Marvelous Math Tricks Transcript

Marvelous Math Tricks Transcript Hello. My name is Susan C. Anthony and this transcript is of my workshop "Marvelous Math Tricks" which will cover lots of fun but practical tricks to help kids get excited about basic calculation and gain confidence in their abilities. The handouts for this workshop are available in portable document format on my web site: www.SusanCAnthony.com. The purpose of this workshop is to share ideas that made a big difference in my students’ attitudes about math. When kids begin to make measurable progress toward objective goals, everyone is encouraged. That positive energy translates into more effort. Both my husband and I spent several years as sixth grade teachers. There was always a problem with kids "knowing" how to do the problems (or thinking they knew), but still making too many careless mistakes. One thing about math, right is right and wrong is wrong, regardless of whether you misunderstand the concept or you just miss one math fact. There’s no such thing as "almost" right. When I moved to a fourth grade classroom from sixth grade, I was surprised that the math texts were almost identical. The sequence of chapters was the same. The sixth grade texts had a few more chapters at the end, but I rarely was able to get to them in sixth grade because most of the kids couldn’t add, subtract, multiply or divide competently! It struck me that perhaps that was the problem. The kids had been taught basic math, but they didn’t learn it to mastery, so the first months of each year had to be devoted to review. Sometimes it almost seemed like they’d had just enough to be immunized, so they were resistant to having it again. But you can’t move forward in math without a strong foundation. In my strong opinion, there is pressure to teach too many concepts too soon in math. Especially if you have primary grade children, take the time to build strong foundations in addition and subtraction, even if a second-grader next door is already being taught multiplication. Your kids will make up for lost time when other kids without a strong foundation begin to flounder. There is a lot of pressure. Once in November, when I was struggling to help my fourth graders master subtraction, I walked past the second grade classroom of a brand new teacher. She was teaching multiplication! I talked with her later that day and told her the problems I was having with prolonged review. She was surprised, and mentioned that she did not really feel the kids were ready for multiplication, but some of the parents were pressuring her. The race to cover more in less time is not based on the best interests of children. In truth, it is based on pride. We all want our kids to be smarter and better than other kids. But pride is a sin. Every child is different. Every child learns in a slightly different way at a slightly different speed. Don’t succumb to pressure to hurry your kids. The pressure comes from everywhere. Here’s an excerpt from an article my mother sent me which was printed in the Denver Post a few years ago: How to tell if your child is behind

The founder of a tutoring center says problems start "way before algebra." If we’re going to require algebra for graduation, shouldn’t we require a child to know the multiplication tables to pass fourth grade?" Samuel says students should master the following skills by the end of each school year to be ready for ninth-grade algebra. •

First grade: addition and subtraction facts through 20.

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Second grade: addition and subtraction of three-digit numbers; counting by 2s and 3s.

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Third grade: multiplication of three- and four-digit numbers; short division; beginning long division.

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Fourth grade: long division; beginning word problems, fractions, decimals and percentages.

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Fifth grade: conversion of percentages to fractions; beginning geometry.

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Sixth grade: word problems, decimals, percentages; introduction to positive and negative numbers, algebraic expressions.

I was not introduced to multiplication until fourth grade, but we mastered it in fourth grade and there was little need for review the following year. Although I went to a very small, impoverished

school in the Colorado Rockies, two of the twenty-five seniors in my class were awarded National Merit scholarships, so going slower didn’t hurt in the long run. The sixth grade teachers at our school once met with several junior high school math teachers. They urged us to focus on teaching the basics in math: addition, subtraction, multiplication and division. If kids have mastered that, they will likely succeed in junior high math. They were spending most of their time on review. The ideas in this workshop are to give you tools to strengthen foundations, to review the basics in an interesting way, and to improve accuracy.

