How Many Ways
How Many Ways TEACHING NUMBER SENSE AND PLACE VALUE How Many W ays
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Power of Ten
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How Many Ways
Teaching Notes
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Power of Ten
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TEA CHING NUMBER SENSE EACHING - THR OUGH HOW MANY WAYS HROUGH The wonder of mathematics is that
AND AND
PLA CE VAL UE LACE ALUE CALEND AR TIME ALENDAR
In grade-one, or on occasion even in kindergarten, ask students: “How many groups of
mathematical solutions may be reached in numerous and varied ways. For primary and early intermediate children, calendar time presents endless opportunities to make meaning of mathematics. Creative calendar instruction yields useful problem-solving strategies while
5 are there?” Students will no doubt respond with: “There are 3 groups of 5”, whereupon you can then rewrite the equation as 3 × 5 + 4. This may seem much too difficult for kindergarten students, but the
it infuses fun into the teaching of mathematics.
equation is easily demonstrated where 19 cubes are placed into Power of Ten egg cartons.
I had occasion to visit Wendy Payne’s kindergarten class in Victoria, B. C. It was June 19th
(Each Power of Ten egg carton has only 10 receptacles available, as 2 have previously been removed from the end of the carton).
and what follows are examples of the equations I saw printed on her blackboard. Wendy had developed a routine in which the students counted the number of the date using tallies of five. Wendy used four lines with a slash through them to indicate a group of 5. This is an efficient method of grouping, and it is very useful in a graphing situation. However, I suggest, instead, that students draw a circle around five lines. My reasoning should become clear later. The equation is then further reinforced with Power of Ten cards.
The equation generated was: 1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1
This is the most common equation generated in classes. It is a useful equation, as it shows the relationship of “one more”. However, the equation will read far more easily if items are grouped into sets of 5. Brackets are then employed in a context of meaning. The teacher follows by asking how the equation relates to the tally. (1+1+1+1+1) + (1+1+1+1+1) + ( 1+1+1+1+1 ) +1+1+1+1
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Power of Ten
The teacher models what the children say regarding the equation, and then demonstrates a mathematical way of saying it. Attempting this on the 19th of a month may in some cases prove challenging. However, the 10th, 20th, 30th or the 8th of each month would present profitable opportunities for establishing groups of 2 or 4.
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In kindergarten (perhaps somewhere in April) introduce the word “even”. Inform students
Early in the year begin looking for ways to “make equal shares” for two people and then follow by trying to discover which dates present opportunities to make those equal shares. In the late fall, on November the 10th or 20th, relate “groups of” to multiplication using Power of Ten cards, pointing out that the short way to write “two groups of five is 10” is: 2 × 5 = 10.
that when something can be made into equal shares with “nothing left over”, we say the number is even. Several days, or perhaps a few weeks later, one student will invariably ask how one describes days that are not even. I usually enjoy replying: “Oh, haven’t I told you that yet? That’s odd! Well, of course, we call the numbers that can’t be made into two equal shares odd!”
Avoid using the word “times”, as it has no real meaning. However, as students often choose to employ this term, accept its use without
However, let us return to the remaining
quibbling, while nevertheless modeling the words “groups of” in class. Where students do employ the
number expressions that students in Wendy Payne’s class suggested on June 19th. The following
term “times”, ask students to “say it with meaning”, in which case you should expect to
demonstrate the characteristic of numbers called the commutative property.
hear: “groups of”. You may be surprised to discover that young students soon grasp the concept of grouping. This then provides you with an exciting new vehicle for exploring the date. The learning is in no way threatening or confusing for students, as ideas are always presented in a rich context
10 + 9
9 + 10
Children comprehend this property quite readily, but it is important that we ask students to express a verbal understanding of the property, or to “say how you know it is true”. Children who
of meaning. At some point introduce the related division equation: (10 ÷ 2 = 5). It is often astounding how
knows this rule probably already know how to conserve numbers, and they are now ready for practice in formal addition.
quickly some students begin employing the division notation, and how soon some are even able to
The following examples demonstrate that
come up with further division equations independently. The fraction format showing division
students soon learn how to “break up” numbers.
should be used occasionally, as this is the only format children will see in high school. The
“breaking up” numbers in the Problem Solving
Japanese teach division very early in the primary years, as they understand that young
such as those that follow, it is important to ask how students know their solutions are true. Often
children are able to relate the division operation to daily experience early in life. Very young children have no trouble making equal shares of 2,
children will refer to the tallies above. Where truncated egg cartons and Power of Ten cards
Power of Ten
Note: Look for further strategies regarding section of this manual. When generating equations
4, 6, 8 and 10 and should therefore be taught the notation to describe these operations.
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and
have been employed, most of the solutions to these equations are obvious, as they have been clearly demonstrated visually.
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A teacher can reinforce both the relationship
5+5+9 10 + 6 + 3 5+5+5+4
to the tally and the more important relationship to the “tenness” of the number system by placing brackets around the fives or the tens that have been broken up. The abacus and the Power of Ten cards are based on five for a good reason. For example:
4+5+5+5 (3 + 2) + (3 + 2) + (3 + 2) + 3 + 1
3+2+3+2+3+2+3+1 6 + 3 + (5 + 5)
6+3+5+5
If your classroom is equipped with Power of Ten cards (see the Appendix for a master) it is profitable to have children relate the numbers and the equations generated
to the corresponding visual representations shown on the cards. This presents a wonderful opportunity to teach place value. Note: This activity also reinforces the brain principles of learning already examined in the introduction to this manual.
Review number recognition by flashing the Power of Ten cards. My preferred order is: 1, 2, 10, 9, repeat the 10; then 8, 5, 6, 4; repeat the 5; then 7 and 3. Then ask students to show the number of colored squares shown on each card with corresponding fingers. Discourage yelling out. Ask students “How many fingers do you have up?” Then ask: “How many fingers do you have down?” The two answers given correspond to what you should then refer to as the friendly numbers. Watch for children who continue to rely on counting, as this demonstrates that these students need further practice and as yet are unable to conserve numbers. As you study each number, ask students to place corresponding big cards on the blackboard ledge in the correct order. This draws a parallel to the number line and gives you an opportunity to talk about right, left, middle, more, and less.
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The next step is to tackle the date. In the original example studied, the date is June 19th, and students are engaged in “making 19”. To accomplish this we need a ten card and a nine card. Hold up the cards and ask two students to
First we added 1+1
Then we added 3, giving us a total of 7 in the seven shape I know so well !
Then we added 2
x
stand at the front of the classroom and hold up 19 fingers: one student holds up ten fingers and the
x
other nine fingers. Follow by asking students how one might “make 29”. Ask one more student to
x
stand at the front of the classroom holding up ten fingers as she stands in the tens column. Then ask students how one may “make 39”. Yet another student stands in the tens column at the front of the classroom. In each case, model specific numbers by indicating the visual representations shown on the cards as well. Continue in this way until you sense that students are beginning to lose interest. You have now modeled the following: • The tenness of the number system.
• Adding ten to a given number. • Counting by tens starting at a given number. At this point it is profitable to make a connection with the Hundreds Chart, which you probably have displayed on a class bulletin board. Begin with 19 and repeat: “nineteen, twenty-nine, thirty-nine,” etc. It is essential that you make these different connections, as students who fail to understand a concept in one context may well understand it in another.
Always make connections between given
x
x
x
x
x
x
x
x
x
x
The question was “How many do we still need to make 10?” She readily replied with: “Three.” Note that this is another example of friendly numbers. The next question was, “How many do we need to make 20?”. Again she readily replied with: “Ten.” The equation now stood at: 1 + 1 + 2 + 3 + 3 + 10 = 20. It was the shape of the number that presented this student with a surefire method for “getting out of her fix”. It helped her to get “unstuck”. Getting “unstuck” is a major skill in problem solving. In this case the student learned to strategize by using a diagram. It would also have proved productive to have asked this student to make further connections by reviewing the use of tallies when “making 20”. Many teachers feel that multiple methods confuse students. Initially, this may be so. However, students make their own meanings in
numbers and the Power of Ten shapes whenever your students are adding. On June 20th a student
their own unique ways, whether we like it or not. Memory is inevitably linked to previous experience,
was struggling to “make 20”. She began with “1 + 1 + 2 + 3” before announcing that she was
and every student brings different experiences to the learning situation. Employing several different
“stuck”. We then worked together and started to fill the ten-frames as follows:
methods is advantageous, as students eventually understand numbers in a variety of ways, and they then soon begin developing their own personal preferences.
