8 Grade F6
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ADDITION
Comparing Numbers
Symbol
Meaning
Example in Symbols
Example in Words
>
Greater than More than Bigger than Larger than
7>4
7 is greater than 4 7 is more than 4 7 is bigger than 4 7 is larger than 4
<
Less than Fewer than Smaller than
4<7
4 is less than 7 4 has fewer than 7 4 is smaller than 7
=
Equal to Same as
7= 7
7 is equal to 7 7 is the same as 7
Ordering Six Digit Numbers Numbers have an order or arrangement. The number two is between one and three. Three or more numbers can be placed in order. A number may come before the other numbers or it may come between them or after them. Example: If we start with the numbers 4 and 8, the number 5 would come between them, the number 9 would come after them and the number 2 would come before both of them. The order may be ascending (getting larger in value) or descending (becoming smaller in value).
Comparing Seven Digit Numbers Symbol
Meaning
Example in Symbols
Example in Words
>
Greater than More than Bigger than Larger than
7>4
7 is greater than 4 7 is more than 4 7 is bigger than 4 7 is larger than 4
<
Less than Fewer than Smaller than
4<7
4 is less than 7 4 has fewer than 7 4 is smaller than 7
=
Equal to Same as
7= 7
7 is equal to 7 7 is the same as 7
Ordering Seven Digit Numbers
Numbers have an order or arrangement. The number two is between one and three. Three or more numbers can be placed in order. A number may come before the other numbers or it may come between them or after them. Example: If we start with the numbers 4 and 8, the number 5 would come between them, the number 9 would come after them and the number 2 would come before both of them. The order may be ascending (getting larger in value) or descending (becoming smaller in value).
Comparing Decimals and Fractions A decimal number and a fractional number can be compared. One number is either greater than, less than or equal to the other number. When comparing fractional numbers to decimal numbers, convert the fraction to a decimal number by division and compare the decimal numbers. If one decimal has a higher number on the left side of the decimal point then it is larger. If the numbers to the left of the decimal point are equal but one decimal has a higher number in the tenths place then it is larger and the decimal with less tenths is smaller. If the tenths are equal compare the hundredths, then the thousandths etc. until one decimal is larger or there are no more places to compare. It is often easy to estimate the decimal from a fraction. If this estimated decimal is obviously much larger or smaller than the compared decimal then it is not necessary to convert the fraction to a decimal
Ordering Decimal Numbers Numbers have an order or arrangement. The number two is between one and three. Three or more numbers can be placed in order. A number may come before the other numbers or it may come between them or after them. Example: If we start with the numbers 4.3 and 8.78, the number 5.2764 would come between them, the number 9.1 would come after them and the number 2 would come before both of them. Example: If we start with the numbers 4.3 and 4.78, the number 4.2764 would come before both of them, the number 4.9 would come after them and the number 4.5232 would come between them. The order may be ascending (getting larger in value) or descending (becoming smaller in value).
Comparing One Digit Integers
Positive Integers Symbol
Meaning
Example in Symbols
Example in Words
>
Greater than More than Bigger than Larger than
7>4
7 is greater than 4 7 is more than 4 7 is bigger than 4 7 is larger than 4
<
Less than Fewer than Smaller than
4<7
4 is less than 7 4 has fewer than 7 4 is smaller than 7
=
Equal to Same as
7= 7
7 is equal to 7 7 is the same as 7
Comparing Two Digit Integers Positive Integers Symbol
Meaning
Example in Symbols
Example in Words
>
Greater than More than Bigger than Larger than
73 > 43
73 is greater than 43 73 is more than 43 73 is bigger than 43 73 is larger than 43
<
Less than Fewer than Smaller than
43 < 73
43 is less than 73 43 has fewer than 73 43 is smaller than 7
=
Equal to Same as
73 = 73
73 is equal to 73 73 is the same as 73
Negative Integers Symbol
Meaning
Example in Symbols
Example in Words
>
Greater than More than Bigger than Larger than
-43 > -73
-43 is greater than -73 -43 is more than -73 -43 is bigger than -73 -43 is larger than -7
<
Less than Fewer than Smaller than
-73 < -43
-73 is less than -43 -73 has fewer than -43 -73 is smaller than -43
=
Equal to Same as
-73 = -73
-73 is equal to -73 -73 is the same as -73
Comparing Three Digit Integers Positive Integers
Symbol
Meaning
Example in Symbols
Example in Words
>
Greater than More than Bigger than Larger than
732 > 432
732 is greater than 432 732 is more than 432 732 is bigger than 432 732 is larger than 432
<
Less than Fewer than Smaller than
432 < 732
432 is less than 732 432 has fewer than 732 432 is smaller than 732
=
Equal to Same as
732 = 732
732 is equal to 732 732 is the same as 732
Negative Integers Example in Symbols
Example in Words
>
Greater than More than Bigger than Larger than
-432 > -732
-432 is greater than -732 -432 is more than -732 -432 is bigger than -732 -432 is larger than -732
<
Less than Fewer than Smaller than
-732 < -432
-732 is less than -432 -732 has fewer than -432 -732 is smaller than -432
=
Equal to Same as
-732 = -732
-732 is equal to -732 -732 is the same as -732
Symbol
Meaning
Addition equations with 3 digit numbers An equation is a mathematical statement that has an expression on the left side of the equals sign (=) with the same value as the expression on the right side. An example of an equation is 222 + 222 = 444. One of the terms in an equation may not be known and needs to be determined. The unknown term may be represented by a letter such as x (e.g. 222 + x = 444). The equation is solved by finding the value of the unknown x that makes the two sides of the equation have the same value. Use the subtractive equation property to find the value of x in addition equations. The subtractive equation property states that the two sides of an equation remain equal if the same number is subtracted from each side. Example: 500 + x = 1200 500 + x - 500 = 1200 - 500 0 + x = 700 x = 700 Check the answer by substituting (700) for x in the original equation. The answer is correct if the expressions on each side of the equals sign have the
same value. 500 + 700 = 1200
Adding Four Digit Numbers How to add four digit numbers (for example 4529 + 6733): •
Place one number above the other so that the thousands', hundreds', tens' and ones' places are lined up. Draw a line under the bottom number.
• • • • •
4529 6733 Add the ones' place digits (9 + 3 = 12). This number is larger than 10 so place a one above the tens' place column and place the two below the line in the ones' place column.
• • • • • •
1 4529 6733 2
Add the tens' place digits (1 + 2 + 3 = 6) and place the answer below the line and in the tens' place column.
• • •
•
4529 6733 62 Add the numbers in the hundreds' place column (5 + 7 = 12) and place the 2 below the line and before the other number below the line. Place the 1 from the twelve above the thousands' place column.
• •
1
• •
•
4529 6733 262 Add the digits in the thousands' place column (1 + 4 + 6 = 11) and place the answer below the line in the thousands' place column.
