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Mental Math “PLAN” for Grade 6 Week 0: Quiz each day first thing to pre-test kids to find out what math facts they know. (5 pre-tests, 20 questions each, 5 mins, 012) Week 1: (Harry Boen’s stuff could be done Days 1, 2, & 3 for one strategy and Day 4, 5 & 6 for the next strategy, instead of one per week) Here is our task, to learn one-hundred multiplication facts. Let's set up a game plan to get this accomplished!" At the time this activity is commenced, students should have a solid understanding of the concept of multiplication based on the manipulation of concrete materials using array patterns. A main thrust of this activity is to help bridge the gap from the concrete stage (building arrays) to the symbolic stage (memorization of the facts). Mark off an eleven by eleven grid on a bulletin board using square decimeters. Use a strip of paper to cover up the first two columns and the bottom two rows of the grid as shown below. Place a dot in the square in the lower left-hand corner. The Twos: (Mental Math: Doubling) Because students already know facts like 2 fours make 8, the twos are a good starting point. Prepare a grid to be placed on the overhead. Provide students graph paper with squares the same size as those on the grid. Have them cut out as many different array patterns as they can in which one side is two units long and no more than eighteen total squares are included. Several examples are pictured here. Some interesting discussions can take place as to whether the number of arrays that fit the conditions will be even or odd (odd because of the 2 by 2 array) and what the total number will be(17) As students begin to cut out such array patterns the questions will usually arise as to whether the common shaded arrays above (4 by 2 and 2 by 4 and 1 by 2 and 2 by 1) are the same or different. Discuss the pros and cons of this issue. Decide (temporarily) to consider such pairs as being different...one showing the problem 2 x 4 = 8 and the other that 4 x 2 = 8. To help distinguish between them, have students place a dot in the lower left hand corner of each array. Have students continue cutting out array patterns until all seventeen have been found. Have a student hold up on of the arrays. "What are the dimensions of your array pattern?" (4 x 2) "How many squares in a four by two array pattern?" (8) Say, "Four twos make eight. Four times two equals eight." Have the student place the array pattern on the overhead grid with the dot on the dot, fold back the corner opposite the dot, and write the numeral 8 in that square as you say, "Four twos make eight. Four times two equals eight." Have another student hold up a different array pattern. "What are the dimensions of your array pattern?" (2 x 5) "How many squares in all?" (10) "Five twos make ten. Two times five equals ten." Have the student hold this array pattern on the grid with the dot on the dot. Fold back the corner opposite the dot and record the numeral 10 as you repeat, "Five twos make ten. Two times five equals ten." Continue until all possible arrays have been utilized, then fill in the facts on the permanent board. The finished product appears below on the left. Say, "We have started today to fill in the multiplication tables. Remember the addition tables you had to learn?" Show a picture of the addition table. Return to the bulletin board and say, "We sure have a lot of facts left to learn, don't we? Was there a big idea that helped make it easier to learn the addition tables?" By studying the addition tables, lead the class to a verbalization of the commutative property of addition ... that 3 + 4 = 4 + 3, 8 + 6 = 6 + 8, etc. Show how this property cut the number of addition facts we had to learn almost in half. Ask, "Do you think this idea also works with multiplication?" Show that it does work using array patterns. Have students demonstrate on the grid how the same array pattern could be used to show both 2 x 4 and 4 x 2. Repeat for several other array patterns. Say, "This idea will also cut the number of multiplication facts we have to learn almost in half." Place a large sheet of colored paper over the lower part of the table as shown below. Leave this partially completed bulletin board up for students to use to practice on the twos. Emphasize the fact that they really know facts like 2 x 7 because they know 7 + 7. Be sure to cover the permanent bulletin board when fact mastery work is in progress. Note: All facts should be taught in context, i.e., presented in verbal problems made up by teacher and students alike. Your FIRST QUIZ WILL BE 20 FACTS FROM THE 2s!! Mental Math Test #1 Name:__________________ 1. 