Ideas beyond Number
Activity worksheets
Activity sheet 1
Task 1
Some students started to solve this equation in different ways:
For each statement tick True or False:
+ =
=
+
+ =
+
=
+
+
+ =
=
+
+
+ =
+
+ =
+
+ =
+ =
=
+
− =−
Task 2: Counter-examples – The exception disproves the rule
Find a COUNTER-EXAMPLE to show that each of these CONJECTURES is FALSE:
1. Square numbers only end in , ,
or .
2. Cube numbers can end in any digit except .
3. The product of two numbers is greater than either of the two numbers.
4. The square of the number is greater than the number.
5. Adding two numbers and then squaring them gives the same result as squaring them and then adding
them.
6. Division always results in a smaller number.
7. Every whole number is either a cube number or is the sum of , ,
8. The sum of two numbers is greater than their difference.
9. The sum of two numbers is always greater than zero.
10. The product of two numbers is greater than their sum.
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or
cube numbers.
Task 3: Developing algebraic reasoning
Decide whether each of the following are:
♦
always true
♦
sometimes true
♦
never true.
Convince a friend
1. Three consecutive numbers sum to a multiple of three.
2. Three consecutive numbers sum to a multiple of 6.
3. For three consecutive numbers, the product of the first and the last is 1 less than the middle squared.
4. The sum of two consecutive triangle numbers is a square number.
5. An odd number added to an odd number always gives an even number.
6. When you add together consecutive odd numbers you get a square number.
7. Add 1 to the square of an odd number. You always get an odd number.
Hint: the technique is:
A.
Try some examples and see if you can find cases that comply – then the conjecture is true sometimes.
B.
Then to look for a counter example and if you find one then the conjecture is not always true.
C.
If you cannot find a counter example fairly quickly, then in these cases algebra should provide a more
formal PROOF.
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Activity sheet 2
Further development of algebraic reasoning
Task 1
represents a square number;
is an integer.
1. Think about the expression
. Explain how you know there are values of
for which this expression
does not represent a square number.
2. Explain why the expression
must represent a square number.
Task 2
Kate is solving the inequality
. She says: “
whenever
is less than .”
Kate is not correct. Explain why.
Task 3: Teacher-led activity
Your teacher will give you a statement – investigate it! Is it true? If not, can you find a counterexample? If so,
can you prove it?
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Activity sheet 3
Task1: The Wason Test
Your teacher will introduce you to the Wason Test.
How did you do on the Wason Test? How did your classmates do?
What do you think this shows about human beings and logic?
Task 2: Three digits
1. Take any 3 digits.
2. From these digits make six different number pairs.
3. Add them up.
4. Add the three original digits.
5. Divide the larger of the last two answers by the smaller.
6. Write down the result.
7. Choose another set of three digits and repeat.
8. Write down briefly what you find.
9. Try to prove it will always be true.
Task 3: Consecutive sums
Investigate.
Are there some whole numbers that cannot be
expressed as the sum of two or more
consecutive numbers?
Which are they? Why?
etc
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Activity sheet 4
Consecutive sums revisited
What is the next line?
What is the
th
line?
Can you prove it?
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Activity sheet 5
However you slice it… (challenging proof activity)
Draw a circle, and place two points on its circumference. Join these points with a line. The interior of the circle is
then divided into two regions.
Now draw another, and place three points on its circumference. Join each of these points to the other two. The
interior of the circle is then divided into four regions.
3
1
1
2
2
4
How many regions are formed if you place four points on the circumference? Five? Six? ? Make a conjecture.
Test your conjecture.
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Refine your conjecture
First, some definitions:
An external node is a point on the circumference of the circle.
A cut is a line joining two external nodes.
external node
external node
cut
An internal node is the intersection of two cuts.
An external arc is part of the circumference of the circle joining two external nodes.
An internal arc is any other line joining two nodes.
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external arc
internal arc
internal node
internal arc
A region is an enclosed area.
Draw up a table relating the number of external nodes to the number of external arcs, cuts, internal nodes,
internal arcs and regions. Look for patterns in this table. Frame some conjectures. Can you prove them?
Things you might find useful along the way
1. Getting choosy
The number of ways of choosing two objects from three is written
, and said “three choose ”. Its value is .
Similarly, the number of ways of choosing two objects from four is written
, and said “four choose ”. Its value
is .
.
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Questions
1. What are the values of
choose
and
2. What are the values of 6 choose ,
choose ?
choose
and
choose ?
2. Pascal’s triangle
This is Pascal’s triangle:
To make each number in Pascal’s triangle, add together the two directly above it: for example,
.
3. What’s the connection between Pascal’s triangle and the “choose” numbers?
4. What’s the connection between Pascal’s triangle and the “nodes, arcs and regions” table?
3. Euler’s formula for plane graphs
A plane graph is a collection of points (nodes) connected by lines (arcs) and dividing the page into regions.
2
1
3
In this example, there are 4 nodes, 6 arcs and 3 regions. Euler’s formula says that if there are
and R regions, then in every case,
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.
nodes,
arcs
5. Does Euler’s formula apply to this example?
6. Check Euler’s formula on some plane graphs of your own. Does it work?
7. Our “nodes and regions” diagrams are plane graphs. Does Euler’s formula work for them?
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