• • •
4529 6733 11262 Addition equations with 4 digit numbers
An equation is a mathematical statement that has an expression on the left side of the equals sign (=) with the same value as the expression on the right side. An example of an equation is 2222 + 2222 = 4444. One of the terms in an equation may not be known and needs to be determined. The unknown term may be represented by a letter such as x (e.g. 2222 + x = 4444). The equation is solved by finding the value of the unknown x that makes the two sides of the equation have the same value. Use the subtractive equation property to find the value of x in addition equations. The subtractive equation property states that the two sides of an equation remain equal if the same number is subtracted from each side. Example: 5000 + x = 12000 5000 + x - 5000 = 12000 - 5000 0 + x = 7000 x = 7000 Check the answer by substituting (7000) for x in the original equation. The answer is correct if the expressions on each side of the equals sign have the same value. 5000 + 7000 = 12000
Adding Five Digit Numbers Adding two five digit numbers (for example 94,529 + 76,733) is illustrated
111 1 94529 76733 171262 Addition equations with 5 digit numbers An equation is a mathematical statement that has an expression on the left side of the equals sign (=) with the same value as the expression on the right side. An example of an equation is 22222 + 22222 = 44444. One of the terms in an equation may not be known and needs to be determined. The unknown term may be represented by a letter such as x (e.g. 22222 + x = 44444). The equation is solved by finding the value of the unknown x that makes the two sides of the equation have the same value. Use the subtractive equation property to find the value of x in addition equations. The subtractive equation property states that the two sides of an equation remain equal if the same number is subtracted from each side. Example: 50000 + x = 120000 50000 + x - 50000 = 120000 - 50000 0 + x = 70000 x = 70000 Check the answer by substituting (70000) for x in the original equation. The answer is correct if the expressions on each side of the equals sign have the same value. 50000 + 70000 = 120000
Adding Six Digit Numbers Adding two six digit numbers (for example 694,529 + 476,733) is illustrated
1111 1 694529 476733
1171262 Addition equations with 6 digit numbers An equation is a mathematical statement that has an expression on the left side of the equals sign (=) with the same value as the expression on the right side. An example of an equation is 222222 + 222222 = 444444. One of the terms in an equation may not be known and needs to be determined. The unknown term may be represented by a letter such as x (e.g. 222222 + x = 444444). The equation is solved by finding the value of the unknown x that makes the two sides of the equation have the same value. Use the subtractive equation property to find the value of x in addition equations. The subtractive equation property states that the two sides of an equation remain equal if the same number is subtracted from each side. Example: 500000 + x = 1200000 500000 + x - 500000 = 1200000 - 500000 0 + x = 700000 x = 700000 Check the answer by substituting (700000) for x in the original equation. The answer is correct if the expressions on each side of the equals sign have the same value. 500000 + 700000 = 1200000
Return to Top
Adding Seven Digit Numbers Adding two seven digit numbers (for example 8,694,529 + 3,476,733) is illustrated
11111 1 8694529 3476733
12171262 Subtracting five Digit Numbers Subtracting two five digit numbers (for example 94,529 - 76,733) is illustrated
834 94529 76733 17796 Subtraction equations - 5 digit An equation is a mathematical statement such that the expression on the left side of the equals sign (=) has the same value as the expression on the right side. An example of an equation is 60000 - 40000 = 20000. One of the terms in an equation may not be know and needs to be determined. Often this unknown term is represented by a letter such as x. (e.g. x - 40000 = 20000). The solution of an equation is finding the value of the unknown x. To find the value of x we can use the additive equation property which says: The two sides of an equation remain equal if the same number is added to each side. Example: x - 50000 = 70000 x - 50000 + 50000 = 70000 + 50000 x - 0 = 120000 x = 120000 Check the answer by substituting the value of x (120000) back into the equation. 120000 - 50000 = 70000
Subtracting six Digit Numbers Subtracting two six digit numbers (for example 694,529 - 476,733) is illustrated
834 694529 476733 217796 Subtraction equations - 6 digit An equation is a mathematical statement such that the expression on the left side of the equals sign (=) has the same value as the expression on the right side. An example of an equation is 600000 - 400000 = 200000. One of the terms in an equation may not be know and needs to be determined. Often this unknown term is represented by a letter such as x. (e.g. x - 400000 = 200000). The solution of an equation is finding the value of the unknown x. To find the value of x we can use the additive equation property which says: The two sides of an equation remain equal if the same number is added to each side. Example: x - 500000 = 700000 x - 500000 + 500000 = 700000 + 500000 x - 0 = 1200000 x = 1200000 Check the answer by substituting the value of x (1200000) back into the equation. 1200000 - 500000 = 700000
Subtracting seven Digit Numbers Subtracting two seven digit numbers (for example 8,694,529 - 3,476,733) is illustrated
834 8694529 3476733
5217796 Multiplying two digit by one digit numbers How to multiply a two digit number by a one digit number (for example 59 + 7). •
Place one number above the other so that the ones' place digits are lined up. Draw a line under the bottom number.
• • • • •
59 7 Multiply the two ones' place digits (9 * 7 = 63). This number is larger than 9, so place the six above the tens' place column and place the three below the line in the ones' place column.
• • • • • •
6 59 7 3
Multiply the digit in the tens' place column (5) by the second number (7). The result is 5 * 7 = 35. Add the 6 to the 35 (35 + 6 = 41) and place the answer below the line and to the left of the 3.
• • •
59 7 413 Multiplying three digit by one digit numbers
How to multiply a three digit number by a one digit number (e.g. 159 * 7).
•
Place one number above the other so that the ones' place digits are lined up. Draw a line under the bottom number.
• • • • •
159 7 Multiply the two ones' place digits (9 * 7 = 63). This number is larger than 9, so place the six above the tens' place column and place the three below the line in the ones' place column.
• • • • • •
6 159 7 3 Multiply the digit in the tens' place column (5) by the other number (7). The result is 5 * 7 = 35. Add the 6 to the 35 which equals 41. Place the one from the number 41 below the line and to the left of the other number. Place the 4 above the hundreds' place column.
• • • •
•
46 159 7 13 Multiply the digit in the hundreds' place column (1) by the digit in the ones' place of the second number (7). The result is 1 * 7 = 7. Add the 4 to the 7 (4 + 7 = 11). Place this below the line and to the left of the other digits.
• • • •
46 159 7
1113 Multiplying Three Numbers How to multiply three numbers: • •
Multiply the first number by the second number. Multiply the product of the first multiplication by the third number.
Multiplication of Two and Three Digit Numbers How to multiply a three digit number by a two digit number (e.g. 529 * 67). •
Place one number above the other so that the hundreds', tens' and ones' places are lined up. Draw a line under the bottom number.
• • • • •
529 67 Multiply the two numbers in the ones' places. (9 * 7 = 63). This number is larger than 9 so place a 6 above the tens' place column and place 3 below the line in the ones' place column.
• • • • • •
6 529 6 7 3 Muliply the digit in the top tens' place column (2) by the digit in the lower ones' place column (7). The answer (2*7=14) is added to the 6 above the top tens' place column to give an answer of 20. The 0 of 20 is placed below the line and the 2 of the 20 is placed above the hundreds' place column.
• • •
26 529
• • •
6 7 03 The hundreds' place of the top number (5) is multiplied by the ones' place of the multiplier (5*7=35). The two that was previously carried to the hundreds' place is added and the 37 is placed below the line.
• • • • • •
26 529 6 7 3703 After 529 has been multiplied by 7 as shown above, 529 is multiplied by the tens' place of the multiplier which is 6. The number is moved one place to the left because we are multiplying by a tens' place number. The result would be 3174:
• • • • • • • •
15 529 67 3703 3174 A line is drawn under the lower product (3174) and the products are added together to get the final answer of 35443.
• • • • • • •
15 529 67 3703 3174 35443
•
Multiplication of Five Digit Numbers. Multiplying a five digit number by a one digit number (for example 52639 * 7) is illustrated below. •
Place one number above the other so that the one's places are lined up. Draw a line under the bottom number.
• • •
52639 7
•
•
368473
Multiplication of Five Digit Numbers. Multiplying a five digit number by a two digit number (for example 52639 * 67) is illustrated below. •
Place one number above the other so that the hundred's, ten's and one's places are lined up. Draw a line under the bottom number.
• • •
52639 67
•
368473 •
315834 •
3526813
• •
Multiplication of Five Digit Numbers. Multiplying a Five digit number by a three digit number (for example 24639 * 687) is illustrated below. •
Place one number above the other so that the hundred's, ten's and one's places are lined up. Draw a line under the bottom number.
• • •
24639 687
•
172473 •
197112 •
147834 •
16926993 • •
Multiplication of Five Digit Numbers. Multiplying a Five digit number by a 4 digit number (for example 24639 * 3687) is illustrated below. •
Place one number above the other so that the thousand's, hundred's, ten's and one's places are lined up. Draw a line under the bottom number.
• • •
24639 3687
•
172473 •
197112 •
147834 •
73917 •
90843993 • •
Multiplication of Six Digit Numbers. Multiplying a six digit number by a two digit number (for example 524639 * 67) is illustrated below. •
Place one number above the other so that the hundred's, ten's and one's places are lined up. Draw a line under the bottom number.
