How I got cancer. Rami Blekt Intro to number four...
Math is hard
But I will say with respect -
Math needed
Everyone without exception!
12 d e to a bry.
To la ss naya p a bot.
11 – 8
15 – 8
Charging for the mind
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TOPIC OF THE LESSON:
ADDITION AND SUBTRACTION OF TWO-DIGITAL NUMBERS
need help
I doubt
I am confident and I can do it
Recall what is important for the lesson
50 – 7 = 80 + 5 =
43 – 21 = 34 + 45 =
60 – 4 = 76 – 6 =
We remember what is important for the lesson.
What do you know?
- Addition and subtraction table
- Add Action Component Names
- Names of Subtract Action Components
An algorithm for adding two-digit numbers, when the sum is a round number.
- Algorithm for subtracting from a round two-digit number
- Have you considered all the ways to solve expressions?
- Are there any problems, and what are they?
- Algorithm for solving expressions in a column for addition with a transition through a discharge.
- Algorithm for solving expressions in a column for subtraction with a transition through the category.
- Group work:
- 26+18=?
- 44-18=?
Stacking units...
14 ones is 1 ten and 4 ones
I write 4 under the units, and I write 1 ten over the tens
Adding dozens...
I add 1 ten, which was obtained from the addition of units
In total it worked out…
I write under tens ...
Reading...
I write tens under tens, and units under units.
subtract units. 4
I take one ten. (I put a dot over the number ...)
I'm counting 10 minus...
Under the units I write a number ...
Subtract tens. There were ... dozens. They took a dozen. Remained … tens. I count ... tens minus ... tens
I write under tens ...
Reading...
Examination
Choose and solve expressions for subtraction with transition through the category. What is the next expression?
Examination
I know
1. Addition and subtraction table.
I want to know
1. We have considered all cases of addition and subtraction.
found out
2. The name of the action components.
1. To find the value of the sum, you need to add the units, and if there are more than ten of them, then write down only the units, and remember the ten and add it when adding the tens.
3. Algorithm for adding two-digit numbers, when the sum is a round number
2. Are there any difficulties in solving expressions, and which ones.
2. To find the value of the subtraction, you must first subtract the units from the units, but there are cases when the values of the units to be reduced are less than the value of the units to be subtracted, then it is necessary to take one ten. And when subtracting, strictly know that the number of tens has become one less.
3.Algorithm for adding two-digit numbers in a column with the transition through the discharge
4. Algorithm for subtracting from a round two-digit number
4. Algorithm for subtracting into a column with the transition through the discharge
3. Algorithm for addition in a column with a transition through a discharge
4. Algorithm for subtracting into a column with the transition through the discharge
Number Magic [Instant mental calculations and other mathematical tricks] Benjamin Arthur
Chapter 1 A Little Pleasure: Oral Addition and Subtraction
A little exchange of pleasantries: verbal addition and subtraction
For as long as I can remember, it has always been easier for me to add and subtract from left to right than from right to left. In doing so, I found that I could shout out the answer to a math problem before my classmates wrote down the terms.
And I didn't even have to write it down!
In this chapter, you will learn the left-to-right method used for mental addition and subtraction of most of the numbers we encounter every day. These mental skills are important not only for doing the math tricks in this book, but also indispensable during school, work, and other situations where you need to manipulate numbers. Before long, you'll be able to put your calculator on a well-deserved rest and start using your brain to its full potential by adding and subtracting two-digit, three-digit, and even four-digit numbers at lightning speed.
ADDITION LEFT TO RIGHT
Most of us are trained to do written calculations from right to left. And this is normal for a paper account. But I have quite a lot of convincing arguments explaining why it is better to do this from left to right in order to count in my mind(i.e faster than on paper). After all, you read numerical information from left to right, and you pronounce numbers from left to right, so it's more natural to think about (and count) numbers from left to right. By evaluating the answer from right to left, you are generating it backwards. This is what makes mental calculations so difficult. Also, in order to simply evaluate the result of a calculation, it is more important to know that it is "just over 1200" than that it "ends in 8".
So, applying the method from left to right, you start the solution with the most significant digits of your answer. If you are used to working on paper from right to left, then this new approach may seem unnatural to you. But with practice, you will come to understand that this is the most efficient way for mental calculations. Although, perhaps the first set of problems - addition of two-digit numbers - will not convince you of this. But be patient. If you follow my recommendations, you will soon realize that the only easy way to solve three-digit (and more "digital" addition problems, all subtraction, multiplication, and division problems is the left-to-right method. The sooner you train yourself to act this way, the better.
Two-digit addition
First of all, I assume that you know how to add and subtract numbers with one digit. We'll start by adding two digit numbers, although I suspect you're pretty good at doing this mentally. However, the following exercises will still be good practice for you, as the two-digit addition skills you eventually acquire will be needed to solve more difficult addition problems, as, indeed, for almost all multiplication problems presented in the following chapters. This illustrates a fundamental principle of mental arithmetic, which is: "Simplify a problem by breaking it down into smaller, easier-to-solve problems." This is the key to almost every method presented in this book. To paraphrase the old adage, there are three ingredients to success: simplify, simplify, simplify.
