Two numerical mathematical expressions connected by the "=" sign are called equality.
For example: 3 + 7 = 10 - equality.
Equality can be right and wrong.
The point of solving any example is to find the meaning of the expression that turns it into a true equality.
To form ideas about true and false equalities in the 1st grade textbook, examples with a window are used.
For example:
Using the selection method, the child finds suitable numbers and checks the correctness of the equality by calculation.
The process of comparing numbers and designating relationships between them using comparison signs leads to inequalities.
For example: 5< 7; б >4 - numerical inequalities
Inequalities can also be true and false.
For example:
Using the selection method, the child finds suitable numbers and checks the correctness of the inequality.
Numeric inequalities are obtained by comparing numeric expressions and numbers.
For example:
When choosing a comparison sign, the child calculates the value of the expression and compares it with a given number, which is reflected in the choice of the corresponding sign:
10-2> 7 5 + K7 7 + 3> 9 6-3 = 3
Another way of choosing the comparison sign is possible - without reference to the calculation of the value of the expression.
Nappimep:
The sum of the numbers 7 and 2 will certainly be greater than the number 7, which means that 7 + 2> 7.
The difference between the numbers 10 and 3 will certainly be less than the number 10, which means that 10 is 3< 10.
Numeric inequalities are obtained by comparing two numeric expressions.
To compare two expressions is to compare their values. For example:
When choosing a comparison sign, the child calculates the values of the expressions and compares them, which is reflected in the choice of the corresponding sign:
Another way of choosing the comparison sign is possible - without reference to the calculation of the value of the expression. For example:
For setting comparison signs, one can carry out the following reasoning:
The sum of 6 and 4 is greater than the sum of 6 and 3, since 4> 3, which means 6 + 4> 6 + 3.
The difference between the numbers 7 and 5 is less than the difference between the numbers 7 and 3, since 5> 3, which means 7 - 5< 7 - 3.
The quotient of 90 and 5 is greater than the quotient of 90 and 10, because when dividing the same number by a larger number, the quotient is smaller, which means 90: 5> 90:10.
To form ideas about true and false equalities and inequalities in the new edition of the textbook (2001), tasks of the form are used:
For verification, the method of calculating the value of expressions and comparing the resulting numbers is used.
Inequalities with a variable are practically not used in the latest editions of the stable mathematics textbook, although they were present in earlier editions. Inequalities with variables are actively used in alternative mathematics textbooks. These are inequalities of the form:
+ 7 < 10; 5 - >2; > 0; > O
After the introduction of a letter to denote an unknown number, such inequalities take on the usual form of inequality with a variable:
a + 7> 10; 12-d<7.
The values of the unknown numbers in such inequalities are found by the selection method, and then each matched number is checked by substitution. The peculiarity of these inequalities is that several numbers can be selected that fit them (giving the correct inequality).
For example: a + 7> 10; a = 4, a = 5, a = 6, etc. - the number of values for the letter a is infinite, any number a> 3 is suitable for this inequality; 12 - d< 7; d = 6, d = 7, d = 8, d = 9, d = 10, d = 11, d = 12 - количество значений для буквы d конечно, все значения могут быть перечислены. Ребенок подставляет каждое найденное значение переменной в выражение, вычисляет значение выражения и сравнивает его с заданным числом. Выбираются те значения переменной, при которых неравенство является верным.
In the case of an infinite set of solutions or a large number of solutions to an inequality, the child is limited to selecting several values of the variable for which the inequality is true.
In this lesson, you and the frog will become familiar with the mathematical concepts of equality and inequality, as well as comparison signs. Use fun and interesting examples to learn how to compare groups of shapes using pairing and compare numbers using a number ray.
Theme:Familiarity with basic concepts in mathematics
Lesson: Equality and Inequality
In this lesson we will get acquainted with mathematical concepts: "equality" and "inequality".
Try to answer the question:
There are tubs against the wall
Each has exactly one frog.