• • •
524639 67
•
3672473
•
3147834 •
35150813 • •
Multiplication of Six Digit Numbers. Multiplying a six digit number by a three digit number (for example 524639 * 687) is illustrated below. •
Place one number above the other so that the hundred's, ten's and one's places are lined up. Draw a line under the bottom number.
• • •
524639 687
•
3672473 •
4197112 •
3147834 •
360426993 • •
Multiplication Equations An equation is a mathematical statement such that the expression on the left side of the equals sign (=) has the same value as the expression on the right side. An example of an equation is 9 * 8 = 72.
One of the terms in an equation may not be known and needs to be determined. The unknown term may be represented by a letter such as x. (e.g. x * 8 = 72). The solution of an equation is finding the value of the unknown x. Use the division property of equations to find the value of x. The division property of equations states that the two sides of an equation remain equal if both sides are divided by the same number Example: x * 5 = 10 x * 5 ÷ 5 = 10 ÷ 5 x*1=2 x=2 Check the answer by substituting the answer (2) back into the equation. 2 * 5 = 10
Multiplication Equations An equation is a mathematical statement such that the expression on the left side of the equals sign (=) has the same value as the expression on the right side. An example of an equation is 12 * 11 = 132. One of the terms in an equation may not be known and needs to be determined. The unknown term may be represented by a letter such as x. (e.g. x * 11 = 132). The solution of an equation is finding the value of the unknown x. Use the division property of equations to find the value of x. The division property of equations states that the two sides of an equation remain equal if both sides are divided by the same number Example: x * 50 = 1000 x * 50 ÷ 50 = 1000 ÷ 50 x * 1 = 20 x = 20 Check the answer by substituting the answer (20) back into the equation. 20 * 50 = 1000 How to divide a three digit number by a one digit number (e.g 413 ÷ 7). •
Place the divisor before the division bracket and place the dividend (413) under it.
• • •
7)413
• •
Examine the first digit of the dividend(4). It is smaller than 7 so it can't be divided by 7 to produce a whole number. Next take the first two digits of the dividend (41) and determine how many 7's it contains. In this case 41 holds five sevens (5*7=35) but not six (6*7=42). Place the 5 above the division bracket.
• • • • •
5 7)413 Multiply the 5 by 7 and place the result (35) below the 41 of the dividend.
• • • • • •
5 7)413 35 Draw a line under the 35 and subtract it from 41 (41-35=6). Bring down the 3 from the 413 and place it to the right of the 6.
• • • • • • •
5 7)413 35 63 Divide 63 by 7 and place that answer above the division bracket to the right of the five.
• • • • • •
59 7)413 35 63
•
Multiply the 9 of the quotient by the divisor (7) to get 63 and place this below the 63 under the dividend. Subtract 63 from 63 to give an answer of 0. This indicates that there is nothing left over and 7 can be evenly divided into 413 to produce a quotient of 59.
• • • • • • • •
59 7)413 35 63 63 0 Division
How to divide a three digit number by a one digit number (e.g. 416 ÷ 7). •
Place the divisor before the division bracket and place the dividend (416) under it.
• • • • •
7)416 Examine the first digit of the dividend(4). It is smaller than 7 so it can't be divided by 7 to produce a whole number. Next take the first two digits of the dividend (41) and determine how many 7's it contains. In this case 41 holds five sevens (5*7=35) but not six (6*7=42). Place the 5 above the division bracket.
• • • • •
5 7)416 Multiply the 5 by 7 and place the result (35) below the 41 of the dividend.
• • • • • •
5 7)416 35 Draw a line under the 35 and subtract it from 41 (41-35=6). Bring down the 6 from the 416 and place it to the right of the other 6.
• • • • • • •
5 7)416 35 66 Divide 66 by 7 and place that answer above the division bracket to the right of the five.
• • • • • • •
59 7)416 35 66 Multiply the 9 of the quotient by the divisor (7) to get 63 and place this below the 66. Subtract 63 from 66 to give an answer of 3. The number 3 is called the remainder and indicates that there were three left over.
• • • • •
59 R 3 7)416 35 66
• • •
63 3 Dividing a 4-digit by 2-digit numbers
How to divide a four digit number by a two digit number (e.g. 4138 ÷ 17): •
Place the divisor before the division bracket and place the dividend (4138) under it.
• • • • •
17)4138 Examine the first digit of the dividend(4). It is smaller than 17 so can't be divided by 17 to produce a whole number. Next take the first two digits of the dividend (41) and determine how many 17's it contains. In this case 41 holds two seventeens (2*17=34) but not three (3*17=51). Place the 2 above the division bracket.
• • • • •
2 17)4138 Multiply the 2 by 17 and place the result (34) below the 41 of the dividend.
• • • • • •
2 17)4138 34 Draw a line under the 34 and subtract it from 41 (41-34=7). Bring down the 3 from the 4138 and place it to the right of the 7.
• •
2
• • • • •
17)4138 34 73 Divide 73 by 17 and place that answer above the division bracket and to the right of the two.
• • • • • • •
24 17)4138 34 73 Multiply the 4 of the quotient by the divisor (17) to get 68 and place this below the 73 under the dividend. Subtract 68 from 73 to give an answer of 5. Bring down the 8 from the dividend 4138 and place it next to the 5
• • • • • • • • •
24 17)4138 34 73 68 58 Divide 58 by 17 and place that answer (3) above the division bracket and to the right of the four.
• • • • •
243 17)4138 34 73
• • • •
68 58 Multiply the 3 of the quotient by the divisor (17) to get 51 and place this below the 58 under the dividend. Subtract 51 from 58 to give an answer of 7.
• • • • • • • • • • •
243 17)4138 34 73 68 58 51 7 There are no more digits in the dividend to bring down so the 7 is a remainder. The final answer could be written in several ways. 243 remainder 7 or sometimes 243r7 or as a mixed number 243 7/17
Division Dividing a three digit number by a one digit number (for example 416 ÷ 7) involves several steps. •
Place the divisor before the division bracket and place the dividend (416) under it.
• • • • •
7)416 Examine the first digit of the dividend(4). It is smaller than 7 so can't be divided by 7 to produce a whole number. Next take
the first two digits of the dividend (41) and determine how many 7's it contains. In this case 41 holds five sevens (5*7=35) but not six (6*7=42). Place the 5 above the division bracket. • • • • •
5 7)416 Multiply the 5 by 7 and place the result (35) below the 41 of the dividend.
• • • • • •
5 7)416 35 Draw a line under the 35 and subtract it from 41 (41-35=6). Bring down the 6 from the 416 and place it to the right of the other 6.
• • • • • • •
5 7)416 35 66 Divide 66 by 7 and place that answer above the division bracket to the right of the five.
• • • • • • •
59 7)416 35 66 Multiply the 9 of the quotient by the divisor (7) to get 63 and place this below the 66. Subtract 63 from 66 to give an answer
of 3. The number 3 is called the remainder and indicates that there were three left over. • • • • • • •
59 R 3 7)416 35 66 63 3 Division
Dividing a three digit number by a one digit number (for example 416 ÷ 7) involves several steps. •
Place the divisor before the division bracket and place the dividend (416) under it.
• • • • •
7)416 Examine the first digit of the dividend(4). It is smaller than 7 so can't be divided by 7 to produce a whole number. Next take the first two digits of the dividend (41) and determine how many 7's it contains. In this case 41 holds five sevens (5*7=35) but not six (6*7=42). Place the 5 above the division bracket.
• • • • •
5 7)416 Multiply the 5 by 7 and place the result (35) below the 41 of the dividend.
• •
5
• • • •
7)416 35 Draw a line under the 35 and subtract it from 41 (41-35=6). Bring down the 6 from the 416 and place it to the right of the other 6.
• • • • • • •
5 7)416 35 66 Divide 66 by 7 and place that answer above the division bracket to the right of the five.
• • • • • • •
59 7)416 35 66 Multiply the 9 of the quotient by the divisor (7) to get 63 and place this below the 66. Subtract 63 from 66 to give an answer of 3. The number 3 is called the remainder and indicates that there were three left over.
• • • • • • • •
59 R 3 7)416 35 66 63 3
Division Dividing a three digit number by a one digit number (for example 416 ÷ 7) involves several steps. •
Place the divisor before the division bracket and place the dividend (416) under it.