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Recently I visited a grade one/two class where the teacher, Sandra Glassel, had the courage to introduce division and multiplication algorithms. It was February 5th, and students were therefore studying the number 5. This is a sample of some of the more complex mathematical sentences the children brainstormed.
100 ÷ 10 + 5 – 10 = 5 5+5–5=5 25 ÷ 5 = 5 5÷1=5 20 – 20 + 5 = 5 30 – 25 = 5 The patterns discussed previously can
45 ÷ 9 = 5
readily be built on and extended in a grade-one class. Many teachers of children in grade-one hesitate to introduce the formal operations of multiplication and division, as they feel these
One year ago, Sandra did not believe that
operations are confusing at this early level. Exactly the opposite is true. There is no expectation that grade-one children must learn these operations. However, children exposed to these operations are immersed in meaning and exposed to higher-order thinking. Some children will learn the operations (multiplication and division) and then model them for others. This is invariably demonstrated at
grade-one students could produce such complex equations, but having had the courage to try generating such equations, she was astounded at the level of her students’ abilities. Sandra’s students now had many effective tools available to them when thinking about numbers and making equations. More importantly, the students felt empowered.
calendar time, when children attempt to find multiple different ways to represent a specific date.
Furthermore, Sandra’s students demonstrated an impressive understanding of the power of one (5 ÷ 1) and the mystery of zero (100 ÷ 10 - 10; 5 - 5; 20 - 20). Students who fully grasp the properties of one and zero have already established two important frameworks for the later understanding of algebraic equations. Most algebraic equation solving involves:
• Using the properties of zero to eliminate terms. • Creating ones to eliminate coefficients of terms. • Balancing each side of an equation (performing the same operation on each side of an equation, or on both parts of a term).
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This sense of student empowerment is best illustrated by two related stories. Kara was
By the end of the year, Kara was unstoppable, and on June 14th she presented
an excellent math student, and she was fortunate enough to have Mary Nall as her teacher. Mary
the equation 1,000,000 - 999, 986 = 14.
used the calendar in grade-one and was often impressed with the skill Kara demonstrated at
Understandably, Kara was henceforth dispatched to the principal’s office to show off her skills. I was that lucky principal. And I have been talking about it ever since…
calendar time. Early in the year, Kara was making patterns, and on October 6th she
The following year, Kara was tackling square roots. Kara had mathematical ability
produced the following pattern:
and supportive parents who encouraged her mathematical endeavors. Several other students
1+5=6
that year began to model Kara’a example. Later in the year, on May 25th , Robby burst through the front
2+4 = 6 First I made
3+3 = 6
this pattern.
4+2 = 6
door of the school beaming with excitement. As the school principal, I often greeted students at the front
5+1 = 6
door each morning in order to establish positive contact with students each day. I asked Robby what
6+0 = 6
he was smiling about.
Robby was beside himself: “It’s the twenty-fifth!” he announced excitedly. When I ran out of additions, I switched to
I was puzzled and asked, “What’s so important about the twenty-fifth?” Robby wasn’t impressed with me at all. “Because,” he replied scathingly,
7-1=6
subtractions!
8-2 = 6
And I can keep on going!
9-3 = 6
“it’s the square root of six hundred and twenty-five, of course!”
I was delighted, and I followed him to his classroom where Janine Roy and the rest
Kara continued until she reached 23 - 17 = 6, whereupon she made a huge leap in her ability to generate complex equations and
of her fortunate students were about to enjoy the daily fun of
produced: 123 - 117 = 6.
calendar time. Clearly, Janine had followed up on Mary Nall’s previous good work.
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When assessing student progress, note
These are only a few of the memorable stories I could tell that illustrate not only the
the quality and the complexity of the thinking
power of teaching calendar, but the enormous advantage to remaining open when students
shown. Look for the following: • Patterns • Equations using the four main operations individually • Equations with more than one operation in the number sentence • Equations involving large numbers • Equations that illustrate the properties of zero and one
proffer a wide variety of potential equations. Brainstorming provides an effective tool at calendar time. However, it is also important to ask students to write and record all equations generated in class at their desks. The 10th, 20th or 30th of each month are profitable dates to attempt individual brainstorming activities. In the intermediate grades you may wish instead to focus on the 50th or 100th day of the year. The examples that follow illustrate the advanced thinking eventually demonstrated by some students during calendar time.
Equations showing each or all of the above clearly demonstrate developing numeracy, and they provide wonderfully fertile grounds for discussion during later parent-teacher conferences. (See this section, pages 20-21, for a complete set of criteria.)
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TEA CHING NOTES EACHING
HOW MANY WAYS (CALEND AR) SECTION ALENDAR FOR THE
Calendar Time is of particular value where teachers remain attuned to potentially teachable moments, mathematical connections and inherent investigations occurring throughout. A list of connections is provided below. As you proceed through the months of the year, note which of the connections or teachable moments you have successfully employed. However, resist the temptation to make all these connections at once, unless you are either approaching the end of year one, or you are working with a mid-to-late primary grade level.
Topics and Connections:
Problem solving: making generalizations, predicting, reasoning, patterning
Communication: explaining how to get
Suggestions and Connections for Calendar Time: Note: The number at the beginning of each paragraph refers to the day of the month or year.
an answer
Numeracy: adding, subtracting, multiplying, dividing, regrouping, factoring
1
A number divided by itself is 1. Any two adjacent numbers on a number line or a ruler are only 1 apart. This is also true on a hundreds chart.
Brain and Learning Principles Emphasized:
2 This is how many people share even numbers fairly.
Learning is both a social and an individual process (brainstorming and working individually).
The search for meaning occurs through patterning.
Emotions are critical to patterning. (Brainstorming in a cooperative group is non-threatening.)
The brain processes parts and wholes simultaneously.
The natural spatial memory is emphasized when connections are made.
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directly, as students must discover this concept themselves through appropriate questioning. Numbers on a number line and a hundreds chart may be shown to be two apart. When you color in a hundreds chart counting by twos, you color in every other square, or half the numbers. Two is a rectangular number (using multilink cubes you can only make two rectangular prisms: 1 × 2 or 2 × 1, which are equivalent). In grade four discuss that 2 is a prime number. However, wait until
Learning is enhanced by challenge.
Power of Ten
14 etc. ) 7 However, resist teaching this concept (4÷2, 6÷3, 8÷4, 10÷5, 12÷6,
Learning is active.
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you reach the 13th of any month before you ask: “What is true about 2, 3, 5, 7, 11,
4 This is both a rectangular (1 × 4) and a square (2 × 2) number.
and 13?” You can then demonstrate that these are all rectangular numbers and can
Coloring the shapes on graph paper will demonstrate this. Four can be shared fairly
only be “made one way”. Thus, prime numbers have only two factors. When you
by two people, and it is therefore even. 8 (8 ÷ 2 = 4, or = 2) 4 Four is the number of vertices on the base
organize the Power of Ten cards for the game Concentration, you can organize 20
of the square pattern block (orange), the trapezoid pattern block (red), the rhombus
cards into 2 rows, which presents a division sentence with a quotient of 2: (20 ÷ 10 = 2).
pattern (tan, sometimes blue, sometimes described as diamond) blocks. All these
The game Friendly Concentration presents 18 ÷ 9 = 2.
blocks are quadrilaterals. Eventually consider describing these two-dimensional
3 This is a triangular number, which is best
shapes as square quadrilateral or square rectangle, trapezoid quadrilateral, rhombus
illustrated by putting two pennies side by side and setting the third penny above the initial two to create a triangular shape.
quadrilateral, diamond quadrilateral, etc. Evidence shows that these descriptors
Counting by threes on a hundreds chart illustrates that the digits of the numbers you
assist students to understand the hierarchical nature of quadrilaterals as
are counting all add to 3, 6 or 9.
shapes. Some figures have two names.
