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The Nature of Multiplication А.V. Shevyakova
© «Home school»
Textbook + workbook For kids of age
7-10
Foreword Dear parents! We are presenting to you a textbook for children in grades 2-5. This book is devoted to the topics of multiplication and division. It book is different from other textbooks. Its material is aimed at the development of deep, intuitive understanding of the operations of multiplication and division. This is achieved by employing different approaches, and by solving many problems using each approach. The first part of the textbook is devoted to understanding the nature of multiplication. The second part is about division and its close connection to multiplication. It also features problems and practice examples.
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Part 1. Multiplication Lesson 1. Visual understanding of multiplication
From a conversation of a father with his daughter: - Papa, what is multiplication? How is it different from addition? - Addition is when we add numbers to each other. For example, if you have three candies, and I give you four more. You put them all together - three and four, and you get seven candies. - I understand that... How about multiplication? - Imagine yourself a box, filled with candies. I put my hand in it and take out three candies. Then I put my hand in again and take out three more. We have TWO times THREE candies. How many will we have? - Three plus three is six. That is addition, not multiplication! - So, if we take three candies two times, that is, two times three – it is the same, as three plus three:
3х2=3+3=6 - What if you take out three more candies out of the box? - Then we will have THREE times THREE candies.
3х3=3+3+3=9 - Now I understand. Thanks, Dad! - Do you have any more questions about multiplication? 3
- Yes. Was multiplication invented by people, or is it found in nature? - Good question. What do you think? - I think it was invented by people, so that we don’t have to write many plus signs. So, we could write the problem
2+2+2+2+2 in a shorter way – as two times five:
2х5 Dad smiled and said: - Let’s draw things that are found in nature, and the we can see if there is multiplication in nature, or it was made by people. - OK! What should we start with? - First, let’s draw some things that have two similar parts. What can you make up? - A butterfly, a man, a bicycle…
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- Good. Let’s draw two dots on each butterfly wing, three dots on each of the man’s legs, and four dots on each bicycle wheel.
- Hmmm. That reminds me of something. Is this something to do with multiplication? - Yes! But in what way? - Just wait. Look:
2х2=4
4х2=8 3х2=6
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- Good job! You figured it out yourself, and it was not easy! Can you think of pictures for muliplication by three? - That’s easy. Like this:
- I am going to put dots on them, and you have to figure out what am I multipling them by! - Sure!
4 х 3 = 12
1х3=3
- Let’s make up other pictures that help us multiply. - Sure!
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3х3=9
And Papa and his dauter started drawing other multiplicator pictures. Problem 1. Count the dots.
4х4=
3 х 5 = 15
2х6=
1 х 10 = 2 х 11 = 7
Lesson 2. Multiplication by TWO - Papa, are we going to draw today, as we did yesterday? I really liked drawing and solving problems about multiplication. - If you want, let’s do it. But we can also play with multiplication a different way. Would you like to make a multiplication robot? - Of course! But how? - Look: this is our multiplication robot. It has an input, in which we put objects, and two outputs, since we are mutiplying by two. Out of this multiplication robot come two identical sets of objects, that is, the robot is doubling the given number.
Multiplication robot
3х2=3+3=6 х2
In the multiplication
3 х 2 = 6,
the answer 6 is called the product of 2 and 3.
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Problem 2. Draw the dots at the output, count them, and solve the problems.
2х2=2+2= х2
5х2= х2
5+5=
х 2 = х2
+
9
=
Lesson 3. The second way of multiplication by TWO - Yesterday, we had a good time playing with the multiplication robot. Right, Papa? - Yes, I also liked it. Today I want to show you a second way of multiplying by two. Look at the picture: you can think of multiplication as working with rows and columns. For example, if we have two rows of tree dots in each row:
3
х
2
=
6
- I see. It looks like a rectangle, in which the length of one side is multiplied by the length of the other side. - Yes, you are right. That’s how area is calculated. Remember what the area is? - The area is how many squares is taken up by a shape. - Yes, you can think of it that way. Look: now we have three rows with two dots in each row.
2
х
3
=
6
- The result is the same! So it does not matter if we take two dots three times or three dots two times…
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Problem 5. Draw the dots, count them, and fill in the blanks.
4
х
2
=
2
х
2
=
6
х
2
=
х
=
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Lesson 13. Word problems for multiplication. Problem 51. There are three rows with five stars in each row. How many stars are there in total?
х3=
There are
stars.
Problem 54. How many Christmas decorations are there in FIVE boxes, if each box contains EIGHT Christmas decorations?
There are
х
= 12
Christmas decorations.
Problem 57. How many petals do four flowers have, if each one has five petals?
There are
х
=
pedals.
Problem 58. A girl picked some flowers in the field. At home, she put them into FIVE vases, so that each vase has SIX flowers. How many flowers did she have in total? (You can may the flowers in the vases.)
There are
х
= 13
flowers.
Lesson 16. Calculating areas of complex figures. Problem 67. Calculate the area of the bounded shape.
S=
х
= S=
х
=
+
Problem 68. Calculate the area of the shaded shape.
The non-shaded squares inside the shaded area.
Area:
S=
х
14
=
Problem 70. Calculate the total area of the shapes according to their colors.