If there were five tubs,
How many frogs would there be? (fig. 1)
Rice. 1
The poem says that there were 5 tubs, in each tub there is 1 frog, no one was left without a pair, which means the number of frogs is equal to the number of tubs.
Let's denote the tubs by the letter K, and the frogs by the letter L.
We write the equality: K = L. (Fig. 2)
Rice. 2
Compare the number of two groups of shapes. There are many figures, they are of different sizes, arranged without order. (fig. 3)
Rice. 3
Let's make pairs of these figures. We connect each square with a triangle. (fig. 4)
Rice. 4
Two squares were left without a pair. This means that the number of squares is not equal to the number of triangles. Let us denote the squares by the letter K, and the triangles by the letter T.
We write the inequality: K ≠ T. (Fig. 5)
Rice. 5
Output: You can compare the number of items in two groups by pairing. If all the elements have enough pairs, then the corresponding numbers are equal, in this case we put between numbers or letters =... This entry is called equality... (fig. 6)
Rice. 6
If there is not enough pair, that is, extra items remain, then these numbers not equal... We put between numbers or letters unequal sign... This entry is called inequality.(fig. 7)
Rice. 7
The elements left without a pair show which of the two numbers is greater and by how much. (fig. 8)
Rice. eight
The method of comparing groups of shapes using pairing is not always convenient and takes a lot of time. You can compare numbers using the number ray. (fig. 9)
Rice. nine
Compare these numbers using the number beam and put a comparison sign.
We need to compare the numbers 2 and 5. Let's look at the number beam. The number 2 is closer to 0 than the number 5, or they say the number 2 on the number ray is more to the left than the number 5. This means that 2 is not equal to 5. This is an inequality.
The sign "≠" (not equal) only fixes the inequality of numbers, but does not indicate which of them is greater and which is less.
Of the two numbers on the number ray, the smaller is to the left and the larger to the right. (fig. 10)
Rice. ten
This inequality can be written differently, using less sign "< » or greater than sign ">" :
On the number ray, the number 7 is to the right than the number 4, therefore:
7 ≠ 4 and 7> 4
The numbers 9 and 9 are equal, so we put the = sign, this is equality:
Compare the number of dots and the number and put the appropriate sign. (fig. 11)
Rice. eleven
In the first picture, we need to put the sign = or ≠.
Compare two points and the number 2, put an = sign between them. This is equality.
We compare one point and the number 3, on the number ray the number 1 is to the left of the number 3, we put the sign ≠.
Compare the four points and 4. Put the = sign between them. This is equality.
Compare three points and number 4. Three points - this is number 3. On the number beam it is to the left, put the sign ≠. This is inequality. (fig. 12)
Rice. 12
In the second figure, between the dots and numbers, you need to put the signs =,<, >.
Let's compare five points and the number 5. Between them we put an = sign. This is equality.
Let's compare three dots and the number 3. Here you can also put the = sign.
Let's compare five points and the number 6. On the number ray, the number 5 is to the left of the number 6. Put the sign<. Это неравенство.
Let's compare two points and one, the number 2 is to the right of the number ray than the number 1. We put the> sign. This is inequality. (fig. 13)
Rice. 13
Insert a number in the box to make the equality and inequality correct.
This is inequality. Let's look at the number beam. Since we are looking for a number less than the number 7, then it must be to the left of the number 7 on the number ray. (fig. 14)
Rice. fourteen
Several numbers can be inserted into the window. The numbers 0, 1, 2, 3, 4, 5, 6 are suitable here. Any of them can be substituted in the window and get several correct inequalities. For example 5< 7 или 2 < 7
On the number ray, find numbers that are less than 5. (Fig. 15)
Rice. 15
These are the numbers 4, 3, 2, 1, 0. Therefore, any of these numbers can be substituted in the window, we get several correct inequalities. For example, 5> 4, 5> 3
You can substitute only one number 8.