2 X 2 = __________ 2. 3 X 2 =____________ 3. 4 X 2 = ___________ 4. 5 X 2 =____________ 5. 6 X 2 =____________ 6. 7 X 2 = ____________ 7. 8 X 2 =____________ 8. 9 X 2 =____________ 9. 10 X 2 =___________ 10. 11 X 2 =__________ 11. 12 X 2 = __________ 12. 13 X 2 = __________ 13. 14 X 2 = _________ 14. 15 X 2 = __________ 15. 16 X 2 = __________ Week 2: The Fives (Mental Math: use the minutes on the clock) Many teachers next work on the fives. This is probably because they can be easily attained by skip-counting. Have students cut out array patterns in which one side is five units long using no more than 45 total squares. Remind the class that we now think of the two arrays below as being the same. Show with several examples how one can be rotated to look like the other. Ask, "How many different arrays can we come up with?" (9) Do not put dots in the corners from this point on. Have a student hold up one of the arrays. "What are the dimensions of your array pattern?" (5 by 6) "How many squares in all?" (30) "What multiplication problem does it show?" (6 times 5 = 30) Place the array pattern on the bulletin board, fold back the appropriate corner, and say, "Six fives make 30 - six times five equals 30." If the student places the array on the grid so the square under the turned down corner is black, have them rotate the array the other way. Continue until all of the fives have been placed on the grid. Various games should be used to make the twos and fives automatic WEEK 2: QUIZ will be on the FIVES and the TWOS!! Mental Math Test #2 Name:______________________ 1. 2 X 2 = _______________ 2. 2 X 3 = _______________ 3. 5 X 3 = _______________ 4. 5 X 2 = _______________ 5. 5 X 5 = _______________ 6. 5 X 7 =_______________ 7. 2 X 4 = _______________ 8. 2 X 16 = _______________ 9. 5 X 8 = _______________ 10. 5 X 9 =_______________ 11. 5 X 1 = _______________ 12. 2 X 17 = _______________ 13. 2 X 5 = _______________ 14. 5 X 6 =_______________ 15. 5 X 10 =_______________ Week 3: The Fours (Mental Math: Double the 2 times tables for the 4 times tables) Next, the fours should be attacked. The fours are attainable by using prior knowledge of the twos and doubling twice. For example, 4 x 7 is twice as much as 2 x 7. This should be shown concretely as pictured below. Since 2 sevens make 14, 4 sevens should be twice as much or 28. This basic strategy can be applied any time one of the factors is even. Although they have already covered 4 x 5, a helpful strategy is to use the perfect squares as below. For some reason, students seem to find the perfect squares easier to remember. Have students cut out arrays for the fours and add them to the multiplication table. Week 3’s quiz will be on the twos, fives, and fours. Mental Math Test #3 Name:_________________ 1. 2 X 18 = _______ 2. 4 X 3 = ________ 3. 5 X 5 = ________ 4. 5 X 9 = ________ 5. 4 X 10 =________ 6. 4 X 6 =_________ 7. 4 X 9 =_________ 8. 2 X 20 = ________ 9. 5 X 20 =_________ 10. 4 X 8 = _________ 11. 5 X 2 = ________ 12. 4 X 2 = ________ 13. 4 X 4 = ________ 14. 4 X 5 =_________ 15. 