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Mathematical reasoning

Mathematical reasoning is about various data representation, its interpretation and use. We’ll explain how to solve problems using simple math.

Reading and interpreting data

In every aspect of life, we are dealing with massive amount of informations and data. We always strive to read those informations as easily as we can. That is why we are labelling them, drawing all kinds of graphs or pictures that show us everything we need to know without wasting any time on searching for information in messy lists.

The most popular way of interpretation data is the bar chart. It consists of two axes with values of things we are observing.

For example we’ll make a simple chart. For the start we’ll need some kind of informations and values in areas we are interested in.

Let’s say every day you take apples to school. On Monday you took 1 apple, Tuesday 2, Wednesday 5, Thursday 3, Friday 4. And you want to make a chart of it.

First you draw two perpendicular lines like this. Then you want to observe the number of apples a day you took through 5 days. You have 5 days, which means that you’ll draw 5 segments representing your days onto horizontal axis. On the vertical axis you’ll draw 5 segments (5 is the max number of apples you took).

Then you should get something like this:

graph showing apples

Now comes the fun part. For every day you mark how many apples you took and you draw rectangles.

First on Monday you took 1 apple, this means that 1 will be the height of your rectangle.

apples on monday

Next is Tuesday with 2 apples:

apples on tuesday

Now continue till you finish. This is the final result:

number of apples in a week

And with this you completed your chart. From this you can easily read out the information you need, no matter how complex and messy something is.

Next thing is the show how to make a pie chart. Pie charts usually represents a share of a whole expressed relatively. it is named like that because it is a circle which you cut like a pie.

For example if you want to draw a pie chart of the things you do in your free time, and you know that 25% of the time you read, 25% you play with your friends and 50% you play video games. This is how you would do it. One circle is one whole, you know that 25% is \frac{1}{4}, and 50% is \frac{2}{4} which means that you’ll divide a circle into four parts. One part will be for reading, one for playing with friends and two for playing video games (two because it is 50% = \ 2 * \frac{1}{4}).

Also, a pie chart should always have a legend beside it. A legend is a list of your informations and matching colours so you can easily manoeuvre your pie chart.

pie chart

Why don’t you try it backwards? From every of these charts you can also retrieve informations.

There are many more charts that are used.
The most common chart is known as the line chart. If we imagine our informations and its values as points in a plane and connect them with a line. We get a function – like chart.

For example. Let’s say you are selling chocolate bars to your neighbours. This would be the line chart of your sales:

line chart number of sold cookies

In which month did you sell the most cookies? You simply look at the growth of the function, if your sales grows, so does your function. This means that all you have to do is find the highest point which would be in December where you sold 15 cookies. The lowest point is in October where you sold only one cookie.

Constructing numerical expressions

In this lesson you’ll learn how to transform simple math problems from words into numerical expressions.

First we’ll start with the easy one. What is the quotient of ten minus two and two?

This, written as a fraction is:

\frac{(10-2)}{2} = \frac{8}{2} = 4
These kinds of tasks can be a little tricky, so read the task carefully, and watch which operation comes first.

Second:

What is the quotient of 10 times 5 – 2 and 3?

\ 10 * \frac{(5 - 2)}{3} = 10
Now let’s get to the simple word problems.
With these kind of tasks it is very important to carefully read what you have and what you have to get. Be sure to read the task a couple of times before writing anything down.
Second thing you do is to write information you have one by one, and be sure to mark them appropriately.
Third and last is to simply solve your math problem.

Example 1.
Martha loves chestnuts. It was fall and she was walking down the street every day collecting them. First day she collected 5 chestnuts. Second day she collected 3, but on her way home lost 2 of them. Her father came and gave her exactly the amount of chestnuts she already had. How many does she have now?

1. day – collected 5
2. day – collected 3, lost 2

Before the day three she had 5 + 3 – 2 = 6. This means that her father gave her 6 more, which means she has 12 chestnuts.