In this lesson, we got acquainted with the mathematical concepts: "equality" and "inequality", learned how to correctly place comparison signs, practiced comparing groups of figures using pairing and comparing numbers using a number ray, which will help in the further study of mathematics.
Bibliography
- Alexandrova L.A., Mordkovich A.G. Grade 1 mathematics. - M: Mnemosina, 2012.
- Bashmakov M.I., Nefedova M.G. Maths. 1 class. - M: Astrel, 2012.
- Bedenko M.V. Maths. 1 class. - M7: Russian Word, 2012.
- Igraem.pro ().
- Slideshare.net ().
- Iqsha.ru ().
Homework
1. What comparison signs do you know, in what cases they are used? Write down the comparison signs for the numbers.
2. Compare the number of items in the figure and put the sign “<», «>"Or" = ".
3. Compare the numbers by putting the sign “<», «>"Or" = ".
1. The concept of equality and inequality
2. Properties of equalities and inequalities. Examples of solving equalities and inequalities
Numerical Equalities and Inequalities
Let be f and g- two numeric expressions. Let's connect them with an equal sign. We will receive an offer f= g which is called numerical equality.
Take, for example, the numerical expressions 3 + 2 and 6 - 1 and connect them with the equal sign 3 + 2 = 6-1. It is true. If we connect 3 + 2 and 7 - 3 with an equal sign, then we get a false numerical equality 3 + 2 = = 7-3. Thus, from a logical point of view, numerical equality is a statement, true or false.
Numeric equality is true if the values of the numeric expressions on the left and right sides of the equality are the same.
Properties of equalities and inequalities
Let us recall some properties of true numerical equalities.
1. If we add the same numerical expression that makes sense to both sides of the true numerical equality, then we also get the true numerical equality.
2. If both sides of a true numerical equality are multiplied by the same numerical expression that makes sense, then we also get a true numerical equality.
Let be f and g- two numeric expressions. Let's connect them with a ">" (or "<»). Получим предложение f > g(or f < g), which is called numerical inequality.
For example, if you combine the expression 6 + 2 and 13-7 with the ">" sign, we get the true numerical inequality 6 + 2> 13-7. If you connect the same expressions with the sign "<», получим ложное числовое неравенство 6 + 2 < 13-7. Таким образом, с логической точки зрения числовое неравенство - это высказывание, истинное или ложное.
Numerical inequalities have a number of properties. Let's consider some.
1. If we add the same numerical expression that makes sense to both sides of a true numerical inequality, then we also get a true numerical inequality.
2. If both sides of a true numerical inequality are multiplied by the same numerical expression that has meaning and a positive value, then we also get a true numerical inequality.
3. If both sides of a true numerical inequality are multiplied by the same numerical expression that has meaning and a negative value, and also change the sign of the inequality to the opposite, then we also get a true numerical inequality.
Exercises
1. Determine which of the following numeric equalities and inequalities are true:
a) (5.05: 1/40 - 2.8 5/6) 3 + 16 0.1875 = 602;
b) (1/14 - 2/7): (-3) - 6 1/13: (-6 1/13)> (7- 8 4/5) 2 7/9 - 15: (1/8 - 3/4);
c) 1.0905: 0.025 - 6.84 3.07 + 2.38: 100< 4,8:(0,04·0,006).
2. Check if the numerical equalities are true: 13 93 = 31 39, 14 82 = 41 28, 23 64 = 32 46. Can you say that the product of any two natural numbers will not change if you rearrange the numbers in each factor?
3. It is known that x> y - true inequality. Will the following inequalities be true:
a ) 2x> 2y; v ) 2x-7< 2у-7;
b) - x/3<-y/ 3; G ) -2x-7<-2у-7?
4. It is known that a< b - true inequality. Replace * with ">" or "<» так, чтобы получилось истинное неравенство:
a) -3.7 a * -3,7b; G) - a/3 * -b/3 ;
b) 0.12 a * 0,12b; e) -2 (a + 5) * -2(b + 5);
v) a/7 * b/ 7; f) 2/7 ( a-1) * 2/7 (b-1).