Handouts Just a quick overview of the handouts. We’ll use the first page for practice. If you don’t practice what I show you today, you will forget it quickly. Without practice, you’ll remember only 75% twenty-four hours from now. If you don’t practice within forty-eight hours, your recall will be only 25%. Page two shows the Gelosia method of multiplication, step by step. You can keep this for reference in the future. Same with the last two pages, which explain casting nines, step by step. Krypto Krypto is a card game which encourages kids to think flexibly about math. It’s currently for sale by Dale Seymour. You deal out five cards, then an additional card below. The object is to be first to figure out a way to combine the first five numbers using addition, subtraction, multiplication and division in order to come up with the last number dealt. For example, if the first cards were 6 / 1 / 20 / 7 / 8 and the last card was 5, you could combine them in at least two ways: 6 - 1 = 5 (the 6 and the 1 are used, you proceed with the 5) 20 ÷ 5 = 4 (20 from the original numbers and the 5 from the first step) 8 + 4 = 12 (8 from the original numbers and the 4 from step 2) 12 - 7 = 5 (12 from the previous step and 7 from the original numbers) 8-7=1 6-1=5 20 ÷ 5 = 4 4+1=5

Now try one for yourself: The first five numbers are 1 / 3 / 2 / 8 / 8 and the last is 10.

Place Value It is critically important for kids to understand place value. One of the best ways to teach it is with base 10 blocks. If you have no other math manipulatives, get some of these. I have half-inch cubes to represent units. Rods 1/2 x 1/2 x 5" represent tens. Flats 1/2" x 5" x 5" represent hundreds. I also had ten cubes made, 5" x 5" x 5" to represent thousands (and stacked, ten thousands). The pattern is ones, tens, hundreds, one thousands, ten thousands, hundred thousands, one millions, ten millions, hundred millions, etc. One, ten, hundred, one ten hundred, each with a different "surname" (thousands, millions, billions, etc.). Note that the "ones" are always cubes, but much larger cubes each time. The "tens" are always rods, but much larger rods each time. The "hundreds" are always flats. As soon as kids get the concept, I introduce a blank place value chart and we fill it in together. The next day we review and fill in a new chart. This goes on until the kids are quite good at filling in the values. Then they fill in the chart first and I dictate numbers to be written in the squares below. 356: 3 in the hundreds, 5 in the tens, 6 in the ones. Introduce the concept of zero as a place holder. For 206, 2 in the hundreds, 6 in the ones. There are no tens, so a zero holds that place. Dictate larger and larger numbers, with more and more place holders, and have kids tell you how many thousands, how many tens, etc. Finally, have kids round numbers to the nearest hundred, thousand, etc.

This same idea can be used with a decimal place value chart. When you begin to work with decimals, you redefine the unit cube as the larger block (formerly used to represent thousands). The flats become tenths, the rods become hundredths, the cubes become thousandths. Have the kids fill in a place value chart day after day as before, then dictate. Five and three-tenths. Five and three-hundredths (zero as place holder in the tenths place). Have kids write expanded numeration and round. Teach them to drop the zero at the end of a decimal number (.13 = .130; in most cases .13 is preferable).

Math Facts in Five Minutes a Day Before your kids will be able to do the tricks I’ll soon be teaching you, they have to know their facts. They need to practice day after day after day until these facts are mastered, second nature. Until they don’t have to think, let alone count on their fingers. I finally developed one system that worked well for my kids. The first day they did 100 facts, just zeros and ones, with a goal to complete them with 100% accuracy within 3 minutes. Pretty simple, most kids passed. The next day they did 100 facts, just twos. Many kids needed a couple of days to get their speed to 3 minutes or less. Soon they spread out, and each worked at his/her own level of challenge. The goal was to master addition facts by the end of second grade, subtraction facts by the end of third grade, multiplication by the end of fourth and division by the end of fifth grade. And I mean master. Kids who met that goal, and most did, received a Susan B. Anthony dollar, which could be spent or saved. The beauty of this was that kids saw progress toward their goal almost every day. On their off days, I encouraged them to persist. We all have ups and downs and we can’t let them stop us from achieving our goals. This can be a great lesson in persistence and character.

Adding by Endings This is the only math trick I’ll be showing you that I didn’t learn from my husband. A friend taught this to me when he was 50. He still remembers his second grade teacher teaching it. It’s an easy way to do column addition, one that your kids can master in short order if they know their facts. It may not be easy for you, because you’ve learned another system, but if you teach kids this from the start, they’ll have less trouble learning and they’ll be able to do much bigger problems. I used this with fourth graders who needed lots of review in addition, to make the review more interesting. Here’s a column addition problem: 8,265 1,978 4,307 6,957 8,821 What you do is start at the beginning of the ones column. 5 + 8 = 13. 13 is a two-digit number. You need to keep the 3 in your head and put a light line under the 8 to represent the tens. Proceed with the 3. 3 + 7 = 10. Keep the zero in your head and put a light line under the 7 to represent the 10. 0+7=7 7 + 1 = 8. Write down the 8 and count the lines for the number to carry (2).