• • • • •
7)416 Examine the first digit of the dividend(4). It is smaller than 7 so can't be divided by 7 to produce a whole number. Next take the first two digits of the dividend (41) and determine how many 7's it contains. In this case 41 holds five sevens (5*7=35) but not six (6*7=42). Place the 5 above the division bracket.
• • • • •
5 7)416 Multiply the 5 by 7 and place the result (35) below the 41 of the dividend.
• • • • • •
5 7)416 35 Draw a line under the 35 and subtract it from 41 (41-35=6). Bring down the 6 from the 416 and place it to the right of the other 6.
• • • •
5 7)416 35
• • •
66 Divide 66 by 7 and place that answer above the division bracket to the right of the five.
• • • • • • •
59 7)416 35 66 Multiply the 9 of the quotient by the divisor (7) to get 63 and place this below the 66. Subtract 63 from 66 to give an answer of 3. The number 3 is called the remainder and indicates that there were three left over.
• • • • • • • •
59 R 3 7)416 35 66 63 3 Division
Dividing a three digit number by a one digit number (for example 416 ÷ 7) involves several steps. •
Place the divisor before the division bracket and place the dividend (416) under it.
• • • •
7)416
•
Examine the first digit of the dividend(4). It is smaller than 7 so can't be divided by 7 to produce a whole number. Next take the first two digits of the dividend (41) and determine how many 7's it contains. In this case 41 holds five sevens (5*7=35) but not six (6*7=42). Place the 5 above the division bracket.
• • • • •
5 7)416 Multiply the 5 by 7 and place the result (35) below the 41 of the dividend.
• • • • • •
5 7)416 35 Draw a line under the 35 and subtract it from 41 (41-35=6). Bring down the 6 from the 416 and place it to the right of the other 6.
• • • • • • •
5 7)416 35 66 Divide 66 by 7 and place that answer above the division bracket to the right of the five.
• • • • • •
59 7)416 35 66
•
Multiply the 9 of the quotient by the divisor (7) to get 63 and place this below the 66. Subtract 63 from 66 to give an answer of 3. The number 3 is called the remainder and indicates that there were three left over.
• • • • • • • •
59 R 3 7)416 35 66 63 3 Division
Dividing a three digit number by a one digit number (for example 416 ÷ 7) involves several steps. •
Place the divisor before the division bracket and place the dividend (416) under it.
• • • • •
7)416 Examine the first digit of the dividend(4). It is smaller than 7 so can't be divided by 7 to produce a whole number. Next take the first two digits of the dividend (41) and determine how many 7's it contains. In this case 41 holds five sevens (5*7=35) but not six (6*7=42). Place the 5 above the division bracket.
• • • • •
5 7)416 Multiply the 5 by 7 and place the result (35) below the 41 of the dividend.
• • • • • •
5 7)416 35 Draw a line under the 35 and subtract it from 41 (41-35=6). Bring down the 6 from the 416 and place it to the right of the other 6.
• • • • • • •
5 7)416 35 66 Divide 66 by 7 and place that answer above the division bracket to the right of the five.
• • • • • • •
59 7)416 35 66 Multiply the 9 of the quotient by the divisor (7) to get 63 and place this below the 66. Subtract 63 from 66 to give an answer of 3. The number 3 is called the remainder and indicates that there were three left over.
• • • • •
59 R 3 7)416 35 66
• • •
63 3 Division
Dividing a three digit number by a one digit number (for example 416 ÷ 7) involves several steps. •
Place the divisor before the division bracket and place the dividend (416) under it.
• • • • •
7)416 Examine the first digit of the dividend(4). It is smaller than 7 so can't be divided by 7 to produce a whole number. Next take the first two digits of the dividend (41) and determine how many 7's it contains. In this case 41 holds five sevens (5*7=35) but not six (6*7=42). Place the 5 above the division bracket.
• • • • •
5 7)416 Multiply the 5 by 7 and place the result (35) below the 41 of the dividend.
• • • • • •
5 7)416 35 Draw a line under the 35 and subtract it from 41 (41-35=6). Bring down the 6 from the 416 and place it to the right of the other 6.
• • • • • • •
5 7)416 35 66 Divide 66 by 7 and place that answer above the division bracket to the right of the five.
• • • • • • •
59 7)416 35 66 Multiply the 9 of the quotient by the divisor (7) to get 63 and place this below the 66. Subtract 63 from 66 to give an answer of 3. The number 3 is called the remainder and indicates that there were three left over.
• • • • • • • •
59 R 3 7)416 35 66 63 3 Division
Dividing a three digit number by a one digit number (for example 416 ÷ 7) involves several steps.
•
Place the divisor before the division bracket and place the dividend (416) under it.
• • • • •
7)416 Examine the first digit of the dividend(4). It is smaller than 7 so can't be divided by 7 to produce a whole number. Next take the first two digits of the dividend (41) and determine how many 7's it contains. In this case 41 holds five sevens (5*7=35) but not six (6*7=42). Place the 5 above the division bracket.
• • • • •
5 7)416 Multiply the 5 by 7 and place the result (35) below the 41 of the dividend.
• • • • • •
5 7)416 35 Draw a line under the 35 and subtract it from 41 (41-35=6). Bring down the 6 from the 416 and place it to the right of the other 6.
• • • • • • •
5 7)416 35 66 Divide 66 by 7 and place that answer above the division bracket to the right of the five.
• • • • • • •
59 7)416 35 66 Multiply the 9 of the quotient by the divisor (7) to get 63 and place this below the 66. Subtract 63 from 66 to give an answer of 3. The number 3 is called the remainder and indicates that there were three left over.
• • • • • • • •
59 R 3 7)416 35 66 63 3 Division Equations
An equation is a mathematical statement such that the expression on the left side of the equals sign (=) has the same value as the expression on the right side. An example of an equation is 72 ÷ 8 = 9. One of the terms in an equation may not be known and needs to be determined. The unknown term may be represented by a letter such as x. (e.g. x ÷ 8 = 9). The solution of an equation is finding the value of the unknown x. Use the multiplication property of equations to find the value of x. The multiplication property property of equations states that the two sides of an equation remain equal if both sides are multiplied by the same number Example: x÷5=2 x÷5*5=2*5 x ÷ 1 = 10 x = 10
Check the answer by substituting the answer (10) back into the equation. 10 ÷ 5 = 2
Division equations with 2 digit numbers An equation is a mathematical statement such that the expression on the left side of the equals sign (=) has the same value as the expression on the right side. An example of an equation is 132 ÷ 12 = 11. One of the terms in an equation may not be known and needs to be determined. The unknown term may be represented by a letter such as x. (e.g. x ÷ 12 = 11). The solution of an equation is finding the value of the unknown x. Use the multiplication property of equations to find the value of x. The multiplication property property of equations states that the two sides of an equation remain equal if both sides are multiplied by the same number Example: x ÷ 50 = 20 x ÷ 50 * 50 = 20 * 50 x ÷ 1 = 1000 x = 1000 Check the answer by substituting the answer (1000) back into the equation. 1000 ÷ 50 = 20
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Factors A number may be made by multiplying two or more other numbers together. The numbers that are multiplied together are called factors of the final number. All numbers have a factor of one since one multiplied by any number equals that number. All numbers can be divided by themselves to produce the number one. Therefore, we normally ignore one and the number itself as useful factors. The number fifteen can be divided into two factors which are three and five. The number twelve could be divided into two factors which are 6 and 2. Six could be divided into two further factors of 2 and 3. Therefore the factors of twelve are 2, 2, and 3. If twelve was first divided into the factors 3 and 4, the four could be divided into factors of 2 and 2. Therefore the factors of twelve are still 2, 2, and 3.
There are several clues to help determine factors.
• • •
Any even number has a factor of two Any number ending in 5 has a factor of five Any number above 0 that ends with 0 (such as 10, 30, 1200) has factors of two and five.
To determine factors see if one of the above rules apply (ends in 5, 0 or an even number). If none of the rules apply, there still may be factors of 3 or 7 or some other number.
