Math 103 Lecture 3 page 1

Math 103 Lecture 3 The results of a traditional Math Education: students' weakness in number sense, estimation, and reasoning. For example, students can add up three given prices, but can't choose three items from a menu that total less than $4.00. They can find common denominators to add two fractions, such as 5/6 and 4/5, but don't know just by looking that the sum will be a little less than 2.

Current research suggests that the math curriculum in the elementary school must contain a balance of study in at least four areas: number, statistics or data analysis, geometry, and what we call mathematics of change.

To have "sense" about number means to understand how numerical quantities are constructed and how they relate to each other. Consider a practical example: I have 1/2 cup of flour and need 1 1/4 coups of flour; how much more flour do I need? If I have a good sense of these familiar fractions, their magnitudes, and their relationships to each other and to 1, I would be unlikely to use the traditional subtraction algorithm, (1 1/4 - 1/2) which requires finding common denominators, transforming the mixed number into an improper fraction, then subtracting. Rather, I immediately "see" that if I needed 1 cup of flour, I would need 1/2 cup more, but I need 1/4 cup more than 1, so in fact I need 1/2 cup and 1/4 cup, or 3/4 cup. Here's another example: Suppose we ask you to add 58 and 57 in your head, Some people will try to use the traditional "carrying" algorithm, but people with good number sense are more likely to do it another way: "Well, 50 and 50, that's 100, and 8 and 7 is 15, so it's 115." Others with good number sense might think: "Both numbers are close to 60. I know 60 and 60 is 120, then I subtract the 5 that I added on, and I get 115." People who use these methods can mentally figure this problem quickly and efficiently. Their calculations are based on sound knowledge of the number system, and they typically have good strategies for estimating and for double-checking for accuracy.

Traditionally, students have learned to calculate with fractions and decimals in the elementary grades without necessarily understanding how fractions and decimal numbers represent quantities less than 1. For example, if an athlete scores 5.73, is her performance better or worse than that of someone who scores 5.8? Many students believe that 5.73 is the better score, because 73 is bigger than 8. These same students may be correctly adding, subtracting, and multiplying with such numbers because they have learned memorized procedures. They are not able to apply their knowledge, because they have little idea about what these numbers mean. Finding one's way around the number system and understanding its intricacies is an essential goal of elementary school mathematics.

Third graders' knowledge of numbers in the hundreds are often not deep. They can tell you that the number 342 is composed of "3 hundreds, 4 tens, and 2 ones" but they cannot make much used of what they recite and really don't have an understanding of the number's magnitude or relationship to other numbers. That is, they don't know that 342 is ten less than 352, ten more than 332, or a hundred more than 242. Many students will not know that 342 is between 300 and 350, and cannot tell you easily how far 342 is from 400 or 1000. Deep understanding begins with understanding "anchor points" or "landmark numbers" such as 100, 200, 500, and 1000. These are referents from which we build our knowledge about the number system.

To understand "landmark numbers", focus student work on taking apart and putting together 100 and multiples of 100, using 100's to build other numbers, and how numbers such as 20, 25 and 50 are related to 100 and multiples of 100. This means considerable work with multiplication and division as students answer questions such as, "how many 40's are there in 400? In 800? In 520?"

Math 103 Lecture 3 page 2

Translated into specifics, the following are major goals in the third-grade unit on "landmark" numbers: 1. Develop familiarity with factors of 100 and their relationships to 100 (ex. There are twenty 5's in 100 and five 20's in 100) 2. Use knowledge about factors of 100 to understand the structure of multiples of 100 (ex. If there are four 25's in 100 there are twelve 25's in 300) 3. Develop strategies to solve problems in multiplication and division situations using knowledge of factors and multiples. To reach these goals, activities might include creating 100's charts, building interlocking cubes, or dividing amounts of money.

Numeration Systems A numeration system can be characterized as consisting of a finite set of symbols for certain numbers together with a set of rules governing the use of the symbols. The set of numbers represented by particular symbols is known as the digits of the system. Within each system, combinations of the digits represent larger numbers and are interpreted according to established rules.

Upper elementary children can study other ancient numeration systems with a view to developing a better understanding and appreciation of our own system.

Ancient Number Systems: Babylonian Numeral decimal value  1  10 used place value of repeated groupings of 60 ex:   represents 20•60 + 1 (or 1201)    represents 11•60•60+11•60+1 (or 40,261) 3 2     represents 1•60 +11•60 +11•60+1 (or 256,261)

What decimal number is this?   