The easiest two-digit addition problems are those that don't require you to keep any digits in mind (i.e. when the first two digits add up to 9 or less, or the last two digits add up to 9 or less). For example:
To add 47 + 32, first add 30 to 47, and then add 2 to the resulting amount. After adding 30 and 47, the task simplified: 77 + 2 equals 79. Let's illustrate this as follows:
The diagram shown is a simple way to represent the thought processes involved in getting the right answer. While you should be reading and understanding these diagrams throughout the book, you do not need to write anything down.
Now let's try a calculation in which you need to keep the numbers in mind:
By adding from left to right, you can reduce the problem to 67 + 20 = 87 and then adding 87 + 8 = 95.
Now try it yourself, and then check with how we did it.
Well, did it work? You added 84 + 50 = 134 and then 134 + 7 = 141.
If holding numbers in your mind is causing you to make mistakes, don't worry. This is probably your first attempt at a systematic mental calculation and, like most people, you will need time to memorize the numbers. However, with experience, you will be able to keep them in your mind automatically. As a practice, try solving one more problem orally, and then check again how we did it.
You should have added 68 + 40 = 108 and 108 + 5 = 113 (final answer). Was it easier for you? If you want to test your skills on more two-digit addition problems, refer to the examples below. (Answers and calculations are given at the end of the book.)
Addition of three-digit numbers
The strategy for adding three digit numbers is exactly the same as adding two digit numbers: you add from left to right, and after each step move on to a new, easier addition problem.
Let's try:
First, we add 300 to 538, then 20, then 7. After adding 300 (538 + 300 = 838), the problem is reduced to 838 + 27. After adding 20 (838 + 20 = 858), the problem is simplified to 858 + 7 = 865. This kind of thought process can be represented as the following diagram:
All mental addition problems can be solved in this way, gradually simplifying the problem until it remains just to add a one-digit number. Note that the example 538 + 327 requires six digits to be kept in mind, while 838 + 27 and 858 + 7 require only five and four digits, respectively. If you simplify a problem, it becomes easier to solve it!
Try mentally solving the following addition problem before looking at our solution
Did you simplify it by adding the numbers from left to right? After adding hundreds (623 + 100 = 723), it remains to add tens (723 + 50 = 773). Simplifying the problem to 773 + 9, we get 782 in total. In the form of a diagram, the solution to the problem looks like this:
When I solve problems like this in my mind, I don't visualize the numbers, I try to hear them. I hear the example 623 + 159 as six hundred and twenty three plus one hundred and fifty nine. Highlighting the word one hundred for myself, I understand where to start. Six plus one equals seven, so my next problem is seven hundred and twenty-three plus fifty-nine, and so on. When solving such problems, also do it out loud. Reinforcement in the form of sounds will help you master this method much faster.
Three-digit addition problems are actually no more difficult than the following:
Take a look at how it's done:
At each step, I hear (rather than see) a new addition problem. In my head it sounds something like this:
858 plus 634 equals 1458 plus 34,
equals 1488 plus 4 equals 1492.
Your inner voice may sound different than mine (it is possible that it is more convenient for you to see the numbers rather than hear them), but, be that as it may, our goal is to “reinforce” the numbers on their way so as not to forget which stage of solving the problem we are and do not start all over again.
Let's get some more practice.
First add in your head, then check the calculations.
This example is a little more complicated than the previous one, as it requires you to keep the numbers in mind for all three steps.
However, it can use an alternative method of counting. I'm sure you'll agree that it's much easier to add 500 to 759 than 496. So try adding 500 and then subtracting the difference.
So far, you have been sequentially splitting the second number to add it to the first. In fact, it does not matter which number to break into parts, it is important to follow the order of actions. Then your brain won't have to decide which way to go. If the second number is much easier to remember than the first, then they can be swapped, as in the following example.
Let's finish the topic by adding three-digit numbers to four-digit ones. Since the average person's memory can only hold seven or eight digits at a time, this is just the right thing for you to do without resorting to artificial memory devices (such as fingers, calculators, or mnemonics from Chapter 7). In many addition problems, one or both numbers end in 0, so let's pay attention to examples of this type. Let's start with the easiest:
Since 27 hundreds + 5 hundreds equals 32 hundreds, we just add 67 to get 32 hundreds and 67, i.e. 3267. The solution process is identical for the following tasks.
Since 40 + 18 = 58, the first answer is 3258. In the second example, 40 + 72 adds up to more than 100, so the answer is 33 hundreds with a tail. So 40 + 72 = 112, so the answer is 3312.
These problems are easy because significant numbers (other than zero) are added only once and the examples can be solved in one step. If significant figures are added twice, then two actions will be required. For example:
The task in two steps is schematically as follows.
Practice the three-digit addition exercises below until you can easily do them in your head without looking back. (Answers are at the end of the book.)
Carl Friedrich Gauss: a mathematical prodigy
A child prodigy is a very talented child. He is usually referred to as "precocious" or "gifted", as he is almost always ahead of his peers in development. German mathematician Carl Friedrich Gauss (1777–1855) was one of those children. He often boasted that he learned to calculate before he could speak. At the age of three, he corrected his father's payroll, stating, "The calculations are wrong." A further check of the record showed that little Carl was right.