Comparing Numbers
Symbol
Meaning
Example in Symbols
Example in Words
>
Greater than More than Bigger than Larger than
7>4
7 is greater than 4 7 is more than 4 7 is bigger than 4 7 is larger than 4
<
Less than Fewer than Smaller than
4<7
4 is less than 7 4 has fewer than 7 4 is smaller than 7
=
Equal to Same as
7= 7
7 is equal to 7 7 is the same as 7
Ordering Six Digit Numbers Numbers have an order or arrangement. The number two is between one and three. Three or more numbers can be placed in order. A number may come before the other numbers or it may come between them or after them. Example: If we start with the numbers 4 and 8, the number 5 would come between them, the number 9 would come after them and the number 2 would come before both of them. The order may be ascending (getting larger in value) or descending (becoming smaller in value).
Comparing Seven Digit Numbers Symbol
Meaning
Example in Symbols
Example in Words
>
Greater than More than Bigger than Larger than
7>4
7 is greater than 4 7 is more than 4 7 is bigger than 4 7 is larger than 4
<
Less than Fewer than Smaller than
4<7
4 is less than 7 4 has fewer than 7 4 is smaller than 7
=
Equal to Same as
7= 7
7 is equal to 7 7 is the same as 7
Ordering Seven Digit Numbers
Numbers have an order or arrangement. The number two is between one and three. Three or more numbers can be placed in order. A number may come before the other numbers or it may come between them or after them. Example: If we start with the numbers 4 and 8, the number 5 would come between them, the number 9 would come after them and the number 2 would come before both of them. The order may be ascending (getting larger in value) or descending (becoming smaller in value).
Comparing Decimals and Fractions A decimal number and a fractional number can be compared. One number is either greater than, less than or equal to the other number. When comparing fractional numbers to decimal numbers, convert the fraction to a decimal number by division and compare the decimal numbers. If one decimal has a higher number on the left side of the decimal point then it is larger. If the numbers to the left of the decimal point are equal but one decimal has a higher number in the tenths place then it is larger and the decimal with less tenths is smaller. If the tenths are equal compare the hundredths, then the thousandths etc. until one decimal is larger or there are no more places to compare. It is often easy to estimate the decimal from a fraction. If this estimated decimal is obviously much larger or smaller than the compared decimal then it is not necessary to convert the fraction to a decimal
Ordering Decimal Numbers Numbers have an order or arrangement. The number two is between one and three. Three or more numbers can be placed in order. A number may come before the other numbers or it may come between them or after them. Example: If we start with the numbers 4.3 and 8.78, the number 5.2764 would come between them, the number 9.1 would come after them and the number 2 would come before both of them. Example: If we start with the numbers 4.3 and 4.78, the number 4.2764 would come before both of them, the number 4.9 would come after them and the number 4.5232 would come between them. The order may be ascending (getting larger in value) or descending (becoming smaller in value).
Comparing One Digit Integers
Positive Integers Symbol
Meaning
Example in Symbols
Example in Words
>
Greater than More than Bigger than Larger than
7>4
7 is greater than 4 7 is more than 4 7 is bigger than 4 7 is larger than 4
<
Less than Fewer than Smaller than
4<7
4 is less than 7 4 has fewer than 7 4 is smaller than 7
=
Equal to Same as
7= 7
7 is equal to 7 7 is the same as 7
Comparing Two Digit Integers Positive Integers Symbol
Meaning
Example in Symbols
Example in Words
>
Greater than More than Bigger than Larger than
73 > 43
73 is greater than 43 73 is more than 43 73 is bigger than 43 73 is larger than 43
<
Less than Fewer than Smaller than
43 < 73
43 is less than 73 43 has fewer than 73 43 is smaller than 7
=
Equal to Same as
73 = 73
73 is equal to 73 73 is the same as 73
Negative Integers Symbol
Meaning
Example in Symbols
Example in Words
>
Greater than More than Bigger than Larger than
-43 > -73
-43 is greater than -73 -43 is more than -73 -43 is bigger than -73 -43 is larger than -7
<
Less than Fewer than Smaller than
-73 < -43
-73 is less than -43 -73 has fewer than -43 -73 is smaller than -43
=
Equal to Same as
-73 = -73
-73 is equal to -73 -73 is the same as -73
Comparing Three Digit Integers Positive Integers
Symbol
Meaning
Example in Symbols
Example in Words
>
Greater than More than Bigger than Larger than
732 > 432
732 is greater than 432 732 is more than 432 732 is bigger than 432 732 is larger than 432
<
Less than Fewer than Smaller than
432 < 732
432 is less than 732 432 has fewer than 732 432 is smaller than 732
=
Equal to Same as
732 = 732
732 is equal to 732 732 is the same as 732
Negative Integers Example in Symbols
Example in Words
>
Greater than More than Bigger than Larger than
-432 > -732
-432 is greater than -732 -432 is more than -732 -432 is bigger than -732 -432 is larger than -732
<
Less than Fewer than Smaller than
-732 < -432
-732 is less than -432 -732 has fewer than -432 -732 is smaller than -432
=
Equal to Same as
-732 = -732
-732 is equal to -732 -732 is the same as -732
Symbol
Meaning
Addition equations with 3 digit numbers An equation is a mathematical statement that has an expression on the left side of the equals sign (=) with the same value as the expression on the right side. An example of an equation is 222 + 222 = 444. One of the terms in an equation may not be known and needs to be determined. The unknown term may be represented by a letter such as x (e.g. 222 + x = 444). The equation is solved by finding the value of the unknown x that makes the two sides of the equation have the same value. Use the subtractive equation property to find the value of x in addition equations. The subtractive equation property states that the two sides of an equation remain equal if the same number is subtracted from each side. Example: 500 + x = 1200 500 + x - 500 = 1200 - 500 0 + x = 700 x = 700 Check the answer by substituting (700) for x in the original equation. The answer is correct if the expressions on each side of the equals sign have the
same value. 500 + 700 = 1200
Adding Four Digit Numbers How to add four digit numbers (for example 4529 + 6733): •
Place one number above the other so that the thousands', hundreds', tens' and ones' places are lined up. Draw a line under the bottom number.
• • • • •
4529 6733 Add the ones' place digits (9 + 3 = 12). This number is larger than 10 so place a one above the tens' place column and place the two below the line in the ones' place column.
• • • • • •
1 4529 6733 2
Add the tens' place digits (1 + 2 + 3 = 6) and place the answer below the line and in the tens' place column.
• • •
•
4529 6733 62 Add the numbers in the hundreds' place column (5 + 7 = 12) and place the 2 below the line and before the other number below the line. Place the 1 from the twelve above the thousands' place column.
• •
1
• •
•
4529 6733 262 Add the digits in the thousands' place column (1 + 4 + 6 = 11) and place the answer below the line in the thousands' place column.