4 X 30 = ________ Week 4: The Threes (Mental Math: Double and add a set) The threes are also usually introduced using skip counting. However, since skip counting by threes is not as easy for students as is skip counting by twos and fives, some of the thinking strategies below might be utilized. All such situations are free rides, i.e., using what you do know to get at what you do not know. Splitting a Product Into Known Parts This strategy, which has several sub-categories, can be used to get any unknown fact from a known fact using the free ride strategy. As at the right, you might use prior knowledge of 2 x 3 to make sense out of 3 x 3. Provide a 2 by 3 array and add a third column of 3. "If 2 threes are 6, what would 3 threes be? In this example, knowledge of 3 x 5 can help make sense of 3 x 6. "If 5 threes make 15, what would 6 threes be?" This same strategy can be used in reverse by starting with the known fact, in this case 4 x3 and eliminating a column. "If 4 threes make 12, what would 3 threes be?" Here, some students might get at 7 x 3 by thinking, "If 5 threes are 15 and 2 threes are 6, then 7 threes should be 15 + 6 or 21." As in the examples above, one should start the thinking with manipulatives and move on to doing the same thinking mentally. The threes should now be added to the table and included in the various fact games and activities. Each time a new set of facts is added to the chart, be sure to stop and point out how the number of facts left to be learned has decreased. (We are now down to 15.) Week 4s test will be on the fours, fives, threes, and twos. Mental Math Test #4 Name:_____________________ 1. 4 X 3 = ________ 2. 5 X 5 = ________ 3. 3 X 7 =_________ 4. 2 X 7 = _________ 5. 3 X 9 =__________ 6. 3 X 10 = ________ 7. 3 X 6 = ______ 8. 3 X 5 = ________ 9. 3 X 4 = _________ 10. 5 X 7 = ________ 11. 4 X 4 =__________ 12. 4 X 20 =_________ 13. 5 X 30 = ________ 14. 2 X 120 = _______ 15. 2 X 70 = ________ Week 5: The Ones At this point, the ones may be added to the table. Although the ones are very easy, it is not a good idea to include them until the concept of multiplication has been established. Array patterns including the ones should be used to place these products on the table. Now only ten additional facts remain. Table With 3's and 4's Added---------------Table With 1's Added Week 5s test will be the threes, twos, fours, fives, and ones. Mental Math Test #5 Name:______________________ 1. 1 X 31 = _______ 2. 2 X 40 = _______ 3. 3 X 5 = ________ 4. 4 X 8 =________ 5. 5 X 6 = ________ 6. 3 X 7 = ________ 7. 5 X 20 = ________ 8. 4 X 40 = ________ 9. 3 X 70 = ________ 10. 1 X 1000 = _______ 11. 2 X 8 = _______ 12. 3 X 8 = _______ 13. 4 X 9 = _______ 14. 5 X 8 = ______ 15. 1 X 9 = _______ Week 6: In the 9 times table, the digits in the answer add up to 9 (from 1 x 9 to 10 x 9) and the answer is always 1 less than the other factor in the tens place and the missing addend to make 9 for the ones place.) The Nines (Mental Math: Because of some interesting patterns, the remaining nines should be covered at this time. Write the nines that have already been covered on the chalkboard as shown below. Allow students to study this information looking for patterns. Some of the patterns usually quickly discovered include: 1. The sum of the digits of each product is nine. 2. As the ones digit of each product is decreasing by one, the tens digit is increasing by one. 3. The tens digit of each product is one less than the number by which nine is being multiplied. Discovery of and discussion about these patterns should lead students to the finding the products of the remaining nines. The discussion should center around why these patterns exist. Some of the explanations develop from knowing how to multiply by ten (10 x 3 = 30 so 9 x 3 should be in the twenties. 10 x 3 = 30 so 6 x 3 should be 3 less or 27.). Although multiplication by ten has not been discussed, it should follow from place value work (3 tens make 30). Discussion should also lead to a method for determining a fact such as 7 x 9 without having to list all of the combinations. If 7 tens is seventy, then 7 nines should be in the sixties. If the answer is sixty-something and the sum of the digits must be nine, then the product must be sixty-three. Array patterns should be used to add the new nines to the table. Now only six facts remain! Week 6s quiz will be on the nines, fives, fours, threes, twos, and ones. Mental Math Test #6 Name:______________________ 1. 9 X 3 = ____ 2. 5 X 4 = ______ 3. 4 X 4 = _______ 4. 3 X 2 = _______ 5. 