At the age of ten, the student Gauss received the following mathematical problem in the lesson: what is the sum of the numbers from 1 to 100? As his classmates frantically made calculations with paper and pencil, Gauss immediately imagined that if he wrote down the numbers from 1 to 50 from left to right, and from 51 to 100 from right to left directly below the list of numbers from 1 to 50, then each sum of the numbers standing one under the other, will be equal to 101 (1 + 100, 2 + 99, 3 + 98…). Since there were only fifty such sums, the answer was 101 x 50 = 5050. To everyone's amazement (including the teacher), young Carl got the answer, not only outperforming all the other students, but also calculating it entirely in his mind. The boy wrote the answer on his chalkboard and threw it on the teacher's desk with the cheeky words, "Here is the answer."
The teacher was so amazed that he bought the best arithmetic textbook available with his own money and gave it to Gauss, declaring: "This is beyond my limits, I can't teach him anything else."
Indeed, Gauss began to teach mathematics to others and eventually reached unprecedented heights, having a reputation as one of the greatest mathematicians in history, whose theories still serve science. His desire to better understand nature through the language of mathematics was summed up in his motto, taken from Shakespeare's King Lear (replacing "law" with "laws"): "Nature, you are my goddess! In life, I only obey your laws.
SUBTRACT LEFT TO RIGHT
For most of us, addition is easier than subtraction. But if you subtract from left to right and start breaking down calculations into simpler steps, subtraction can become almost as easy as addition.
Subtraction of two digit numbers
When subtracting two-digit numbers, you should simplify the task by reducing it to the subtraction (or addition) of single-digit numbers. Let's start with a very simple example.
After each action, you move on to a new, easier subtraction step. First we subtract 20 (86–20 = 66), then 5, having a simple action of 66 - 5, we end up with 61. The solution can be schematically represented as:
Of course, subtraction is much easier if you do not need to take one from the most significant digit (this happens when a larger digit is subtracted from a smaller one). However, I want to reassure you: difficult subtraction problems can usually be turned into easy addition problems. For example:
There are two ways to solve this example in your mind.
1. First subtract 20, then 9:
But for this task, I propose a different strategy.
2. First subtract 30, then add 1
To determine which method is best to use, the rule will help you:
in a two-digit subtraction problem, if the digit being subtracted is greater than the digit being reduced, round it up to ten.
Next, subtract the rounded number from the reduced number, and then add the difference between the rounded number and the original. For example, in problem 54–28, the subtracted 8 is greater than the reduced 4. Therefore, we round 28 to 30, calculate 54–30 = 24, after which we add 2 and get the answer - 26.
And now we will consolidate knowledge on the example of 81–37. Since 7 is greater than 1, we round 37 to 40, subtract that number from 81 (81-40 = 41), and then add the difference of 3 to get the answer:
Just a little practice - and you can easily solve problems in both ways. Use the rule above to decide which method is best.
Subtraction of three-digit numbers
Now let's subtract three-digit numbers.
This example does not require rounding numbers (each digit of the second number is at least one less than the corresponding digits of the first), so the task should not be too difficult. Just subtract one digit at a time, making it easier with each step.
Now consider a three-digit subtraction problem that requires rounding.
At first glance, it seems rather complicated. But if we first subtract 600 (747–600 = 147) and then add 2, we get 149 (147 + 2 = 149).
Now try it yourself.
Did you subtract 700 from 853 first? If so, then we got 853–700 = 153, right? Since you subtracted 8 more than the original number, did you add 8 to get the answer 161?
Now I can confess that we managed to simplify the subtraction process, because the numbers being subtracted were almost multiples of 100. (Did you notice?) What about other problems, like this one?
What happens if you round the subtracted to 500?
Subtracting 500 is easy: 725-500 = 225. But you subtracted too much. The trick is to define exactly what "too much" is.
At first glance, the answer is not obvious. To find the difference between 468 and 500. The answer can be obtained with the addition - a neat trick that will simplify most tasks for subtracting three-digit numbers.
Computing Complements
Quickly, how far from 100 are these numbers?
Here are the answers:
Note that for each pair of numbers that add up to 100, the first digits (on the left) add up to 9, and the last ones (on the right) add up to 10. You could say that 43 is the complement of 57, 32 is the complement of 68, and so on. .
Now find the additions to the following two-digit numbers:
To find the complement of 37, first determine how much you need to add to 3 to get 9. (The answer is 6.)
Then figure out how much to add to 7 to get 10. (The answer is 3.) Therefore, 63 is the complement of 37.
Other additions: 41, 7, 56, 92, respectively. Note that as a mathmag you are looking for additions like everything else, from left to right. As we have already found out, we increase the first digit to 9, the second to 10. (The exception is if the numbers end in 0 - for example, 30 + 70 = 100 - but such additions are easy to calculate!)
What is the relationship between additions and verbal subtraction?
They allow you to convert complex subtraction problems into simple addition problems. Consider the last problem, which gave us some difficulties.
So, first subtract the number 500 from 725 instead of 468 and get 225 (725-500 = 225). However, since we have subtracted too much, we need to figure out how much to add now. Using add-ons allows you to instantly give an answer. How many digits is 468 from 500? The same distance as 68 is from 100. If you look for the addition of 68 in the way shown above, you get 32. Add 32 to 225 and get 257.
Try another three-digit subtraction problem:
Here is another example:
Check your answer and solution progress:
Subtracting a three-digit number from a four-digit number is not much more difficult, as the following example illustrates.
By rounding, we subtract 600 from 1246. We get 646.
Then we add the complement for 79 (i.e. 21). Answer: 646 + + 21 = 667.