For example: 12 (1 + 2 = 3), 15 (1 + 5 = 6),
(Clements and Sarama, 2000)
36 (3 + 6 = 9). However, 39 and 48 are tricky:
5 This cannot be shared equally, and it is
(3 + 9 or 4 + 8 = 12, and then 1 + 2 = 3). Thus the groundwork for the divisibility rule for three is established with students. Three
therefore not even. Do not introduce the term odd for at least two weeks. Five is
is a prime number. However, do not introduce this terminology before grade-
rectangular one way only, and therefore it is a prime number. Five is the number of
three or four. Late in the grade-one year, use Friendly Concentration to establish a
fingers on one hand and the number of toes on one foot. Five is the basis for an abacus.
division sentence (18 ÷ 6 = 3). The following picture shows all the triangular numbers you
It is a pentagonal shape. A pentagon has five vertices. This is the shape shown on the Chrysler logo and the shape of some stars.
will encounter in a month
A nickel is worth 5 pennies. Use the game Concentration to establish a division
(3, 6, 10, 15, 21, 28).
sentence (20 ÷ 4 = 5 or 20 grouped into
(3)
(15)
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Power of Ten
(6)
(21)
4 rows yields 5 in each row).
(10)
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8 Rectangular four ways (1 × 8, 2 × 4 and
6 Even, triangular (3 pennies sit on the bottom row, 2 sit on the second row, 1 sits on the top). Six is also rectangular
the reversals of each). Eight can be shared by two and therefore it is even (8 ÷ 4 = 2 or
in four ways (1 × 6, 2 × 3, and the reversals of each). Investigate these
8 ÷ 2 = 4). Ask students to make 8 into a cube using multilink cubes. Eight is the
multiplication sentences using the language of groups: one group of six
number of squares in one row or column of a chess board.
equals six, and two groups of three equals six. Coloring the shapes on graph paper easily demonstrates these groupings. Six is also hexagonal and corresponds to the number of vertices on the base of the yellow hexagonal-shaped pattern block. Many stars have six vertices. Use the game Friendly Concentration to establish a division sentence (18 cards organized into 3 rows with 6 in each row, 18 ÷ 3 = 6).
9 Rectangular and square (1 × 9, 3 × 3). When counting by nines, the sum of the digits in each multiple always adds to 9 (18 yields 1 + 8 = 9; 27 yields 2 + 7 = 9). This operation is somewhat similar to the addition of digits in the multiples of three, although more powerful. See Friendly Concentration for a division sentence: 18 ÷ 2 = 9. Some magic squares have nine frames. Ask students: Can you arrange the numbers from 1 to 9 in a 3-by-3 magic square, so that the sum is 15 in all horizontal, vertical and diagonal rows?
7 Rectangular two ways only; therefore 7 is prime. Note that numbers on the pages of calendars are printed in such a way that the differences between two vertically adjacent numbers always equals 7.
6
7
2
1
5
9
8
3
4
For example: 21 will appear directly below 14 on most calendars, and 14 will appear directly below 7.
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10 Rectangular two ways, and triangular
14 Rectangular four ways. Fourteen may be
(using pennies build in rows of 4 + 3 + 2 + 1). Ten is the basic shape for ten frames and
shared equally, and it yields two division sentences. It is the number of days in two weeks. Fourteen is the difference between
the Power of Ten cards. Ten is the sum of all the friendly numbers, which may be
any two numbers in alternate rows on a calendar. Some examples are: 17th to the 3rd,
illustrated by employing Power of Ten cards (5 + 5, 6 + 4, 7 + 3, 8 + 2, 9 + 1).
21st to the 7th, 29th to the 15th.
Ten is the sum of fingers on two hands or the number of toes on two feet (5 + 5 or
15 Rectangular four ways (1 × 15, 3 × 5
2 × 5). When two people share twenty things, each person receives 10 things. In
and the reversals of each). A three-by-
the game Concentration, twenty cards may be placed in two rows of ten (20 ÷ 2 = 10).
1, 2, 3, 4, 5, 6, 7, 8 and 9 has a sum of 15
three magic square using the numbers
A hundreds chart has ten rows of ten
in each row, each column and each diagonal. Ask your grade-three class: Can
(100 ÷ 10 = 10). A dime is worth 10 pennies. In Friendly Concentration (using 18 cards) the
you create this magic square using these nine numbers? (See number 9 above.)
goal number is 10 (see number 20 for details).
Students will achieve eight different number combinations, each with a sum of 15.
11 Rectangular, one way only. This is the
Fifteen is a multiple of 3 and 5.
number of sides shown on a loonie.
16 Square (4 × 4), rectangular four ways (1 × 16, 2 × 8 and reversals of each). Sixteen is even. Sixteen squares constitute one quarter of an entire chess board.
12 Rectangular many ways: therefore 12 has many divisors. This is probably why 12 was chosen as the basis for a foot
17 Rectangular and prime.
(See number 13)
(1 × 12, 2 × 6, 3 × 4 and reversals of each). Twelve is the number in a dozen. Twelve may be shared fairly, and so it is even.
18 Rectangular six ways (1 × 18, 2 × 9, 3 × 6 and reversals of each). Eighteen is the number of cards in Friendly
Twelve is a multiple of 2, 3, and 4, and therefore it will appear in the list when counting by those numbers. Twelve is also
Concentration (20 - 2). Eighteen is a multiple of 3. See number 20 for Concentration
a multiple of 6 and 12, although at the early
game ideas.
primary level we rarely teach counting by 6 or 12.
13 Rectangular two ways only. In grade-three
19 Rectangular and prime:
(See numbers 13, 17)
or four, discuss the similarity between the numbers 2, 3, 5, 7 and 11 and introduce the word prime. Ask: Can you predict the next prime number? Thirteen is the number of cards in each suit of a deck of cards.
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20 The 20th of the month is an opportune
25 Rectangular (1 × 25), and square
moment to remind students of cards in a set of Power of Ten cards. When playing Concentration, place cards on the floor in four rows of five. This leads to a
(5 × 5). Twenty-five is a multiple of five.
26 Rectangular in four ways (1 × 26, 2 × 13 and the reversals of each). Twenty-six equals two suits of a deck of cards or half
discussion comparing 5 + 5 + 5 + 5 or four groups of five, which is written 4 × 5 = 20.
of a full deck. Twenty-six is a multiple of two.
However, when seen in columns of four, there are five groups of four, which is written 5 × 4 = 20. Twenty cards may be placed in two rows of ten, yielding 2 × 10,
27 Rectangular (1 × 27) and cubic (3 × 3 × 3). Twenty-seven is a multiple of three.
or 10 × 2. Concentration may also be employed to introduce the concept of division. Ask: If twenty cards are placed in rows of a five, how many rows are there? Thus 20 ÷ 5 = 4, 20 ÷ 4 = 5, 20 ÷ 10 = 2, 20 = 10. 2 In Friendly Concentration, the teacher removes two cards leaving 18 (which also presents a potential equation for the 18th of a given month). Friendly Concentration produces two rows of nine,
28 Rectangular several ways (1 × 28,
or three rows of six, which yields:
2 × 14, 4 × 7 and the reversals of each). Twenty-eight is a multiple of four.
3 × 6, 6 × 3, 2 × 9, or 9 × 2.
Four weeks are comprised of 28 days
21 Rectangular four ways (1 × 21, 3 × 7
(4 groups of 7). There are usually twentyeight days in the month of February.
and reversals of each). Twenty-one is a multiple of three.
29 Rectangular (1 × 29) and prime.
22 Rectangular four ways (1 × 22, 2 × 11
There are twenty-nine days in the month
and reversals of each). Twenty-two is a multiple of two.
of February each Leap Year.
30 Rectangular in several ways (1 × 30,
23 Rectangular and prime.
2 × 15, 3 × 10, 5 × 6 and the reversals of each). April, June, September and
24 Rectangular many ways (1 × 24,
November each have 30 days.
2 × 12, 3 × 8, 4 × 6). Twenty-four is a multiple of three. (See 12 above for more connections.) Twenty-four equals the
31 Rectangular and prime.
number of eggs found in two dozen. Twenty-four is a multiple of 2, 3 and 4.
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A SAMPLE ONE-YEAR PLAN FOR “28 28 28” The previous section may lead one to believe that all the suggestions for a given number should be covered in a single session. While this may be true in a grade-three classroom it definitely wouldn't be true in a kindergarten or grade-one classroom. The best strategy is to try to find examples of addition, subtraction, multiplication and division each time the number is investigated and then look for opportunities to identify student thinking that allows the teacher to extend to problem solving or to the "All the Facts" sheets. The following represents a hypothetical sequence for the number 28 over the course of a school year.