2 X 80 = _________ 6. 1 X 500 = ________ 7. 9 X 9 = ________ 8. 9 X 4 = _________ 9. 9 X 1 = __________ 10. 9 X 50 = ___________ 11. 4 X 60 = ___________ 12. 3 X 8 = _________ 13. 2 X 9 = _______ 14. 4 X 9 = ________ 15. 5 X 80 = ______ Week 7: The Perfect Squares Next, the remaining "doubles" should be added to the table (6 x 6, 7 x 7 and 8 x 8). These facts can be covered using the "splitting a product into known parts" and "twice as much as a known fact" strategies discussed earlier. 7x7=? Since 5 x 7 = 35, and 2 x 7 = 14 7 x 7 should = 49 6x6=? Since 5 x 6 = 30, 6 x 6 should be 6 more or 36. 8x8=? Since 4 x 8 = 32, 8 x 8 should be twice that or 64. Any of this type of reasoning is a free ride, using what you do know to solve a problem you don't know. Students should be given the opportunity to make such math connections on their own. Adding the doubles to the table leaves only three facts to be covered! Week 7s test will be on perfect squares, nines, twos, threes, fours, fives, ones. Mental Math Test #7 Name:___________________ 1. 4 X 4 =_______ 2. 5 X 5 = ______ 3. 3 X 3 = _______ 4. 2 X 2 = _______ 5. 1 X 1 = _______ 6. 9 X 9 = _______ 7. 8 X 8 = _______ 8. 7 X 7 = ________ 9. 6 X 6 = _______ 10. 10 X 10 = ________ 11. 11 X 11 = __________ 12. 12 X 12 = ________ 13. 9 X 2 = ________ 14. 3 X 9 = _______ 15. 5 X 1 = _______ Week 8: The Last Three Mental Math: Double the 3 times table for the 6 timetables To fill in the remaining three facts (6 x 7, 6 x 8, and 7 x 8) in addition to "splitting" and "twice as much" strategies used earlier, students can also make use of the doubles. These, of course are all free rides. 6x7=? Since 6 x 6 = 36, so 7 x 6 should 6 more or 42 7x8=? Since 5 x 8 = 40, and 2 x 8 = 16, so 7 x 8 should be 56 6x8=? 3 x 8 = 24, so 6 x 8 should be twice that or 48 Week 8s test will be on the twos, fives, threes, sixes, sevens, eights, nines, ones. Mental Math Test #8 Name:___________________ 1. 1 X 1 = _______ 2. 2 X 5 = _______ 3. 3 X 6 = _______ 4. 4 X 6 = _______ 5. 5 X 9 = _______ 6. 6 X 7 = _______ 7. 7 X 10 = ______ 8. 8 X 8 = _______ 9. 9 X 4 = _______ 10. 10 X 2 = ______ 11. 6 X 3 = _______ 12. 6 X 5 = ______ 13. 7 X 8 = _______ 14. 8 X 4 = _______ 15. 8 X 6 = _______ Week 9: Multiplying by Zero Finally, multiplication with zero as a factor can be covered. Although these facts may actually be dealt with earlier, it is better not to put them on the chart until all other facts have been covered. Now the strips covering the first two columns and rows can be removed to reveal a completed multiplication table. Week 9s test will be on the zeros, ones, twos, threes, fours, fives, sixes, sevens, eights, and nines. Mental Math Test #9 Name:________________ 1. 0 X 99 = __________ 2. 1 X 99 = __________ 3. 2 X 5 = __________ 4. 3 X 8 = __________ 5. 4 X 4 = ________ 6. 5 X 5 = ________ 7. 0 X 9 = ________ 8. 6 X 7 = ________ 9. 7 X 7 = _________ 10. 8 X 10 = __________ 11. 9 X 5 = _________ 12. 1000 X 0 = __________ 13. 8 X 9 = ________ 14. 9 X 1 = ________ 15. 10 X 8 = _______ Week 10: Multiplication by 10, 100, and 1000. Show students by using manips (base-ten) why you can add “trailing zeros” (See Mental Math grade 4) For example: 41 x 100 = 4100 Week 10’s quiz will be on multiplication facts 0-10, 100, and 1000. Mental Math Test #10 Name:_________________ 1. 1 X 100 = _________ 2. 2 X 25 = __________ 3. 3 X 1000 = __________ 4. 75 X 10 = __________ 5. 4 X 80 = __________ 6. 5 X 7 = _________ 7. 2 X 17 = _______ 8. 8 X 9 = ________ 9. 88 X 100 = ________ 10. 9 X 8 = _________ 11. 0 X 67 = __________ 12. 9 X 9 = __________ 13. 12 X 12 = ________ 14. 76 X 1000 = __________ 15. 75 X 2 = __________ Week 11: Multiplication of two single-digit numbers in powers of 10. Break the numbers so to separate the multiples of ten. Then rearrange the factors to show how multiplication can be done easier. Ex. 30 x 400 = [(3 x 10 x 4 x 100) ] = [3 x 4 x 10 x 100] = 12 x 1000 = 12000 After the students see where it comes from, then we can tell them to multiply the digits and place the “trailing zeros”. Week 11’s test will be on multiplication facts 0-10, 100, 1000, and this strategy. Mental Math Test #11 Name:______________ 1. 