PR-6816IRE Essential Study Guides - Number Algebra Strategies
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Order of operation<br />
This is a rule for performing<br />
operations in expressions<br />
which have more than<br />
one operation, to ensure<br />
calculations are handled in<br />
the same way.<br />
Some calculators use an<br />
‘algebraic operating system’<br />
(AOS). This is used to follow<br />
the Rule of Order.<br />
Note:<br />
Multiplication and division are equally powerful operations,<br />
completed left to right in order as they appear, as are<br />
addition and subtraction.<br />
Patterns<br />
Patterns are repeated designs. Patterns occur in<br />
mathematics in many different ways and always follow a rule.<br />
Simple number patterns such as odd and even numbers are<br />
easily identifi ed. Here are some more complex patterns.<br />
Triangular numbers<br />
A triangular number is a number that can form the shape of a<br />
triangle.<br />
1 3 6 10<br />
May also be shown as<br />
Square numbers<br />
A square number is a number that can form the shape of a<br />
square.<br />
1 4<br />
9<br />
16<br />
Word problems as equations<br />
Sequences<br />
A sequence is a list of items. Any item in the list can be<br />
named by its position: first, second, third, fourth and so on.<br />
Some lists have patterns which define the position of each<br />
item. There are two kinds of sequences:<br />
Mathematical terms<br />
A word problem needs to be changed into an equation before it can be solved. ‘Key words’ and phrases are used in word<br />
problems to tell what type of operation (addition, subtraction, multiplication, division) should be used to solve the problem.<br />
Look at the<br />
table. It shows<br />
some common<br />
key words and<br />
phrases, together<br />
with the correct<br />
operation needed<br />
to solve the<br />
problem.<br />
Brackets ( )<br />
Index notation 2 3<br />
Multiplication x<br />
Division ÷<br />
Addition +<br />
Subtraction –<br />
Operation Key word Word problem Equation<br />
addition<br />
subtraction<br />
multiplication<br />
division<br />
<strong>Algebra</strong> facts<br />
sum The sum of my age and 15 equals 32. a + 15 = 32<br />
total<br />
The total of my age and my brother’s age, who is 11<br />
years old, is 24.<br />
a + 11 = 24<br />
more than<br />
Fifteen more than my age equals 32.<br />
(Can also be subtraction.)<br />
a + 15 = 32<br />
difference<br />
The difference between my age and my younger<br />
sister’s age, who is 9 years old, is 3 years.<br />
a – 9 = 3<br />
less than Twelve less than my age equals 49. a – 12 = 49<br />
product The product of my age and 21 is 252. a x 21 = 252<br />
times Five times my age is 60. 5 x a = 60<br />
group A number grouped into lots of 6 is 5. a ÷ 6 = 5<br />
shared<br />
equally<br />
A ‘finite’ sequence is a list made up of<br />
a limited number of items.<br />
For example,<br />
9, 0, 9, 0, 9, 0, 9, 0, 9, 0, 9, 0 is the sequence of 6<br />
alternating 9s and 0s.<br />
1, 3, 5, 7, 9, 11, 13, 15, 17, 19 is the sequence of the<br />
fi rst 10 odd numbers.<br />
An ‘infinite’ sequence is a list that<br />
continues without end.<br />
For example,<br />
d, e, f, d, e, f, d, e, f, d, e ... is the sequence of the<br />
letters d, e, f, repeating in this pattern forever.<br />
2, 4, 6, 8, 10, 12, 14, 16 ... is the sequence of even<br />
whole numbers. The 50th number in this sequence<br />
is 100.<br />
constant<br />