How would you write this in a single mathematical expression?

number of chestnuts = \ (5 + 3 - 2) * 2 = 12

Example 2.
Pier was a famous painter and had a display of his paintings. He had only one week to prepare 8 paintings. By Tuesday he had already prepared 2 paintings, but he knew that Wednesday and Thursday are the days when he could not paint. How many paintings minimum must he need to paint in Friday, Saturday and Sunday?

Solution:
One week has 7 days. If we know he can’t work on Wednesday and Thursday, that means that he has only 5 days. If he finished two paintings by Tuesday, this means that:

8 – 2 = 6 – he is left with 6 paintings
7 – 4 = 3 – and he has 3 days to do it
This means that he has to paint \frac{6}{3} -> two paintings a day minimum.
How would you write this problem in one expression?

painting task

Multi-step word problems

Multi-step word problems usually include few parentheses.
You solve them step by step. First you always write down what you have, all informations you can extract from the task. This will help you easily see what to do and how to do it.

Example 3.
Ella and Martin are going to buy some ice cream for their families and themselves. One ice cream costs 2 dollars. Ella will buy 5 ice creams, and Martin 4. How much money did Ella spend more than Martin?

1. Ice cream = 2 dollars
2. Ella bought 5 ice creams – she spent \ 5 * 2 dollars = 10 dollars
3. Martin bought 4 ice creams – he spent \ 4 * 2 dollars = 8 dollars
4. Their difference – 10 dollars – 8 dollars = 2 dollars

The formula: \ 5 * 2 - 4 * 2 = 10 - 8 = 2
Or if you already know that all ice creams cost 2 dollars this can also be written as: \ 2 * ( 5 - 4) = 2

Next thing that may occur in your problem solving is inequality. If you are not sure about what it is or how it is solved you can look it up in inequalities lesson.

Example 4.:
Maria and Peter are competing in gathering postcards. Maria has 2 postcards more than Peter. If Maria is getting two postcards a week, and Peter three, will Maria still have more postcards than Peter in three weeks?

Let’s mark Peter’s postcards with a ‘P’, and Maria’s with an ‘M’.
M = P + 2 (Maria has two more postcards than Peter)

In the second week their numbers will be:
\ M = P + 2 + 3 * 2 = P + 8 (she receives two postcards for three weeks)
\ P = P + 3 * 3 = P+ 9 ( P is the start value, and he receives three postcards for three weeks)
\ P + 8 < P + 9 this means that Peter will have one postcard more.

In an algebraic expression:
‘O’ will represent our missing inequality sign.

Maria O Peter

The start values

\ P + 2 > P

After three weeks

\ P + 2 + 3 * 2 O P + 3 * 3
\ P + 8 O P + 9
\ P < P + 1
\ M < P
 

Cross topic arithmetic

Word problems can also be given by ratios, rational numbers and percentages. They are solved in the same way you’d solve any other word problem, you just have to be careful of what you take percentage of and what you add it with. It isn’t hard but can be messy.

To remind yourself how to calculate with percentages, read lesson about percents.

Example 5.
You found two cubes. One weighs 4 pounds and second one weighs 9 pounds. What is the ratio between their weights?

This is one of fundamental tasks. You simply write it in a fraction and shorten the fraction if possible.

\frac{(weight1)}{(weight2)} = \frac{4}{9}

The ratio of their weights is 4 : 9.

Example 6.
If Anna bought 25 chocolate bars and Michael came and ate 10% of them, how many does she have left?

The starting value is 25. This is how many chocolate bars Anna had in the beginning.

Now we want to know how many of them Michael ate: 10% from \ 25 = \frac{10}{100} * 25 = 2,5

Michael ate 2,5 bars. Since Anna had 25, she now has \ 25 - 2,5 = 22,5 chocolate bars.

Example 7. (slightly more complicated task)
Martha had a big bowl of strawberries. This bowl contained 50 strawberries. First she ate 14% of them. After one hour she came back and ate 10%. If she had to leave 50% from the whole bowl, how many more strawberries is she allowed to eat?

The starting value is 50 strawberries.

If she ate 14% of them she ate: \frac{14}{100} * 50 = 7 which means she had \ 50 - 7 = 43 strawberries left.

