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ILLUSTRATIVE SCHOOL PROGRAM B – Sample Unit B Topic 5: Saving and Borrowing Subtopic 5.1: Financial Institutions Key Questions and Key Ideas Why invest and borrow money? Where can money be invested or borrowed? Banks Credit unions Investment companies. What does it cost to invest and borrow money? Government fees and charges Financial institution fees and charges. How often is the interest calculated and charged or paid? Term deposits Personal loans Savings and cheque accounts. Under what conditions are various rates of interest offered for particular periods and amounts of investment? Rates of interest Terms of loans. Teaching and Learning Activities and Assessment Tasks Class discussion used to explore the reasons people invest and borrow money, and where these transactions can occur. Directed Investigation 1 (Part A) As a follow-up to the class discussion above, pairs of students will select different financial institutions to research. This will be done personally, by phone, or on the internet. Information is to be sought about: three different savings accounts – interest rates offered, fees & charges, services available and any conditions that apply. three different credit options (personal loan, mortgage, and a credit card) – interest rates charged, frequency of payments, conditions, other fees and charges. The results of this research are to be shared with the whole class in short oral presentations by each pair of students. The information is then to be collated by the class in an appropriate form for all to use in future class exercises. Subtopic 5.2: Investing for Interest Why invest and borrow money? (revisited) Learning task - Initial „motivational‟ questions to be formulated by the class in scenarios of their own choosing with teacher guidance, eg if you had $4000 dollars to invest what questions might you want to know the answers to? If you wanted to go on a holiday some time in the future what questions might you be asking yourself now? If you want to own a stereo system how could you finance it? etc. Answers to these questions to be sought at the appropriate time during the teaching of simple and compound interest using current information collected by students in the Directed Investigation. SSABSA Support Materials: 6ca4c744-3d99-4e87-8e54-30ac8c347deb.doc, 3 November 2001, page 1 How does simple interest work? Effects of changing the principal, interest rate, and time. How is simple interest calculated, and in which situations is it used? Use the formula SI = PRT to find the: 100 Review of the calculation of simple interest from the previous year. Spreadsheet application used to answer “What if..?” questions and examine, graphically, the linear nature of the growth of a simple interest investment. Rearrangement of the simple interest formula to solve the “What if…?” questions directly. Practice of skills using textbook questions. Extension of more able students – deriving the slightly more complex formula required to find P if only R, T and the final amount (P+I) are given. simple interest; principal; interest rate; time invested in years How does compound interest work? Effects of changing the principal, interest rate, and time. How is compound interest calculated? How is the formula A = P(1 + i)n derived? Use of the formula to find the amount accrued principal. Initial investigation of compound interest using a spreadsheet. Solving “What if…?” problems to find i, n and P as well as A and I. (These include the problems originally posed by the students in the opening lesson of the topic.) Derivation of the compound interest formula and its various rearrangements. Use the formulae to confirm the spreadsheet answers to “What if…?” questions and practise skills on textbook questions. Graphing calculators used by students to compare the effects of changing the compounding period and to find the „breakeven‟ point between simple and compound interest at different rates. Directed Investigation 1 (Part B) Which is the better option: simple or compound Use some of the information researched in Part A to calculate the answers to three of your own interest? „hypothetical‟ investment questions. Comment on the answers. Using „local knowledge‟ investigate the potential return of investing money in some other way, eg buying shares, land, coins, an old car and doing it up, etc.