YEAR 10 MATHS INVESTIGATIONS
TOPIC AREA: SURDS
Starting Point - Lesson 1: “Modelling Surds”
This Starting Point must be satisfactorily completed before commencing Investigations.
Equipment needed: Ruler, pencil & 90° square
This exercise requires good geometry skills and careful measurement. Follow the instructions to produce a series of right-angled triangles. The hypotenuse of each triangle will give an estimate for , , and .
Instructions:
v Start by collecting a blank sheet of A4 paper. You will also need the drawing equipment listed above.
v Turn the paper so that it is in landscape orientation. Place a dot in the bottom left hand corner exactly 1 cm from the bottom edge and 1 cm from the left edge.
v Starting from the dot, draw lines exactly 10.0 cm up and 10.0 cm across as shown. Complete the triangle by drawing in the hypotenuse. Measure and mark the length of the hypotenuse. (Be accurate – measure to the exact mm!)
v Now use the hypotenuse of the first triangle as the base for the next one. Draw a perpendicular line exactly 10.0 cm long as shown. Again measure and mark the hypotenuse.
v Repeat the whole process two more times.
Questions:
1) Stick your completed geometric construction into your workbook.
2) Compare your construction with the diagram at left. Notice that your construction is magnified . Draw a table in your book with the following headings “Surd” “Predicted Value” “Actual Value”. Complete the table to an accuracy of 2 decimal places.
3) Comment on the accuracy of your results.
4) A picture of a nautilus shell is shown at right. What does this have to do with surds?
Starting Point - Lesson 2: “Distance Between Two Points”
1) “Orthogonal distances”
a. “Vertical distances”
i. Plot the two points P (-3, 0) and Q (-3, 6) on a set of axes on 1 cm graph paper.
ii. What is the distance between these two points?
iii. Repeat for the following pairs (on the same graph):
o R (-4, -6) & S (-4, 2)
o T (3, 5) & U (3, -1)
o V (0, 4) & W (0, -3)
b. “Horizontal distances”
i. Plot the two points G (3, 5) and H (-2, 5) on a new set of axes on 1 cm graph paper.
ii. What is the distance between theses two points?
iii. Repeat for the following pairs (on the same graph):
o I (4, -1) & J (6, -1)
o K (-3, 2) & L (2, 2)
o M (5, 0) & N (-3, 0)
2) “Diagonal distances”
i. Plot the three points A (0, 0), B (0, 4) and C (-2, 4) on a new set of axes on 1 cm graph paper, and join the points.
ii. Using a ruler, measure , and .
iii. What type of triangle is formed?
iv. What is the theorem for this type of triangle & its side lengths?
v. Is it true for this triangle?
vi. Repeat for the following sets of points (on the same graph):
o D (4, 2), E (4, -3) and F (-8, -3)
o G (-3, 4), H (3, 4) and I (-3, -4)
3) Using what you have discovered in 1. and 2. above find the distance between:
o A (0, 0) and B (9, 12)
o C (2, 3) and D (3, 6)
o E (-2, 3) and F (3, -5)
4) Extension:
o The coordinate of A is (….., …..) and B is (.…., …..).
o What is the horizontal distance between A & B? (Hint: if x1 = 2 and x2 = 6, how do you find it?)
o What is the vertical distance between A & B?
o Draw point C which forms the right-angled triangle ABC. What is C’’s coordinate?
o Use Pythagoras’ Theorem to calculate the length of AB, complete:
(AB)2 = (AC)2 + (AB)2
(AB)2 = (….. - …..)2 + (….. - …..)2
AB =
Investigation A - “Paper Sizes “
Although we are supposed to be moving towards a “paperless society”, there has never been more paper sold in Australia for “office use”. With the progress and innovation in home computing printers and office photocopiers, the average Australian uses lots of different types of paper. In this investigation, we discover the meanings behind the different paper sizes.
1) The paper you are looking at is A4 size.
a) Measure the length and width (in mm, be very accurate) of this sheet.
b) Calculate the area (in mm2) of this sheet.
c) What is the area in cm2 and m2?
2) Let’s investigate other “A” sizes. Collect other samples of A2, A3 and A5.
a) Measure the length and width (in mm, be very accurate) of each sheet.
b) Calculate the area (in mm2) of each sheet.
c) What are the areas in cm2 and m2?
3) Now organise your results (in some order) in a table. This will assist you in looking for patterns and connections between the sizes of the sheets.
4) What patterns can you discover? (Hint: one of the patterns involves a surd).