September 28 • 1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+ 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 = 28
• • • • •
• • • •
(Whenever this one arises show the multiplication as 28 × 1 = 28)
October 15
10 + 10 + 8 = 28
Have students work in homogenous groups of three or four with egg cartons, Power
8 + 10 + 10 = 28 10 + 8 + 10 = 28 5 + 5 + 5 + 5 + 5 + 3 = 28 5 × 5 + 3 = 28 (This is related to the previous question, 5 + 5 + 5 + 5 + 5 + 3 = 28, by writing the sentence: five groups of five plus three equals twenty-eight and then translating the sentence from English to mathematics 5 × 5 + 3 = 28. Notice the use of the word groups rather than times – ‘times’ has no meaning while ‘groups of’ does have meaning to young children.)
of Ten cards and other manipulatives to construct and to write all the equations they can for 15 and then present their results to the class. Assign one recorder (someone who can print) to each group. It is useful when doing this to put together groups of equal ability so that gifted children do not dominate the group. Use the criteria sheets on pages 20-21of this section to check for the different types of thinking. You may want
20 + 8 = 28 8 + 20 = 28
to repeat this process once per week so that students get non-threatening practice before
30 - 2 = 28 28 ÷ 2 = 14 (This is accepted even though it doesn’t equal 28; later students will learn to go beyond the goal number for subtraction and division sentences.)
they have to do a “How Many Ways” sheet on their own.
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October 28 Expect much the same as September 28 except for subtraction which is illustrated by using egg cartons for 28 then noticing the holes. Add an empty egg carton and now there are 12 holes not filled, add two empty egg cartons and there are three holes not filled and so on.
• 1+1+1+1+1+1+1+1+1+1+1+1+1+1+ 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 = 28
• • • • • • • •
Also look in the egg cartons for examples of students mixing the colours. If a student has the cubes arranged in the egg cartons as 1 red plus 4 gray in the first column and 4 red plus 1 gray in the second column, then 2 green plus 2 yellow and 1 blue and 3 green plus 2 blue and 3 yellow plus 2 blue and ending with 3 orange. The equation would be:
10 + 10 + 8 = 28
(1 + 4) + (4 + 1) + (2 + 2 + 1) + (3 + 2) + (3 + 2) + 3 = 28 (brackets are used as punctuation to show the fives)
8 + 10 + 10 = 28 10 + 8 + 10 = 28 5 + 5 + 5 + 5 + 5 + 3 = 28 5 × 5 + 3 = 28 20 + 8 = 28 8 + 20 = 28
30 - 2 = 28 (This is related to the previous question, 30 - 2 or 5 + 5 + 5 + 5 + 5 + 3 = 28,
by writing the sentence: six groups of five subtract two 2 6 × 5
Look for a student who might be counting by twos and model: (2 + 2 + 2 + 2 + 2) + (2 + 2 + 2 + 2 + 2) + 2 + 2 + 2 + 2 = 28 (brackets are used as punctuation to show the tens)
Now write that as a multiplication and translate: (five groups of two) plus (five groups of two) (5 × 2) + (5 × 2) plus four groups of two equals twenty-eight. + 4 × 2 = 28
equals twenty-eight = 28
• 40 - 12 = 28 • 50 - 22 = 28 • 28 ÷ 2 = 14 (This is accepted even though it doesn’t equal 28; later students will learn to go beyond the goal number for subtraction and division sentences.)
Now collect like terms and translate into mathematics again: fourteen groups of two equals twenty-eight 14 × 2 = 28 In early late October or early November do a “How Many Ways Can You Make 20” sheet. Allow ten minutes. Then have students check their own sheets for examples of an adding, subtracting, multiplying and dividing. Record their mark out of four. The “How Many Ways Can you Make ____” sheet should be done once per term. The goal number should be changed to 25 or 30. In late primary grades the numbers can be changed to 40, 50 or 100.
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Power of Ten
How Many Ways
page 16
How Many Ways
November 28 Expect what was given in previous sessions but start to look for more opportunities to show the use of brackets as punctuation for tens. Be sure to find examples of all four operations and sentences that include multiple operations with more than two terms and mixed operations (4 × 5 + 8 = 28) Examples:
• (1 + 3 + 3 + 3) + (2 + 3 + 2 + 3) + 1 + 3 + 3 + 1 • (1 + 4 + 3 + 2) + (2 + 3 + 1 + 4) + 4 + 4 Also start to look for examples of the use of ones and zeros:
• 28 × 1 = 28 • 28 + 0 = 28 Look for more subtraction patterns: • 40 - 12 = 28 • 50 - 22 = 28
• 60 - 32 = 28 Also multiplication related to groups of two, fourteen or seven:
• • • •
14 × 2 = 28 2 × 14 = 28 7 + 7 + 7 + 7 = 28
January 28 Accept previous examples and start to look for multiplication. Give students the multilink cubes without the egg cartons this month and ask them to create arrays. See how many different arrays you can get. 4 × 7 = 28 and 7 × 4 = 28 are related but not identical. A real life example would be apartment blocks with seven floors and four apartments or four floors and seven apartments. Which apartment building would you want to live in if there was no elevator? Storage cubbies in the classroom have the same issue. Cubbies that are seven high would not work in a primary classroom. 2 × 14 and 14 × 2 = 28 1 × 28 and 28 × 1 = 28 Have students draw diagrams of the arrays on cm graph paper and cut the arrays out. Tell them that each rectangle has an area of 28 square cm. Establishing the area model for multiplication sets a solid base for multi-digit multiplication later. This also establishes a visual tool for learning multiplication facts. If a student knows 2 × 7 = 14 and can visualize an array then 4 × 7 = 28 will be easy. So will 8 × 7 = 56 if the students have been taught to add horizontally using the meaning of numbers (20 + 8 + 20 + 8 = 40 + 16) or (40 + 10 + 6 = 56)
4 × 7 = 28 Do not teach multiplication tables, instead teach thinking in groups of two, four and seven.
December 28
Don’t forget to have students work independently or in small groups on a regular basis to create number sentences for the day of the month or day of the year.
December 28 is a holiday. Next month reinforce this model on days like: 6, 8, 10, 12, 16, 20, 24, 27, and 30.
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Power of Ten
How Many Ways
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How Many Ways
April 28
February 15 Give students ten minutes to do a “How Many Ways” sheet for 30. Assess using the criteria on pages 20-21 of this section and compare to last terms example for growth. Use the data for reporting purposes.
If area has been sufficiently explored start to explore the concept of borders or perimeter. Use multilink cubes to explore building fences of 28 cubes around an enclosure. Put animals inside. Will different configurations of perimeters of 28 allow more animals?
• • • • • •
February 28
1 + 13 + 1 + 13 = 28 2 + 12 + 2 + 12 = 28 3 + 11 + 3 + 11 = 28 4 + 10 + 4 + 10 = 28 5 + 9 + 5 + 9 = 28 6 + 8 + 6 + 8 = 28 or 2(6 + 8) = 28
Expect previous examples for each of the four operations and start focusing on multiple operations, mixed operations, examples of one (28 ÷ 1 = 28 or for the sophisticated grade-three student 10 ÷ 10 + 27) and zero (100 - 100 + 28).
or 2 × 6 + 2 × 8 = 28
• 7 + 7 + 7 + 7 = 28
Introduce triangular numbers. Illustrate using bingo chips on the overhead. The triangular numbers available to study in February include 3, 6, 10, 15, 21 and 28. Reinforce the area model for multiplication.
March 28 Look for examples of zeros and ones and reinforce that these concepts will be very useful later on in algebra. An example taken from a Sandra Glassel’s grade-one class on February 5 one year was: 100 ÷ 10 + 5 - 10 = 5
• 15 - 15 + 28 = 28 • 115 - 115 + 28 = 28 • 28 ÷ 28 + 27 = 28
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Power of Ten
May 28 Reinforce earlier models. Explore threedimensional views of 28 and have students make interesting shapes for which they then create addition and multiplication sentences. Tell them they are measuring the volume of the figure but do not expect early primary students to use this word. 2 × 2 × 7 = 28 is the only regularly shaped prism (it can be argued that 1 × 1 × 28 is another but this one is very difficult for young students to understand). In grade-three and four it is neat, but frustrating to try to learn how to draw such shapes (grid paper is useful). There are many irregularly shaped figures that have a volume of 28 cm cubed.
How Many Ways
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How Many Ways
Using the How Many Ways Sheet
May 30
Have students create a “How Many Ways to Make 25” sheet. Mark using the criteria from pages 20-21of this chapter. Use the results in conjunction with the sheets for 20 and 30 done in November and February to show growth over the year.