5. Given inequality 5> 3. Multiply both sides by 7; 0.1; 2.6; 3/4. Is it possible, on the basis of the results obtained, to assert that for any positive number a inequality 5a> 3a is it true?
6. Complete the tasks that are intended for primary school students, and draw a conclusion about how the concepts of numerical equality and numerical inequality are interpreted in the elementary mathematics course.
This article has collected information that shapes the concept of equality in the context of mathematics. Here we will find out what equality is from a mathematical point of view, and what they are. We'll also talk about notation for equalities and the equal sign. Finally, we list the main properties of equalities and give examples for clarity.
Page navigation.
What is Equality?
Equality is inextricably linked with comparison - the juxtaposition of properties and attributes in order to identify similarities. And comparison, in turn, presupposes the presence of two objects or objects, one of which is compared with the other. If, of course, you do not compare the object with itself, and then, this can be considered as a special case of comparing two objects: the object itself and its "exact copy".
From the above reasoning, it is clear that equality cannot exist without the presence of at least two objects, otherwise we simply will have nothing to compare. It is clear that you can take three, four or more objects for comparison. But it naturally boils down to comparing all kinds of pairs made up of these objects. In other words, it boils down to comparing two objects. So equality requires two objects.
The essence of the concept of equality in the most general sense is most clearly conveyed by the word "identical". If we take two identical objects, then we can say about them that they equal... As an example, we will give two equal squares and. The differing objects, in turn, are called unequal.
The concept of equality can refer both to objects in general and to their individual properties and characteristics. Objects are generally equal when they are equal in all their inherent parameters. In the previous example, we talked about the equality of objects in general - both objects are squares, they are the same size, the same color, and in general they are completely the same. On the other hand, objects may be unequal in general, but may have some of the same characteristics. As an example, consider such objects and. Obviously, they are equal in shape — they are both circles. And they are unequal in color and size, one of them is blue and the other is red, one is small and the other is large.
From the previous example, we note for ourselves that you need to know in advance what equality we are talking about.
All the above reasoning applies to equalities in mathematics, only here equality refers to mathematical objects. That is, studying mathematics, we will talk about the equality of numbers, the equality of the values of expressions, the equality of any quantities, for example, lengths, areas, temperatures, labor productivity, etc.
Equal notation, equal sign
It's time to dwell on the rules for writing equalities. For this it is used =(it is also called an equal sign), which has the form =, that is, it is two identical dashes located horizontally one above the other. The equal sign = is generally accepted.
When writing equalities, equal objects are written down and an equal sign is put between them. For example, writing the equal numbers 4 and 4 would look like 4 = 4, and it can be read as “four equals four”. Another example: the equality of the area S ABC of a triangle ABC seven square meters will be written as S ABC = 7 m 2. By analogy, you can give other examples of writing equalities.
It is worth noting that in mathematics, the considered notation of equality is often used as a definition of equality.
Definition.
Records that use an equal sign separating two mathematical objects (two numbers, expressions, etc.) are called equalities.
If it is required to indicate in writing the inequality of two objects, then use sign is not equal≠. We can see that it represents a strikethrough equal sign. As an example, let's take the notation 1 + 2 ≠ 7. It can be read like this: "The sum of one and two does not equal seven." Another example | AB | ≠ 5 cm. - the length of the segment AB is not equal to five centimeters.
True and False Equality
Recorded equalities can correspond to the meaning of the concept of equality, or they can contradict it. Depending on this, the equalities are subdivided into true equalities and false equalities... Let's figure it out with examples.
We write the equality 5 = 5. The numbers 5 and 5 are undoubtedly equal, so 5 = 5 is a true equality. But the equality 5 = 2 is incorrect, since the numbers 5 and 2 are not equal.
Equality properties
The introduction of the concept of equality naturally implies its characteristic results — the properties of equalities. The main ones are three properties of equalities:
- Reflexivity, which states that an object is equal to itself.