Egyptian Numeral description decimal value vertical staff 1 heel bone 10 scroll 100 lotus flower 1000 pointing finger 10,000 polliwog or burbot 100,000 astonished man 1,000,000 4. value of a number was sum of the face values of the numerals 5. numerals written in decreasing order

Mayan Numeral decimal value • 1 5 0 1. numbers written vertically with greatest value on top Math 103 Lecture 3 page 3

2 3 4 2. almost base 20 system: 1, 20, 20 •18, 20 •18, 20 •18, … ex: ••• = 13•20 • = 16•20 •• = 12•1 0•1 271 320

Roman Numeral decimal value i or I 1 v or V 5 x or X 10 l or L 50 c or C 100 d or D 500 m or M 1000

Roman numerals use additive property - if smaller value is to right, add values. ex: CCCXXVIII = 328 and subtractive property - when only one smaller symbol is to left, subtract from numeral on right. ex: ix = 9 CD = 400 A bar over numerals means multiply by 1000. DCLIX

In early days people often counted on a scale of 3, 4, or 12 instead of 10. Because we have 10 toes and 10 fingers, some people used a number scale of 20. The French in early times counted by 20 (vignt). Even now they say “four 20’s” (quatre vignt) for 80, and “four 20’s and 10” (quatre vignt dix) for 90. In the English language, this plan was also used for a long time; as when Lincoln said, “Four score and 10 years ago . . “ the word “score” means 20. Whatever scale we use, we need as many digits as the scale contains. On our scale of 10, we need 10 digits. If we used the scale of 9, we should need 8 digits an would write eight as 10. If we were brought up with such a system, it would be just as easy as our scale of 10 – maybe easier because 8 is more easily divided into fourths and eights than 10.

When people first began to use numbers, they knew only one way to work with them; to count. Little by little, they found out how to add, subtract and multiply; but his was slow work and in some countries special devices were invented to make computation easier, especially with large numbers. The Romans used a counting table, or abacus, in which units, fives, tens and so on were represented by beads which could be moved in grooves. The beads were called calculi, which is the plural of calculus, or pebble. Here we see the origin of the word “calculate.” Since the syllable calc means lime, and marble is a limestone, a calculus was a small piece of marble. The Chinese abacus is called a suan-pan. The Japanese abacus is called a soroban. In Russia, it’s known as the s’choty, in Turkey the coulba, in Amenia the choreb.

- demonstration of abacus -

You have often bought things over a “counter” in a store, but did you know that the “counter” tells part of the story of addition and subtraction? Roman numerals were in common use for nearly 2000 years. It was difficult to write large numbers with them. For example, 98,549 might be written lxxxxviiijMDXLVIIII. Merchants invented an easy way of writing big numbers. They drew lines on a board with spaces between the lines, and used disks to count with. Math 103 Lecture 3 page 4

1000’s 500’s 100’s 50’s 10’s 5’s 1’s

Because these disks were used in counting they were called “counters” and the board was called a “counter board.” When European countries gave up using counters of this kind (about 400 years ago) they called the boards used in the shops and bands “counter” and this name has since been commonly used for the bench on which goods are shown in stores. The expression “counting house” is still used in some places to designate the room in which accounts are kept. Subtraction done on counting boards used the terms “carry” and “borrow” and had more meaning than at present, because a counter was actually lifted up and carried to the next place. If one was borrowed from the next place, it was actually paid back.

The Western world adapted aspects of a numeration system used by the Hindus and one developed by the Arabs, thus the name Hindu-Arabic numeration system. The Hindu-Arabic numeration system can be described by the following five characteristics: 1. base-ten – the number of objects used in the grouping process. Ten is the number that designates a first grouping. Therefore, there is no symbol for 10 . . the largest number with a symbol is 9, the number which precedes the grouping size. What would be the largest number for Base Five? What are the number symbols for Base Five? What would be the largest number for Base Twelve? What are the number symbols for Base Twelve? What would be the largest number for Base Two? What are the number symbols for Base Two? 2. Positional or place value – means a digit takes on a value determined by the place it occupies in a number. The units digits occupy the place furthest to the right. Because ten is the size of the grouping, ten is the value of the place to the immediate left of the ones place. One more than nine groups of ten and nine (99) requires a regrouping and is represented by the numeral 100. Place value enables one to distinguish between the face value of a digit (5 as five) and its value because of its particular position in a numeral (the digit 5 has the value of fifty in 52 and five hundred in 543) 3. Multiplicative principle – is used in decoding the value of each digit in a numeral. For example, in the base-ten numeral 152, the value of 1 is 1x100, and the value of 5 is 5x10, and the value of 2 is 2x1. When children first decode numbers, they are not likely to use the multiplication tern “times.” Rather, they talk about “groups,” as 1 group of hundred or ten tens, and 5 groups of ten, and 2 ones. 4. Additive principle – means the numbers are the sum of the products of each digit and its place value in a numeral. For example, 765 = 7x100) + (6x10) + (5x1) In a numeration system without a multiplicative principal, the value of a number is determined by the sum of the digits in a numeral. For example, in Roman numeration, the number XXVII is determined by adding 10 + 10 + 5 + 1 + 1 to make 27. When they have learned multiplication, children can describe numbers as, for example, 36 means “3 times 10 plus 6.”