0 X 990 = _________ 2. 80 X 20 = _________ 3. 30 X 50 = _________ 4. 40 X 40 = _________ 5. 80 X 20 = _________ 6. 40 X 90 = _________ 7. 50 X 70 =_________ 8. 20 X 10 = _________ 9. 60 X 50 = _________ 10. 11 X 100 = _________ 11. 12 X 10 = _________ 12. 1 X 44 = _________ 13. 9 X 4 = _________ 14. 2 X 14 = _________ 15. 70 X 30 = _________ Week 12: Multiplication by 25 Divide by 4. If no remainder add 00 If remainder = 1 add 25 If remainder = 2 add 50 If remainder = 3 add 75 Ex 25 x 36 = (36 ÷ 4) = 9 R 0 place 00 at the end = 900 25 x 27 = (27 ÷ 4) = 6 R 3 place 75 at the end= 675 25 x 33 = (33 ÷ 4) = 8 R 1 place 25 at the end = 825 25 x 42 = (42 ÷ 4) = 10 R 2 place 50 at the end = 1050 Week 12’s test will be on all ‘mental math’ stuff we’ve learned so far and THIS. Mental Math Test #12 Name:_______________ 1. 25 X 4 = ________ 2. 25 X 100 = _________ 3. 25 X 6 = __________ 4. 14 X 10 = ___________ 5. 3 X 7 = __________ 6. 7 X 8 = _______ 7. 8 X 9 = ________ 8. 0 X 100 =_______ 9. 28 X 10 = ________ 10. 9 X 10 = _________ 11. 25 X 8 = ________ 12. 25 X 3 = _________ 13. 30 X 60 = ________ 14. 6 X 6 = _________ 15. 5 X 25 = ____________ Week 13: Double / Halving You double one number and half the other. ** It works well when you can double the 5 to a 10. Ex. 4 x 16 = 2 x 32 = 64 24 x 5 = 12 x 10 = 120 Week 13’s test will be on all strategies learned up to this point, including this one. Mental Math Test #13 Name:______________ 1. 4 X 16 = _______ 2. 24 X 5 = ________ 3. 48 X 5 = ________ 4. 5 X 1000 = _______ 5. 80 X 60 = _________ 6. 4 X 18 = __________ 7. 9 X 9 = _________ 8. 71 X 10 = _______ 9. 25 X 6 = _________ 10. 8 X 4 = ________ 11. 5 X 84 = ________ 12. 2 X 81 = _______ 13. 9 X 7 = _______ 14. 5 X 8 = _______ 15. 50 X 40 = _______ Week 14 Front-end solving When the numbers are easy to multiply with no or very few regroupings. You do the multiplication from left to right. You can add the numbers as you go. Ex. 3124 x 3 = 9000 + 300 + 60 + 12 = 9372 A student should say 9000 …300… 60.. 72 (avoid saying the “plus”) Mental Math Test #14 Name:___________________ 1. 25 X 9 = _______ 2. 4 X 80 = ________ 3. 541 X 3 = _______ 4. 3124 X 3 = ___________ 5. 80 X 20 = ___________ 6. 524 X 5 = __________ 7. 313 X 4 = ___________ 8. 8000 X 0 = ________ 9. 2 X 90 = ________ 10. 24 X 4 = ________ 11. 51 X 0 = _______ 12. 1 X 13 = ______ 13. 5 X 7 = _______ 14. 6 X 8 = _______ 15. 7 X 3 = __________ Week 15 Compensate Strategy Multiply by a multiple of 10 and subtract the extra. Ex. 4 x 29 = Think 4 x 30 = 120 then subtract 4 = 116. Multiply by one- or two-digit numbers ending in 8 or 9 Multiply by a multiple of 10 and compensate. Ex. 29 x 6 = Think 30 x 6 is 180 but this is 1 extra group of 6 so you need to subtract 6. 180 – 6 = 174. Ex. 49 x 24 = Think 50 x 24 or 100 x 12 (Double half) = 1200 but this is an extra group of 24 so you need to take away 24. 1200 – 24 = 1176. Mental Math Test #15 Name:_________________ 1. 0 X 989 = ________ 2. 1 X 87 = ________ 3. 2 X 45 = _________ 4. 3 X 60 = _________ 5. 4 X 29 = _________ 6. 5 X 900 = _________ 7. 310 X 3 = _________ 8. 6 X 9 = _________ 9. 7 X 100 = ________ 10. 80 X 50 = ________ 11. 9 X 701 = _______ 12. 10 X 623 = _______ 13. 3 X 49 = ________ 14. 4 X 50 = _______ 99 X 6 = __________ Week 16 Multiply by 11 Write the first digit and the second digit with a space inbetween. In that space write the sum of the 2 digits. Please note that if the sum of the 2 digits is more than 9, you have to increase the first digit. Ex. 11 x 23 = 2 (2+3) 3 = 253 11 x 48 = 4 (4+8) 8 = 528 The 4 changes to 5 because 4+8 = 12. Show the students why this will always work. Use a vertical multiplication by 11. Mental Math Test #16 Name:______________ 1. 0 X 11 = ______ 2. 1 X 11 = ______ 3. 2 X 91 = _______ 4. 11 X 56 = _______ 5. 11 X 32 = ________ 6. 79 X 4 = _______ 7. 9 X 60 = ________ 8. 5 X 5 = _______ 9. 11 X 99 = _________ 10. 24 X 4 = _________ 11. 8 X 3 = ________ 12. 80 X 60 = _________ 13. 32 X 100 = _________ 14. 25 X 7 = __________ 15. 