# Mental Math Addition and Subtraction

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3 Skill 2: Subtracting 2 from an Even Number i) Finding the preceding one-digit even number: Name a one-digit even number and ask your students to give the preceding number in the sequence. For instance, the number that comes before 4 is 2 and the number that comes before 0 is 8. (Remember: the sequence is circular.) ii) Finding the preceding two-digit number: CASE 1: Numbers that end in 2, 4, 6, or 8 Write a two-digit number that ends in 2, 4, 6, or 8 on the board. Ask students to name the preceding even number. Students should recognize that if a number ends in 2, then the preceding even number ends in 0; if it ends in 4 then the preceding even number ends in 2, etc. For instance, the number 78 has ones digit 8 so the preceding even number has ones digit 6. Name the preceding even number: a) 48 : b) 26 : c) 34 : d) 62 : e) 78 : CASE 2: Numbers that end in 0 Write the number 80 on the board and ask your students to name the preceding even number. Students should recognize that if an even number ends in 0 then the preceding even number ends in 8 (but the ones digit is one less). So the even number that comes before 80 is 78. Name the preceding even number: a) 40 : b) 60 : c) 80 : d) 50 : e) 30 : ii) Subtracting 2 from an even number: Point out to your students that subtracting 2 from an even number is equivalent to finding the preceding even number: Example: 48 2 = 46, 46 2 = 44, etc. Subtract: a) 58-2 = b) 24-2 = c) 36-2 = d) 42-2 = e) 60-2 = A-32 Mental Math Teacher s Guide for AP Book 4.1

7 Mental Math Further Strategies 1. Your students should be able to explain how to use the strategies of rounding the subtrahend (the number you are subtracting) up to the nearest multiple of ten. Examples: a) = b) = c) = Subtrahend Subtrahend rounded to the nearest tens You must add 1 because 20 is 1 greater than 19 You must add 2 because 30 is 2 greater than 28 Practice Questions: a) = 27 + b) = 52 + c) = 76 + d) = 84 + e) = 61 + f) = 42 + NOTE: This strategy works well with numbers that end in 6, 7, 8, or Your students should be able to explain how to subtract by thinking of adding. Examples: Count by ones from 45 to the nearest tens (50) Count from 50 until you reach the first number (62) a) = = 17 b) = = 23 c) = = 56 Practice Questions: The sum of counting up to the nearest ten and the original number is the difference What method did we use here? a) = + = b) = + = c) = + = d) = + = e) = + = f) = + = 3. Your students should be able to explain how to use doubles. Examples: Minuend a) 12 6 = 6 If you add the subtrahend to itself and = 12 b) 8 4 = 4 the sum is equal to the minuend then the subtrahend is the same as the difference Practice Questions: a) 6 3 = b) 10 5 = c) 14 7 = d) 18 9 = e) 16 8 = f) = Same value as minuend Minuend plus itself A-36 Mental Math Teacher s Guide for AP Book 4.1

8 Mental Math Exercises Note to teacher: Teaching the material on these worksheets may take several lessons. Students will need more practice than is provided on these pages. These pages are intended as a test to be given when you are certain your students have learned the materials fully. Skills 1, 2, 3, and 4 1. Name the even number that comes after the number. Answer in the blank provided: a) 32 b) 46 c) 14 d) 92 e) 56 f) 30 g) 84 h) 60 i) 72 j) Name the even number that comes after the number: a) 28 b) 18 c) 78 d) 38 e) Add. Remember: adding 2 to an even number is the same as finding the next even number: a) = b) = c) = d) = e) = f) = g) = h) = i) = j) = k) = l) = 4. Name the even number that comes before the number: a) 38 b) 42 c) 56 d) 72 e) 98 f) 48 g) 16 h) 22 i) 66 j) Name the even number that comes before the number: a) 30 b) 70 c) 60 d) 10 e) Subtract. Remember: subtracting 2 from an even number is the same as finding the preceding even number: a) 46 2 = b) 86 2 = c) 90 2 = d) 14 2 = e) 54 2 = f) 72 2 = g) 12 2 = h) 56 2 = i) 32 2 = j) 40 2 = k) 60 2 = l) 26 2 = 7. Name the odd number that comes after the number: a) 37 b) 51 c) 63 d) 75 e) 17 f) 61 g) 43 h) 81 i) 23 j) Name the odd number that comes after the number: a) 69 b) 29 c) 9 d) 79 e) 59 Mental Math Teacher s Guide for AP Book 4.1 A-37