Adding, subtracting,<br />
multiplying or dividing by the<br />
same amount each time.<br />
equivalent<br />
Having the same value.<br />
equation<br />
A statement of equality<br />
between two expressions;<br />
e.g. 3 x 4 = 6 + 6<br />
132 lollies shared equally among a number of<br />
children is 11 lollies each.<br />
growing<br />
A pattern that is becoming<br />
larger.<br />
inequality<br />
Not having the same value.<br />
repeating<br />
The pattern uses the same<br />
symbols or pictures over<br />
and over.<br />
e.g. 3, 4, 3, 4, 3 …<br />
132 ÷ a = 11<br />
To make mental maths easier, use your known number facts first. Here are some<br />
ideas to help you …<br />
Counting on<br />
3 + 5 =<br />
(change the order) 5 + 3 =<br />
1 2 3<br />
5 6 7 8<br />
6 + 11 =<br />
(change the order) 11 + 6 =<br />
1 2 3 4 5 6<br />
11 12 13<br />
Doubling<br />
3 + 3 = 6<br />
5 + 5 = 10<br />
13 + 13 = 26<br />
Ask<br />
clarifying<br />
questions<br />
What is the<br />
question<br />
asking?<br />
Do I know<br />
anything about<br />
this topic?<br />
What does …<br />
really mean?<br />
What else<br />
could I find out<br />
about to help<br />
me answer the<br />
question?<br />
14 15 16 17<br />
Make<br />
assumptions<br />
Using my<br />
knowledge<br />
of … I can<br />
assume …<br />
I think …<br />
because I<br />
know …<br />
If I know …<br />
then it could<br />
be …<br />
Mental strategies<br />
25 + 25 = 50<br />
33 + 33 = 66<br />
70 + 70 = 140<br />
Estimate<br />
Can I use my<br />
judgment<br />
to make<br />
a suitable<br />
guess?<br />
What<br />
strategies<br />
could I use<br />
to estimate<br />
a solution to<br />
the problem?<br />
Near doubles<br />
5 + 6 =<br />
(5 + 5) + 1 =<br />
10 + 1 = 11<br />
Survey<br />
Can I find out<br />
information<br />
by asking a<br />
sample of<br />
people?<br />
Do I need to<br />
investigate<br />
similar data<br />
and compare<br />
it to my<br />
information?<br />
OR<br />
Bridging a ten<br />
Example 1<br />
9 + 6 =<br />
(9 + 1) + 5 =<br />
10 + 5 = 15<br />
Partitioning<br />
5 + 8 =<br />
(5 + 5) + 3 =<br />
10 + 3 = 13<br />
Problem-solving strategies<br />
Locate<br />
information<br />
Where<br />
could I find<br />
information to<br />
help me solve<br />
the problem?<br />
Internet<br />
Library<br />
Things I<br />
already know<br />
Ask an expert<br />
Friends<br />
Adults<br />
5 + 6 =<br />
(6 + 6) – 1 =<br />
12 – 1 = 11<br />
Example 2<br />
www .prim-ed.com<br />
7 + 5 =<br />
(7 + 3) + 2 =<br />
10 + 2 = 12<br />
Present<br />
findings<br />
What is the<br />
best way to<br />
show what I<br />
have found?<br />
Diagrams<br />
Tables<br />
Graphs<br />
Calculations<br />
Explanations<br />
It is important to check your work to make sure answers to problems are<br />
correct and sensible. Checking your work can be done in many ways; some<br />
are shown below.<br />
Question your<br />
answer<br />
Ask yourself if the answer sounds<br />
right. A question you might ask<br />
yourself is …<br />
‘Is the answer way too big or way<br />
too small?’<br />
If you think the answer does<br />
not seem right, try some of the<br />
following checking strategies.<br />
Addition Subtraction Multiplication<br />
odd + odd = even<br />
odd + even = odd<br />
even + even = even<br />
Consider the size of the answer …<br />
When you read a question, do you ever consider how big the answer will be? Think<br />
about if the answer will be in tens, hundreds, thousands … or bigger! There is a<br />