• • •
4529 6733 11262 Addition equations with 4 digit numbers
An equation is a mathematical statement that has an expression on the left side of the equals sign (=) with the same value as the expression on the right side. An example of an equation is 2222 + 2222 = 4444. One of the terms in an equation may not be known and needs to be determined. The unknown term may be represented by a letter such as x (e.g. 2222 + x = 4444). The equation is solved by finding the value of the unknown x that makes the two sides of the equation have the same value. Use the subtractive equation property to find the value of x in addition equations. The subtractive equation property states that the two sides of an equation remain equal if the same number is subtracted from each side. Example: 5000 + x = 12000 5000 + x - 5000 = 12000 - 5000 0 + x = 7000 x = 7000 Check the answer by substituting (7000) for x in the original equation. The answer is correct if the expressions on each side of the equals sign have the same value. 5000 + 7000 = 12000
Adding Five Digit Numbers Adding two five digit numbers (for example 94,529 + 76,733) is illustrated
111 1 94529 76733 171262 Addition equations with 5 digit numbers An equation is a mathematical statement that has an expression on the left side of the equals sign (=) with the same value as the expression on the right side. An example of an equation is 22222 + 22222 = 44444. One of the terms in an equation may not be known and needs to be determined. The unknown term may be represented by a letter such as x (e.g. 22222 + x = 44444). The equation is solved by finding the value of the unknown x that makes the two sides of the equation have the same value. Use the subtractive equation property to find the value of x in addition equations. The subtractive equation property states that the two sides of an equation remain equal if the same number is subtracted from each side. Example: 50000 + x = 120000 50000 + x - 50000 = 120000 - 50000 0 + x = 70000 x = 70000 Check the answer by substituting (70000) for x in the original equation. The answer is correct if the expressions on each side of the equals sign have the same value. 50000 + 70000 = 120000
Adding Six Digit Numbers Adding two six digit numbers (for example 694,529 + 476,733) is illustrated
1111 1 694529 476733
1171262 Addition equations with 6 digit numbers An equation is a mathematical statement that has an expression on the left side of the equals sign (=) with the same value as the expression on the right side. An example of an equation is 222222 + 222222 = 444444. One of the terms in an equation may not be known and needs to be determined. The unknown term may be represented by a letter such as x (e.g. 222222 + x = 444444). The equation is solved by finding the value of the unknown x that makes the two sides of the equation have the same value. Use the subtractive equation property to find the value of x in addition equations. The subtractive equation property states that the two sides of an equation remain equal if the same number is subtracted from each side. Example: 500000 + x = 1200000 500000 + x - 500000 = 1200000 - 500000 0 + x = 700000 x = 700000 Check the answer by substituting (700000) for x in the original equation. The answer is correct if the expressions on each side of the equals sign have the same value. 500000 + 700000 = 1200000
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Adding Seven Digit Numbers Adding two seven digit numbers (for example 8,694,529 + 3,476,733) is illustrated
11111 1 8694529 3476733
12171262 Subtracting five Digit Numbers Subtracting two five digit numbers (for example 94,529 - 76,733) is illustrated
834 94529 76733 17796 Subtraction equations - 5 digit An equation is a mathematical statement such that the expression on the left side of the equals sign (=) has the same value as the expression on the right side. An example of an equation is 60000 - 40000 = 20000. One of the terms in an equation may not be know and needs to be determined. Often this unknown term is represented by a letter such as x. (e.g. x - 40000 = 20000). The solution of an equation is finding the value of the unknown x. To find the value of x we can use the additive equation property which says: The two sides of an equation remain equal if the same number is added to each side. Example: x - 50000 = 70000 x - 50000 + 50000 = 70000 + 50000 x - 0 = 120000 x = 120000 Check the answer by substituting the value of x (120000) back into the equation. 120000 - 50000 = 70000
Subtracting six Digit Numbers Subtracting two six digit numbers (for example 694,529 - 476,733) is illustrated
834 694529 476733 217796 Subtraction equations - 6 digit An equation is a mathematical statement such that the expression on the left side of the equals sign (=) has the same value as the expression on the right side. An example of an equation is 600000 - 400000 = 200000. One of the terms in an equation may not be know and needs to be determined. Often this unknown term is represented by a letter such as x. (e.g. x - 400000 = 200000). The solution of an equation is finding the value of the unknown x. To find the value of x we can use the additive equation property which says: The two sides of an equation remain equal if the same number is added to each side. Example: x - 500000 = 700000 x - 500000 + 500000 = 700000 + 500000 x - 0 = 1200000 x = 1200000 Check the answer by substituting the value of x (1200000) back into the equation. 1200000 - 500000 = 700000
Subtracting seven Digit Numbers Subtracting two seven digit numbers (for example 8,694,529 - 3,476,733) is illustrated
834 8694529 3476733
5217796 Multiplying two digit by one digit numbers How to multiply a two digit number by a one digit number (for example 59 + 7). •
Place one number above the other so that the ones' place digits are lined up. Draw a line under the bottom number.
• • • • •
59 7 Multiply the two ones' place digits (9 * 7 = 63). This number is larger than 9, so place the six above the tens' place column and place the three below the line in the ones' place column.
• • • • • •
6 59 7 3
Multiply the digit in the tens' place column (5) by the second number (7). The result is 5 * 7 = 35. Add the 6 to the 35 (35 + 6 = 41) and place the answer below the line and to the left of the 3.
• • •
59 7 413 Multiplying three digit by one digit numbers
How to multiply a three digit number by a one digit number (e.g. 159 * 7).
•
Place one number above the other so that the ones' place digits are lined up. Draw a line under the bottom number.
• • • • •
159 7 Multiply the two ones' place digits (9 * 7 = 63). This number is larger than 9, so place the six above the tens' place column and place the three below the line in the ones' place column.
• • • • • •
6 159 7 3 Multiply the digit in the tens' place column (5) by the other number (7). The result is 5 * 7 = 35. Add the 6 to the 35 which equals 41. Place the one from the number 41 below the line and to the left of the other number. Place the 4 above the hundreds' place column.
• • • •
•
46 159 7 13 Multiply the digit in the hundreds' place column (1) by the digit in the ones' place of the second number (7). The result is 1 * 7 = 7. Add the 4 to the 7 (4 + 7 = 11). Place this below the line and to the left of the other digits.
• • • •
46 159 7
1113 Multiplying Three Numbers How to multiply three numbers: • •
Multiply the first number by the second number. Multiply the product of the first multiplication by the third number.
Multiplication of Two and Three Digit Numbers How to multiply a three digit number by a two digit number (e.g. 529 * 67). •
Place one number above the other so that the hundreds', tens' and ones' places are lined up. Draw a line under the bottom number.
• • • • •
529 67 Multiply the two numbers in the ones' places. (9 * 7 = 63). This number is larger than 9 so place a 6 above the tens' place column and place 3 below the line in the ones' place column.
• • • • • •
6 529 6 7 3 Muliply the digit in the top tens' place column (2) by the digit in the lower ones' place column (7). The answer (2*7=14) is added to the 6 above the top tens' place column to give an answer of 20. The 0 of 20 is placed below the line and the 2 of the 20 is placed above the hundreds' place column.
• • •
26 529
• • •
6 7 03 The hundreds' place of the top number (5) is multiplied by the ones' place of the multiplier (5*7=35). The two that was previously carried to the hundreds' place is added and the 37 is placed below the line.
• • • • • •
26 529 6 7 3703 After 529 has been multiplied by 7 as shown above, 529 is multiplied by the tens' place of the multiplier which is 6. The number is moved one place to the left because we are multiplying by a tens' place number. The result would be 3174:
• • • • • • • •
15 529 67 3703 3174 A line is drawn under the lower product (3174) and the products are added together to get the final answer of 35443.
• • • • • • •
15 529 67 3703 3174 35443
•
Multiplication of Five Digit Numbers. Multiplying a five digit number by a one digit number (for example 52639 * 7) is illustrated below. •
Place one number above the other so that the one's places are lined up. Draw a line under the bottom number.
• • •
52639 7
•
•
368473
Multiplication of Five Digit Numbers. Multiplying a five digit number by a two digit number (for example 52639 * 67) is illustrated below. •
Place one number above the other so that the hundred's, ten's and one's places are lined up. Draw a line under the bottom number.
• • •
52639 67
•
368473 •
315834 •
3526813
• •
Multiplication of Five Digit Numbers. Multiplying a Five digit number by a three digit number (for example 24639 * 687) is illustrated below. •
Place one number above the other so that the hundred's, ten's and one's places are lined up. Draw a line under the bottom number.
• • •
24639 687
•
172473 •
197112 •
147834 •
16926993 • •
Multiplication of Five Digit Numbers. Multiplying a Five digit number by a 4 digit number (for example 24639 * 3687) is illustrated below. •
Place one number above the other so that the thousand's, hundred's, ten's and one's places are lined up. Draw a line under the bottom number.
• • •
24639 3687
•
172473 •
197112 •
147834 •
73917 •
90843993 • •
Multiplication of Six Digit Numbers. Multiplying a six digit number by a two digit number (for example 524639 * 67) is illustrated below. •
Place one number above the other so that the hundred's, ten's and one's places are lined up. Draw a line under the bottom number.