What’s great about this is that the kids go in order. They are much less likely to lose their place and forget what they’ve done and what needs to be done in a large problem. Let’s go ahead with the second column: 2 (from the carry) + 6 = 8 8 + 7 = 15 (draw a line) 5+0=5 5 + 5 = 10 (draw a line) 0 + 2 = 2. Write down the two and count the lines, carry 2.

Now the third column. Watch what happens when we get to the bottom of this column: 2 (from the carry) + 2 = 4 4 + 9 = 13 3+3=6

6 + 9 = 15 5 + 8 = 13

This is one place kids are likely to make a mistake unless they’re careful. They must put a line under the last 8 in this column, just as they did before. Then count all the lines and carry 3. Your handout has a sample problem. Try it now. 35,263 82,914 74,635 65,033 91,943 28,602 15,355 Click here for the answer. This is a great activity for reviewing addition with older students. I filled a blackboard with a problem and we solved it over a period of days. I handed out a worksheet with several problems in column addition, one of which was huge. Students chose the problems they wanted to do until they were comfortable enough to try the big one. Several of them got it right. I made an overhead of the problem and the principal was proud to honor the students who had successfully done that problem. One year, I was working with a second grade home schooler who "hated math." I offered to show him how to do this. He was reluctant, but an older girl said, "I didn’t think I could do math either, but this is easy. Let me show you." He did a huge problem correctly and came back the next week asking for another one. He felt a new sense of confidence in his mathematical abilities. Even adults in my workshop have had that experience at times!

Gelosia Multiplication This is a method of doing multiplication that actually predates the common algorithm. I don’t recommend that you teach it instead of the regular way, but rather show this to kids once they’re good at regular multiplication. They can do HUGE problems correctly using this method, although it takes a bit longer. First, multiply 48 x 35 the regular way. (Answer is 1,680.) Now, on 1/2" graph paper, draw a quadrangle 2 squares high and 2 squares wide. Put slashes through the squares from bottom left to top right as shown. Then write 48 across the top, with one digit for each square, and 35 down the right side, one digit per square. You are now 1/3 done with the problem. The second step is to multiply. 4 x 3 = 12. Write the answer in the intersection between 4 and 3, with the 1 on top of the slash mark and the 2 under it. Continue multiplying until the grid is filled in. You are now 2/3 done with the problem. The last step is to add diagonally. Put a 0 below the 0 in the right column. Add 4 + 4 + 0 and write 8 under the left column. Then add 2 + 2 + 2 = 6. Write the 6 to the left of the bottom row, and write 1 to the left of the top row. Then copy the digits wrapped from top left to bottom right, 1,680. You got the same answer and the problem is done. Of course, it is a lot easier to do this problem the regular way. Not so if you have a really big multiplication problem. Let’s try 6,292 x 435. This time your quadrangle should be four squares across and three down. The answer will be 2,737,020. Practice by doing the problem on your handout: 4,265 x 921 Click here for the answer.

Casting Nines

I’ve already introduced a couple of important tricks. If math has never been your strong subject and you’re beginning to feel like your brain is stretching, you have my permission to tune me out for the rest of this workshop. If you practice and use the ideas I’ve already introduced, you’ll have some excellent tools you didn’t have before. Casting nines can be confusingly similar to adding by endings. Be sure your kids master adding by endings before you introduce casting nines. I won’t have the opportunity to talk to you again in a few weeks, so I’m going to introduce this now for those of you who still feel fresh and able to learn something new. The step-by-step directions are in your handout, and I have a book which teaches casting nines at a much slower pace, so there’s no need to learn it today. Casting nines is an old-fashioned method of checking math computation. I’ve had grandparents watch me teach this and express surprise that everyone doesn’t know it, because they learned it in elementary school. I don’t know why good methods fall by the wayside in the name of progress, but sometimes they do. There are two parts of casting nines, how you do it and why you do it. When I show people how, they wonder why. When I show them why, they wonder how. So bear with me, we’ll get to both. First, how to cast nines. Add the digits and throw away any nines. For 37, 3 + 7 = 10. Throw away the nine and 1 is left. That’s the result of casting nines. In 25, add 2 + 5 = 7. There are no nines, so 7 is the result of casting the nines. In 81, add 8 + 1 = 9. Throw away the 9 and you get 0, the result of casting the nines. With bigger numbers, find combinations that make nines and throw them away. For example, in 6,241, 6 + 2 + 1 = 9. Throw it away and 4 is the result of casting the nines. Try casting the nines from the numbers on your handout: 47 255 819 3,212 4,531 9,306 38,267 846,787 Click here for the answers. Casting Nines in Addition If you add 871 + 46, you get 917. Cast nines from 871 and get 7. Cast nines from 46 and get 1. Add 7 + 1 to get 8. This is your check number. Cast nines from your sum (917). The result is 8. It matches the check number.

