01 Natural numbers
ACTIVITY
BAR CODE
Every item we find in a market has a number to identify it quickly. This is the bar code.
The bar code of every thing has 13 digits. The bars are a way to introduce these numbers through a laser reader.
THE MEANING OF A BAR CODE
We are going to do this activity with the above example. Fill the gaps with the bar code.
Country Company Product Control
WHAT IS THE INFORMATION OF A BAR CODE?
The first two figures is the country code.
The following five digits is the information about the company. The following five digits are about the product and, finally,
the last digit is the control digit which is the interesting subject for us in this chapter.
WHAT IS THE PURPOSE OF THE CONTROL DIGIT?
The purpose of the control digit is to locate possible reading or writing mistakes by a machine.
The control code has a formula that links its value with the other numbers.
FORMULA
The control digit formula is the next. It is the result of doing the following operations: To add the digits which are in odd position from left to right not including the last digit because it is the control digit.
To add up the even digit from left to right and after that to multiply the result by 3. After that we add the two results.
Finally, the control digit is the number we have to add up in order to get a multiple of ten.
In this case it is 7.
5 9 0 1 2 3 4 1 2 3 4 5
⇓ ⇓ ⇓ ⇓ ⇓ ⇓
+ + + + + =
⇓ ⇓ ⇓ ⇓ ⇓ ⇓
+ + + + + x3 =
There are other numbers to identify something or someone. For instance, the identity card, the bank account number, the international standard book number (ISBN)…
Item number 1
Introduce the bar code in the chart. You can attach the bar code in the empty space:
Country Company Product Control
What do you think about the country? Write the country and next to it the identification number.
What do you think about the company? Write the company and next to it the identification number.
What do you think about the product? Write the product and next to it the identification number.
Finally we are going to check the control digit doing the right operations to calculate it:
⇓ ⇓ ⇓ ⇓ ⇓ ⇓
+ + + + + =
⇓ ⇓ ⇓ ⇓ ⇓ ⇓
+ + + + + x3 =
Item number 2
Introduce the bar code in the chart. You can attach the bar code in the empty space:
Country Company Product Control
What do you think about the country? Write the country and next to it the identification number.
What do you think about the company? Write the company and next to it the identification number.
What do you think about the product? Write the product and next to it the identification number.
Finally we are going to check the control digit doing the right operations to calculate it:
⇓ ⇓ ⇓ ⇓ ⇓ ⇓
+ + + + + =
⇓ ⇓ ⇓ ⇓ ⇓ ⇓
+ + + + + x3 =
Bar code
LESSON SUMMARY
Complete the gaps with the words below.
Vocabulary
associate; associative; brackets; commutative; count; decimal; digit; distributive; division; taking away; infinite; inverse; left; number; order (2); positional; quantity; represent; symbolic; union.
1. NUMBERS
A number is the complete the gap expression of a complete the gap.
A complete the gap is each symbol we use to make numbers. Our number system is named complete the gap because it has 10 digits: 0…9
Our number system is also complete the gap because the value of a digit depends on the position that it has.
We represent the numbers in a line with two values: the zero and the unit. This is the complete the gap line.
Numbers are used to describe quantities and order. The symbols we use to express complete the gap are ‘<’ and ‘>’
2. NATURAL NUMBERS
Natural numbers are the easiest numbers. They are useful to complete the gap. We represent these by the capital letter N: 1, 2, 3… They are complete the gap because they never end.
3. ADDITION AND MULTIPLICATION
The addition is the operation that gives the value of the complete the gap of two quantities.
The addition has two properties: complete the gap and complete the gap.
The first property means that the complete the gap of the quantities doesn’t matter to the result.
The second property means if we have to do two additions we can complete the gap numbers in every way.
The subtraction is the complete the gap operation. It is the result of complete the gap some quantity from another.
The multiplication is the way to complete the gap a multiple addition. This operation is commutative and associative too.
The inverse operation for the multiplication is the complete the gap. Then it is the result of a multiple subtraction.
And finally we have a property that links the addition and multiplication. It is the complete the gap property.
The meaning of this property is the following: we get the same result by multiplying a number by a sum than multiplying the number by each member and adding all together.
