + 1 (b) n + 1 (c) & (d) We have done them in class.
Q6 f = 10, v = 9, e = 16 and f + v - e = 3. So Euler's formula does not work in this case. This object is not a polyhedron and Euler's formula is only valid for all convex polyhedra.
Tutorial 4 answer keys (note: you still have work to submit; refer to the requirements indicated in red) Q1 (a) 5, (b) 4, (c) 4 Requirement: Provide a reason for each part.
Q2 (a) True, (b) False, (c) True, (d) False, (e) False, (f) True Requirement: Provide reasons for (b), (d) and (e).
Q3 - Q5Requirement: Do it yourself.
Q6 Pentagon: 5, 2, 10 Hexagon: 6, 3, 18 Requirement: Complete the table for Heptagon and n-gon.
Q7 Left solid: 7, 12, 7 for V, E, F respectively (thanks to Noel for pointing out the error). Requirement: Find V, E, F for the right solid.
Q8n + 2, 2n, 3n for F, V, E respectively. Yes for (d). Requirement: Give reasons to support (a) - (d).
Return of tutorials (apologies for the delay): TG 2 - please collect from my pigeonhole (block 7, level 3) on Wednesday (10 Sep) at any time. TG 4 - please collect from my pigeonhole (block 7, level 3) on Thursday (11 Sep) from 11am. Look out for a hanging string to retrieve the tutorial envelopes. Thanks!
Answers to the quiz 1 of last year, for your reference.
Week 7
Main points:
Quiz 1 on chapters 1, 2 & 4. We encountered the vertex and total deficit in the quiz. This site gives us a brief description of the concepts.
Please cover chapter 5 pages 1 - 4 on your own first.
I'll put up the rest of chapter 5 for your advance reading. We'll probably do hands-on activities during lesson after the recess week.
Worksheet to prepare for Quiz 2 (put up in advance, though we'll look at them after completing chapters 5 & 6).
Have a good rest during recess week!
Week 9
Main points:
Answers to some questions in tutorial 3: Q2 True, since angle at the centre is twice the angle at the circumference.
Q3 Angles inscribed in the same or congruent arcs are congruent.
Q4 Opposite angles in a cyclic quadrilateral are supplementary.
Q5 Value = 0 if angle is 90 degrees (Pythagoras' Theorem). Value > 0 if angle < 90 degrees. Value < 0 if angle > 90 degrees. The results come from the Cosine Rule, whose RHS is actually 2(AB)(AC) cos (angle BAC).
Q6 Height remains the same, so area is unchanged.
Outline of chapter 5 1. 4 stages of statistical investigations 2. Representations of data (notes for reference): - pictographs - dot plots - stem-and-leaf plots - back-to-back stem-and-leaf plots - frequency tables - histograms and bar graphs - line graphs - scatter plots - circle graphs (or pie charts)
Outline of chapter 6 (notes for reference) 1. Measures of central tendency, i.e. mean, mode, median 2. Measures of spread, i.e. range, lower quartile, upper quartile, interquartile range 3. Outliers, comparing sets of data 4. Variance and standard deviation 5. Misuse of statistics
I plan to make lessons on chapter 5 & 6 hands-on, with you doing work on the spot with my guidance. You can print copies of graph paper (on which most of your statistical graphs will be drawn) here, to save cost of buying it :)
I will use these examples for our coming 2 statistics lessons (where I will focus on important skills) before I go for in-camp training between 13 & 25 Oct. I will make some copies in case you have not printed before lessons.
Source of examples: Elementary Statistics 9th Edition by Mario F. Triola, NIE library call number QA276.12 Tri 2005
Wishing all muslim students a very Happy Hari Raya Aidilfitri!
Weeks 10 - 13
Main points:
Summary of main concepts of chapter 6: 1. Mean, mode, median. - Mean = (sum of data points) / (number of data points)
- If 2 modes occur, the data set is bimodal.
- If there is an even number of data points (arranged in ascending order), we take the average of 2 data points in between. For example, we consider the average of 5th and 6th data points for a data set of 10 items.
- If there is an odd number of data points (arranged in ascending order), we take the only data point in the middle. For example, we consider the 5th data point for a data set of 9 items.
- It is important to be mindful that mean, mode and median are not necessarily good measures. Say we have 0, 100 as test scores. We see that the mean of 50 is not representative at all.
2. Range, upper quartile (Q3), lower quartile (Q1), interquartile range (IQR). - Range = maximum value - minimum value
- It is interesting to know that there is "not universal agreement on a single procedure for calculating quartiles, and different computer programs often yield different results". I'm quoting it from the text entitled "Elementary Statistics 9th Edition by Mario F. Triola", pages 94, 95.