5) Use your results to predict the size of:
a) A1 and A0
b) A6, A7 and A8, etc…
6) Compare your values with the official values. (http://home.inter.net/eds/paper/papersize.html)
7) Final Challenge:
a) All school photocopier machines have another paper size, B size. What are the dimensions of B sizes and what are the connections?
Investigation B - “Lacing Patterns”
Have you ever tried tying your shoe laces and found the laces are too short? Could there be a way of lacing up your shoes in a different manner, so there enough left to tie the bows?
There are several different ways of lacing up a pair of shoes. Have a look around at peoples’ shoes to see various styles. Which is most common?
Here are some common patterns:
The American style can be depicted like this:
1) Draw the patterns for the other examples.
2) If we assume that a normal pair of shoes:
· has 5 eyelets on each side;
· the horizontal distance is 4 cm;
· and the vertical distance is 2 cm.
How much lace is needed to thread each pattern (to the nearest mm)?
3) A shoe manufacturer wants to put the shortest lace into each shoe to save money. Which pattern should she choose?
4) Final Challenge:
a) How much shoelace is needed to tie a bow?
b) If shoelaces are available in 5 cm increments (e.g. 45cm, 50 cm, 55 cm…), which length is needed for each pattern?
Investigation C - “Triathlon Tactics ”
It is the start of a gruelling triathlon race. The first part of the race is the swim leg. The competitors are lined up at the top of the beach and have to run down the beach, then swim out to the first marker buoy. A diagram of this is shown:
· A tri-athlete is able to run at 6 m/s in the sand and swim in the water at 4 m/s.
1) Competitor # 1 decides to run straight to the water (A) then swim directly to the buoy.
a) Draw a diagram (to scale);
b) Calculate the distances;
c) Use the formula:, to calculate the time it would take for each part.
d) What is the total time for competitor #1?
2) Competitor #2 decides to run directly to B, then swim straight to the buoy.
a) Complete the above steps to calculate competitor #2’s total time.
3) Try all other points between A & B in 20 m increments and calculate the times.
4) Plot a graph of TIME versus POSITION.
a) What tactics would you recommend to the competitors?
5) Final Challenge:
Can you calculate the exact distance and time for the quickest route?
Investigation D - “What’s Behind The Ö Button?”
· How does a calculator actually calculate square root values?
· There are actually many different techniques that can be used.
· This investigation looks at 2 different methods:
1. Divide and average;
2. Subtraction;
1) Write down the calculator value for correct to 4 decimal places.
2) Finding
a) Method 1: “Divide and average”
i) Step 1: “Guess” – Let’s guess 2.
ii) Step 2: “Divide” – Divide 5 by 2, equals 2.5.
iii) Step 3: “Average” – (2 + 2.5)/2 = 2.25. This value becomes our next guess.
iv) Step 4: Return to Step 2: ie divide 5 by 2.25 ….
v) Continue until you obtain the same accuracy as the calculator. How many steps?
b) Method 2: “Subtraction”
i) Step 1: “Multiply by 5” – The number x 5 (let it be called ‘ a ’) and pair with ‘ b ’ (which always starts at 5) (ie (a, b) )
(a) for ,therefore we get (25, 5)
ii) Step 2: “Repeated Steps”
(1) If a ³ b then replace a with a – b and then add 10 to b.
OR
(2) If a b then add two zeros to the end of a and add a zero to b just before the final digit(which is always 5).
(a) for then we get (20, 15) 2A
(b) then (5, 25) (step 2A)
(c) then (500, 205) 2B
(d) then (295, 215) 2A
(e) then (80, 225) 2A
(f) then (8000, 2205) 2B ….
iii) The number in the second position is your estimate. Continue until you obtain the same accuracy as the calculator. How many steps?
3) Repeat both methods for and .
4) As you may have noticed, the calculations will continue forever. Numbers like this are called IRRATIONAL. Some important irrationals are and e. There are different methods to find each of these constants. Using your calculator state the value of each to five decimal places. ( Phi, the golden ratio, can be calculated as and e can be calculated by e^1).
5) Final challenge:
· Since Archimedes (287 – 212 B.C) mathematicians have been approximating. Some examples:
Greeks: ; Chinese: (400 AD): ; Modern India: ;
Irish: ; English: .
· Calculate all the values for. In the last two calculations, if you keep going does the approximation get better?
· . Calculate the values for when n = 2, 3, 5, 7 and compare to the calculator’s value. This method is called a series expansion.
· Now place the following irrationals on a number line: ,,,,,,,,,and .