Evaluation: Using Calendar Time, the “How Many Ways Sheet”.
Schools and districts may wish to collect this data and develop local benchmarks for each grade level.
Month
Whole Class
September
September 30
October
October 28
November
November 8
December
December 15
January
January 18
February
February 15
March
March 28
April
April 24
(see pages 20 and 21 of this chapter).
Teacher Evaluates and Collects Data for Reporting
Individual Student
November 10
School or District Collects Data
Yes
(use 20 as the number)
February 20
Yes
(use 30 as the number)
Yes May
May 25
May 25
Yes
(use 25 as the number)
Districts and schools may wish to extend this concept into the intermediate years. The data would be collected from students by teachers prior to reporting each November, February and May and then benchmarks would be established in June using the June data.
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Power of Ten
(teachers turn in individual data to be used for benchmarks for school or district data)
The numbers could be changed as follows:
Grade
November "How Many Ways to Make ________"
February "How Many Ways to Make ________"
May "How Many Ways to Make ________"
4
20
50
100
5
50
100
200
6
50
200
500
7
100
500
1000
How Many Ways
page 19
How Many Ways
STUDENT-CRITERIA SHEET FOR
HOW MANY WAYS Date Assessed:
Criteria: Uses addition. Uses subtraction. Uses multiplication. Uses division. Has an equation with two or more terms. Has an equation with two or more operations. Has an equation with a term much greater than the goal number. Uses the principle of zero (119 - 119 + 7 = 7). Uses the principle of one (119 ÷ 119 + 17 = 18). Uses the commutative principle (6 + 4 = 10 and 4 + 6 = 10). Has three equations that are linked in a pattern. Uses the standard form of a number. Uses brackets to clarify order of operations. Implements a doubling or halving strategy.
Has an equation that has a special factor.
Do not share this list with students until most of the criteria above has been developed by the class. The blank spaces are for new criteria brought in by your students. See page (3)21 for possible criteria to be developed.
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Power of Ten
How Many Ways
page 20
Suggestions for Using the Sheet: “How Many Ways Can You Make a Number?” How Many Ways Practice this exercise often during calendar time in the primary grades and at least twice a month in the intermediate grades. Have students mark their own work. Marks are awarded as follows:
Mark
Criteria
1
Where any equation contains the addition operation.
1
Where any equation contains the subtraction operation.
1
Where any equation contains the multiplication operation.
1
Where any equation contains the division operation.
1
Where any equation contains more than two terms. e.g. 2 + 3 + 5 = 10
1
Where any equation contains more than two operations. e.g. 2 x 3 + 4 = 10
1
Where any equation contains a number more than the goal number. (at least double the goal number)
1
Where any equation contains a number substantially greater than the goal number. (the numbers exceed expectations)
1
Where any group of equations shows evidence of a pattern. e.g. 1 + 9, 2 + 8, 3 + 7
1
Where any equation shows knowledge of the power of zero (uses fancy zeros). e.g. 6 - 6 + 10 = 10 or 123 + 13 - 136 + 17 = 17
1
Where any equation uses doubling or halving to generate new questions. e.g. 4 x 6 = 24, 2 x 12 = 24, 1 x 24 = 24
1
Where any equation shows knowledge of the power of one. e.g. 6 ÷ 6 + 9 = 10 or
( 14
3
+ 4 ) x 15 = 15
1
Where any equation shows knowledge of the commutative principle.
1
Where any equation uses the idea of “tenness”.
1
Where any equation uses the idea of “hundredness”.
1
Where any equation shows knowledge of the standard form of the number.
e.g. 6 + 4 = 10 and 4 + 6 = 10, or 110 + 99 = 99 + 110
e.g. 7 + 6 + 4 + 3 + 17 = 37 e.g. 67 + 41 + 33 = 141
Note: This applies only for numbers greater than 10, such as 24. e.g. 20 + 4 = 24 and 2 x 10 + 4 = 24 In upper intermediate grades, also award marks for exponential notation.
1
Where any sentence contains brackets, such as: (3 + 2) + (3 + 2) + (3 + 2) + (3 + 2) + 4 = 24.
2
Where any sentence contains exponents, square roots, decimal fractions or common fractions. Note: there should be no expectation of the demonstration of exponents, square roots or factorials before grade six, but their use should be acknowledged and rewarded in earlier grades where a student chooses to employ such operations.
1 1.
2.
©
Has a special factor not included in any of the above.
Always remember that specific scores provide far less instructive data regarding acquired number sense than do the strategies and mathematical thinking employed to reach solutions. Where the criteria listed above are a focus of classroom brainstorming sessions students become conversant with different mathematical operations and varied mathematical vocabulary. Encourage your students to focus upon a few of the criteria listed above as they complete activity sheets or as they create problems and write related number sentences. For example, ask students to check if they have employed all four operations in their work, or if they have written any sentences with mixed operations. Use examples of new ideas discovered in student work.
Power of Ten
3.
4.
Attach the criteria sheet to student journals involving “How Many Ways” then the teacher can assess progress as the class writes equations. Try to visit each student once per week in primary, and once per month in intermediate. Schools may wish to collect data from the “How Many Ways to Make...” sheets on an annual basis and to use this data for numeracy performance standards.
Note: Remove the suggestions listed beneath the table, and enlarge the print before distributing to your students.
How Many Ways
page 21
How Many Ways
In the late-primary grades you may wish to create a table showing all numbers to 100 and to investigate their respective shapes. Ask students to investigate any numbers not yet examined (as in those above) by using this table.
Rectangular Number
Prime
Square
Triangle
Cubic
1 2 3
4 5
1+2 2x2
6 7
1+2+3
8
2x2x2
9
3x3
10 11
1+2+3+4
12 13
14 15
1+2+3+4+5
16 17
4x4
18 19
20 21
1+2+3+4+5+6
22 23
24 25
5x5
26 27
3x3x3
28
100
29
1 + 2 + 3 + 4 +5 + 6 + 7
30
(Every rectangle has a reverse image so double the number)
Total
1x1
1
1x2
2
1x3
3
1x4
3
1x5
2
1 x 6, 2 x 3
5
1x7
2
1 x 8, 2 x 4
5
1x9
3
1 x 10, 2 x 5
5
1 x 11
2
1 x 12, 2 x 6, 3 x 4
6
1 x 13
2
1 x 14, 2 x 7
4
1 x 15, 3 x 5
5
1 x 16, 2 x 8
5
1 x 17
2
1 x 18, 2 x 9, 3 x 6
6
1 x 19
2
1 x 20, 2 x 10, 4 x 5
6
1 x 21, 3 x 7
5
1 x 22, 2 x 11
4
1 x 23
2
1 x 24, 2 x 12, 3 x 8, 4 x 6
8
1 x 25
3
1 x 26, 2 x 13
4
1 x 27, 3 x 9
5
1 x 28, 2 x 14, 4 x 7
7
1 x 29
2
1 x 30, 2 x 15, 3 x 10, 5 x 6
8
(continued next page)
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Power of Ten
How Many Ways
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How Many Ways
In the-late primary grades you may wish to create a table showing all numbers to 100 and to investigate their respective shapes. Ask students to investigate any numbers not yet examined (as in those above) by using this table.
Rectangular Number
Prime
31
Square
Triangle
Cubic
(Every rectangle has a reverse image so double the number)
Total
1 x 31
2
32
1 x 32, 2 x 16, 4 x 8
6
33
1 x 33, 3 x 11
4
34
1 x 34, 2 x 17
4
1 x 35, 5 x 7
4
1 x 36, 2 x 18, 3 x 12, 4 x 9
10
1 x 37
2
38
1 x 38, 2 x 19
4
39
1 x 39, 3 x 13
4
40
1 x 40, 2 x 20, 4 x 10, 5 x 8
8
1 x 41
2
1 x 42, 2 x 21, 3 x 14, 6 x 7
8
1 x 43
2
1 x 44, 2 x 22, 4 x 11
6
1 x 45, 3 x 15, 5 x 9
7
1 x 46, 2 x 23
4
1 x 47
2
1 x 48, 2 x 24, 3 x 16, 4 x 12, 6 x 8
10
1 x 49
3
50
1 x 50, 2 x 25, 5 x 10
6
51
1 x 51, 3 x 17
4
52
1 x 52, 2 x 26, 4 x 13
6
1 x 53
2
1 x 54, 2 x 27
4
1 x 55, 5 x 11
5
56
1 x 56, 2 x 28, 7 x 8
6
57
1 x 57, 3 x 19
4
58
1 x 58, 2 x 29
4
1 x 59
2
1 x 60, 2 x 30, 3 x 20, 4 x 15, 5 x 12, 6 x 10
12
35 36 37
41
6x6
1+2+3+4+ 5+6+7+8
42 43
44 1+2+3+4+ 5+6+7+8+9
45 46 47
48 49
53
7x7
54 1+2+3+4+ 5 + 6 + 7 + 8 + 9 + 10
55
59
60
(continued next page)
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Power of Ten
How Many Ways
page 23
How Many Ways In the-late primary grades you may wish to create a table showing all numbers to 100 and to investigate their respective shapes. Ask students to investigate any numbers not yet examined (as in those above) by using this table.