70 X 40 = _________ Week 17 . Division involving facts Division of a two-digit number by a single digit divisor: Think multiplication! Ex. 56 ÷ 8 = think “ 8 x what number = 56” 75 ÷ 3 = think “3 groups of what number = 75” try: 18 ÷ 6 or 84 ÷ 4 Mental Math Test #17 Name:_____________ 1. 56 ÷ 7 =_________ 2. 72 ÷ 9 = ____________ 3. 8 X 9 = _____________ 4. 4 X 25 = __________ 5. 16 ÷ 4 = ___________ 6. 100 ÷ 4 = __________ 7. 63 ÷ 7 = __________ 8. 75 ÷ 1 = ________ 9. 45 X 4 = __________ 10. 11 X 61 = ________ 11. 569 X 10 =________ 12. 31 X 4 = ________ 13. 301 X 9 = _______ 14. 24 ÷6 = _______ 15. 50 ÷ 5 = _______ Week 18 Divide by 10, 100, 1000 Use base-ten to show the sharing or grouping for division. Students will see that in a two-digit number, the digit in the tens place is the number of rods I can have so if I divide by 10, this digit has to be the whole number and the unit place will be only a part of a rod, therefore creating a decimal. Ex. 45 ÷ 10 = 4 rod and 5 units left = 4.5 567 ÷ 10 = 56 rods and 7 units left = 56.7 1240 ÷ 10 = 124 rods and 0 units left = 124 The same thing happens when you divide by 100. Ex. 394 ÷ 100 = 3 flats and 94 units left = 3.94 2865 ÷ 1000 = 28 flats and 65 units left = 28.65 63 ÷ 100 = no flats and 63 units left = 0.63 After all this practice and seeing the pattern, the students should understand the short cut and why it works. Moving the decimal to the left the same number of spaces as there are zeros in the divisor. Technically and on my paper I say that I move the decimal to the left. In some of our resources you will find that on the place value chart to show the students, they say that you move the entire number to the right. Therefore changing the value of each digit. Example: Hundred 3 Tens Ones 9 4 3 Mental Math Test #18 Name:______________ 1. 72 ÷ 9 = ______ 2. 78 ÷10 = ________ 3. 11 X 33 = _______ 4. 70 X 20 = _______ 5. 3 X 3 = _______ 6. 8900 ÷ 100 = _______ 7. 25 X 7 = _______ 8. 789 ÷10 = _______ 9. 24 X 4 = _______ 10. 23 ÷100 = _______ 11. 88 ÷11 = _______ 12. 8 X 9 = ______ 13. 8X8 = _______ 14. 334 X 2 = _________ 15. 501 X 3 = _______ Tenths 9 Hundredths Thousandths 4 Week 19 Multiply whole numbers by 0.1, 0.01 and 0.001 1. Show the students how these multiplications relate to divisions by 10, 100, and 1000. Use manipulatives and comparison to other examples of the same concept. Example: When you multiply a number by 2, you have 2 of that number. When you multiply a number by 0.1, you have 0.1 (one 1 tenth) of that number. When you divide by 10, you have (one 10 tenth) of that number. 2 x 8 = 16 0.1 x 8 = 0.8 8 8 ÷ 10 = = 0.8 10 2. Use patterns. Mental Math Test #19 Name:________________ 1. 89 X 0.1 = ________ 2. 85 X 0.001 = __________ 3. 7 X 7 = ___________ 4. 3456 X 0.01 = _______ 5. 11 X 88 = __________ 6. 20 X 60 = _________ 7. 602 X 3 = _________ 8. 56 X 0.01 = _________ 9. 61 X 4 = _________ 10. 64 ÷ 8 = _________ 11. 54 ÷ 6 = __________ 12. 895 ÷10 = __________ 13. 3 X 4 = _________ 14. 6 X 7 =_________ 15. 90 X 20 = ________ Week 20 SCO B9: Estimate products and quotients involving whole numbers only, whole numbers and decimals, and decimals only. Facility with basic facts and mental computation skills are required for estimation. Estimation should not be considered an exercise one does only when called upon to do so, but an integral part of doing any computation, whether it is with a calculator or with pencil and paper. Strategies that may be used for estimation include: 1. Rounding: Multiplication - Change the problem to one that is easier to work with mentally by substituting “nicer numbers” of similar magnitude (Multiples of 10). For example: 213 x 48 = 210 x 50 = 10 500 or 213 x 48 = 200 x 50 = 10 000 Mental Math Test #20 Name:______________ 1. 0 X 78956 = _______ 2. 1 X 97 = _______ 3. 2 X 65 = _______ 4. 3 X 3 = _______ 5. 4 X 9 = ________ 6. 5 X 8 = _______ 7. 8 X 9 = ________ 8. 9 X 3 = ________ 9. 10 X 9989 =_________ 10. 4567 ÷ 100 = _________ 11. 34 X 0.1 = __________ 12. 11 X 74 = __________ 13. 25 X 2 = _______ 14. 27 X 4 = ________ 15. 