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TI-Nspire Exemplary Activities

Activity 1: The Falling Ladder

(Suitable for CAS extension)What does it feel like to be at the top of a ladder as the bottom begins to slide away? Do you fall at a steady rate? If not, then what is the nature of your motion - and when are you falling fastest?

This modelling problem is suitable for students across the secondary school, from consolidation of work on Pythagoras' Theorem in the early years, to optimization using differential calculus in the senior years. At all levels, it is a realistic and valuable task, which links a variety of mathematical skills and understandings with a practical real-world context.

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(Suitable for CAS extension)What is the chance of sharing a birthday with someone in your class? This simple question offers a rich context for mathematical modeling, which is potentially accessible to students from the early years of secondary school to seniors. Using TI-Nspire CAS, students are offered the tools by which they can investigate the problem and build a meaningful model, which will deepen their understanding of the problem, and help them to further appreciate the applications of mathematics to their world.

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(Suitable for CAS extension)This beach race begins from a point 4 kilometres out to sea from one end of a 6 kilometre beach, and finishes at the opposite end. Contestants must swim to a point along the beach, and then run to reach the finish line first. I can swim at 4 km/h and run at 10 km/h - where should I aim to land on the beach so as to minimize my total time for the race?

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Activity 4: The Diminishing Square

(TI-Nspire CAS recommended)Study the diagram provided. A smaller square has been constructed inside a larger square, as shown.

A point x is located on the base of the larger square. (As shown) The smaller square is constructed using similar points on each of the remaining sides of the larger square. If x is the midpoint of the base, what is the ratio between the area of the larger square and the smaller square?

Explore the relationship between the position of this point and the area of the smaller square.

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My friend and I agree to meet during our lunch hour. If we each decide to wait for 15 minutes, what is the probability that we will meet? How long should we agree to wait in order to have a 50% chance of meeting? How long for an 80% chance?

A version of this problem was set as the final question for the 2005 New South Wales Higher School Certificate examination in Mathematics. Copyright is held by the New South Wales Board of Studies.

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Activity 6: Exploring the Parabola

This activity explores the key features of the parabola, both geometrically and algebraically. A variety of interactive representations support student learning as they build their understanding of this important curve and its real world applications.

The primary objective in the study of parabolas in many high school curricula, tend to be algebraic, moving quickly to the study of the quadratic function. Key defining features of this function are geometric in nature. Students often misrepresent other curves as 'parabolic' simply because they have a similar appearance. It is therefore important for students to understand some of the properties of a parabola, features that make this curve both unique and important. This activity supports students in actively linking some of the geometric and algebraic properties of a parabola.

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(TI-Nspire CAS recommended)This activity explores the equivalence of algebraic expressions in expanded and factored form, using patterning with CAS to expose commonly held student misconceptions.

The big algebraic mathematical ideas this activity explores are equivalence and symbol sense. More precisely, the activity speaks to the following curriculum expectations: expand and simplify second-degree polynomial expressions involving one variable that consist of the product of two binomials [e.g., (2x + 3)(x + 4)] or the square of a binomial [e.g., (x + 3)

^{2 }], using a variety of tools (e.g., algebra tiles, diagrams, computer algebra systems, paper and pencil) and strategies (e.g. patterning)Top of Page

What variables characterize a rectangle? What kind of relationships exists between these variables? In this activity you will explore this, examining patterns and forms using tables, graphs and equations.

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(TI-Nspire CAS recommended)What is the optimal position for a rugby player attempting a conversion?

The aim of this activity is to determine this optimal position and then to study the variations in the angle q obtained depending on the position where the try was scored.

This task enables us to evaluate the advantage gained by positioning oneself as close as possible to the posts before scoring the try.

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Activity 10: The Witch of Agnesi

(Suitable for CAS extension)This activity involves a curve "with history"; an excellent example of combining geometry and algebra. This function, the so-called "Witch of Agnesi" is defined by a geometric description. After implementing the construction, students are then challenged to find the equation of the constructed curve. This equation, in turn, allows further investigations and generalisations, including some from the field of analysis - an alternative to conventional curve sketching.

The first and second parts of the task are suitable for students of secondary school age with knowledge of the theorems of intersecting lines and the laws of similarity, as well as the Pythagorean theorem. Methods of differential calculus are required only for the final task.

Activity 10 TNS files (zipped)

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Activity 11: The Art Gallery Problem

(Suitable for CAS extension)The question of where patrons should stand to enjoy the best view of a painting is one art gallery curators must consider on a regular basis. In this particular case we will consider a painting that is two meters tall that has been placed 1 meter above the average person's eye level. Where should the average person stand in order to get the best view?