Rectangular Number
Prime
61
(Every rectangle has a reverse image so double the number)
Total
1 x 61
2
62
1 x 62, 2 x 31
4
63
1 x 63, 3 x 7, 7 x 9
6
1 x 64, 2 x 32, 4 x 16
8
1 x 65, 5 x 13
4
1 x 66, 2 x 33, 3 x 22, 6 x 11
9
1 x 67
2
68
1 x 68, 2 x 34
4
69
1 x 69, 3 x 23
4
70
1 x 70, 2 x 35, 5 x 14, 7 x 10
8
1 x 71
2
1 x 72, 2 x 36, 3 x 24, 4 x 18, 6 x 12, 8 x 9
12
1 x 73
2
74
1 x 74, 2 x 37
4
75
1 x 75, 3 x 25, 5 x 15
6
76
1 x 76, 2 x 38, 4 x 19
6
77
1 x 77, 7 x 11
4
1 x 78, 2 x 39, 3 x 26, 6 x 13
9
1 x 79
2
1 x 80, 2 x 40, 4 x 20, 5 x 16, 8 x 10
10
1 x 81, 3 x 27
6
1 x 82, 2 x 41
4
1 x 83
2
84
1 x 84, 2 x 42, 3 x 28, 4 x 21, 6 x 14, 7 x 12
12
85
1 x 85, 5 x 17
4
86
1 x 86, 2 x 43
4
87
1 x 87, 3 x 29
4
88
1 x 88, 2 x 44, 4 x 22, 8 x 11
8
1 x 89
2
1 x 90, 2 x 45, 3 x 30, 5 x 18, 6 x 15, 9 x 10
12
64
Square
Triangle
Cubic
8x8
4x4x4
65 1+2+3+4+5+6+ 7 + 8 + 9 + 10 + 11
66 67
71
72 73
1+2+3+4+5+6+ 7 + 8 + 9 + 10 + 11 + 12
78 79
80 81
9x9
3x3x3x3
82 83
89
90
(continued next page)
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Power of Ten
How Many Ways
page 24
How Many Ways In the-late primary grades you may wish to create a table showing all numbers to 100 and to investigate their respective shapes. Ask students to investigate any numbers not yet examined (as in those above) by using this table.
Rectangular Number
Prime
Square
Triangle
Cubic
(Every rectangle has a reverse image so double the number)
Total
1 x 91, 7 x 13
5
1 x 92, 2 x 46, 4 x 23
6
1 x 93
2
94
1 x 94, 2 x 47
4
95
1 x 95, 5 x 19
4
96
1 x 96, 2 x 48, 3 x 32, 4 x 24, 6 x 16, 8 x 12
12
1 x 97
2
98
1 x 98, 2 x 49, 7 x 14
6
99
1 x 99, 3 x 33, 9 x 11
6
1 x 100, 2 x 50, 4 x 25, 5 x 20
9
1+2+3+4+5+6+7+ 8 + 9 + 10 + 11 + 12 + 13
91 92 93
97
100
10 x 10
When Calendar Time is taught consistently throughout the primary years, the results near the end of the primary experience are often phenomenal. For a more challenging experience in grades three and four, it is more productive to employ the day of the school year, rather than the day of the month, as shown in the chart to 100 above.
the class to make change employing a hundreds sheet. I was astonished that almost all students were able to subtract sixty-nine cents from one thousand dollars using only mental arithmetic by the end of the lesson. This demonstrated that Heidi’s class was accustomed not only to facing challenges in mathematics, but also to searching for patterns and taking risks when solving problems.
Heidi Rensing, a teacher and writer, taught grade three/four in an inner-city school at one time. On several occasions I visited Heidi’s class, and I always left astounded at the general level of numeracy demonstrated by young students in her care. During one visit I taught
These are only a few of the equations produced by Heidi’s class on the 94th and 95th day of the school year.
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Power of Ten
How Many Ways
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How Many Ways
The most exciting aspect of my visit to Heidi’s classroom was the clear sense of number evidenced by her students’ thinking as they readily used doubling and halving to get equivalent dividing questions. These students were well prepared for equivalent fractions. These students also demonstrated a sound sense of how 0 and 1 operate, and how these numbers may be employed to make an equation appear different but “stay the same”. Clearly these students were also well prepared for algebra. They demonstrated a good sense of “tenness”, “hundredness” and “thousandness” of numbers: They understood the base-ten system and showed extensive knowledge of place value.
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How Many Ways
page 26
How Many Ways
SUGGESTIONS
FOR
TEA CHING EACHING
AND
REPOR TING ON: EPORTING
HOW MANY WAYS
Preparation Some teachers use a special “How Many Ways” booklet. Simply take ten blank sheets of paper and staple them together with the studentcriteria sheet for “How Many Ways” stapled to the back (see page 20 of this chapter) . The students create sentences each day (or
The third number can be the result of a draw or related to the study of place value and probability. It can be given by the special student of the day or the week. It can be a special day, event, or birthday. The following game can be used to obtain the third number.
several times a week or even once a week in intermediate classes). Usually allow ten minutes. Other teachers have students write their sentences in a scribbler or folder. Students are asked to write about three sentences a day, then to proceed to another activity. Again, the criteria sheet should be available for student reference –
Place Value Probability The teacher or student of the day declares that today’s number will be a three-digit number and each student draws three blank squares on a piece of paper.
it can be stapled into the scribbler.
Routine The teacher writes up to three numbers on the board and students choose one of the numbers for the ‘goal number’ for their written equations. The advantage of three numbers is that students feel they have a choice. One number is an imposition; two numbers are “either – or”, and three numbers provide a choice. Choice gives a feeling of commitment and control.
As cards are drawn (or dice are rolled), using one number at a time, students are told to make their choice for placement of the number. (All students play but the strategies are shown for three players.) Round One: A 7 is rolled and each student places it in what they believe is the best place.
Round Two: A 2 is rolled.
students who are interested can plan their equations ahead of time. They can collaborate with parents and siblings and, as a result, they often bring new ideas into the class that exceed curricular expectations.
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Power of Ten
A B
7
C
Student Hundreds A B
7
C
At lease two of the numbers should be predictable. Most teachers use the ‘date’ as one of the numbers and the ‘number of days in school’ as the second number. This means that
Student Hundreds
Student Hundreds
Round Three: A 1 is rolled.
A B C
1 7 1
Tens
7
Ones
7 Tens
Ones
7
2 2 2
Tens
Ones
7
7 1 7
2 2 2
Student B has the greatest number so 712 becomes the third number of the day. Another option is to roll the die again. If the number is odd, the greatest number wins, and if the number is even the least number wins and becomes the third number of the day.
How Many Ways
page 27
How Many Ways
Now students choose from the three numbers. Their choice often reflects their
Establishing School and District Norms
confidence, ability and number sense and this alone can be invaluable in assessing an individual student. Sometimes the teacher has to encourage students to make a more difficult number choice and apply the same strategies that they are using for two-digit numbers on to threedigit numbers. In the intermediate grades this process is extended to multi-digit numbers which
Earlier in this section it was suggested that the school or district create “Performance Standards” for the How Many Ways activities. A set of standards normed in one school district has been provided in the Evaluation section of this manual (see District Norms, pages 91-96).
meet higher curriculum expectations.