• • •
524639 67
•
3672473
•
3147834 •
35150813 • •
Multiplication of Six Digit Numbers. Multiplying a six digit number by a three digit number (for example 524639 * 687) is illustrated below. •
Place one number above the other so that the hundred's, ten's and one's places are lined up. Draw a line under the bottom number.
• • •
524639 687
•
3672473 •
4197112 •
3147834 •
360426993 • •
Multiplication Equations An equation is a mathematical statement such that the expression on the left side of the equals sign (=) has the same value as the expression on the right side. An example of an equation is 9 * 8 = 72.
One of the terms in an equation may not be known and needs to be determined. The unknown term may be represented by a letter such as x. (e.g. x * 8 = 72). The solution of an equation is finding the value of the unknown x. Use the division property of equations to find the value of x. The division property of equations states that the two sides of an equation remain equal if both sides are divided by the same number Example: x * 5 = 10 x * 5 ÷ 5 = 10 ÷ 5 x*1=2 x=2 Check the answer by substituting the answer (2) back into the equation. 2 * 5 = 10
Multiplication Equations An equation is a mathematical statement such that the expression on the left side of the equals sign (=) has the same value as the expression on the right side. An example of an equation is 12 * 11 = 132. One of the terms in an equation may not be known and needs to be determined. The unknown term may be represented by a letter such as x. (e.g. x * 11 = 132). The solution of an equation is finding the value of the unknown x. Use the division property of equations to find the value of x. The division property of equations states that the two sides of an equation remain equal if both sides are divided by the same number Example: x * 50 = 1000 x * 50 ÷ 50 = 1000 ÷ 50 x * 1 = 20 x = 20 Check the answer by substituting the answer (20) back into the equation. 20 * 50 = 1000 How to divide a three digit number by a one digit number (e.g 413 ÷ 7). •
Place the divisor before the division bracket and place the dividend (413) under it.
• • •
7)413
• •
Examine the first digit of the dividend(4). It is smaller than 7 so it can't be divided by 7 to produce a whole number. Next take the first two digits of the dividend (41) and determine how many 7's it contains. In this case 41 holds five sevens (5*7=35) but not six (6*7=42). Place the 5 above the division bracket.
• • • • •
5 7)413 Multiply the 5 by 7 and place the result (35) below the 41 of the dividend.
• • • • • •
5 7)413 35 Draw a line under the 35 and subtract it from 41 (41-35=6). Bring down the 3 from the 413 and place it to the right of the 6.
• • • • • • •
5 7)413 35 63 Divide 63 by 7 and place that answer above the division bracket to the right of the five.
• • • • • •
59 7)413 35 63
•
Multiply the 9 of the quotient by the divisor (7) to get 63 and place this below the 63 under the dividend. Subtract 63 from 63 to give an answer of 0. This indicates that there is nothing left over and 7 can be evenly divided into 413 to produce a quotient of 59.
• • • • • • • •
59 7)413 35 63 63 0 Division
How to divide a three digit number by a one digit number (e.g. 416 ÷ 7). •
Place the divisor before the division bracket and place the dividend (416) under it.
• • • • •
7)416 Examine the first digit of the dividend(4). It is smaller than 7 so it can't be divided by 7 to produce a whole number. Next take the first two digits of the dividend (41) and determine how many 7's it contains. In this case 41 holds five sevens (5*7=35) but not six (6*7=42). Place the 5 above the division bracket.
• • • • •
5 7)416 Multiply the 5 by 7 and place the result (35) below the 41 of the dividend.
• • • • • •
5 7)416 35 Draw a line under the 35 and subtract it from 41 (41-35=6). Bring down the 6 from the 416 and place it to the right of the other 6.
• • • • • • •
5 7)416 35 66 Divide 66 by 7 and place that answer above the division bracket to the right of the five.
• • • • • • •
59 7)416 35 66 Multiply the 9 of the quotient by the divisor (7) to get 63 and place this below the 66. Subtract 63 from 66 to give an answer of 3. The number 3 is called the remainder and indicates that there were three left over.
• • • • •
59 R 3 7)416 35 66
• • •
63 3 Dividing a 4-digit by 2-digit numbers
How to divide a four digit number by a two digit number (e.g. 4138 ÷ 17): •
Place the divisor before the division bracket and place the dividend (4138) under it.
• • • • •
17)4138 Examine the first digit of the dividend(4). It is smaller than 17 so can't be divided by 17 to produce a whole number. Next take the first two digits of the dividend (41) and determine how many 17's it contains. In this case 41 holds two seventeens (2*17=34) but not three (3*17=51). Place the 2 above the division bracket.
• • • • •
2 17)4138 Multiply the 2 by 17 and place the result (34) below the 41 of the dividend.
• • • • • •
2 17)4138 34 Draw a line under the 34 and subtract it from 41 (41-34=7). Bring down the 3 from the 4138 and place it to the right of the 7.
• •
2
• • • • •
17)4138 34 73 Divide 73 by 17 and place that answer above the division bracket and to the right of the two.
• • • • • • •
24 17)4138 34 73 Multiply the 4 of the quotient by the divisor (17) to get 68 and place this below the 73 under the dividend. Subtract 68 from 73 to give an answer of 5. Bring down the 8 from the dividend 4138 and place it next to the 5
• • • • • • • • •
24 17)4138 34 73 68 58 Divide 58 by 17 and place that answer (3) above the division bracket and to the right of the four.
• • • • •
243 17)4138 34 73
• • • •
68 58 Multiply the 3 of the quotient by the divisor (17) to get 51 and place this below the 58 under the dividend. Subtract 51 from 58 to give an answer of 7.
• • • • • • • • • • •
243 17)4138 34 73 68 58 51 7 There are no more digits in the dividend to bring down so the 7 is a remainder. The final answer could be written in several ways. 243 remainder 7 or sometimes 243r7 or as a mixed number 243 7/17
Division Dividing a three digit number by a one digit number (for example 416 ÷ 7) involves several steps. •
Place the divisor before the division bracket and place the dividend (416) under it.
• • • • •
7)416 Examine the first digit of the dividend(4). It is smaller than 7 so can't be divided by 7 to produce a whole number. Next take
the first two digits of the dividend (41) and determine how many 7's it contains. In this case 41 holds five sevens (5*7=35) but not six (6*7=42). Place the 5 above the division bracket. • • • • •
5 7)416 Multiply the 5 by 7 and place the result (35) below the 41 of the dividend.
• • • • • •
5 7)416 35 Draw a line under the 35 and subtract it from 41 (41-35=6). Bring down the 6 from the 416 and place it to the right of the other 6.
• • • • • • •
5 7)416 35 66 Divide 66 by 7 and place that answer above the division bracket to the right of the five.
• • • • • • •
59 7)416 35 66 Multiply the 9 of the quotient by the divisor (7) to get 63 and place this below the 66. Subtract 63 from 66 to give an answer
of 3. The number 3 is called the remainder and indicates that there were three left over. • • • • • • •
59 R 3 7)416 35 66 63 3 Division
Dividing a three digit number by a one digit number (for example 416 ÷ 7) involves several steps. •
Place the divisor before the division bracket and place the dividend (416) under it.
• • • • •
7)416 Examine the first digit of the dividend(4). It is smaller than 7 so can't be divided by 7 to produce a whole number. Next take the first two digits of the dividend (41) and determine how many 7's it contains. In this case 41 holds five sevens (5*7=35) but not six (6*7=42). Place the 5 above the division bracket.
• • • • •
5 7)416 Multiply the 5 by 7 and place the result (35) below the 41 of the dividend.
• •
5
• • • •
7)416 35 Draw a line under the 35 and subtract it from 41 (41-35=6). Bring down the 6 from the 416 and place it to the right of the other 6.
• • • • • • •
5 7)416 35 66 Divide 66 by 7 and place that answer above the division bracket to the right of the five.
• • • • • • •
59 7)416 35 66 Multiply the 9 of the quotient by the divisor (7) to get 63 and place this below the 66. Subtract 63 from 66 to give an answer of 3. The number 3 is called the remainder and indicates that there were three left over.
• • • • • • • •
59 R 3 7)416 35 66 63 3
Division Dividing a three digit number by a one digit number (for example 416 ÷ 7) involves several steps. •
Place the divisor before the division bracket and place the dividend (416) under it.
