Homework 3 Can you divide 36 balls into 9 groups such that each group has odd number of balls? 36 ÷...
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Transcript of Homework 3 Can you divide 36 balls into 9 groups such that each group has odd number of balls? 36 ÷...
Homework 3
• Can you divide 36 balls into 9 groups such that each group has odd number of balls?
• 36 ÷ 9 = 4, 4 is even• What if we change things around a little bit?• 5, 3, 5, 3, … etc. Would it work?• Note that 9 odd numbers added together
must be odd.• Impossible.
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Homework 4
• are 9 squares in a grid that each has a coin in it as below.
• Can you remove 4 coins such that each row and each column has odd number of coins?
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Homework 4
• are 9 squares in a grid that each has a coin in it as below.
• Can you remove 3 coins such that each row and each column has even number of coins?
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Olympiad Math III
Lesson 10Area of patterns on grids
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Purpose
• Calculating area of patterns on a grid improves your understanding of the patterns
• Prepare you for formal geometry and analytical geometry
• It will be fun
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Setup
• All patterns are drawn on a square grid.• All area is measured by the size of the unit
square.
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Calculate simple areas
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2 x 4 = 8
2 x 4 = 8
4×2÷2=4
2×(4 + 2)÷2 = 6
Area Formulas
• Square: a2
• Rectangle: a × b• Parallelogram: b × h• Triangle: b × h ÷ 2• Trapezoid: (b1 + b2) × h ÷ 2
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Trapezoid Formula Covers All
• Trapezoid: (b1 + b2) × h ÷ 2
• Square: a2 (b1 = b2 = h = a)
• Rectangle: a × b (b1=b2=a, h=b)
• Parallelogram: b × h (b1 = b2 = b)
• Triangle: b × h ÷ 2 (b2 = 0, b1 = b)9
b1
b2
h
What is this area?
4 x 5 – 3 – 5 – 4 = 8
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(1)
(2)
(3)
(4)
Calculate areas• Hat: 3 + 4 + 2 = 9• Goose: 1 + 2 + 4 + 1 = 8
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Cowboy• Area = ½ + ½ + 1 + 3 + ½ + ½ + 1 = 7
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Area of a square
• Area = 5 x 5 – 4 x (2 x 3 / 2) = 25 – 12 = 13
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Length of a side
• The square’s area is 13, what is the length of its side?
• The length times itself is 13• It must be more than 3 and less than 4• The value is 3.60555127546….• It is called the square root of 13 with this
notation: 13
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There is a relationship
• How are the three areas of squares related?
• C = A + B15
A
B
C
Does the same relationship hold?
• A = 4, B = 4, is C 8?
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A
BC
Pythagorean Theorem
• The area of the slanted square is always the sum of the area of two straight squares
• In a right angled triangle the square of the hypotenuse (the longest side of a right triangle) is equal to the sum of the squares of the other two sides.
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Can you explain this?
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Let’s solve the puzzle
• What is the area of the big triangle we put together?
• 13 x 5 / 2 = 32.5
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Solving the puzzle
• What s the area of each colored tiles?• Red: 2 x 5 / 2 = 5• Green: 7• Yellow: 8• Blue: 8 x 3 / 2 = 12• Totally: 5 + 7 + 8 + 12 = 32
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Solving the puzzle
• The top figure covers 32• The lower figure should cover 33 because of
the blank
• Bottom line: both figures are not triangles. The “hypotenuses” are not straight lines.
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Can you explain this?
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Another rectangle
• This rectangle’s perimeter is 20 and the area is 24. Another rectangle’s area is 20 and perimeter is 24. What is the length and width?
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Thoughts on the right track
• You notice that the area is getting smaller but the perimeter is bigger
• The rectangle has to be “skinnier” to have this• The length plus width is 12• So the length is 10 and width is 2
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Magic
• Here is an interesting number trick done by David Copperfield
http://www.youtube.com/watch?v=nZTR7kyz3g8
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If we still have time
• A 4 x 4 grid is divided into 5 pieces. Fill in the numbers 1, 2, 3, 4 to each of the squares such that the 4 numbers in each column or row are all different and the sum of all digits in each piece are the same.
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Solution
• Since each column has non repeating numbers each column or row sum to 10
• All five pieces have the same sum of digits, they must be all 8
• Here is one of the solutions
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3 4 1 2
41 3 2
14 2 3
32 1 4