Pattern Spotting - the 'Number Machine'
Number machines are a good way of introducing students to the concept of
algebra. The basic principle is that a number goes into the machine,
something happens to it (a 'function') and another number comes out.
Decide before the lesson what the ''functions' of the number machine are
going to be.
Teachers should start off with easy functions which get progressively more
difficult:
+4 (easy)
-4 (easy)
x 2 then + 1 (moderate)
÷ 2 then -1 (moderate)
© Education Umbrella, 2014 1 In each case above, at least five numbers should 'be put into' the number
machine.
Students should record all input and output numbers on a table like the one
on the right (in this case, the function is '+4'). Every time the teacher gives
an 'output' number, the students should think about what the possible
function may be.
In the example on the right, a student may first think that the function is
'double', but they will soon realise this isn't the case (3 x 2= 6, not 7)
For the 'moderate' functions, get students to write out numbers in order. This
makes pattern spotting easier. Teachers should give the output number 1 by
1. Students should construct a graph like the one below:
© Education Umbrella, 2014 2 Questions
•
What pattern do you notice going down the output column?
•
Subtract 1 from each number in the output column. What do you
notice?
•
Can you come up with a function that will get you from the input
number to the output number?
•
Can you use this function to work out the output number if the input
number is 15/20?
Missing Number Problems
Questions
•
What is the answer to the sum 4+3?
•
What is the question mark in the sum 4+? =7.
•
Can you rewrite this as a subtraction?
•
Can you find the identity of the question marks in each equation? In
each case, rewrite the question to make the question mark the
'answer.'
© Education Umbrella, 2014 3 Worked Examples:
The following illustration should help students grasp the concept of
rearranging an equation. Real objects such as pens, erasers or counters could
be used to make the activity more engaging for students (donuts may be a
bad idea!).
The following image explains the process pupils should go through to work
out the more complicated questions (in yellow):
© Education Umbrella, 2014 4 Balancing Equations
Students should think of a pair of weighing scales such as the one below.
Students should understand that, in order to balance the scales, the total
weight in each bowl should be the
same:
Students should attempt the following questions, replacing the question
marks with apples and/or pears:
Things students should consider:
•
What is the total weight of the blue bowl?
© Education Umbrella, 2014 5 •
What is the current weight of the yellow bowl?
•
What weight will I need to add to the yellow bowl so that it equals
weight of the blue bowl?
•
What combination of apples and pears could you use to achieve this
weight?
•
Can you write balanced number equations for each set of weighing
scales?
Students should have a look at the following questions. They require the
identity of two unknowns. The total that the bowls should add up to are
shown beneath the bowls.
In question h) students can decide themselves what the total weight can be.
Students should be aware that , in each case, there are various combinations
that will enable the scales to balance, students should try and come up with
all possible combinations.
The Language of Algebra:
Sadly, not every algebraic equivalence can be easily represented by apples
and pears. Therefore, it is important that pupils are able to express these
balanced equations algebraically. Although this lesson plan uses question
marks to represent the unknown, students should soon get used to seeing 'x'
to represent the unknown. The same goes for functions.
© Education Umbrella, 2014 6 Questions:
•
Can you express the functions that you determined in the number
machine algebraically? (Remember to use brackets)
•
For each of the 'balancing questions,' can you represent each one
algebraically, in the simplest form, using x as the unknown?
For example:
b) 400 = 100+x
e) 100 + x = 200 + x = 400
© Education Umbrella, 2014 7