- A symmetry property that states that if the first object is equal to the second, then the second is equal to the first.
- And finally, the transitivity property, which states that if the first object is equal to the second, and the second to the third, then the first is equal to the third.
Let's write the sounded properties in the language of mathematics using letters:
- a = a;
- if a = b, then b = a;
- if a = b and b = c then a = c.
Separately, it is worth noting the merit of the second and third properties of equalities - the properties of symmetry and transitivity - in that they allow us to talk about the equality of three or more objects through their pairwise equality.
Double, triple equalities, etc.
Along with the usual notation of equalities, examples of which we gave in the previous paragraphs, the so-called double equalities, triple equalities and so on, which are, as it were, chains of equalities. For example, 1 + 1 + 1 = 2 + 1 = 3 is a double equality, and | AB | = | BC | = | CD | = | DE | = | EF | - an example of a quadruple equality.
With double, triple, etc. equalities it is convenient to write the equality of three, four, etc. objects respectively. These records inherently denote the equality of any two objects that make up the original chain of equalities. For example, the above double equality 1 + 1 + 1 = 2 + 1 = 3 essentially means the equality 1 + 1 + 1 = 2 + 1, and 2 + 1 = 3, and 1 + 1 + 1 = 3, and in by virtue of the symmetry property of the equalities and 2 + 1 = 1 + 1 + 1, and 3 = 2 + 1, and 3 = 1 + 1 + 1.
In the form of such chains of equalities, it is convenient to formulate a step-by-step solution of examples and problems, while the solution looks brief and the intermediate stages of transformation of the original expression are visible.
Bibliography.
- Moro M.I.... Maths. Textbook. for 1 cl. early shk. At 2 o'clock, Part 1. (First half of the year) / M. I. Moro, S. I. Volkova, S. V. Stepanova. - 6th ed. - M .: Education, 2006 .-- 112 p .: ill. + App. (2 separate l. Ill.). - ISBN 5-09-014951-8.
- Maths: textbook. for 5 cl. general education. institutions / N. Ya. Vilenkin, V. I. Zhokhov, A. S. Chesnokov, S. I. Shvartsburd. - 21st ed., Erased. - M .: Mnemosina, 2007 .-- 280 p .: ill. ISBN 5-346-00699-0.
EQUALITIES WITH QUANTITIES.
After the child gets acquainted with the cards-quantities from 1 to 20, you can add the second stage to the first stage of learning - equality with quantities.
What is Equality? This is an arithmetic operation and its result.
You begin this phase of the study with Addition.
Addition.
To display two sets of card counts, you add equalities for addition.
It is very easy to teach this operation. In fact, your child has been ready for this for several weeks now. After all, every time you show him a new card, he sees that one additional point has appeared on it.
The kid does not yet know what it is called, but already has an idea of what it is and how it works.
You already have material for addition examples on the back of each card.
Equality display technology looks like this: You want to give the child equality: 1 +2 = 3. How can you show it?
Before starting the lesson, place three cards face down on your lap, one on top of the other. As you pick up the top card with one knuckle spoke, say "one", then put it aside, say "a plus", show a card with two knuckles, say "two", postpone it after the word "will", show the three-knuckle card while saying "three".
You teach three equalities per day and show three different equalities in each lesson. In total, the baby sees nine different equalities per day.
The child understands without any explanation what the word means "a plus", he himself deduces its meaning from the context. By performing actions, you thereby demonstrate the true meaning of addition faster than any explanations. Always use the same language when talking about equalities. Having said "One plus two is three" do not speak later "Add two to one is three." When you teach facts to your child, he draws his own conclusions and comprehends the rules. If you change the terms, then the child has every reason to think that the rules have also changed.
Prepare in advance all the cards necessary for this or that equality. Do not think that your child will sit quietly and watch as you rummage through the stack of cards, picking up the right ones. He will simply get away and be right, because his time is worth no less than yours.
