1. Generating sequences: Arithmetic progressions a. Create a dynamic AP in a spreadsheet with columns for n and u (enter the value for a in B1, and =B1+d in B2, where a and d are to be chosen, and Fill down). b. Plot these points (a scatter graph is best for this and you may find a split-screen helpful. i. How do a and d effect the terms? ii. What formula would describe the sequence in terms of n ? iii. Can you find APs with different values of a and d that have the same 10th term?
2. Generating sequences: Geometric progressions a. Create a dynamic GP in a spreadsheet with columns for n and u (enter the value for a in B1, and =B1*r in B2, where a and r are to be chosen, and Fill down). b. Plot these points (a scatter graph is best for this and you may find a split-screen helpful. i. How do a and r effect the terms? ii. What formula would describe the sequence in terms of n ? iii. Can you find GPs with different values of a and r that have the same 5th term?
3. Summing series: Geometric progressions a. Add a graphs page and create sliders for a and r . b. Add a lists and spreadsheet page and create columns for a and u (enter the value for a in B1, and =B1*r in B2, where a and r are to be chosen, and Fill down). A split-screen is useful for this. c. Plot these points using a scatter graph. d. Add a column for the cumulative sum (enter =B1 in C1, =C1+B2 in C2, and fill down). NB you can’t name the column sum as this is a built-in function so choose a name such as csum or s . e. Plot the cumulative sum against n on a scatter graph. f. Change the slider for a so that it ranges from 0 to 2. i. What happens to the sum when r <1? ii. What would happen if r <0?
4. Investigating sequences: comparing recursive relations and general terms a. To enter a sequence recursively on a graphing page use: Graph Type >
Sequence > Sequence and enter the recursive definition and the initial term. e.g. u1 (n)= u1 (n-1)+3
Initial Terms:= 2 b. To enter a sequence defined by its general term on a graphing page use: Graph
Type > Sequence > Sequence and enter the general term. e.g. u2 (n)= –1+3n c. For some recursive definitions can you find the general term, e.g.:
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i. u1 (n)= u1 (n-1)+1, Initial Terms:= 4 ii. u2 (n)= u1 (n-1)*3, Initial Terms:= 2 iii. u3 (n)= u1 (n-1)*.25, Initial Terms:= 10 d. For some recursive definitions can you find the general term, e.g.: i. u4 (n)=5+4n ii. u5 (n)=4*2^n iii. u6 (n)=0.5^n
5. Generating sequences a. Generate the first 6 terms of 5 sequences i. At least one AP ii. At least one increasing GP iii. At least one decreasing GP iv. At least one sequence that is neither an AP or GP b. Give the terms to another student and ask them to find: i. the 20th term ii. a formula for the n th
term (this may be difficult for sequences that are neither APs or GPs)
To set the angle measure to degrees for solving trigonometric functions:
Handheld: Home > Settings > Settings > General > Angle: Degrees
File > Settings > Document settings > Angle: Degrees
To set the graphing angle to degrees:
Handheld: Home > Settings > Settings > Graphs and Geometry > Graphing Angle:
Degrees
Software: File > Settings > Document settings > Graphs and Geometry > Graphing Angle:
Degrees
6. Solving trigonometric equations a. Plot a trigonometric graph, e.g. f1(x)=sin(x) . You may need to change the graphing angle to degrees and use Zoom – Trig to show the full curve. b. Add a slider for k and a line f2(x)=k . c. Find the Intersection points of the line and the curve. i. How many solutions will there be in a given range? ii. How are these solutions related? d. What would be the effect of adding parameters a and b in f1(x)= a*sin(b*x) ?
7. Transformations of trigonometric graphs a. Plot a trigonometric graph, e.g. f1(x)=sin(x) . You may need to change the graphing angle to degrees and use Zoom – Trig to show the full curve. b. Add sliders for a and b . Plot f2(x)= sin(x+a)+b . c. What is the effect of changing a and b ? You may need to change the settings on the sliders.
8. Introduction to circular measure.
You should initial set the angle to degrees.
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a. Draw a circle with a fixed centre through a point (centre at the origin and a point on the x-axis is the most useful). It may be helpful to make the circle dashed or a different colour. b. Add two further points on the circle. Draw a circular arc (Points and lines: Circle arc) through all 3 points. Add a line segment for the radius. c. Make the following measurements: i. The length of the arc and store it as a ii. The radius and store it as r . iii. The angle at the centre made by the arc. d. Add a textbox for a/r and calculate its value. i. What is the relationship between the values of a , r and the angle? ii. Does this relationship hold for different sized circles? iii. Change the Graphing angle to radians. What is the relationship between the values of a , r and the angle?
9. Trig identities a. Construct an interactive file that you can use to explain tanθ ≡ sinθ/cosθ to someone else. b. Construct an interactive file that you can use to explain sin²θ + cos²θ ≡ 1 to someone else.
10. Sine and cosine rules a. Construct an interactive file that you can use to explain the cosine rule to someone else. b. Construct an interactive file that you can use to explain the sine rule to someone else. c. Use your sine rule file to demonstrate the ambiguous case of the sine rule.
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