⋅ + = ⋅ + ⋅
5 (2 4) 5 2 5 4 . That is, 5·6 = 5·2 + 5·4
6 5
2 4
The formula is the following; that is, for whatever numbers:
a b c
⋅ + = ⋅ + ⋅
(
)
a b
a c
4. ORDER OF OPERATIONS
To do combined operations we have to follow the next order: First. We do the complete the gap.
After. Multiplication and division After. Addition and subtraction.
Operations with same hierarchy we do from complete the gap to right.
EXERCISES AND PROBLEMS
1.
Put the result of these operations in each square:Five by
four add 4
divide by 3
half of
that times 4
minus 1 divide by three times itself minus 6
2.
Write the answer to the question in each cell:Three plus four times two minus five divided by three
by five plus one
half of this
half of this
add on six
3.
Do the same as above:nine by three
plus nine
share into six equal parts by seven three more divide into nine double it by seven
half of it
4.
Obtain the result of each step for the next chain.5.
Calculate mentally, do not write anything. You have to spend less than one minute on getting the final result.Twelve subtract
eight by six plus one
divide by five
plus
two by four plus two
half of this
minus six
6.
Work out in your mind following the operations in each step. You have to get the final result in less than one minute.Seventeen take away five divided by four eight times this double it add one divide by seven times eight plus four half of it
7.
Do the same as the previous ones:Thirty nine plus nine divide by eight times itself minus one divide into five by four remove one share into three by seven
8.
Obtain the result of each step for the next chain:Nine By three Plus nine Divided by 6 Times
seven Add 3
Divided by 9 Double it Times 7 Half of it
9.
Calculate mentally. This is without using the pen. After that, write the final result.33 add fifteen ¼ of this one sixth of this by 9 times 2 6 more one over seven of this
times nine
minus 5
10.
Do the following multiplication: 37569 x 47811.
Do the following multiplication: 402987 x 67312.
Calculate the quotient and the remainder for the division: 257284 ÷ 73413.
Calculate the quotient and the remainder for the division: 520687 ÷ 65914.
Find the quotient and the remainder of this division: Listnumber730415:86415.
Do the following division, that is, calculate its quotient and its remainder. N54604:729 N is your list number.16.
Work out: N702454 : 76517.
Do the following combined operation: 23 + 2 · (18 – 6 : 3) – 5 · 4=18.
Do step by step: 13 · 2 · 5 – 4 · (8 – 6 : 2) – 3 · 519.
Work out the following operation: 3 + 2 · (15 – 5 · 2) – 2 · 320.
Calculate doing the parenthesis first:a)
7 (3 9)
⋅ + =
; b)12 (5 8)
⋅ + =
; c)12 (10 8)
⋅
+ =
a)
7 (3 9)
⋅ + =
; b)12 (5 8)
⋅ + =
; c)12 (10 8)
⋅
+ =
22.
Look at this way to calculate using the distributive property: 15·18 = 15 · (10 + 8) = 150 + 120 = 270.Do the same to calculate:
a) 12 · 15 =; b) 17 · 24 =; c) 25 · 32 =
23.
What is the control digit for the following bar code: 84 12376 28795 __24.
A tailor bought 850 trousers 9 € each. He sells all of them getting 12750€ for the sale. What is the profit per trouser?25.
A tailor bought 1200 trousers 35 € each. He sells all of them getting 44400€ of profit. What is the gain per trouser?26.
A woman carries 15 vases to sell them in a market for 12 € each. On the way she has an accident and 3 vases break. How much must be the price to get the same money?27.
A dealer carries 12 vases to sell them in a market for 30 € each one. On the way he has an accident and 2 vases break. How much must be the price to get the same money?28.
A carrier has 20 mirrors that he plans to sell at 24 € each. 4 mirrors are broken in an accident. At what price does he have to sell the remainder mirrors to obtain the same amount of money?29.
A farmer buys a cow for 1100 €. The cow consumes a sack of feed per day which costs 2 €. Every day, on average, the farmer gets 40 liters of milk, which sells for € 0.5 per liter.He sells it for 900 € after 180 days. What benefit has he obtained?