Try not to compose equalities that would have something in common and would allow the child to predict them in advance (such equalities can be used later). Here is an example of such equalities:
It is much better to use these:
1 +2 = 3 5+6=11 4 + 8 = 12
The child must see the mathematical essence, he develops mathematical skills and ideas. After about two weeks, the baby discovers what addition is: after all, during this time you showed him 126 different equalities for addition.
Examination.
Checking at this stage is a solution of examples.
How is an example different from equality?
Equality is an action with a result shown to a child.
An example is an action to be taken. In our case, you show the child two answers, and he chooses the correct one, i.e. solves the example.
Example You can lay out after a normal lesson with three equations for addition. You show an example in the same way as you demonstrated equality before. That is, you shift the cards in your hands, saying each out loud. For example, "would twenty plus ten be thirty or forty-five?" and show the kid two cards, one of which is with the correct answer.
The answer cards should be kept at the same distance from the baby's eyes and no prompting actions should be allowed.
With the right choice of a child, you violently express your delight, kiss and praise him.
If you choose the wrong answer, without expressing your grief, you push the card with the correct answer to the kid and ask the question: "There will be thirty, won't it?" The child usually answers this question in the affirmative. Be sure to praise your child for this correct answer.
Well, if out of ten examples your kid solves at least six correctly, then it’s time for you to move on to equalities for subtraction!
If you do not consider it necessary to check the child (and rightly so!), Then after 10-14 days you still go to the equalities for subtraction!
Consider subtraction.
You stop doing addition and completely switch to subtraction. Do three daily lessons with three different equalities each.
Sound the equalities for subtraction like this: "Twelve minus seven is five."
At the same time, you simultaneously continue to show quantity cards (two sets, five cards each) also three times a day. In total, you will have nine daily very short lessons. So you work for no more than two weeks.
Examination
The check, as in the case of addition, can be a solution of examples with a choice of one answer out of two.
Consider Multiplication.
Multiplication is nothing more than multiple addition, so this action will not be a big revelation for your child. As you continue to study the number cards (two sets of five cards each), you have the option of drawing up equalities for multiplication.
Sound equalities for multiplication like this: "Two times three is six."
The child will understand the word "multiply" as quickly as he understood before this word "a plus" and "minus".
You still have three lessons a day, each with three different equalities per multiplication. This work lasts no more than two weeks.
Continue to avoid predictable equalities. For example such as:
It is necessary to constantly keep your child in a state of surprise and expectation of something new. The main question for him should be: "What's next?"- and at each lesson he should receive a new answer to it.
Examination
You carry out the solution of examples in the same way as in the topic "Addition" and "Subtraction". If the kid liked the games-check-boxes with cards-numbers, you can continue to play them, thus repeating new, large numbers.
By adhering to the scheme we have proposed, by this time you can already complete the first stage of teaching mathematics - study the quantities within 100. Now it is time to get acquainted with the card that children like best.
Consider the concept of zero.
They say that mathematicians have been studying the idea of zero for five hundred years. True or not, but children, having barely learned the idea of quantity, immediately understand the meaning of its complete absence. They just adore zero, and your journey into the world of numbers will be incomplete if you don't show your kid a card that doesn't have any dots at all (i.e. it will be a completely blank card).
To make the kid's acquaintance with zero be fun and interesting, you can accompany the display of the card with a riddle:
At home - seven squirrels, On a plate - seven honey agarics. All the mushrooms have eaten the proteins. What's left on the plate?
Saying the last phrase, we show the card "zero".
You will be using it almost every day. It will be useful to you for operations of addition, subtraction and multiplication.
You can work with the "zero" card for one week. The child learns this topic quickly. As before, during the day, you have three sessions. In each lesson, you show the kid three different equalities for addition, subtraction and multiplication with zero. In total, you will get nine equalities per day.
Examination
Solving examples with zero follows a familiar scheme.
Consider -Division.
When you have gone through all the quantity cards from 0 to 100, you have all the necessary material for the division examples with quantities.