Do the three-digit subtraction exercises below, and then try to come up with your own addition (or subtraction?) examples.
This text is an introductory piece.
From the book Seekers of Extraordinary Autographs author Levshin Vladimir ArturovichA LITTLE HAJJ IN HISTORY - Everything in the world is made of something. A pencil, for example, is a bit of wood and a bit of graphite. Or walnut cake. This is a little crushed crackers, a lot of crushed nuts and a lot of cream. But if you want to explain what
From the book Alice's Adventures in Puzzleland author Smullyan Raymond MerrillChapter 1 graphics46 Who is John? To find out which of the two brothers is John, ask one of them, "Is John truthful?" If he answers yes, it must be John, whether he lied or told the truth. If he answers "no", then he is not John. And here is how it is confirmed.Answering
From the book Alice in the Land of Wit author Smullyan Raymond MerrillChapter 3 graphics50 14. Caterpillar and Bill the Lizard The caterpillar is convinced that both she and Bill the Lizard are out of their minds. If the Caterpillar were sane, then her judgment that both of them were out of their minds would be false. If so, then the Caterpillar (being in his mind) could hardly be seriously
From the book Puzzles. Release 1 author Perelman Yakov IsidorovichChapter 5 graphics51 42. Revealing the First Spy It is definitely not possible to be a knight, since no knight could slander himself by calling himself a spy. Therefore, B is either a crook or a spy. Suppose B is a spy. Then statement A is false, and in this case A is a crook (he
From the book Fun Tasks. Two hundred puzzles author Perelman Yakov IsidorovichChapter 1 Who is John? In order to find out which of the two twin brothers is called John, you need to ask one of them: "Is John telling the truth?". If the answer to this question is “yes”, then regardless of whether the questioned twin lies or always tells only the truth, he must
From the book Cryptography and Freedom author Maslennikov MikhailChapter 2 1. The first story. Essentially, the Hatter claimed that either the March Hare or the Dormouse had stolen the jam. If the Hatter lied, then neither the March Hare nor the Dormouse stole the jam. But then the March Hare, since he did not steal the jam, gave truthful testimony.
From the book The Magic of Numbers [Instant calculations in the mind and other mathematical tricks] author Benjamin ArthurChapter 4 26. How many pretzels does each one have? Let's call one portion all the pretzels that Sonya got, no matter how many there are. Then Sonya got 1 serving. The March Hare got twice as many pretzels as the Dormouse (because the Hatter put the Dormouse in a place where
From the author's bookChapter 5 42. The appearance of the first spy. C obviously cannot be a knight, since no knight would lie and claim that he is a spy. Therefore C is either a knave or a spy. Suppose C is a spy. Then A is false, so A is a spy (A cannot be a spy, so
From the author's bookChapter 6 52. First question. Alice made a mistake writing eleven thousand eleven hundred and eleven as 11111, which is wrong! The number 11111 is eleven thousand one hundred and eleven! In order to understand how to write the dividend correctly, let's add eleven thousand,
From the author's bookChapter 7 64. First round (Red and black). If the brother who suddenly spoke spoke the truth, then his name would be Tweedledum and he would have a black card in his pocket. But the one who has a black card in his pocket cannot speak the truth. Therefore, he is lying. So in his pocket
From the author's bookChapter 9 In all the decisions of this chapter, A stands for the first defendant, B for the second, and C for the third.78. Who is guilty? From the conditions of the problem it is known that the perpetrator gave false testimony. If B were guilty, he would have told the truth when he pleaded guilty. Therefore, B cannot
From the author's bookChapter 11 88. Just one question. Really follow. Consider first statement 1. Suppose someone is convinced that he is awake. In reality he is either awake or he is not awake. Let's assume he's awake. Then his conviction is correct, but anyone
From the author's book6. Addition and multiplication You, no doubt, have already paid attention more than once to a curious feature of the equalities: 2 + 2 \u003d 4.2? 2 = 4. This is the only example where the sum and product of two integers (and, moreover, equal) are the same. However, you may not know that there are fractional
From the author's book26. Addition and multiplication You, no doubt, have already paid attention more than once to a curious feature of equalities: 2 + 2 = 42 x 2 = 4 This is the only example when the sum and product of two integers (and, moreover, equal) are the same. However, you perhaps it is not known that there are fractional
From the author's book From the author's bookChapter 7 A memorable chapter for memorizing numbers The most frequently asked question is about my memory. No, I’ll tell you right away, it’s not phenomenal for me. Rather, I am using a system of mnemonics that can be learned by anyone and described on the following pages.
The topic of this video lesson is "Written techniques for adding two-digit numbers with a transition through a dozen of the form 37 + 48." Often you have to perform addition when both terms are from the first ten, and the sum is from the second ten. Such calculations are called action with the transition through the ten.
Lesson:Written tricks for adding two-digit numbers with a transition through a dozen of the form 37 + 48
We need to find the sum of two numbers 37 and 48. First, we will do it verbally, presenting the numbers in the form of models. (Fig. 1.)
There are 3 tens and 7 ones in the number 37. There are 4 tens and 8 ones in the number 48. When we execute , we concatenate both numbers.
Let's combine the units. We add 2 units to 8 units and we get a dozen. We can represent ten as a model of the number 10. (Fig. 2.)
What number did we get?
There are 8 tens and 5 ones in this number. This number is 85.