(choose something that interests you). Convert the return to an effective annual compound interest rate. Compare this investment with a bank account, discussing potential return, convenience, risk etc. Subtopic 5.3: Costs of Borrowing Why do many people use credit to buy expensive items rather than save for them? Investigate the various forms of credit available, and discuss their advantages and disadvantages. For example, interest-free periods on credit card accounts can substantially increase the interest charged on outstanding balances. SSABSA Support Materials: 6ca4c744-3d99-4e87-8e54-30ac8c347deb.doc, 3 November 2001, page 2 What types of credit are available? Calculation of the total cost of flat-rate loans and comparison with paying cash. Investigation of the cost of buying something on hire purchase. Discussion of the extra costs to the purchaser of this type of credit. Reexamine the government and financial institution fees and charges associated with loans from the research done earlier in the topic. Learning task - Comparing credit options using local retail examples (payment on terms) with credit options from financial institutions researched in the D.I. Class discussion of the benefits and drawbacks of the various options for people in different situations. Class discussion of when is it better to borrow than save. Skills and Applications Task 1 – Test on skills learnt in this topic. Topic 8: Geometry and Mensuration Subtopic 8.1: Measuring Instruments Key Questions and Key Ideas Choosing an appropriate instrument for taking a given measurement. Using the instrument correctly. Knowing the degree of accuracy of a measurement made with a specific instrument. Subtopic 8.2: Right-angled Triangle Geometry What tools are there for solving problems involving right-angled triangles? Pythagoras‟ theorem Trigonometric ratios. Are these tools sufficient to solve any such problem, given appropriate data? What is the minimum data required? Learning task - estimation of the height of various tall objects in the local vicinity using estimation by eye, measuring angles of elevation (trigonometry) and measuring shadows (similar triangles). Class collation of the results to look at variation and to stimulate discussion of reliability of the answers, sources of error and simplifying assumptions made when applying mathematical models to a real situation. Textbook questions for skills practice with Pythagoras rule and right-angled triangle trigonometry. Teaching and Learning Activities and Assessment Tasks Preliminary practical measurement exercises outside the classroom using estimation, measuring tapes, trundle wheel, clinometers, protractors and a thoedolite correctly. Discussion comparing the accuracy measurement and appropriateness of the use of the instruments in different situations. SSABSA Support Materials: 6ca4c744-3d99-4e87-8e54-30ac8c347deb.doc, 3 November 2001, page 3 Subtopic 8.3: Areas of Non-right Triangles and Related Compound Shapes How do you find the area of a non-right triangle if the perpendicular to a side cannot be measured easily or accurately? What alternative measurement data should be used in this case? What other more complex areas can be calculated using triangles? Derivation of the formula A = 1/2 a.b.sin C , using right triangles. Practical and contextual problems that require students to decide which measurements to take in order to find a specified area will be posed. Students carry out the exercise and confirm their result by comparing it with an alternative calculation (where possible). Textbook questions for skills practice with calculation of areas. Learning task - individual student estimation of the area and perimeter of a large irregular shape (with curved boundary), followed by radial surveys carried out by small groups of students using only four or five points (theodolite to remain in the same position throughout). Students will calculate the approximate area of the shape using the irregular polygon from their survey. Results will be compared in class then combined to refine the approximation to the best possible estimate. Class discussion of errors as with the previous exercise. Subtopic 8.4: Solving Problems Involving Non-right Triangles How do you solve problems in which the triangles involved are not right-angled? How much information about a triangle is needed to determine all its measurements? Solving triangles where two sides and the included angle are known. The cosine rule. The need for tools to deal with non-right triangles will be emphasised by posing problems in surveying or in a surveying context. Students will be asked how they would find the answers to these problems using the skills they have learnt so far. Methods such as scale drawing and trial and error will be discussed and their validity and/or shortcomings discussed. Textbook questions for skills practice with calculation using the cosine rule. Recognition that the cosine rule is a „generalised‟ version of Pythagoras‟ theorem with a „correction factor‟ for angles that are larger or smaller than 90. Learning task – Continuation of the earlier surveying task. Students calculate the approximate perimeter of the shape using the irregular polygon from their survey. Results to be compared in class then combined to show how using smaller steps refines the approximation. Answers and initial estimates to be compared with perimeter as measured with a trundle wheel. Class discussion of errors as with the previous exercise. Justification of the sine rule by direct measurement. Derivation of the sine rule from the area formula. The solution of contextual problems drawn from recreation and industry for an unknown side or angle, using the sine rule. Textbook questions for skills practice with the sine rule. Discussion of ambiguous or impossible cases, and how they arise in practical situations. SSABSA Support Materials: 6ca4c744-3d99-4e87-8e54-30ac8c347deb.doc, 