The technology for showing the equalities of this topic is the same. You have three sessions every day. In each lesson, you show your child three different equalities. It is good if the passage of this material does not exceed two weeks.
Examination
Checking is a solution of examples with a choice of one answer out of two.
When you have passed all the quantities and are familiar with the four rules of arithmetic, you can diversify and complicate your studies in every possible way. First, show the equalities where one arithmetic operation is used: only addition, subtraction, multiplication, or division.
Then - equalities, where addition and subtraction or multiplication and division are combined:
20 + 8-10=18 9-2 + 26 = 33 47+11-50 = 8
In order not to get confused in the cards, you can change the way of conducting classes. Now it is not necessary to show every card of knitting needles, you can show only the answer, and the actions themselves can only be pronounced. As a result, your sessions will be shorter. You just tell the child: "Twenty-two divided by eleven, divided by two is one",- and show him the "one" card.
In this topic, you can use equalities between which there is some kind of regularity.
For example:
2*2*3= 12 2*2*6=24 2*2*8=32
When combining four arithmetic operations in equality, remember that multiplication and division must be placed at the beginning of the equality:
Do not be afraid to demonstrate more than a hundred equalities, for example,
intermediate result in
42 * 3 - 36 = 90,
where the intermediate result is 126 (42 * 3 = 126)
Your baby will do a great job with them!
Checking is a solution of examples with a choice of one answer out of two. You can show an example by showing all the equality cards and two answer cards, or you can simply say the whole equality by showing the baby only two answer cards.
Remember! The longer you study, the faster you need to introduce new topics. As soon as you notice the first signs of a child's inattention or boredom, move on to a new topic. After a while, you can return to the previous topic (but to get acquainted with the equalities not yet shown).
Sequences
The sequences are the same equalities. Parents' experience with this topic has shown that sequences are very interesting for children.
Plus sequences are ascending sequences. Minus sequences are decreasing.
The more varied the sequences are, the more interesting they are for the baby.
Here are some examples of sequences:
3,6,9,12,15,18,2 (+3)
4, 8, 12, 16, 20, 24, 28 (+4)
5,10,15,20,25,30,35 (+5)
100,90,80,70,60,50,40 (-10)
72, 70, 68, 66, 64, 62, 60 (-2)
95,80,65,50,35,20,5 (-15)
Technology showing sequences can be like this. You have prepared three plus sequences.
Announce the topic of the lesson to the kid, lay out the cards of the first sequence one after the other on the floor, voicing them.
Move with the child to another corner of the room and lay out the second sequence in the same way.
In the third corner of the room, you lay out the third sequence, while voicing it.
You can lay out the sequences under each other, leaving gaps between them.
Try to always go forward, moving from simple to complex. Vary the activity: sometimes saying out loud what you show, and sometimes show the cards in silence. In any case, the child sees the sequence unfolded in front of him.
For each sequence, you need to use at least six cards, sometimes more, so that it is easier for the child to determine the very principle of the sequence.
Once you see the sparkle in the child's eyes, try adding an example to the three sequences (i.e. check his knowledge).
You show an example like this: first you lay out the entire sequence, as you usually do, and at the end you pick up two cards (one card is the one that goes next in the sequence, and the other is random) and ask the child: "Which one is next?"
At first, lay out the cards in sequences one after another, then the forms of laying out can be changed: put the cards in a circle, around the perimeter of the room, etc.
As you get better and better, don't be afraid to use multiplication and division in your sequences.
Examples of sequences:
4; 6; eight; ten; 12; 14 - in this sequence, each next number is increased by 2;
2; 4; 7; fourteen; 17; 34 - this sequence alternates between multiplication and addition (x 2; + 3);
2; 4; eight; 16; 32; 64 - in this sequence, each next number is doubled;
22; eighteen; fourteen; ten; 6; 2 - in this sequence, each next number is reduced by 4;
84; 42; 40; twenty; eighteen; 9 - this sequence alternates between division and subtraction (: 2; - 2);
Greater than, less than signs
These cards are included in 110 cards of numbers and signs (the second component of the ANASTA methodology).