When she came back she ate 10%, but now your value you take the 10% of is 43 because she already ate the first 14%.

This means that she ate 10% from \ 43 = \frac{10}{100} * 43 = 4,3. She now has \ 43 - 4,3 = 38.7 strawberries left.

Now we have to see how many of them she can’t eat. If she can’t eat 50% of the starting value, she has to leave 25 strawberries.

She has 38.7 and has to leave 25 -> she can eat \ 38.7 - 25 = 13.7 strawberries.

Number patterns

Number pattern is a property of a continuous sequence of numbers which helps us anticipate how a certain sequence will behave after a lot of steps.

Here we’ll be observing a few simple patterns that you may encounter.

First are the repeating numerals.

These patterns are easy to recognize and continue.

For example:
The pattern with only one or more repeating numerals:

repeating numerals

Task 1. Continue the sequence with two more numerals: 8 7 6 5 8 7 6 8
If we examine this sequence we’ll see that the numbers 8, 7, 6 are repeating. This means that the next two numerals will be 7 and 6.

Second are patterns that involve addition.

This sequence is made by simply adding one fixed number onto another. How will you recognize this pattern? You’ll simply subtract two adjacent numbers, the left one from the right one, and do this few times and see if you always get the same number. In case you do get the same number, this will be the number you’ll always add onto your start value. This number is called their common difference.

You already know sequence:

\ 1, 2, 3, 4, 5, 6, 7...

This is a sequence that is made by the start value 1 and with constantly adding number 1.
The start value is always the first number in your sequence, and the number you’re constantly adding is
\ 2 - 1 = 1; 3 - 2 =1; 4 - 3 = 1 and so on, this means that our number is 1.

Of course, this number we’re adding can be any real number. For example examine the following sequence and find the missing element.

\ 1, \frac{3}{2}, 2, ... , 3, \frac{7}{2}, 4, ...

The first thing to do is to determine which kind of sequence this is. Because we only learned one type, which are the repeating decimals, which this clearly isn’t, this has to do something with addition.
Again, we’ll find their difference.

\frac{3}{2} - 1 = \frac{1}{2}, 2 - \frac{3}{2} = \frac{1}{2}, \frac{7}{2} - 3 = \frac{1}{2}, 4 - \frac{7}{2} = \frac{1}{2} ...

This means that this sequence is made by adding 1/2 on our start value 1.

This means that the missing value is \ 2 + \frac{1}{2} = \frac{5}{2}

We can check it by subtracting \frac{5}{2} from 3, if we get \frac{1}{2} our number is correct.

\ 3 - \frac{5}{2} = \frac{1}{2}
But things don’t have to be necessarily so easy. Sequences can be mixed up:

\ 1, 4, 2, 5, 3, 6, 4, 7, 5, 8, 6...

This sequence can be broken into two separate sequences:

\ 1, 4, 2, 5, 3, 6, 4, 7, 5, 8, 6...

One sequence 1, 2 ,3, 4, 5, 6… where our start value is 1, and we add 1, and second 4, 5, 6, 7, 8 where our start value is 4 but we also add 1.
You can also subtract. So, if your start value is equal to 4, and their common difference is -1:

 

\ 4, 3, 2, 1, 0, -1, -2...

 

One of the most famous sequences of this sort is the Fibonacci sequence.

It goes like this:

\ 1, 1, 2, 3, 5, 8, 13, 21...

These elements have no common difference. But if you go and try to find it:
\ 21 - 13 = 8; 13 - 8 = 5... See the pattern? Every following number in Fibonacci sequence is given as the sum of its 2 predecessors. This pattern can be seen in many examples in nature: the petals in the flowers, proportions of human bodies, growth of any living thing and so on.

The last thing we’re going to mention is the sequence in which the elements are bounded by some sort of multiplication.

For example:

\ 2, 4, 8, 16, 32, 48, 96...\

This sequence is made by the start value 2 that is constantly multiplied with 2. If we want to know which is the number we’re constantly multiplying with, we’d divide a number in a sequence with his predecessor. If that number is the same in every element, we got our common factor.