If the numbers match, the problem is correct (almost all the time, there are a few exceptions if kids make mistakes in two columns that balance each other). Try the problem on your handout: 468 + 327 = 795. Click here to see how you did. Casting Nines in Multiplication The only difference with multiplication is you multiply the numbers left after you cast nines from multiplier and multiplicand instead of adding. For example, 68 x 23. Cast nines from 68 by adding the digits (14). Then add the digits again (5). Note: adding 1 + 4 gives the same result as subtracting 14 - 9. Cast nines from 23 (5). Multiply 5 x 5 = 25. Cast nines to get 7. Then cast nines from the product, 1,564. 5 + 4 = 9, throw it away. 1 + 6 = 7. It matches the check number and is correct.

Try the problem on your handout: 833 x 532 = 443,156. Click here to see how you did. Casting Nines in Subtraction It’s pretty much the same with subtraction, except in some cases you will not be able to subtract. For example: 3,942 - 663 = 3,279 Cast nines from 3,942. The result is 0. Cast nines from 663. The result is 6. Subtract 0 - 6. There’s the problem. To resolve it, add a nine to the minuend. 9 - 6 = 3. 3 is the check number. Then cast nines from the difference, 3,729. The result is 3. It matches the check number and the answer is correct.

Try the two subtraction problems on the handout: 692 - 325 = 367 and 1,243 - 746 = 497. Click here to see how you did. Casting Nines in Division You definitely have my permission to rest your mind at this point. You will rarely need to cast nines in division, but for those of you who are still with me, I want to show you how, since it applies to all functions. It is most useful with very large division problems rather than small ones. For example: 1,434 ÷ 6 = 239. Cast nines from 239. The result is 5. Cast nines from 6. The result is 6. Cast nines from 1,434. The result is 3. To check, use the normal procedure but with the smaller numbers. Normally, you’d multiply 239 x 6 to get 1,434. Now, multiply 5 x 6 = 30, cast nines to get 3. That matches the check number, showing that the answer is correct.

If you have a remainder, you cast nines from all numbers. Multiply the quotient times the divisor, and add the remainder, casting nines every time you get a two-digit number. The final one-digit result should match the check number (the number you got when you cast nines from the dividend). If you wish, try the problem on the handout: 36,225 ÷ 75 = 483. Click here to see how you did. I strongly urge you to choose at least one of these tricks and practice it until you really know it. A good way to learn is to teach it to someone else as soon as you can. By the end of the year, most of my students could accurately do the problems on this math quiz. We’d challenge another class to a contest. Everyone in both classes would do a quiz with ten calculation problems. Scores would be averaged in each class and the class with the highest average would win the contest. The fourth graders challenged sixth graders and won! One fourth grade class in Minnesota challenged eighth graders and won, not because the older kids didn’t know how to calculate, but they didn’t know how to do these big problems systematically, or how to check by casting nines. Even the fourth graders whose scores pulled the class average down got a real boost as far as their confidence and enjoyment of math as a result of this. We offered to share the tricks with the older kids if they were interested. Don’t feel bad if you don’t understand everything I talked about today. If you get one idea from this hour that you actually use, this will have been worthwhile. And you can go to my web page to review this workshop and try more examples if you wish later on, after your brain has had a rest. In closing, I want to urge you to take the time your kids need to build strong foundations, especially in math. It will pay off in spades later on. All my best to you.