This investigation offers opportunities for review and consolidation of key concepts related to measurement of distances and angles, and inverse trigonometric functions.

Activity 11 TNS files (zipped)

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Kepler's Third Law compares the motion of the various planets. Each of them is characterised by two quantities in particular, the mean distance from the sun and the time of revolution around the sun. Are these two quantities related?

The aims of this activity are:

- to establish whether a power function is a good fit for the data
- to discover the power function that "best" matches the data
Activity 12 TNS files (zipped)

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Activity 13: The Impact of "b"

(TI-Nspire CAS recommended)In this activity you will examine the influence parameter b has on the quadratic function. You will use different techniques to visualise this. You will use CAS to find the vertex coordinates. You will plot the vertex points of f(x) for different values of b and look for a pattern. You will use regression to determine a function which fits this pattern. You will use sliders so that you can see dynamically that your conjecture is true. You will use CAS to test your guess.

Activity 13 TNS files (zipped)

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Activity 14: The Chicken Run Problem

(Suitable for CAS extension)A farmer has 60 metres of chicken wire and wants to build the largest rectangular run he can. He has two options. The first option is to build a free standing run in which he will use the fencing at his disposal to enclose the four sides of the run. The second option is to build the run up against an existing fence and use the available fencing to enclose the area with three sides additional to the existing fence. The farmer wishes to know which dimensions will offer the greatest floor space for each model as well as which design will provide the greatest area for the chickens to walk around in.

The question of optimizing the area enclosed by an invariant perimeter is not new but the ideas that the enclosed area will vary with the dimensions of the enclosure is almost anti-intuitive to many students (and some adults).TI-Nspire permits a much more integrated, multi-dimensional view of this problem that is often only investigated using the device of calculus.

Activity 14 TNS files (zipped)

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Activity 15: Building Concrete Foundations to Algebra

(Suitable for CAS extension)Concrete approaches to building student understanding and manipulative skills in early algebra have been well established through research and successful practice. The approach offered in this activity should be used subsequent to pattern building activities and jointly with concrete manipulatives (cardboard shapes are quite suitable).

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(TI-Nspire CAS recommended)Many special properties can be shown to be associated with cubic functions. This activity investigates tangents to the cubic function and their relationship with the roots of the function. The investigation involves functions, solving for roots, calculus and graphing techniques.

Some students have difficulty in interpreting problems presented and analysed using algebraic techniques. The use of dynamic graphing technology allow these students to access problems in a more visual manner and relates the algebra with the visual image.

Activity 16 TNS files (zipped)

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Activity 17: Looking Into Parabolas

(TI-Nspire CAS recommended)A parabolic reflector can be studied visually using a light box found in many science departments. Light rays parallel to the principle axis reflect back through the focal point. Can mathematics be used to model this situation and subsequently determine the location of such a point?

Focus and directrix properties are generally studied jointly and incorporate a study of eccentricity. This activity is aimed at using the physical characteristics of a parabolic reflector to determine the location of the focal point without the necessity to introduce the directrix or eccentricity. TI-Nspire CAS provides an environment where students can use their understanding of reflection to explore a mathematical solution incorporating calculus.

Activity 17 TNS files (zipped)

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Activity 18: Eco-Tourism: Whale Watching

(Suitable for CAS extension)The peak Whale Watching Season in Hervey Bay is from August to October. The crew on board the Whale Explorer have kept records on the number of whales seen in the Bay during the main 90 day period of the previous season.

The data for the number of whales seen in the Bay and how many days they saw this number of whales provide the basis for data analysis and some probability investigation.Activity 18 TNS files (zipped)

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Activity 19: Environmental Mathematics: Black Bear Cubs

(TI-Nspire CAS recommended)The data in this activity was collected in order to better understand the health and growth patterns of black bear cubs in the wild. In their first year of life, these cubs grow from 225 grams to between 22.5 and 31.5 kilograms.

Analysis of this data involves curve fitting and some introductory calculus from first principles.

Activity 19 TNS files (zipped)

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Activity 20: Areas and PaperFolding

(Suitable for CAS extension)Who would have thought that there could be so much mathematics in simply folding a piece of paper? This activity spans the years of secondary school, beginning with measurement, data collection and interpreting scatter plots in the early years, through linear functions, Pythagoras' Theorem and trigonometry, right through to calculus in the senior years.

The mathematical focus at each level is different - from finding the largest area to discovering functional relationships between the sides of a right-angled triangle, and on to optimisation. While algebra and calculus can be used to prove this result, it actually takes some geometry to understand

whythe final result is true.Top of Page

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