9 9. Add. REMEMBER: Adding 2 to an odd number is the same as finding the next odd number: a) = b) = c) = d) = e) = f) = g) = h) = i) = j) = k) = l) = 10. Name the odd number that comes before the number: a) 39 b) 43 c) 57 d) 17 e) 99 f) 13 g) 85 h) 79 i) 65 j) Name the odd number that comes before the number: a) 21 b) 41 c) 11 d) 91 e) Subtract. REMEMBER: Subtracting 2 from an odd number is the same as finding the preceding odd number. a) 47 2 = b) 85 2 = c) 91 2 = d) 15 2 = e) 51 2 = f) 73 2 = g) 11 2 = h) 59 2 = i) 31 2 = j) 43 2 = k) 7 2 = l) 25 2 = Skills 5 and Add 3 to the number by adding 2, then adding 1 (EXAMPLE: = ): a) = b) = c) = d) = e) = f) = g) = h) = i) = j) = k) = l) = 14. Subtract 3 from the number by subtracting 2, then subtracting 1 (EXAMPLE: 35 3 = ): a) 46 3 = b) 87 3 = c) 99 3 = d) 14 3 = e) 8 3 = f) 72 3 = g) 12 3 = h) 57 3 = i) 32 3 = j) 40 3 = k) 60 3 = l) 28 3 = 15. Fred has 49 stamps. He gives 2 stamps away. How many stamps does he have left? 16. There are 25 minnows in a tank. Alice adds 3 more to the tank. How many minnows are now in the tank? A-38 Mental Math Teacher s Guide for AP Book 4.1

10 Skills 7 and Add 4 to the number by adding 2 twice (EXAMPLE: = ): a) = b) = c) = d) = e) = f) = g) = h) = i) = j) = k) = l) = 18. Subtract 4 from the number by subtracting 2 twice (EXAMPLE: 26 4 = ): a) 46 4 = b) 86 4 = c) 91 4 = d) 15 4 = e) 53 4 = f) 9 4 = g) 13 4 = h) 57 4 = i) 40 4 = j) 88 4 = k) 69 4 = l) 31 4 = Skills 9 and Add 5 to the number by adding 4, then adding 1 (or add 2 twice, then add 1): a) = b) = c) = d) = e) = f) = g) = h) = i) = j) = k) = l) = 20. Subtract 5 from the number by subtracting 4, then subtracting 1 (or subtract 2 twice, then subtract 1): a) 48 5 = b) 86 5 = c) 55 5 = d) 69 5 = e) 30 5 = f) 13 5 = g) 92 5 = h) 77 5 = i) 45 5 = j) 24 5 = k) 91 5 = l) 8 5 = Skill Add: a) = b) = c) = d) = e) = f) = 22. Add by thinking of the larger number as a sum of two smaller numbers: a) = b) = c) = d) = e) = f) = Mental Math Teacher s Guide for AP Book 4.1 A-39

11 Mental Math Exercises Skills 12, 13, and a) = b) = c) = d) = e) = f) = g) = h) = 24. a) = b) = c) = d) = e) = f) = g) = h) = 25. a) = b) = c) = d) = e) = f) = g) = h) = i) = j) = k) = l) = 26. a) = b) = c) = d) = e) = f) = g) = h) = i) = j) = k) = l) = 27. a) = b) = c) = d) = e) = f) = g) = h) = i) = j) = k) = l) = Skills 15 and a) = b) = c) = d) = e) = f) = g) = h) = 29. a) = b) = c) = d) = e) = f) = g) = h) = Skills 17 and a) = b) = c) = d) = e) = f) = g) = h) = 31. a) = b) = c) = d) = e) = f) = g) = h) 17-8 = A-40 Mental Math Teacher s Guide for AP Book 4.1

15 Mental Math Checklist #1 Student Name Can Add 1 to Any Number Can Subtract 1 from Any Number Can Add 2 to Any Number Can Subtract 2 from Any Number Knows All Pairs that Add to 5 Can Double 1-Digit Numbers A-44 Mental Math Teacher s Guide for AP Book 4.1

16 Mental Math Checklist #2 Student Name Can Add Near Doubles. Example: = Can Add a 1-Digit Number to Any Multiple of 10. Example: = 36 Can Add Any 1-Digit Number to a Number Ending in 9. Example: = = 36 Can Add 1-Digit Numbers by Breaking them Apart into Pairs that Add to 10. Example: = = Can Subtract Any Multiple of 10 from 100. Example: = 60 Mental Math Teacher s Guide for AP Book 4.1 A-45

17 Mental Math Checklist #3 Student Name Can Mentally Make Change from a Dollar. See: AP Book Sheets on Money. Can Mentally Add Any Pair of 1-Digit Numbers. Can Mentally Subtract Any Pair of 1-Digit Numbers. Student Can Multiply and Count by: A-46 Mental Math Teacher s Guide for AP Book 4.1