Working With Students Who Cannot Get Started Materials Teachers should set up the booklet, journal or file and then have the following materials available: Truncated ten-frame egg cartons
Power of Ten large cards Multi-link cubes
Once teachers have shown the class how to write the equations, students should be expected to work independently at their desks. Obviously this is dependent on grade level. Most students in grade two should be able to write several equations equal to a given number without much instruction. One or two whole-class demonstrations using egg cartons should be sufficient. In kindergarten the process is almost always undertaken as a whole group activity until a few students are able to write numerals. Once students have the fine motor and number sense, and if they are able to, they should be allowed to work independently. Many students in grade one should be able to work independently by the end of the first term and most by the end of the second term. In grade two or three there may be a few students who are new to the concept and need several special demonstrations. The following process (see following page) works well for a whole class (kindergarten/grade one) and then for small groups as needed in the second term of grade one and in grade two, three, or four classes.
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Power of Ten
How Many Ways
page 28
How Many Ways
EXAMPLE: Make Equations for 24
Set up the egg cartons in the standard position and then have students move them to create new positions as the teacher asks appropriate questions.
Multilink Cubes: R = red, Y = yellow, B = blue, G = green, O = orange, W = white Colour gives some students clues for other possibilities.
Position B
Position A R
Y
B
G
R
Y
B
G
R
Y
B
G
R
Y
B
G
R
Y
B
G
R
Y
B
G
R
Y
B
G
0
W
R
Y
0
W
B
G
R
Y
B
G
0
W
R
Y
0
W
B
G
10
+
10
+
10
4
Teachers should ask the following questions:
4
+
10
Teachers should ask the following questions: How many in the first egg carton?
How many in the first egg carton?
Ask the student to write 10.
Ask the student to write 10. How many in the second egg carton?
How many in the second egg carton? Ask the student to write + 4.
Ask the student to write
a “+” sign and another 10.
How many in the third egg carton?
How many in the third egg carton?
Ask the student to write + 4.
Ask the student to write + 10.
The student should have written the equation:
The student should have written the equation:
10 + 10 + 4 = 24.
10 + 4 + 10 = 24.
Instruct the student to change the order (position) of the egg cartons.
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+
Power of Ten
Instruct the student to change the order (position) of the egg cartons.
How Many Ways
page 29
How Many Ways
Position D
Position C R
Y
B
G
G
R
Y
G
Y
B
B
R
R
G
Y
G
Y
B
B
R R
Y
B
G
0
W
R
Y
B
G
G
0
W
0
Y
B
W
R
Y
B
R
G
R
Y
B
G
0
W
4
+
10
+
10
10
Teachers should ask the following questions:
+
10
+
4
Teachers should ask the following questions:
How many in the first egg carton?
How many in the first column of the first egg carton? (red) Ask the student to write 5.
Ask the student to write 4.
How many in the second egg carton?
How many in the second column of the first egg carton? (yellow)
Ask the student to write + 10.
Ask the student to write + 5. How many in the first column of the second egg carton? (blue) Ask the student to write + 5.
How many in the third egg carton? Ask the student to write + 10.
How many in the second column of the second egg carton? (green)
The student should have written the equation:
4 + 10 + 10 = 24.
Ask the student to write + 5. How many in the first column of the third egg carton? (orange) Ask the student to write + 2.
Return to the first position (A) which parallels the standard form of a number.
How many in the second column of the third egg carton? (white) Ask the student to write + 2.
The student(s) should have written the equation:
5 + 5 + 5 + 5 + 5 + 2 + 2 = 24.
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Power of Ten
How Many Ways
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How Many Ways
Position F
Position E R
Y
B
G
R
Y
B
G
R
Y
B
G
R
Y
B
G
R
Y
B
G
R
Y
B
G
R
Y
B
G
0
W
R
Y
B
G
0
W
R
Y
B
G
0
W
R
Y
B
G
0
W
10
+
10
+
10
4
Teachers should ask the following questions:
+
10
+
4
Teachers should ask the following questions: How many egg cartons did you use in making these equations?
How many fives did you write in the last equation?
Answer is 3.
Student says “four”. Have the student write four groups of five.
4x5
How many holes are there in three empty ten-frame egg cartons? Answer is 30.
How many twos did you write in the last equation?
Write 30 on your paper or 3 x 10 (whichever the student says).
Student says “two”. Have the student write two groups of two.
2x2
How many holes are left unfilled when you put twenty-four cubes in the three empty egg cartons? Answer is 6.
Advise the student to use brackets so that it is clear which operation is done first and have the student write the equation:
(4 x 5) + (2 x 2) = 24
Now have the student write the equations:
(3 x 10) - 6 = 24 or 30 - 6 = 24 Now add an empty egg carton and the equation becomes: (4 x 10) - 16 = 24 or 40 - 16 = 24 Now add yet another empty egg carton and the equation becomes:
(5 x 10) - 26 = 24 or 50 - 26 = 24 Ask the student to do the next equation in the pattern without physically adding an empty egg carton.
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Power of Ten
How Many Ways
page 31
How Many Ways
Position G Share the blocks between 2 people.
24 ÷ 2 = 12
R
Y
B
G
R
Y
B
G
R
Y
B
G
R
Y
B
G
W
0
W
2 ○
○
○
○
○
○
○
○
○
○
○
○
○
○
○
○
○
○
○
○
○
○
○
B G R Y ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ 10○ ○ ○ ○ ○ ○ ○ ○ ○ 10 ○ ○ 2○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○
0
R
Y
B
G
R
Y
B
G
R
Y
B
G
R
Y
0
B
G
W
R
Y
0
B
G
W
10
+
2
=
10
12
Teachers should ask the following questions:
2
=
12
Note the other equations that can be written for this diagram:
You have twenty-four cubes in these three egg cartons. If you had to share those cubes with me (sometimes it helps to pretend that these are Halloween candies and you have decided to share them)
2 x 12 = 24 and 12 + 12 = 24 If there is a small group of 3 students, have them share as well to get 24 ÷ 3 = 8
how many would each of us get? Many students, when trying to share, will start sharing one at a time. The teacher demonstrates that larger amounts can be easily shared by each taking one full egg carton of cubes. This leaves four cubes and most students quickly figure out that each person would then get two. Now they can see that each person gets twelve in total which is written:
24 ÷ 2 = 12
+
Note the other equations that can be written for this diagram:
3 x 8 = 24 and 8 + 8 + 8 = 24 If there is a small group of 4 students, have them share as well to get 24 ÷ 4 = 6 Note the other equations that can be written for this diagram:
4 x 6 = 24 and 6 + 6 + 6 + 6 = 24 (continued next page ...)
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Power of Ten
How Many Ways
page 32
How Many Ways
If there is a small group of 5 students, have them share as well to get
24 ÷ 5 = 4 remainder 4 which can be written 4 r 4.
Note the other equations that can be written for this diagram:
(4 x 5) + 4 = 24 5 + 5 + 5 + 5 + 4 = 24 It is important to show students how to deal with remainders or they learn to think that division can only be used when there is no remainder. This situation seldom turns up in life so students think division usually does not work and cannot be done. The important thing is that students who are exposed to division early find it easy. Since the teacher does not have to assess division in the early grades there is no stress associated with learning it. When students learn only addition they can come to believe that the other operations are difficult. The next time the teacher needs to work with a student or a small group, the students should be encouraged to do all the manipulative work with the equations themselves although it may be necessary to give them prompts. Eventually the students need to be able to imagine all the movement of egg cartons in their minds. It is very important to focus on the idea of making fives, tens and doubles. Remember to teach the role of brackets as a tool for avoiding interpretation errors:
6 + 3 x 2 = 12
but can be misinterpreted as
6 + 3 = 9, 9 x 2 = 18
if it is read left to right which is the natural way to read. If it is written with brackets there is no ambiguity.
6 + (3 x 2) = 12 6 + 6 = 12
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Power of Ten
How Many Ways
page 33
How Many Ways
ASSESSING
AND
EVAL UATING ALU
One of the great advantages of the How
The easiest way to assess how well the class
Many Ways activities is that it is easy to assess and also provides a very good sense of a student’s
is doing, and to increase student participation, is to have students periodically share their sentences
number sense. This activity is very good as a beginning of the period activity because once
with the whole class. Once the five to ten-minute writing period is over have students share their
students learn the routine most will love it and get to work quickly. Some teachers have stations that
favourite equations. Students should try to memorize their equations before coming forward to
students can go to when they are finished writing at least three good equations. Other teachers ask
present to the class. Many teachers use a random process for this sharing by pulling students’ names
the students to write as many equations as they can.
from a hat. This keeps the excitement high and everyone listening to each other.