60 X 50 = ______ Week 21: Estimation continued… Division – Change the problem to one that is easier to work with mentally by substituting “nicer numbers” of similar magnitude (Multiples of 10). For example: 789.6 ÷ 89 = think 90 for the divisor and 800 for the dividend. Then use a multiplication with a missing factor to help with estimation: 90 x ? ≈ 800 The estimation would be 9. Mental Math Test #21 Name:________________ 1. 30 X 90 = _________ 2. 77 X 0.01 = ___________ 3. 88 X 0.1 = __________ 4. 81 ÷ 9 = __________ 5. 3 X 7 = __________ 6. 30 X 70 = __________ 7. 123 X 3 = __________ 8. 98765 ÷ 1000 = __________ 9. 20 X 40 = __________ 10. 1000 X 67 = __________ 11. 70 X 4 = _________ 12. 528 X 10 = _________ 13. 11 X 44 = _________ 14. 11 X 33 = _________ 15. 1 X 987 = _________ Week 22: Front-end strategy: Multiplication - Perform operations from left to right. Because these are decimal numbers, the fraction part is not multiplied but estimated. For example: 6.1 x 23 Front: 6 x 20 =120 End: 6 x 3 = 18 For the decimal part, since these are low numbers, we add a bit more and round up the product. Total estimation is 120 + 18 and a little more = 140 Week 12s test will be on estimation, multiplication facts 0-9, and front-end strategy for multiplication. Mental Math Test # 22 Name:______________ 1. 25 X 5 = ________ 2. 11 X 55 = _________ 3. 10 X 94 = __________ 4. 7 X 204 = __________ 5. 8 X 321 = __________ 6. 0 X 65 = _________ 7. 8 X 3 = __________ 8. 76 ÷100 = ________ 9. 3 X 0.01 = ____________ 10. 6 X 702 = __________ 11. 64 X 2 = __________ 12. 54 X 5 = __________ 13. 70 X 30 = __________ 14. 98 X 1000 = __________ 15. 45 X 0.01 = ___________ Week 23: The Front-end strategy for Division Division – You need to do front-end estimation to do paper pencil division problems but before they start, students should have an idea of the quotient. Example: 8 424.53 The first estimation is 8 x ? = 400. They need to know that the quotient will be between 50 and 60 therefore the first digit in the quotient will be 5 tens. Mental Math Test #23 Name:______________________ 1. 0 X 5 = _______ 2. 1 X 23 = ________ 3. 2 X 75 = _______ 4. 3 X 21 = ________ 5. 4 X 14 = _________ 6. 5 X 15 = ________ 7. 6 X 7 = ________ 8. 7 X 81 = _______ 9. 8 X 93 = _________ 10. 9 X 53 =_______ 11. 10 X 88 = ________ 12. 11 X 73 = ________ 13. 14 ÷ 7 = __________ 14. 560 ÷8 = _________ 15. 100 X 34 = ________ Week 24: SCO B10: Divide numbers by 0.1, 0.01, and 0.001 mentally. N.B. The document reminds us that students usually expect the division process to result in a quotient that is smaller than the dividend, and it is important they understand why this is not the case with decimal numbers and fractions. 1. Thinking Fractions: Students need to realize that 0.10 = 0.01 = 1 , 100 and 0.001 = 1 , 10 1 1000 This should lead to an understanding that: 1 25 ÷ 0.10 = 25 ÷ = 25 x 10 = 250 10 Similarly 14 ÷ 0.01 = 14 ÷ 1 100 = 14 x 100 = 1400 The question you should ask to explain the division of fractions or decimal numbers is: “How many groups of one tenth (0.1 1 or ) are there in the dividend?” There will be 10 for every unit 10 therefore increasing the dividend by a factor of 10(ten times the number). This explains the larger quotient and why the decimal will move to the right when you divide by a decimal. Similarly: 56.9 ÷ 0.01 = (56.9 divided into hundredths). When we divide by hundredths it is like multiplying by 100 so the decimal moves 2 places to the right. Therefore 56.9 divided into hundredths = 5690. When we divide by 0.001 (thousandths) we move the decimal 3 places to the right. Example: 76.3 ÷ 0.001 = 76300. Mental Math Test #24 Name:______________ 1. 76.3 ÷ 0.001 = _______ 2. 35 ÷ 0.01 = _______ 3. 7 X 9 = ________ 4. 99 X 4 = ________ 5. 0 X 5678 = ________ 6. 70 X 60 = ________ 7. 45 X 4 = _______ 8. 25 X 6 = _________ 9. 100 ÷ 2 = __________ 10. 785 ÷ 0.1 = __________ 11. 87 X 0.01 = ____________ 12. 332 X 0.1 = ________ 13. 8 X 8 = __________ 14. 1 X 785 = ___________ 15. 