# DEMO LESSON

## Problems involving multiplication.

Dear student,

Recall how long multiplication works. Compare it with multiplication using distributive law of multiplication over addition:

The expression above is called a formula. It means that if we multiply a sum of any two numbers a and b by a third number c, we get the same result as when we multiply each of the numbers a and b separately by number c, and then add these products.

 Compare: 1 2 First we multiply ones by 4 and get 8 ones. 12 4 = (10 + 2) 4 4 Then we multiply tens by 4 and get 4 tens, = 40 + 8 4 8 or 40 ones. 40 + 8 = 48 = 48 1 8 First we multiply ones by 3 and get 24 ones. 18 3 = (10 + 8) 3 3 or 2 tens and 4 ones. Then we multiply tens = 30 + 24 5 4 by 3 and get 3 tens. 3 tens + 2 tens + 4 ones = = 54 50 + 4 = 54

Long multiplication works faster on paper, but when you multiply larger numbers mentally, the second method is very useful. So far we multiplied only by a one-digit number. Multiplication by a two-digit number works the same way as multiplication by one-digit number does.

 32 First we multiply 32 by 4, and get 128. 32 24 = 32 (4 + 20) 24 Then we multiply 32 by 20 and get 640 = 128 + 640 128 Then we add up 128 and 640. = 768 +640 768

Now let's practice! First do the exercises

Exercises.

1. Calculate, using the distributive law of multiplication over addition. Write down all the steps as shown in this example:

 25 6 = (20 + 5) 6 = 20 6 + 5 6 = 120 + 30 = 150

 a) 33 7 = b) 48 3 = c) 57 4 = d) 32 8 = e) 54 5 = f) 36 3 = g) 42 6 = h) 83 5 = i) 71 3 = j) 26 9 = k) 68 2 = l) 59 7 = m) 92 4 = n) 69 6 = o) 43 4 =

2. Multiply:

 a) 57 431 452 648 3 7 6 5 b) 89 563 327 124 4 2 9 8

3. Multiply:

 a) 43 232 357 642 15 24 38 53 b) 58 341 567 737 23 35 61 46

4. In problems below some digits are replaced with shapes. The same shapes in a problem stand for the same digit. Fill in each shape to make a correct multiplication problem. Keep in mind that the same shapes in different problems do not stand for the same digit.

5. If you multiply the number 142,857 by 2 , 3, 4, 5 and 6, you will get products that form a pattern. Find the products and describe the pattern.

### Miscellaneous.

1. Five students in a math competition got the prizes. The first prize was awarded for 20 points, the second for 19 points and the third for 18 points. How many students got first, second and third prize if the sum of all the points they got is 94 ?
2. A clock is set to the correct time at 10 a.m. From that time, for every minute that passes, it loses 5 seconds. What is the actual time when the clock shows 9 p.m. on that day?

## Here is the last part of the lesson. I hope you do well. Good luck!

After you become our student, you will take similar tests, but your answers will be sent to us automatically. After you submit the answers, you will be able to check them on-line.

Question 1:

 Calculate the following:12 9 + 3 =123 9 + 4 =1234 9 + 5 =Do you recognize the pattern? Without calculating, 123456789 9 + 10 = ?

Question 2:

Calculate the following:
9 9 + 7 =
98 9 + 6 =
987 9 + 5=
Do you recognize the pattern? Without calculating, 9876543 9 + 1 = ?

Question 3:

 A club does a five-dollar gift exchange at Christmas each year. Each person buys a five-dollar gift for every other member of the club. How much money will be spent on gifts by all the members of the club together if there are 8 members?

Question 4:

 Use each of the numbers 5,6 and 7 in the boxes. Find the largest answer

Question 5.a:

 What is the last digit of the product of 10 factors 11131517 ...2729?

Question 5.b:

 What is the last digit of the product 101 102103 104105 106107 108109?

Question 5.c:

 What is the last digit of the product of 32 factors if each of the factor equals 3?

Question 5.d:

 What is the last digit of the product of 21 factors if each of the factor equals 2?

Question 5.e:

 What is the last digit of the product of all odd numbers from 1 to 99?

Question 5.f:

 What is the last digit of the product of all odd numbers from 1 to 199?

Question 6:

 7 cubes were glued together as shown in the picture. Then the surface area of the figure was painted. How many squares were painted, including the ones painted in the picture?