Lessons for getting your child to know more-less will be very short. All you need to do is show three cards.
Display technology
Sit on the floor and lay out each card in front of the child so that he can see all three cards at once. You name each card.
You can sound like this: "Six is more than three" or "Six is more than three."
In each lesson, you show the child three different variants of inequalities with
cards "more" - "less". inequalities per day.
Thus, you demonstrate nine different
As before, you only show each inequality once.
After a few days, you can add an example to the three impressions. This is already examination, and it is carried out like this:
Place pre-prepared cards on the floor, such as the card with the number “68” and the card with the “greater than” sign. Ask your baby: "Sixty-eight is more than what number?" or "Is sixty-eight more than fifty or ninety-five?" Invite your child to choose the one you want from the two cards. You (or he) put the card correctly indicated by the kid after the "more" sign.
You can put two number cards in front of the child and give him the opportunity to choose the sign that suits, that is,> or<.
Equality and inequality
Teaching equality and inequality is as easy as teaching more and less.
You will need six arithmetic cards. You will also find them in 110 cards of numbers and signs (the second component of the ANASTA method).
Display technology
You decided to show your child the following two inequalities and one equality:
8-6<10 −7 11-3= 9 −1 55-12^50 −13
You put them on the floor in sequence so that the child can see each of them at once. In this case, you say everything, for example: "Eight minus six is not equal to ten minus seven."
In the same way, you pronounce the remaining equality and inequality during laying out.
At the initial stage of learning this topic, all the cards are laid out.
Then it will be possible to show only the "equal" and "not equal" cards.
One fine day you give the kid the opportunity to show their knowledge. You lay out the cards with the quantities, and you offer him to choose the card with which sign to put: "equal" or "not equal".
Before you start learning algebra with your toddler, you need to introduce him to the concept of a variable represented by a letter.
Usually the letter x is used in mathematics, but since it can be easily confused with the multiplication sign, it is recommended to use y.
You put first a card with five beads - knuckles, then a + plus (+) sign, after it with a y sign, then an equal sign and, finally, a card with seven beads - knuckles. Then you pose the question: "What does u mean here?"
And you yourself answer it: "In this equation means two"
Examination:
After about one - one and a half weeks of classes at this stage, you can give the child the opportunity to choose an answer.
FOURTH STAGE OF EQUALITY WITH NUMBERS AND QUANTITIES
When you have passed the numbers from 1 to 20, it is time to "build bridges" between numbers and quantities. There are many ways to do this. One of the simplest is to use equalities and inequalities, more and less relationships, demonstrated with number cards and dice.
Display technology.
Take the card with the number 12, put it on the floor, then put the "greater than" sign next to it, and then the card-number 10, while saying: "Twelve is more than ten."
Inequalities (equalities) can look like this:
Each (equality) day consists of three lessons, and each lesson consists of three inequalities in numbers and numbers. The total number of daily equalities will be nine. At the same time, you simultaneously continue to study the numbers with the help of two sets of five cards each, also three times a day.
Examination.
You can give your child the choice of cards "greater than", "less", "equal" or make an example in such a way that the child can finish it himself. For example, we put the card-number 7, then the sign "greater than" and give the child the opportunity to finish the example, that is, choose the card-number, for example, 9 or card-number, for example, 5.
After the baby has understood the connection between quantities and numbers, you can start solving the equalities using cards with both numbers and quantities.
Equalities with numbers and quantities.
Using cards with numbers and quantities, you go through familiar topics: addition, subtraction, multiplication, division, sequences, equality and inequality, fractions, equations, equality in two or more actions.
If you carefully look at the example diagram of teaching mathematics (page 20), you will see that there is no end to the classes. Come up with your own examples for the development of the child's oral counting, correlate the quantities with real objects (nuts, spoons for guests, slices of sliced banana, bread, etc.) - in a word, go for it, create, invent, try! And you will succeed!