3 November 2001, page 4 Solving problems where the cosine rule will not work. The sine rule. Are there now sufficient tools to solve any problem involving triangles? Directed Investigation 2 - Crossing the River Students take measurements to determine the width of a river they may not cross. Analysis involves calculations using sine and cosine rules and right-triangle trigonometry, discussion of errors and comparison of methods. Both group and individual work required. Skills and Applications Task 2 – Test on skills learnt in this topic. Topic 7: Statistics Subtopic 7.1: Consideration of how Data Is Represented to Us by Others Key Questions and Key Ideas What are statistics? The „statistical process‟ Identify the problem Formulate the method of investigation Collect data Analyse the data Interpret the results and form a conjecture Consider the underlying assumptions. Subtopic 7.6: Sampling from Populations Why take samples? To provide a small-scale representation of a larger population. What constitutes an „appropriate‟ sample? Simple random sample Convenience (one that is easy) Systematic (e.g. every tenth one). Subtopic 7.2: Organisation of Your Own Data What is the best way to present an overall picture of a given set of data? Exercises with given data sets to practise the production of histograms, stem & leaf plots, and box and whisker diagrams, both by hand and with the aid of a graphic calculator. This work will include discussions of the shapes of distributions, finding the five figure summary and calculating mean and Class exercise to demonstrate how samples can be inadvertently biased and practice in taking a random sample. Circles activity from “Practical Statistics, Rouncefield & Holmes. Class discussion of how unbiased sample could be taken for various purposes, eg ordering hats for the school book-room, counting the grapes in the vineyard, etc Teaching and Learning Activities and Assessment Tasks Class discussion of the key questions and ideas stimulated by questions from contexts in the local and school community. Examples are: “Have the hooping skills of the workers in the local cheese factory improved over the last two years?” “How do we determine a fair ranking of students for the school‟s academic and sporting prizes at the end of the year?” “How does the local vineyard owner know how many bins to order for the next harvest?” etc. SSABSA Support Materials: 6ca4c744-3d99-4e87-8e54-30ac8c347deb.doc, 3 November 2001, page 5 Measurement levels of data Tables, charts, and graphs Shapes of interval distributions Comparison of: Single sets of data with a standard Related sets of data Subtopic 7.3: Centre of a Distribution What is meant by „average‟? Median and mean. How do you decide on the most appropriate measure of „average‟? When can these measures become unreliable or misleading? Subtopic 7.4: Spread of a Distribution Do sets of data with the same „average‟ necessarily tell the same story? Range, IQR, standard deviation. standard deviation. Outliers will be discussed and the effects of removing them explored using electronic technology. In all cases in these exercises the ultimate gaol is to support a conjecture about the data used, either comparing two related data sets or comparing a single data set with a reference or standard. The coverage of this subtopic and the one following will be integrated into the classwork described above in subtopic 7.2. Much of this work will be revising and building on work done in previous years. As above with subtopic 7.3 Subtopic 7.5: Forming and Supporting Conjectures from Interval Data How do the statistical techniques and measures you have learnt so far help you to argue whether a claim is true or false? Putting the pieces together (interval data): Graphical representation Dealing with outliers Shape of the distribution(s) Measures of centre and spread Argument to support conjecture. Subtopic 7.7: Data-based Investigation How meaningful is the data you have used for Project - Students will choose their own area of investigation but it must incorporate the comparison Learning task - Students will carry out a detailed analysis of data provided by a local industry. The study will include sampling from a very large data set from two different years of production in a cheese factory. The data in the samples will be displayed using the various visual forms studied in class. Statistics measuring central tendency and spread will be calculated and students will use these to support their conjectures about what is shown by the data. Skills and Applications Task 3 – Test on skills learnt in this topic. SSABSA Support Materials: 6ca4c744-3d99-4e87-8e54-30ac8c347deb.doc, 3 November 2001, page 6 the investigation? of a measurable characteristic between samples from two different (but related) populations. They will be required to formulate a question which their project will endeavour to answer, supported by their Exploration, calculation, and analysis of statistics. the data Evaluation of the findings Report, summary, or conclusions SSABSA Support Materials: 6ca4c744-3d99-4e87-8e54-30ac8c347deb.doc, 3 November 2001, page 7