Math 103 Lecture 3 page 5

5. zero as a place holder - is the genius of our numeration system – the combination of place value and placeholder symbol, that is, a symbol for the number zero. Have children think of zero as a placeholder. Ex: “I have 8 hundreds and 3 tens. What’s my number?” “I have 3 hundreds and 6 ones. What’s my number?” “I have 1 thousand and 6 ones. What’s my number?” “I have 5 ones and 9 hundreds. What’s my number?” Counting in Base 5: 0, 1, 2, 3, 4, 10, 11, 12, 13, 14, 20, 21, 22, 23, 24, 30, 31, 32, 33, 34, 40, 41, 42, 43, 44, 100, 101, 102, 103, 104, 110, 111, 112, 113, 114, 120, 121, 122, 123, 124, 130, 131, 132, 133, 134, 140, 141, 142, 143, 144, 200, …

Change 324 to base 10 5 How would you write 76 in base 5?

There is evidence from research that elementary-school children do not understand our numeration system. It is recommended that teachers have Grade 1 children engage in grouping activities as foundational experiences for place-value development. Generally, Grade 2 children are expected to develop understanding of two-digit numbers and Grade 3 children work with numbers greater than 99.

When it has been determined that children understand two-digit numbers, the progression to three- digit numbers should be a smooth one.

Five-stage development of place value understanding: 1. Children associate two-digit numerals with the quantity they represent. 2. Children identify the positional names but do not necessarily know what each digit represents. 3. Children can identify the face value of digits in a numeral. 4. A transitional stage when true understanding of place value is constructed. 5. The level of understanding the structure of our numeration system. The child knows that digits in a two-digit numeral represent a partitioning of the whole quantity into tens and ones and that the number represented is the sum of the parts.

Consolidating and reinforcing number skills: - Chisanbop handout - Counting on fingers

WIPE OUT (4th or 5th grade) Direct children to enter a six-digit number on their calculator and ask them to "wipe out" a specified digit. Ex. Enter 345 876. Wipe out the digit 8.

DIE: Players try to create the largest possible multi-digit number using the digits generated by repeatedly tossing a die or drawing a card from a set of cards 0 - 9. After each digit is revealed, players must write it in one of the place-vale positions and cannot later change a position.

Counting in a Foreign Language Make Calendar writing the numbers in a base other than 10. “nines” the easy way

Math 103 Lecture 3 page 6

MENTAL MATH is the process of producing an exact answer to a computation without using external computational aids. Addition 1. Standard Right-to-Left Renaming Algorithm

2. Left-Right Decomposition Strategy

3. Partial Decomposition Strategy

Multiplication 1. Using Area Model

2. Front-End Multiplying

3. Compatible Number

4. Thinking Money

5. Compensating with 8’s and 9’s

6. Compensation for Nice Multiples

39 + 57 = 96 What happens if 1 is added to 39 and 1 is subtracted from 57? 39 x 57 = 2223 What happens if 1 is added to 39 and 1 is subtracted from 57? 57 – 39 = 18 What happens is 1 is added to 57 and 1 is subtracted from 39? What happens if 1 is subtracted from 57 and 1 added to 39? 108 + 48 = 156 Would adding 2 to 108 change the sum as much as adding 2 to 48? 28 x 58 = 1624 Would adding 2 to 28 change the product as much as adding 2 to 58?

Division 1. Breaking up the Dividend

2. Compatible Numbers

3. Shortcuts involving zeros

- examples from UpClose notes - Math 103 Lecture 3 page 7

COMPUTATIONAL EXTIMATION is the process of producing an approximate answer to a numerical problem. Addition 1. Front-End

2. Grouping to nice numbers

3. Clustering

4. Range Strategy

5. Rounding Strategy

Multiplication & Division 1. Front-End

2. Compatible number

In each of the following a student has used a calculator to determine the answer. Estimate whether the answer is plausible. If the answer is implausible, determine how the miscalculation occurred.

1. 900 eggs put into cartons of a dozen each. How many cartons? Answer: 0.013333

2. Item cost for remodeling project were $1,027.99, $396.00, $2,333.67, $27.95 for a total of $6,552.66.

3. A tally of a card game scores of 20, 50, 30, 90, 100 points produced a total of 380 points.

4. Total of 175 years for a football running back who gained 12 yards, 8 yards, 15 yards, and 5 yards on four runs.

What’s Appropriate? Pencil & paper algorithm Calculator math Mental computation Estimation

- examples from UpClose notes -