Let's use another way to add numbers. This method does not require the use of number models.
Look at the expressions:
Let's represent the second number as the sum of 40 and 8.
37 + 48 = 37 + (40 + 8)
Let's group the numbers differently. First we find the sum of the first two numbers, and then we add the third term.
37 + 48 = 37 + (40 + 8) = (37 + 40) + 8 = 77 + 8
In order to make it more convenient to add numbers, you can decompose the number 8 into a sum of terms, one of which will complement the number 77 to a round number. These are numbers 3 and 5.
37 + 48 = 37 + (40 + 8) = (37 + 40) + 8 = 77 + 8 = 77 + 3 + 5 = 80 + 5 = 85
Do you think there is a faster way to add numbers?
Let's use the column addition method.
When adding, the numbers are written one below the other. We start calculations in a column with the smallest digit - the digit of ones.
We add 8 units to 7 units and get 15 units. Under the units place, we can only write units. To do this, we must find out how many ones are in the number 15. The number 15 consists of 1 ten and 5 ones. This means that we write the number 5 under the unit digit.
Ten we send to the category of tens.
Now let's count the tens. 3 + 4 = 7. And 1 more ten, 7 + 1 = 8. We write the number 8 under the tens place.
We added two numbers and got the number 85.
The little fox, the little squirrel and the kitten also learned to add numbers in a column. Let's see if they got it right. Look at the two numbers that the fox has stacked up. (Fig. 3.)
Let's check the correctness of his calculations. Let's find the sum of units. 5 + 7 = 12. Under the ones place, we write the number 2 and pass 1 tens to the tens place. The fox didn't show it. Let's see if he forgot to add it later ?.
Add up dozens. 3 + 2 = 5. We need to add another ten. 5 + 1 = 6. Therefore, you need to change the digit in the tens place. Therefore, let us remind Fox that we should not forget to give a dozen. (Fig. 4.)
Let's look at the calculations of the Kitten. (Fig. 5.)
First add the units. 7 + 6 = 13. The Kitten has the number 1 written, which means that a mistake was made. Now add up the tens. 4 + 1 = 5. And we also add the ten that we gave away from the category of units. 5 + 1 = 6. We see that the Kitten got the wrong answer. Did you guess what the Kitten made a mistake? He messed up the action. He subtracted the number 16 from the number 47. Therefore, we replace the sign and get the correct expression. (Fig. 6.)
Let's check the example of Belchonok. (Fig. 7.)
We add units. 8 + 5 = 13. We write down the number 3 and give 1 ten to the tens place. Now add up the tens. 2 + 1 = 3. And we also add 1 ten, which we gave away from the category of units. 3 + 1 = 4. We must not forget to write down the one, which we give from the discharge of units to the discharge of tens. (Fig. 8.)
do at home
1. Solve expressions:
a) 28 + 43 b) 34 + 17 c) 22 + 69
Solve expressions:
Solve expressions.
UMK "Perspektiva" Subject: Mathematics Author of the textbook: L.G. Peterson. Grade: 2 Type of lesson: ONZ Topic: "Subtraction of two-digit numbers with the transition through the discharge: 41 - 24" Main objectives: 1) To consolidate knowledge of the structure of the I step of educational activity and the ability to perform UUD included in its structure. 2) Build an algorithm for subtracting two-digit numbers with the transition through the category and form the primary ability to apply it. 3) Fix the algorithm for subtracting two-digit numbers (general case), solving equations for finding an unknown term, subtracted, reduced, solving problems for the relationship of part and whole. Mental operations necessary at the design stage: analysis, comparison, generalization, analogy. Demonstration material: 1) separate cards, on which: The class took up the work 2) the standard of subtraction in parts with the transition through a dozen: ; 1 2 - 5 = 10 - 3 = 7 2 3 : 5) reference signal for recognizing the type of example: m - B 6) card with the topic of the lesson: 41 - 24 7) graphic models; 8) an algorithm for subtracting two-digit numbers from a round number (from lesson 2-1-9): 9) cards to clarify the algorithm of lesson 2-1-9: 1 There are not enough units in the reduced one. 10) card I subtract units from all units received: ... to replace zero in the reference signal of lesson 2-1-9. Handout: 9 - 64 1) worksheets for the update stage: 7 - 54 5 - 44 3 - 34 41 - 24; 2) graphic models; 3) a notebook for supporting notes or a corresponding sheet from the “Build Your Math” manual; 4) two halves (cut lengthwise) of a clean sheet of A-4 for the number of groups. Lesson progress: 1. Motivation for learning activities: - What was your goal during the trip in the last lesson? (Find a short way to the island. It turned out to be a convenient verbal technique for adding two-digit numbers with a transition through the discharge - in parts.) - Today you will continue to study actions with two-digit numbers. Your familiar fairy-tale hero - Dunno - found out how interesting you are to study. How will you study a new topic? (First we repeat the necessary, then we perform a trial action, fix our difficulty, identify its cause of the difficulty.) - So, Dunno sent a telegram in verse. Do you want to read it and learn new things about actions with two-digit numbers? 2. Actualization of knowledge and fixation of difficulties in a trial educational action. 1) Repetition of the studied methods of subtracting two-digit numbers. - But since Dunno is a big inventor, he encrypted his telegram. To read, you need to solve examples. Open examples on the board. After the “=” sign, sheets with the words of the first line of the poem are attached with the white side. Sheets cover the written answers. 