Example 8. Find the missing number in a sequence.

\ 5, 15, 75, x, 675

First, we’ll find their common factor:
\ 15 : 5 = 3, 75 : 15 = 3. This means that in this sequence our common factor will be 3.
This leads us to the fact that our element is: \ 75 * 3 = 225.
Also: \ 675 : 225 = 3

Things, of course can get a lot more complicated than this. You can have also powering, roots, mixed everything.
The best thing to learn how to recognize these patterns is to construct your own. Play with it and find new sequences.

Binary and hexadecimal number systems

First what is a number system? A number system tells us how we denote our numbers. The number system we are used to is called the decimal number system or a number system with a base ten. As you can see in the title, this base determines whole system.

Decimal number system has a base 10 because we are working with 10 numbers (0, 1, 2, 3, 4, 5, 6, 7, 8, 9), and every other larger number can be constructed from these 10 main numbers.

We didn’t always use the decimal system, it developed through the ages. There are two main number systems, apart from decimal.

In this lesson we’ll be learning how to transform from one system to the other and what are their uses.

Binary system is a system with base 2, which means that it operates with only base 2.

Hexadecimal system is a system with base 16 which means that it operates with 16 “numbers” – 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, A, B, C, D, E, F.

In decimal A = 10, B = 11, C = 12, D = 13, E = 14, F = 15.

Binary number system

Transforming a decimal number into binary and vice versa.

The process of transforming a decimal number is quite simple. You just divide with two as many times as you want, write on the side the remainders and read them backwards.

When writing a number when dealing with different bases it’s easier and better if you always mark with what base you’re working. The standard is to write the base in brackets in index of a number.

Example: Transform number 25(10) into a binary notation

transform into binary notation

Now don’t forget to read the number upside down:

\ 25_{10} = 11001_{2}

Now, how to do it backwards? Since in transforming from decimal to binary we divided, in opposite case we’ll multiply.

If we want to transform number \ 11001_{2} in decimal first you do the following. On every element of this number we’ll mark numbers from zero, starting from the right:

binary into decimals

Since the base is 2, do the following. You’ll go one by one element and multiply it with 2 to the power of the index above your elements and bound them with addition.

\ 1 * 2^4 + 1 * 2^3 + 0 * 2^2 + 0 * 2^1 + 2^0 = 2^4 + 2^3 + 1 = 16 + 8 + 1 = 25

Addition and subtraction in binary

addition and subtraction in binary

First steps:
Subtract \ 10_{2} and 1_{2}.

\ 0 - 1 = - 11 -> 10 - 1 = 1
Add \ 11_{2} and 1_{2}
\ 1 + 1 = 10 -> 11 + 1 = 100
Example:
Add and subtract numbers \ 11001_{2} and 110_{2}.

add and subtract numbers 110012 and 11021

You can always check these answers by doing opposite:
10011 + 110 = 11001

Hexadecimal number system

Hexadecimal number system is a system with base 16. The procedure of transforming between a decimal system and hexadecimal system is the same as from binary, only you divide, or multiply with 16.

transform number from base 10 to base 16

The middle number is 11 which is equivalent to B.
This means that \ 2489_{10} = 9B9_{16}

Backwards procedure:

\ 9^2 B^1 9^0 -> 9B9_{16} = 9 * (16)^2 + 11 * 16^1 + 9 * (16) ^ 0 = 2304 + 176 + 9 = 2489_{10}

Addition and subtraction in hexadecimal number system is also equivalent to addition and subtraction in any other number system.

Example: Subtract and add numbers \ 15A_{16} and \ 25B_{16}
The most confusing part here is this A + B, so we’ll explain it in detail. A has the value of 10 and B 11. Added together they have the value of 21. You have one 16 in 21 and remainder 5 which means that you’ll write down five and take 1 to the next number. Next are 5 and 5 which is 10, when given 1 we transferred, it is 11 which is B. the last step is 1 and two which is 3.

add and subtract binary numbers

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