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Classroom Activities
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Students start by considering the motion of a point on the wheel of a bicycle. With the aid of interactive diagrams in the TI-Nspire document, students go on to explore the case when the wheel rotating inside another resulting in a hypocycloid. Students generate the parametric form of the equation, use compound and double angle formulas to generate the equivalent Cartesian equations and use calculus to analyse features of the curves, including gradient and arc length.
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0
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Students explore the relationship between the number of discs and moves required to solve the classic “Tower of Hanoi” problem. The TI-Nspire file contains an interactive model produced by Andy Kemp. The interactive model allows you to change the number of disks and automatically record the total number of moves allowing students to focus on solving the problem and identifying patterns. The activity also includes a degree of scaffolding to help student identify and understand the nature of the geometric recursive relationship between the number of disks and the number of moves.
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- TI-Nspire™ CX
- TI-Nspire™ CX CAS
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4
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The classic "Tower of Hanoi" problem involves moving discs of different sizes amongst three columns with the restrictions that large discs can’t be placed on small discs and discs can only be moved one at a time. The challenge is to move all the discs from one column to another. This activity requires students to solve this problem, recognise and use the recursive definition of an arithmetic sequence and explore alternative approaches to solving the problem. The TI-Nspire file contains a virtual Tower of Hanoi.
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6
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Imagine you just arrived home from school, you’re really hungry. You decide the block of chocolate will help you with your homework. Pretty soon half the block is gone, so you quickly put it back in the refrigerator. Your brother arrives home, spots the half eaten block of chocolate and immediately breaks half the remaining block off for himself. Dad’s next, when he visits the refrigerator he too breaks off half the remaining block. How long will the block last you wonder?
This introduction combined with a dynamic representation and accurate mathematical notation is used for the basis of introducing and exploring Geometric sequences and series.
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3
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The modulus and argument form of a complex number can be written in two ways, but how are they connected? ‘CIS’ format is a straight forward application of trigonometry, but what about e ? In this activity students explore the Taylor expansion of ex and compare it with the Taylor expansion of cos(x) + sin(x) revealing a small difference relating to an alternating sequence of negative signs which can be adjusted using i. The activity removes the mystery and replaces it with a beautifully connected piece of mathematics
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4
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This activity uses a combination of simulation and simple probability tree diagrams to explore a set of dice with some very unusual characteristics. Grime Dice, created by Dr. James Grime are used in the initial investigation, however Efron dice and other non-transitive dice can be explored in the extension section of the investigation.
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- TI-Nspire™ CX
- TI-Nspire™ CX CAS
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6
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Division of numerical quantities is a great way to start exploring and understanding division of polynomials. In this activity students start with what they know about division of numerical quantities and then apply this to division of polynomials. This assists with some of the terminology and also developing a deeper understanding. Polynomials can be regenerated dynamically and rapidly including the resultant quotient and, where applicable, the remainder, this helps students identify and explain patterns. The visual representation of polynomial division also helps students understand the factor and remainder theorem, particularly expressing a polynomial as a product of its linear factors.
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6
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Imagine you just arrived home from school, you’re really hungry. You decide the block of chocolate will help you with your homework. Pretty soon half the block is gone, so you quickly put it back in the refrigerator. Your brother arrives home, spots the half eaten block of chocolate and immediately breaks half the remaining block off for himself. Dad’s next, when he visits the refrigerator he too breaks off half the remaining block. How long will the block last you wonder?
This introduction combined with a dynamic representation and accurate mathematical notation is used for the basis of introducing and exploring Geometric sequences and series.
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5
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The AFL Coleman Medal is awarded to the player who kicks the most goals in a season. In 1970 Peter Hudson kicked 146 goals, then backed that effort up with 140 goals in 1971. Gary Ablett (senior) won the Coleman Medal in three successive years totals of 124, 113 and 118. Lance Franklyn was the last player to kick more than 100 goals back in 2008. Since then the average number of goals scored by the Coleman Medallist has averaged just over 70 goals. Has the day of the dominant full forward come to a full stop? In this activity, students use moving averages to explore trends over the history of the Coleman Medal.
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- TI-Nspire™ CX CAS
- TI-Nspire™ CX
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7
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In the game of Blackjack, players continually draw cards until they can gain a score as close to 21 as possible, without going bust. In Calculator-Blackjack the random number generator is used to produce numbers between 0 and 1, these numbers are summed with the aim to keep this sum less than 1. On average how many cards do you think a player might have drawn to end up bust? This question is explored through statistics and sampling distributions. The answer to the question may surprise you!
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- TI-Nspire™ CX CAS
- TI-Nspire™ CX
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6
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Maddi the painter needs to work out the longest ladder that she can get around a tight hallway corner. Formulate an expression for the length of the ladder, differentiate and make equal to zero. Then what? Why does the function minimum align with the longest ladder?
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13
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In this activity students maximize the area of a triangle formed when a piece of A4 paper is folded in a special way. The activity provides an opportunity to explore the problem practically followed by an animated TI-Nspire file that automatically generates data so that students are able to check the validity of their data before proceeding. The unusual general result from this task makes the activity quite intriguing.
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8
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This activity brings together a range of ideas from Activity 1 and 2 using a combination of rotation and dilation matrices and a very powerful visual and algebraic approach. The activity gives function to the use of matrices and highlights how relatively complicated expressions can be determined very easily. Students connect many aspects of the Specialist Mathematics course in a single activity that is sure to engage students.
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107
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Students simulate drawing random samples of 3 marbles, selected with and without replacement, from an urn containing 10 marbles. Comparisons are made between the results obtained for these selection methods. The effect of increasing the number of trials in the simulations is explored. Finally, comparisons are made between the simulation results and the theoretical probability distributions of the random variables associated with these random processes.
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- TI-Nspire™ CX CAS
- TI-Nspire™ CX
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11
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This is an extension of String Graphs Part 1 but can be done independently. In this activity students stitch points on the lines y = x and y = -x. The family of straight lines form an envelope which can be modelled by finding successive points of intersection. The points of intersection can be modelled by a parabola. A range of extension questions are provided including generalising the parabola for stitching points on the lines y = mx and y = -mx.
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12
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