pattern that can help alert you to potential errors.<br />
Addition<br />
Multiplication<br />
<strong>Number</strong> of digits<br />
Answer likely to be in the …<br />
1 digit + 1 digit units or tens<br />
2 digits + 1 digit tens or hundreds<br />
2 digits + 2 digits tens or hundreds<br />
3 digits + 1 digit hundreds or thousands<br />
3 digits + 2 digits hundreds or thousands<br />
3 digits + 3 digits hundreds or thousands<br />
1 digit x 1 digit units or tens<br />
2 digits x 1 digit tens or hundreds<br />
2 digits x 2 digits hundreds or thousands<br />
3 digits x 1 digit hundreds or thousands<br />
3 digits x 2 digits thousands or ten thousands<br />
3 digits x 3 digits ten thousands or hundred thousands<br />
Estimating<br />
Estimating gives you an answer that is<br />
close to the exact answer. It is usually<br />
found by rounding or by using judgment<br />
to make a ‘best guess’.<br />
Front-end rounding<br />
1. Look at the left-most digit in the<br />
number.<br />
2. Consider the place value of the digit.<br />
For example,<br />
3 2 1 5<br />
6 9 1 0<br />
+ 4 3 4 2<br />
3 + 6 + 4 = 13<br />
Viewing Sample<br />
www .prim-ed.com<br />
So the estimate would be 13 000.<br />
Checking strategies<br />
Odd and even numbers<br />
Odd and even numbers follow a pattern. Once you are aware of the pattern, all<br />
you need to do is look at the units digits of the numbers in the problem, and the<br />
answer to determine whether the answer is defi nitely wrong or possibly correct.<br />
Think about the context of the<br />
numbers before rounding.<br />
For example,<br />
When calculating the cost of groceries,<br />
it is probably better to overestimate so<br />
you do not run short of money.<br />
When calculating, you<br />
should:<br />
1. Estimate<br />
2. Calculate<br />
3. Evaluate (How close was your<br />
estimate? Could you improve on<br />
your technique?)<br />
odd – odd = even<br />
odd – even = odd<br />
even – odd = odd<br />
even – even = even<br />
odd x odd = odd<br />
odd x even = even<br />
even x even = even<br />
Divisibility<br />
rules<br />
It is often helpful to know when one<br />
number may be divided into another<br />
without leaving a remainder.<br />
2<br />
3<br />
4<br />
5<br />
6<br />
9<br />
10<br />
11<br />
12<br />
25<br />
100<br />
Any even number;<br />
e.g. 2, 4, 6, 8 or 0<br />
Sum of all digits = multiple of 3;<br />
e.g. 8652 = 8 + 6 + 5 + 2 = 21;<br />
2 + 1 = 3<br />
Last two digits divisible by 4;<br />
e.g. 23 632, 45 656, 13 988<br />
Any number ending in 0 or 5;<br />
e.g. 20, 45, 670, 9845<br />
Any number divisible by 2 and 3;<br />
e.g. 54, 972, 2196<br />
Sum of all digits = multiple of 9;<br />
e.g. 72 567 = 7 + 2 + 5 + 6 + 7<br />
= 27; 2 + 7 = 9<br />
Any number ending in 0;<br />
e.g. 10, 200, 7680, 98 450<br />
Alternate digit sums differ by a<br />
multiple of 11 or 0;<br />
e.g. 76 285 = 7 + 2 + 5 = 14<br />
6 + 8 = 14; 14 – 14 = 0<br />
Any number divisible by 3 and 4;<br />
e.g. 180, 720, 3600<br />
Last two digits divisible by 25;<br />
e.g. 6475, 8950<br />
Last two digits are 00;<br />
e.g. 67 400, 589 780 300<br />
Repeat the calculation:<br />
•<br />
carefully, in exactly the same<br />
way.<br />
• using the inverse operation.<br />
• using a different method.<br />
Maths<br />
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<strong>PR</strong>–6816<br />
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6816RE maths 1.indd 2<br />
8/12/05 11:01:07 AM