122 X 4 = __________ ~Approximately 37 weeks in a school year….last 13 weeks will be review of year. Week 25 SCO A6 Develop and apply divisibility rules for 3,4,6, and 9. Knowledge of the divisibility rules will provide a valuable tool for mental arithmetic and general development of operation sense. The divisibility rules are as follows: A number is divisible by 2 if it is even 3 if the sum of the digits is divisible by 3 4 if the number formed by the last two digits is divisible by 4 5 if it ends in a 5 or a 0 6 if the number is divisible by 3 and even 9 if the sum of the digits is divisible by 9 10 if it ends in a 0 Mental Math Test #25 Name:___________________ 1. 12 divided by 2 = _________ 2. 30 divided by 3 = _________ 3. 25 divided by 5 = _________ 4. 11 X 44 = _________ 5. 25 X 40 = _________ 6. 1200 X 3 = ________ 7. 500 X 20 = ________ 8. 31 X 3 = _______ 9. 500 divided by 2 = _______ 10. 370 divided by 100 = _______ 11. 3700 divided by 10 = _______ 12. 0.81 divided by 0.9 = _______ 13. 7500 divided by 0.1 = _______ 14. 0.42 divided by 0.7 = _______ 15. 0.56 divided by 0.8 = _______ Week 26 N.B. Divisibility tests for 7 and 8 are not as simple as the tests for the other numbers from 1 through 10. Just go ahead and do the division. Example: Try 360 - The number 360 ends in 0, the sum of its digits is 9, and the number formed by its last two digits is 60. The divisibility rules tell you that 2,3,4,5,6,9, and 10 are some of the factors of 360. Try 175: The number 175 ends in 5, the sum of its digits is 13 and the number formed by its last two digits is 75. The divisibility rules tell you that 5 is a factor of 175 and 2,3,4,6,9, and 10 are not. Try this one: There will be 138 people at a party. Can the host fill tables of 5? No, 138 is not divisible by 5. Can the host fill tables of 6? Yes, 138 is divisible by 6. Week 27 Strategies that may be used for estimation include: 1. Rounding: Change the problem to one that is easier to work with mentally by substituting ‘nicer’ numbers (multiples of 10) of similar magnitude; For example, 213 x 48 210 x 50 = 10 500 Week 28 2. Front-end strategy: Perform operations from left to right: For example: $2.39 + $4.56 + $2.97 + $2.28 + $5.78 =? Front 2+4+2+2+5 = $15 End 39 + 56 $1 97 $1 28 + 78 $1 1+1+1 = $3 Total = $15 + $3 = $18.00 Week 29 3. Special number strategy: turn one of the numbers into 1, 10, 100, etc., for easy multiplication or division For example: 324.4 0.97 = ? Since 0.97 is almost 1 then the estimate would be 324 1= 324. Another strategy is to double both the dividend and divisor if dividing by 5 Week 30 4. Clustering strategy: round a quantity of numbers to the same number and multiply the quantity by the rounded number For example: $389.22 + $420.27 + $396.45 + $403.67 + $395.50, think: all of these numbers are about $400.00 so 5 x $400.00 = $2000.00 Week 31 5. Compatible Numbers: Look for number combinations that result in 10, 100, 1000, etc. For example,467+281+241+325, think: 467 + 241 is close to 700 and 281 + 325 is close to 600, 700 + 600 = 1300. Week 32 Use mental math strategies for calculations involving integers and decimal numbers. Many of the strategies for estimation can be used in mentally calculating exact answers. 1. Front-end Strategy: For example, –46 + (-38): Add the tens and ones separately and combine. -40 + (-30) = -70 -6 + (- 8) = -14 -70 +(-14) = -84 Week 33 Compatible Numbers: Analyze the numbers to see if compatible sums resulting in 10, 100, 1000, etc. are present. For example, -28 + 63 + 37 + 33 +(-72) = [-28 + (-72)] + [63 + 37] + 33 = -100 + 100 +33 = 33 Week 34 3. Compatible Factors: Analyze the numbers to see if compatible products resulting in 10, 100, 1000, etc. are present. For example, -8 x 137 x 125 = (-8 x 125) x 137 = -1000 x 137 = -137 000. Week 35 4. Working by Parts: Break a number into two parts and find the missing factor; one (or both) of the parts would be a multiple of 10, 100, 1000, etc. For example, 5472 9, Think: 5472 is 5400 + 72, 5400 = 600 x 9 and 72 = 9 x 8; so, 600 +8 = 608. Week 36 5. Double and Halve: Double one factor and halve the other. (Half as many groups, which are twice as large, result in the same product.) For example: 486 x 500 is the same as 243 x 1000 = 243 000.