Download Problem solving

Problem solving
Choose one or some of the following. Think about how you could use it in a
lesson, bearing in mind the following points: presentation versus investigation,
thinking and discussion, hints and pictures, inter topic links, IT, creativity,
reversing or undoing the problem and extensions.
1. Find integers A, B and C such that A2 + B2 = C3
2. What have the following equations got in common? Investigate
graphically.
x+2y = 3
4x+5y = 6
3x+6y=9
-2x+4y = 10
3. Take an odd square number and express it in the form n +n+1. Evaluate
(n+1)2-n2. Explain what you notice using algebra
4. How would you respond to the following situations?
A pupil says,
a)
b)
c)
“if you do the same to both sides of an equation the solution stays
the same”
“if you do the same to numerator and denominator of a fraction then
the fraction stays the same”
“ 15 ÷ 4 and 18 ÷ 5 are equal because they are both 3 remainder 3”
5. What do you notice about the following?
3 3
3 3
 and 
4 7
4 7
5 5
5 5

and 
8 13
8 13
4 4
4 4

and 
9 13
9 13
2 2
2 2
 and 
5 7
5 7
What about
5 5
5 5
 and  ?
3 8
3 8
Describe the structure and patterns in the above fractions. Explain what
you notice and give an algebraic equation that describes the property.
Calculator extension: Evaluate 3/7 +(3/7)2+(3/7)3+(3/7)4+…..+(3/7)n where n is
a very large integer. Use the ANS key on you calculator to do this
efficiently. What do you notice? If we used 4/9 instead of 3/7 what would our
answer be?
6.
.A
7.
C
Given any old triangle, explain how you would find
points A and B so that going along the perimeter
from A to B is half the perimeter. You can only use
a straight edge and a pair of compasses.
.B
.
Write a thorough explanation of how
this diagram proves Pythagoras’
Theorem. Use triangle AGD as the
right-angled triangle with right-angle at
G
G
B
D
A
E
F
8.
Given any cuboid, form a tetrahedron inside it as above. Sow that the
squares of the area on the right angled triangular faces sum to the square
of the area on the fourth non right-angled triangular face.
9. Take a pair of consecutive odd or even integers and add their reciprocals.
Next, work out √((numerator)2+(denominator)2). Explain thoroughly!
10. Draw a picture to help you investigate which numbers can be expressed
as the difference between two square numbers and then investigate.
What is the smallest square number that needs to be added to 2007 to
produce another square number?
11. Some numbers can be represented as the sum of consecutive integers.
For example 5 = 2+3;
22 = 4 + 5 + 6 + 7; 24 = 7 + 8 + 9
Investigate which numbers can and cannot be represented in this way.
From this year’s UK Intermediate Maths
Challenge.
7
Calculate the area of this shape.
3
9
To extend: Find other sets of integers that
give this quadrilateral integer side lengths.
13.
In this triangle lines are drawn parallel to the
base so that the other two sides are trisected.
What fraction of the area of the triangle is
shaded?
14.
The shorter sides of the right- angled
triangle are a and b. Express in terms
of a and b the radius of
i) the inscribed circle
ii) the circum-scribed circle
Extensions: Finding the in-circle radius
and circum-circle radius for a kite,
initially with a pair of opposite rightangles and then without any rightangles.