20 - You name the answers of the examples, I take off the sheet so that you can check yourself. The teacher records all the answers on the sheets. If there are several of them, the correct answer is revealed on the basis of the standards D-2 and D-3, which are displayed on the board. After agreeing on the answers, the teacher removes the sheets, attaches them separately with the text down in the order of the examples, and the students compare the answers received with the numbers under the sheets. - You did a great job with Dunno's examples, and you can read his telegram. The teacher turns over the sheets. - Read in unison. (The class took up the work ...) - What is it? (The telegram is not finished, it looks like the first line of a poem, ...) 2 - Probably Dunno, due to his forgetfulness, did not send the second line. But nothing, but these examples will help you clarify what calculations you will be interested in today. What do all examples have in common? (They are all for subtraction, a single-digit number must be subtracted from a two-digit number.) - Which example is “extra”? (20 - 8 is an example for subtraction from a round number, and the rest - for subtraction with a transition through a dozen.) - What other subtraction examples can you solve? (For the subtraction of two-digit numbers according to the general rule.) The D-4 standard is set on the board and the corresponding rule is spoken out. 2) Training of mental operations. Distribute worksheets. What is separated by a dotted line is wrapped. The kids don't see it yet. Open the same on the board. 9 - 64 - 54 7 5 - 44 - 34 3 41 - 24 - Look at the task on your papers. It's also written on the board. What's interesting about differences? (In the minuend, one digit is unknown, unknown digits alternate; known digits in the minuend are odd, go in descending order; in the subtrahend, the number of tens decreases by 1, and the number of units does not change.) - Find the unknown digit of the minuend, if it is known that the difference between digits denoting tens and units is 3. One at a time with an explanation. The teacher writes the numbers on the blackboard, the children - on the pieces of paper. (In the first example there are 6 tens, 12 tens is not suitable, since it is a two-digit number; in the second example - 4 e, since 10 e are not suitable; in the third example - 8, since ...; in the fourth - 6 ..., in the fifth - 4…) - What technique will you need to solve these examples? (Subtraction of two-digit numbers according to the general rule.) - Do you know him? (Yes.) - Then solve these examples yourself. Runtime 1 minute. - What is the answer of the first (second, third, fourth) example. (5; 20; 41; 2.) The teacher writes the results in the course of the children's answers. If there are different answers, the calculation method is specified according to the D-4 standard. - What methods of subtraction did I choose to repeat? (According to the general rule, from round, with a transition through a dozen.) - Tell me, what will happen next? (Assignment for trial action.) - What does “task for trial action” mean? (This means that there is something new in it.) - Why am I offering it to you? (We try to do it to understand what we do not know.) 3) Task for a trial action. - Right. Turn away the lower part of the sheet and find the meaning of the expression written there. - Name the result. (17; 23; 27, ...) The teacher writes out all the answers of the children. 3 - What do you see? (Opinions were divided, and someone could not find the result.) - Raise your hand those who did not receive an answer. - What couldn't you do? (We could not solve the example 41 - 24.) - Those who received the answer, prove, using the generally accepted rule, that you have decided correctly. (We cannot prove that we correctly solved example 41 - 24.) - Remind yourself and Dunno what to do when a person fixes a difficulty? (We need to stop and think.) 3. Identification of the place and cause of the difficulty. - Let's think. What numbers did you subtract? (Two-digit.) - Remember the general rule for subtracting two-digit numbers. (When subtracting two-digit numbers from tens, you need to subtract tens, from units - units.) - What prevented you from doing this? (Here, there are not enough units in the minuend.) - What was new for you in this example? (We did not solve examples when there are fewer units in the minuend than in the subtrahend.) Hang a reference signal on the board to determine the type of example: m - B - Well done! You have noticed an important feature of this example, which distinguishes it from the previous ones: there are not enough units in the minuend. - Where have you already met with such a case? (When a one-digit number was subtracted from a two-digit number with a transition through a ten.) - Here are two-digit numbers, so they say "with a transition through the discharge." - Tell us, how did you act, and in what place did you feel that there was not enough knowledge? (...) - What is the reason for your difficulties? (There is no way to subtract two-digit numbers with a transition through the digit.) 4. Building a project for getting out of the difficulty. So what is your goal to set for yourself? (Construct a way to subtract two-digit numbers with a jump through the digit.) - What is the topic of the lesson. (Subtraction of two-digit numbers with the transition through the discharge.) - In the subject, for convenience, we will write briefly. Hang a card on the board with the topic: 41 - 24 - First, let's decide on the means. What tool do you need to visualize how the transition through the discharge occurs? (Graphic models.) - What recording method will be needed? (Entry in a column.) - And what standards known to you can help? (The standard for subtracting a two-digit number from a round one.) - So, you will refine this standard. - Now plan your work: in what order you will move towards achieving the goal. (First, we will solve the example using graphical models, then in a column, and then we will clarify the standard for subtracting a two-digit number from a round number.) It is advisable to fix the plan on the board. 