S=
х
=
S=
х
=
S=S+S S=
+
=
S=
х
=
S=
х
=
S=S+S=
S=
х
=
S=
х
=
S=S+S= 15
Part 2. Division. Lesson 17. Dividing into equal parts. Problem 71. Divide the toys equally between a boy and a girl. This is the number of piles they were divided into.
This is how many rabbits there were.
6 : 2 = This is the answer, how many toys each pile has.
: 2 =
:
16
=
Problem 72. Fill in the blanks. Use color pencils or crayons to divide the given sets of dots into the required number of piles, as shown.
12 24 : 2 = ____ 24 : 3 = ____ 8 6 24 : 4 = ____ 12 : 3 = ____ 12 : 4 = ____ 12 : 2 = ____ 18 : 2 = ____ 18 : 3 = ____ 18 : 9 = ____
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Lesson 19. Word problems requiring division. Problem 77. A clownfish laid 24 eggs equally into 6 anemones. Draw the eggs, and count how many there is under each anemone.
:6=
Under each anemone, there are
eggs.
Problem 78. A herd of cows is grazing in a field. The total number of legs and heads in the herd is 30. How many cows are there?
There are
:
= 18
cows.
Problem 79. There are 6 boxes of eggs, each one containing 6 eggs. How many eggs are there in total?
Fill in the blanks and find the answer:
х There are
=
eggs.
How many cakes can be made out of these eggs, if each cake needs 9 eggs?
: Answer:
=
cakes. 19
Lesson 22. Check your skills! Problem 95. Write answers and compare them (use , =).
5х5…5х6
5х5… 5х6
4х7…9х3
30 : 5 … 36 : 6
24 : 3 … 24 : 4
9:1… 9х0
18 : 2 … 45 : 5
7х9…8х8
Problem 96. Calculate.
289 х 1 = ____
765 : 1 = ____
72 : 72 = ____
591 х 0 = ____
0 : 53 =
____
0 х 0 = ____ 20
Problem 97. Calculate. If in any line, both answers are the same, you made a mistake! If so, find it and correct it, using the right order of operations.
9 х (4 + 4) = ____
9 х 4 + 4 = ____
7 х (6 – 6) = ____
7 х 6 – 6 = ____
12 : (3 + 1) = ____
12 : 3 + 1 = ____
16 : (2 + 2) = ____
16 : 2 + 2 = ____
2 х (9 – 3) = ____
2 х 9 – 3 = ____
18 : (2 – 1) = ____
18 : 2 – 1 = ____
Problem 98. Calculate.
14 х 2 = 10 х 2 + 4 х 2 = ____ 48 : 4 = 40 : 4 + 8 : 4 = ____ 15 х 7 = 10 х 7 + 5 х 7 = ____ 72 : 6 = 60 : 6 + 12 : 6 = ____
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Problem 99. Check whether the problems are being solved correctly. If you find a mistake, correct it. Write down answers. (Hint: there are two mistakes.)
5 х 8 = 5 х 5 + 5 х 3 = ____ 14 х 7 = 10 х 7 + 4 х 7 = ____ 6 х 6 = 6 х 5 – 6 х 1 = ____ 6 х 20 = 6 х 10 + 6 х 10 = ____ 3 х 8 + 3 х 4 = 3 х 12 = ____ 10 х 2 – 4 х 2 = 14 х 2 = ____ Problem 100. Decode the word by solving the problems and putting the letters in the correct places.
49 : 7 = ____
l
48 : 8 = ____
x
20 : 4 = ____
t
64 : 8 = ____
n
81 : 9 = ____
e
16 : 4 = ____
с
9
6
4
9
7 22
7
9
8
5
Contents. Foreword................................................................................................................................................................2
Part 1. Multiplication................................................................................................................................. 3 Lesson 1. Visual understanding of multiplication.............................................................................................3 Lesson 2. Multiplication by TWO ......................................................................................................................8 Lesson 3. The second way of multiplication by TWO ....................................................................................12 Lesson 4. Multiplication by THREE .................................................................................................................15 Lesson 5. Practice of multiplication by TWO and THREE ...........................................................................19 Lesson 6. The multiplication table for TWO and THREE ............................................................................24 Lesson 7. Multiplication by FOUR..................................................................................................................28 Lesson 8. Multiplication by FIVE....................................................................................................................32 Lesson 9. The multiplication table for FOUR and FIVE. .............................................................................36 Lesson 10. Multiplication by TEN...................................................................................................................39 Lesson 11. Multiplication by SIX, SEVEN, EIGHT and NINE......................................................................42 Lesson 12. The multiplication table for SIX, SEVEN, EIGHT and NINE ....................................................45 Lesson 12. The multiplication table for SIX, SEVEN, EIGHT and NINE ....................................................46 Lesson 13. Word problems for multiplication..................................................................................................50 Lesson 14. Multiplication and the Area. ...........................................................................................................54 Lesson 15. Area. ..................................................................................................................................................57 Lesson 16. Calculating areas of complex figures..............................................................................................60
Part 2. Division......................................................................................................................................... 63 Lesson 17. Dividing into equal parts. ...............................................................................................................63 Lesson 18. Dividing into equal parts (continued)............................................................................................66 Lesson 19. Word problems requiring division. ...............................................................................................69 Lesson 20. Relations between multiplication and division. ............................................................................73 Lesson 21. Properties of multiplication and division. .....................................................................................77 Lesson 22. Check your skills!............................................................................................................................83
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