• • • • •
7)416 Examine the first digit of the dividend(4). It is smaller than 7 so can't be divided by 7 to produce a whole number. Next take the first two digits of the dividend (41) and determine how many 7's it contains. In this case 41 holds five sevens (5*7=35) but not six (6*7=42). Place the 5 above the division bracket.
• • • • •
5 7)416 Multiply the 5 by 7 and place the result (35) below the 41 of the dividend.
• • • • • •
5 7)416 35 Draw a line under the 35 and subtract it from 41 (41-35=6). Bring down the 6 from the 416 and place it to the right of the other 6.
• • • •
5 7)416 35
• • •
66 Divide 66 by 7 and place that answer above the division bracket to the right of the five.
• • • • • • •
59 7)416 35 66 Multiply the 9 of the quotient by the divisor (7) to get 63 and place this below the 66. Subtract 63 from 66 to give an answer of 3. The number 3 is called the remainder and indicates that there were three left over.
• • • • • • • •
59 R 3 7)416 35 66 63 3 Division
Dividing a three digit number by a one digit number (for example 416 ÷ 7) involves several steps. •
Place the divisor before the division bracket and place the dividend (416) under it.
• • • •
7)416
•
Examine the first digit of the dividend(4). It is smaller than 7 so can't be divided by 7 to produce a whole number. Next take the first two digits of the dividend (41) and determine how many 7's it contains. In this case 41 holds five sevens (5*7=35) but not six (6*7=42). Place the 5 above the division bracket.
• • • • •
5 7)416 Multiply the 5 by 7 and place the result (35) below the 41 of the dividend.
• • • • • •
5 7)416 35 Draw a line under the 35 and subtract it from 41 (41-35=6). Bring down the 6 from the 416 and place it to the right of the other 6.
• • • • • • •
5 7)416 35 66 Divide 66 by 7 and place that answer above the division bracket to the right of the five.
• • • • • •
59 7)416 35 66
•
Multiply the 9 of the quotient by the divisor (7) to get 63 and place this below the 66. Subtract 63 from 66 to give an answer of 3. The number 3 is called the remainder and indicates that there were three left over.
• • • • • • • •
59 R 3 7)416 35 66 63 3 Division
Dividing a three digit number by a one digit number (for example 416 ÷ 7) involves several steps. •
Place the divisor before the division bracket and place the dividend (416) under it.
• • • • •
7)416 Examine the first digit of the dividend(4). It is smaller than 7 so can't be divided by 7 to produce a whole number. Next take the first two digits of the dividend (41) and determine how many 7's it contains. In this case 41 holds five sevens (5*7=35) but not six (6*7=42). Place the 5 above the division bracket.
• • • • •
5 7)416 Multiply the 5 by 7 and place the result (35) below the 41 of the dividend.
• • • • • •
5 7)416 35 Draw a line under the 35 and subtract it from 41 (41-35=6). Bring down the 6 from the 416 and place it to the right of the other 6.
• • • • • • •
5 7)416 35 66 Divide 66 by 7 and place that answer above the division bracket to the right of the five.
• • • • • • •
59 7)416 35 66 Multiply the 9 of the quotient by the divisor (7) to get 63 and place this below the 66. Subtract 63 from 66 to give an answer of 3. The number 3 is called the remainder and indicates that there were three left over.
• • • • •
59 R 3 7)416 35 66
• • •
63 3 Division
Dividing a three digit number by a one digit number (for example 416 ÷ 7) involves several steps. •
Place the divisor before the division bracket and place the dividend (416) under it.
• • • • •
7)416 Examine the first digit of the dividend(4). It is smaller than 7 so can't be divided by 7 to produce a whole number. Next take the first two digits of the dividend (41) and determine how many 7's it contains. In this case 41 holds five sevens (5*7=35) but not six (6*7=42). Place the 5 above the division bracket.
• • • • •
5 7)416 Multiply the 5 by 7 and place the result (35) below the 41 of the dividend.
• • • • • •
5 7)416 35 Draw a line under the 35 and subtract it from 41 (41-35=6). Bring down the 6 from the 416 and place it to the right of the other 6.
• • • • • • •
5 7)416 35 66 Divide 66 by 7 and place that answer above the division bracket to the right of the five.
• • • • • • •
59 7)416 35 66 Multiply the 9 of the quotient by the divisor (7) to get 63 and place this below the 66. Subtract 63 from 66 to give an answer of 3. The number 3 is called the remainder and indicates that there were three left over.
• • • • • • • •
59 R 3 7)416 35 66 63 3 Division
Dividing a three digit number by a one digit number (for example 416 ÷ 7) involves several steps.
•
Place the divisor before the division bracket and place the dividend (416) under it.
• • • • •
7)416 Examine the first digit of the dividend(4). It is smaller than 7 so can't be divided by 7 to produce a whole number. Next take the first two digits of the dividend (41) and determine how many 7's it contains. In this case 41 holds five sevens (5*7=35) but not six (6*7=42). Place the 5 above the division bracket.
• • • • •
5 7)416 Multiply the 5 by 7 and place the result (35) below the 41 of the dividend.
• • • • • •
5 7)416 35 Draw a line under the 35 and subtract it from 41 (41-35=6). Bring down the 6 from the 416 and place it to the right of the other 6.
• • • • • • •
5 7)416 35 66 Divide 66 by 7 and place that answer above the division bracket to the right of the five.
• • • • • • •
59 7)416 35 66 Multiply the 9 of the quotient by the divisor (7) to get 63 and place this below the 66. Subtract 63 from 66 to give an answer of 3. The number 3 is called the remainder and indicates that there were three left over.
• • • • • • • •
59 R 3 7)416 35 66 63 3 Division Equations
An equation is a mathematical statement such that the expression on the left side of the equals sign (=) has the same value as the expression on the right side. An example of an equation is 72 ÷ 8 = 9. One of the terms in an equation may not be known and needs to be determined. The unknown term may be represented by a letter such as x. (e.g. x ÷ 8 = 9). The solution of an equation is finding the value of the unknown x. Use the multiplication property of equations to find the value of x. The multiplication property property of equations states that the two sides of an equation remain equal if both sides are multiplied by the same number Example: x÷5=2 x÷5*5=2*5 x ÷ 1 = 10 x = 10
Check the answer by substituting the answer (10) back into the equation. 10 ÷ 5 = 2
Division equations with 2 digit numbers An equation is a mathematical statement such that the expression on the left side of the equals sign (=) has the same value as the expression on the right side. An example of an equation is 132 ÷ 12 = 11. One of the terms in an equation may not be known and needs to be determined. The unknown term may be represented by a letter such as x. (e.g. x ÷ 12 = 11). The solution of an equation is finding the value of the unknown x. Use the multiplication property of equations to find the value of x. The multiplication property property of equations states that the two sides of an equation remain equal if both sides are multiplied by the same number Example: x ÷ 50 = 20 x ÷ 50 * 50 = 20 * 50 x ÷ 1 = 1000 x = 1000 Check the answer by substituting the answer (1000) back into the equation. 1000 ÷ 50 = 20
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Factors A number may be made by multiplying two or more other numbers together. The numbers that are multiplied together are called factors of the final number. All numbers have a factor of one since one multiplied by any number equals that number. All numbers can be divided by themselves to produce the number one. Therefore, we normally ignore one and the number itself as useful factors. The number fifteen can be divided into two factors which are three and five. The number twelve could be divided into two factors which are 6 and 2. Six could be divided into two further factors of 2 and 3. Therefore the factors of twelve are 2, 2, and 3. If twelve was first divided into the factors 3 and 4, the four could be divided into factors of 2 and 2. Therefore the factors of twelve are still 2, 2, and 3.
There are several clues to help determine factors.
• • •
Any even number has a factor of two Any number ending in 5 has a factor of five Any number above 0 that ends with 0 (such as 10, 30, 1200) has factors of two and five.
To determine factors see if one of the above rules apply (ends in 5, 0 or an even number). If none of the rules apply, there still may be factors of 3 or 7 or some other number.
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