5. Implementation of the constructed project. - So, first ... (Let's lay out a graphical model of the example.) One student is at the blackboard, the rest are on their desks: - Repeat again, how are two-digit numbers subtracted? (Tens are subtracted from tens, units are subtracted from units.) - What prevents you from using this rule here? (The minuend lacks units.) 4 - Is the minuend less than the subtrahend? (No.) - Where did the units hide? (In the top ten.) - How to be? (Replace 1 ten with 10 ones. - Opening!!!) - Well done! Continue subtracting. - What next? (We act according to the general rule: subtract 2 d from 3 d, we get 1 d; subtract 4 units from 11 units, we get 7 units. Result: 1 d 7 e or 17.) - So, the correct answer is 17. - Well done, guys ! So, you have found a new method of calculation: if there are not enough units in the minuend, then ... (You can split the ten and take the missing units from it). - What will you do next according to the plan? (Let's solve the same example in a column.) - I think you can do it without my help. One at the blackboard with an explanation: (I write units under units, tens under tens. There are fewer units in the minuend, so I take 1 ten, split it into 10 units and add them to the units of the minuend. I subtract the units: 11 - 4 = 7. I write the result under I decrease the number of tens by 1. I subtract the tens: 3 - 2 = 1. I write under the tens. Answer: 17.) - You did it really easily. What algorithm did you use? (There is no required algorithm, we used a similar algorithm for subtracting a two-digit number from a round number.) Open on the board the algorithm for subtracting a two-digit number from a round number (from lesson 2-1-9): - What's next according to the plan? (We need to clarify this algorithm.) Divide the children into groups of 4 people, as is customary in the class. - Consult in groups and refine this algorithm. Give each group two halves of sheet A-4 (cut lengthwise). You have 1-2 minutes to complete the task. - Let's see what you got. Each group presents refinements to the algorithm and indicates the place of these refinements. During the discussions, a new version is agreed upon and placed on the board in the place indicated by the children. As a result, the algorithm should take approximately the following form: :… 5 - How can we change the reference signal of addition into a column? Open the reference signal for subtracting a two-digit number from a round number (from lesson 2-1-9): (You need to replace 0 with a card depicting units.) The teacher makes changes to the lesson 2-1-9 reference signal from the words of the children: - What do you think about What should always be remembered when using this technique? Where is the error possible? (The number of tens decreases by 1, ...) - Well done! You went exactly according to plan. What can you say about achieving the goal? (We have reached the goal, but we still need to practice.) 6. Primary consolidation with pronunciation in external speech. 1) No. 2, page 24. - Open in the textbook No. 2 on page 24. - Read the assignment. Task: Solve examples according to the model. Write and solve the following example: - We solve the first example. One from the place with an explanation. (There are fewer units in the reduced one, so I take 1 ten and split it into 10 units: 10 + 1 = = 11. I subtract the units: 11 - 9 = 2. I decrease the number of tens by 1, I subtract the tens: 7 - 2 = = 5. I write under tens. Answer: 52.) - We decide further. "Chain" from the place with an explanation. Children solve examples until they notice a pattern: the minuend increases by 1, so the difference will increase by 1. When enough hands rise, you can ask the children: - What happened? Is there an error somewhere? (No, you can just write down the answers without calculating.) - Why? (Here, the minuend is increased by 1, but the subtrahend does not change, so the difference will increase by 1.) Great! List your answers below. (55, 56, 57.) - So that's what mathematical laws are for! They are always so helpful! Now compose your last example, taking into account the pattern. (87 - 29.) - Write down the answer without calculating. (58.) 2) No. 3, p. 24. - Well done! Now you can play! The Guess Game. The teacher distributes the columns in rows. - You will work in pairs. Write in your notebook examples of your column in a column. One person from the pair explains aloud to the other the solution to the first example of the column. Then together you try 6 to guess the answer of the second example, understanding and explaining the pattern. Next, the second person from the pair checks the answer of the second example. The teacher provides assistance to individual students as needed. Completion of the task is checked frontally. - Now it's all clear? (You must first work on your own.) 7. Independent work with self-examination according to the standard. - Well, try your hand at independent work: No. 4, p. 24. Task: Choose and solve examples for subtraction with a transition through the category. What is interesting about them? What is the next example? 98 - 19 47 + 38 95 - 20 54 - 17 50 + 30 29 - 9 76 - 18 68 + 23 - Read the task. a) - The task consists of several parts. What should be done first? (Choose examples for a new computational technique.) - Complete this part of the task yourself by ticking the boxes next to the examples you have chosen in the textbook. - Check. Open on the board the standard for this part of the task: m - B - What difficulties did you encounter during the implementation? (Didn't pay attention to the sign, didn't compare the units to find out the type of the example.) - How did you proceed when searching for examples for a new computational trick? (We first looked at the sign, then compared the units. If the number of units of the reduced is less, then put a tick.) - Correct, who incorrectly found examples of the new type. - Who did it right? Put "+" in the margins of the textbook. b) What should be done next? (Solve examples for a new computational technique.) - Solve all the selected examples in the notebook yourself. - Check. Open on the board the standard for solving examples: - What difficulties arose when solving examples? (Forgot to reduce the number of tens by 1, ...) - Who was not mistaken? Put another “+” in the margins of your notebook. - What interesting things did you notice in the examples? (The numbers in the minuends are written in order from 9 to 4; the subtrahends are in decreasing order, etc.) - What will be the next example? (32 - 16.) - How to write down the answer without counting? (Trace the pattern according to the answers: the number of tens decreases by 2, and the number of units - by 1, which means that the answer of the following example is 16.) 