- We will use this procedure (again from the same textbook) to find Q1, Q3: Step 1. Sort the data in ascending order.
Step 2. For Q1 (i.e. 25th percentile), compute L = 0.25(n). For Q3 (i.e. 75th percentile), compute L = 0.75(n). In each case, n = number of data points.
Step 3: If L is a whole number in any case, the quartile is taken to be the average of Lth and (L+1)th data values. Otherwise, round L up to the next larger whole number and the quartile is taken to be the (L+1)th data value.
As an example, say we have 39 data values. For Q1, L = 0.25(39) = 9.75 so we take the 10th element in the data set. To obtain Q3, L = 0.75(39) = 29.25 so we consider the 30th element in the data set.
As a second illustration, suppose we have 10 data values. For Q1, L = 0.25(10) = 2.5 so we take the 3rd element in the data set. To find Q3, L = 0.75(10) = 7.5 so we take the 8th element in the data set.
Naturally, these steps can be applied to find the median too.
- Interquartile range = upper quartile - lower quartile
- Outliers are data points below the lower cutoff point or above the upper cutoff point.
- To draw a boxplot, we need to include the maximum (i.e. the data point that is just within the upper cutoff point), Q3, median, Q1, minimum (i.e. the data point that is just within the lower cutoff point) values. On the boxplot, we also include outliers that are represented as asterisks, i.e. '*'.
- Parallel boxplots are useful for comparisons to be made between 2 or more data sets.
- Refer to this Excel file on how we may make use of Excel to do statistics. Learn practical formulae through the cells I have marked with orange colour.
4. Variance, standard deviation. - Variance = {sum of squares of (x - mean)} / number of data points
- Standard deviation = square root of variance
- It is usually desirable to have a small standard deviation, i.e. less spread or variability from the mean.
- We will go through the solution to example Q9 when we meet next time.
Tutorial questions to be submitted for grading: Q1, Q2, Q3, Q6, Q7, Q9, Q11, Q14, Q15.
For weeks 11, 12 please proceed to LT3 for your lessons with Dr. Shutler, as I will be away for in-camp training. Try to seek hints from him for quiz 2, haha!
On week 13 when I return, we plan to do a review of chapters 5 & 6 with the aid of the worksheet I have posted up previously. Please try the questions in advance.
Note to TG2: We'll miss a lesson on Monday (27 Oct) as it is Deepavali. I plan to do a make-up lesson on Friday (31 Oct) between 12pm and 2pm or between 2.30pm and 4.30pm at TR63.
Preparation for quiz 2
Main points:
Comments on some questions in tutorial 6:
Q8 SD = 0. Approach: Since all numbers are equal, say x, so mean is x. Therefore, sum of squares of every difference between the number and the mean is 0.
Q9 Equal. Approach: Since SD = 0, sum of squares of every difference between the number and the mean is 0. This implies that every number is equal to the mean.
Q12 Mean (i.e. 3416.7) is not representative of this list: 100, 100, 100, 100, 100, 20000. Median (i.e. 50) is not representative of this list: 0, 0, 0, 50, 100, 100, 100.
Q15 1. No title to describe the line graph. 2. Both axes are not equally spaced. 3. No title for the horizontal axis.
Q9(a) mean = 60.9, median = 65, mode = 65 (b) Q3 = 78.5, Q1 = 46 (d) No outliers. (e) v = 494, s = 22.2
Page 2 Q21(b) Year 2000: min = 12900, max = 54000, Q1 = 33850, Q2 = 36450, Q3 = 48450 Year 2001: min = 15000, max = 34900, Q1 = 27250, Q2 = 28950, Q3 = 31000
(c) Year 2000: end-points are min and max above Year 2001: end-points are 25500 and max above, with outlier 15000
(d) Prices in 2000 are generally higher than prices in 2001.
Page 3 Q1(b) Both have same shape in terms of distribution. Histogram only shows frequencies whereas stem-and-leaf plot captures raw data in addition to frequencies.
(c) mean = 9.23, mode = 8.8, 9.4, median = 9.2 Depends on how one argues. See article below.
(d) range = 7.8, s = 1.54 s is most appropriate as it gives the spread about the mean, whereas range does not give any precise indication of distribution.
Page 4 Q2(b) Greater range for brand X than brand Y. Greater IQR for brand X than brand Y. Greater Q1 and Q2 for brand Y than brand X. Greater Q3 for brand X than brand Y.
(c) Brand Y, because it has a much smaller range of lifetime, indicating higher reliability.
(d) mean = 69.3, standard deviation = 18.1, so 83 is within one standard deviation from the mean.
Page 5 Q1(a) Because it has the smallest median (or Q3) among the three.
(b) Because it has the smallest IQR among the three.
(c) Beef, because it has the largest range of sodium content among the three.