18 Mental Math How to Learn Your Times Tables in a Week Trying to do math without knowing your times tables is like trying to play the piano without knowing the location of the notes on the keyboard. Your students will have difficulty seeing patterns in sequences and charts, solving proportions, finding equivalent fractions, decimals and percents, solving problems, etc. if they don t know their tables. Using the method below, you can teach your students their tables in a week or so. (If you set aside five or ten minutes a day to work with students who need extra help, the pay-off will be enormous.) There is really no reason for your students not to know their tables! Day 1: Counting by 2s, 3s, 4s, and 5s If you have completed the Introductory Unit Using Fractions you should already know how to count and multiply by 2s, 3s, 4s, and 5s. If you don t know how to count by these numbers you should memorize the hands below: If you know how to count by 2s, 3s, 4s, and 5s, then you can multiply by any combination of these numbers. For instance, to find the product 3 2, count by 2s until you have raised 3 fingers = 6 Day 2: The Nine Times Table The numbers you say when you count by 9s are called the multiples of 9 (zero is also a multiple of 9). The first ten multiples of 9 (after zero) are: 9, 18, 27, 36, 45, 54, 63, 72, 81, 90. What happens when you add the digits of any of these multiples of 9 (Example: or 6 + 3)? The sum is always 9! Here is another useful fact about the nine times table: Multiply 9 by any number between 1 and 10 and look at the tens digit of the product. The tens digit is always one less than the number you multiplied by: 9 4 = = = 18 3 is one less than 4 7 is one less than 8 1 is one less than 2 You can find the product of 9 and any number by using the two facts given above. For instance, to find 9 7, follow these steps: Step 1: 9 7 = 9 7 = 6 Subtract 1 from the number you are multiplying by 9: 7 1 = 6 Now you know the tens digit of the product. Mental Math Teacher s Guide for AP Book 4.1 A-47

19 Note 1. Make sure your students know how to subtract (by counting on their fingers if necessary) before you teach them the trick for the nine times table. 2. Give a test on Step 1 (above) before you move on. Step 2: 9 7 = = 6 3 These two digits add to 9. So the missing digit is 9 6 = 3 (You can do the subtraction on your fingers if necessary). Practise these two steps for all of the products of 9: 9 2, 9 3, 9 4, etc. Day 3: The Eight Times Table There are two patterns in the digits of the 8 times table. Knowing these patterns will help you remember how to count by 8s. Step 1: You can find the ones digit of the first five multiples of 8, by starting at 8 and counting backwards by 2s Step 2: You can find the tens digit of the first five multiples of 8, by starting at 0 and count up by 1s (Of course you don t need to write the 0 in front of the 8 for the product 1 8.) Step 3: You can find the ones digit of the next five multiples of 8 by repeating step 1: Step 4: You can find the remaining tens digits by starting at 4 and count up by 1s Practise writing the multiples of 8 (up to 80) until you have memorized the complete list. Knowing the patterns in the digits of the multiples of 8 will help you memorize the list very quickly. Then you will know how to multiply by 8: 8 6 = 48 Count by eight until you have 6 fingers up: 8, 16, 24, 32, 40, 48. A-48 Mental Math Teacher s Guide for AP Book 4.1

20 Day 4: The Six Times Table If you have learned the eight and nine times tables, then you already know 6 9 and 6 8. And if you know how to multiply by 5 up to 5 5, then you also know how to multiply by 6 up to 6 5! That s because you can always calculate 6 times a number by calculating 5 times the number and then adding the number itself to the result. The pictures below show why this works for 6 4: Plus one more Plus one more = = = = 24 Similarly: 6 2 = ; 6 3 = ; 6 5 = Knowing this, you only need to memorize 2 facts: ONE: 6 6 = 36 TWO: 6 7 = 42 Or, if you know 6 5, you can find 6 6 by calculating Day 5: The Seven Times Table If you have learned the six, eight, and nine times tables, then you already know: 6 7, 8 7, and 9 7. And since you also already know 1 7 = 7, you only need to memorize 5 facts: = = = = = 49 If you are able to memorize your own phone number, then you can easily memorize these 5 facts! NOTE: You can use doubling to help you learn the facts above. 4 is double 2, so 4 7 (= 28) is double 2 7 (= 14). 6 is double 3, so 6 7 (= 42) is double 3 7 (= 21). Try this test every day until you have learned your times tables: = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = Mental Math Teacher s Guide for AP Book 4.1 A-49

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