7 8. Inclusion in the knowledge system and repetition. - Today at the lesson you showed that you can work one by one, in pairs, and now work again in groups. Divide the class into groups. - What, in your opinion, is the most important skill when working in a group? (The ability to listen, the ability to hear each other, etc.) - You will complete the tasks for repetition in groups: No. 6 (3rd column), page 24; No. 9 (a, b - one task of choice), p. 25. The task is written on the board. 3-4 minutes are given for group work. After that, samples of writing the solved equations and problems are put up on the board. Task number 6, p. 24. Solve the equations and check: x - 9 \u003d 14 x + 25 \u003d 40 63 - x \u003d 27 5 + x \u003d 52 50 - x \u003d 12 x - 48 = 24 - Check the solution according to the sample. If there are errors, correct and write down the correct solution. Solution (3rd column): 63 - x = 27 x = 63 - 27 x = 36 63 - 36 = 27 27 = 27 x - 48 = 24 x = 24 + 48 x = 72 72 - 48 = 24 24 = 24 Task No. 9 (a, b), p. 25: Draw a diagram, put questions to the tasks and answer them: a) There are 5 horses, 4 camels and 2 elephants on the carousel. b) There are 30 dolls in the kindergarten, and there are 2 fewer trucks. - Evaluate your work in the group. Did everything work out? What were the difficulties? (It was difficult to agree on what we would decide, ...) 9. Reflection of educational activities in the lesson. - What is your goal for the lesson? (Construct a way to subtract two-digit numbers with a transition through the discharge.) - Reached the goal? Prove it. (…) - What solution did you come up with? (…) - What did you like? (...) - You know, Dunno remembered that he had sent us only half of the poem, and here is the next telegram: Open the entry on the board: Everything will work out for you! - Was Dunno right? What did you get? (…) - What was difficult? - What else needs to be worked on? - And now let's get back to Dunno's poem. Let's read it again. (I got down to work - everything will work out for you.) - Change the second line so that it contains an assessment of the work of the class. (Everything turned out with us, ...) - Read the entire poem in unison. - Tell me, what qualities helped you, and what hindered you when working in pairs, in a group? (…) Homework: #5 (come up with two examples), p.24; No. 8, 9 (c), p. 25; ☺ No. 11, page 25. 8
Have you ever been amazed by people who easily add and multiply three-digit numbers in their heads or instantly call the root of 729?
In fact, it is not as difficult as it seems, just here, as in any skill, you need knowledge of the technique and regular training. Well, training depends only on you, and we will now analyze the technique.
Let's start with adding two-digit numbers.
Let us need to calculate 37 + 85 + 29 + 42 . To do this, first add all the tens: 3 + 8 + 2 + 4. Note that 8 + 2 = 10, 3 + 4 = 7, together 17. Remember. Now add the units: 7 + 5 + 9 + 2 = 23.
17 tens is 170. 170 + 23 = 193. As you can see, this is faster than adding 37 and 85, then adding 29, and so on.
By the way, the same can be done if we add three-digit numbers.
For example: 228 + 39 + 485 + 91.
Adding tens:
22 + 3 + 48 + 9 = (22 + 48) + (3 + 9) = 70 + 12 = 82.
Now add the units:
8 + 9 + 5 + 1 = (8 + 5) + (9 + 1) = 13 + 10 = 23.
(If two numbers add up to ten, it's always convenient to add them up first.)
Well, now there are 82 tens, i.e. 820 plus 23 is 843.
Now let's move on to a more interesting topic - multiplication of two-digit numbers. Here we will also act unusually. The technique that we will now consider is called multiplication "cross" or Hindu way of multiplication.
We want to multiply 76 by 28. We proceed as follows:
First, we multiply the units: 6 8 = 48,
now we multiply with a “cross” 7 8 + 2 6 \u003d 56 + 12 \u003d 68 tens, and taking into account 4 tens out of 48, we have 72 tens and 8 units or 720 and 8. Now we multiply hundreds: 7 2 \u003d 14 hundreds or, taking into account 7 hundreds out of 720, we have 21 hundreds, 2 tens and 8 units. Answer: 2128.
We have considered a method that works for any two-digit numbers, but often the calculations can be simplified by noticing certain features of our multipliers.
For example, in our case, 76 is nothing more than 75 + 1.
Then: 28 76 = 28 (75 + 1) = 28 75 + 28 = 28 (50 + 25) + 28 = 28 50 + 28 25 + 28 = 2800/2 + 1400/2 + 28 =
1400 + 700 + 28 = 2128
Of course, we do not paint all these calculations, but we do them in our minds. They, like the drawing with a cross, are given in order to show the multiplication algorithm. In fact, all calculations are carried out "in the mind." Yes, at first, calculations using this method may seem complicated, but we do not forget about the second component of success - practice. A little practice and you'll be fine!
Well, for inquisitive "theorists" we will show how this method was invented.
Consider the multiplication of two two-digit numbers in general terms: multiply ab by cd.
We can always write ab as a 10 + b and cd as c 10 + d. Then:
(a 10 + b)(c 10 + d) =
100 a c + 10 a d + 10 b c + b d =
A c 100 + (a d + b c) 10 + b d
It can be seen from the multiplication result that in order to get hundreds, it is necessary to multiply the first digits of our number, to get tens - we multiply them with a cross and add them up, and finally, multiplying the last digits gives us the number of units.
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