Once teachers have almost all students
If primary students come to the front of their
working independently they are able to circulate and work with individual students. This gives the
class without their journals, they have to remember their equations. Their short-term working memory is
teacher an opportunity to ask students about their thinking and give the students feedback and ideas.
not very large at this age and they may have trouble remembering exactly what they wrote. This causes
Teachers can suggest to shy students that this would be a good equation to share later as this
them to pay attention to what others are doing so they can steal ideas. It is during these times of
provides these students with confidence.
class sharing and ‘borrowing’ of ideas that some of the peer-transfer of learning occurs. This is also a
By allowing ten minutes for the writing process, the teacher can talk to as many as four
time when new ideas can be introduced by the students who are keen.
students per day. During this time she can also be assessing by using the How Many Ways
A climate of respect is important as this
student-criteria sheet stapled to the back of the student booklet or stapled into the student’s journal
can become a time when everyone, including the teacher, learns to take risks in math class. Students
(see page 20 of this chapter). One of the most
effective strategies is having the student put check
are allowed to discuss other students’ equations when these are written on the board, but this must
marks in the categories as the teacher sees evidence in his work. The advantage of having
be done with respect and mistakes must be treated as opportunities for learning.
students use a journal is that the teacher can use the work from several days for the evaluation. In an early primary class this process might
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Many students bring concepts from home that would never be taught in a primary classroom. Teachers are often shocked to see students
happen almost every day and, in grade three to five classes, it might happen several times a week.
bringing in equations which involve square roots, decimal fractions, common fractions, percent and
In grades six and up this process would probably occur bi-weekly.
powers. Students obtain these concepts from their parents, their siblings, and their baby-sitters.
Power of Ten
How Many Ways
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How Many Ways
The students who bring these concepts into the classroom tend to use them over and over again until they become reference points for further learning. The key for the teacher is not to overreact and to simply ask the student to explain their understanding of the concept they are introducing. If clarification is needed, the teacher assists. If the student’s explanation is too complex then the teacher just moves on and ignores the explanation. There will be another chance later to explain as the student uses the concept over and over and learns how it works.
Examples of Ideas Brought in by Students Marla Margetts’ grade one class had many students using square roots and negative numbers. The following equation was written by Rhys:
(20 x 50) - 1000 + 70 - 90 + 20 + 1 000 000 - 900 - (8 x a decade) = 20
When asked to explain his equation, Rhys said
Finally, he subtracted 8 x (a decade) which was
he got the ideas for 20 groups of 50 from Nolan’s equation where Nolan had said 10 x 50 = 500 so
not what he had written at his desk. While he was waiting his turn to share he remembered the word
Rhys knew that 20 groups of 50 would be 1000. This is what he wanted because in the equation he
‘decade’ means ten because the librarian had told him this a week earlier when she was making up
had written at his desk he had originally started with 1000 - 1000 but 20 x 50 was more creative.
some puzzles. He finally reached his goal number. This equation received eleven points in the criteria table for the following:
Rhys attributed the 70 - 90 to another student in
(1) (1) (1) (1)
the class who had told the class last week that 70 - 90 was -20. At this point he was back to 0 again after making his second set of ‘tricky zeroes’. Next he told Marla that his mother had told him that the square root of 1 000 000 was 1000 and he knew it was true because 1000 groups of 1000 = 1 000 000. Then he subtracted 900 and ended up with 100.
(1) (1) (1) (1) (1) (1) (1)
Addition Multiplication Subtraction A large number greater than the goal number (at least double) Integers Fancy or tricky zeros (he has two, but he only gets one mark) More than two operations More than three terms Square root (worth 2 in new scoring system) Proper use of brackets to avoid misinterpretation Unusual use of a new concept (decade)
11 marks total
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Power of Ten
How Many Ways
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How Many Ways
On the day I observed this grade one class
On the second day of observation, every child
there were six children using fractions and several children using square roots. None of these
in the class had at least three terms in their equations. Most students had multiple operations
concepts are in the grade one curriculum but Marla had the courage to let them use the concepts and
and most used numbers well over their goal number such as 100 or 200. Over half of the
explain to the other children how the concepts work. When the first child brought in square roots she
children had tricky zeros: 100 + 100 - 200 + 100 + 15 - 115 + 20 = 20
might have drawn squares on cm graph paper and noted the sides. Or she could have shown a pattern: the square root of 9 is 3
One child had a fancy way of making one: (159 ÷ 159) + 20 + 50 + 50 - 100 - 1 = 20
the square root of 16 is 4 the square root of 25 is 5
One child was using percent:
and the square root of 100 is 10.
the square root of 400 plus ten percent of 200 (20 + 20)
When the first child brought in percents Marla remembers having demonstrated a pattern like:
take away twenty-five percent of 40 (20 + 20) - 10
10% of 10 =1 10% of 20 = 2 10% of 30 = 3
take away ten percent of 100 equals 20. (20 + 20) - 10 - 10 = 20
and so on.
Marla was astonished as this was the first day she had observed percent in her grade one class.
Rhea Properzi, who piloted an evaluation
In Marg Penny’s grade two/three class,
system using the “How Many Ways” Criteria Sheet for an AISI project in the Pembina Hills School
one of the students often used 10 x 10 x 10 and 10 x 10 x 10 x 10 so Marg decided to show him
Division in Alberta, found that whenever one child in the class used fractions, there would be five or six
the exponential form of writing 103 and 104. She thought he might implement this format but he
other children using fractions as well. If one child used square roots there would be four or five others
did not. A few weeks later other students were using the 10 x 10 x 10 model so she explained the
also using square roots. The same was true of decimal fractions and exponents. Rhea says,
exponential form to the class during sharing time. She then had many students using it, just as Rhea
“Number sense seems to be contagious.”
Properzi’s research suggested.
Here is Callum’s question in Marg Penny’s class on May 11, 2007: (1010) - 10 000 000 000 + (56) + 100 + 3 + 40 - 4 000 = 11 768 Ethan, in the same class wrote: (57) - 67 000 + 600 + 43 = 11 768
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Power of Ten
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How Many Ways
Marg talks about how a small group of grade two/three students got together once they had the power of ten figured out and created a table for the power of five. Marg required them to do the calculations without using a calculator. They incorporated these powers into their work every day. Marg’s only qualification was that they had to be able to read the numbers correctly. Colleen Howard, a grade two French Immersion teacher, has a student, Callum, who has figured out how to meet all the criteria every day in one equation:
100 + 50 + 1 - 152 + 1 + 10 - 1 + 8 - 17 + (106) + (1 ÷ 1) + (1 x 1) + (1 ÷ 2) + 1 + ( 144 x 144 ) + ( 25 2
x 25
) - 1 000 172 + ‘ x ’
where ‘ x ’ = the goal number
Colleen comments that Calum has one part of his equation that he uses every day. It seems to be his anchor. That part is: 10 - 1 + 8 - 17 which is a fancy way to make 0.
This confirms the work of Marla Margetts who suggested in her video “Exploring Student
curriculum. They help students develop personal confidence by letting each student show his
Thinking” that each child seems to create his or her own referents or benchmarks that she goes
thinking. Many of these teachers describe themselves as former math phobics who have
back to often. These benchmarks seem to be a place of safety that can serve as a springboard
been transformed by the “How Many Ways” activities and now possess new awareness of
for future growth*.
number sense and increasing confidence about number. All suggest that it is student enthusiasm
No doubt the reader is shocked at these kinds of equations – so was I. But after fifteen
that drives them to continue into unchartered waters.
years of doing this, I have seen this replicated many times in at least five provinces. I have seen
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In classrooms where this activity is used
it in rural classrooms, urban classrooms, French Immersion classrooms, inner city classrooms and
regularly, alsmost all students start exceeding the curricular expectations. Many students develop
suburban middle-class classrooms. The only common factor is that the teachers all have the
a deep sense of place value and they learn to use all the operations in a context of meaning.
confidence to let the students lead. They do not worry about students who can exceed the
* Video: “Exploring Student Thinking” was developed by the Greater Victoria School District.
Power of Ten
How Many Ways
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How Many Ways
As this idea has been used in more and more classrooms, the criteria list has exploded
Most students who solve the equation 2x + 5 = 31, subtract 5 from both sides as their first
from under ten criteria to almost twenty. Many of the criteria (square roots, exponents, decimal fractions,
step (some students describe it as taking the 5 to the other side but cannot explain whey they can do
common fractions, percent) were not on the original list but have been added as students present them
that). The key is to remember to “do whatever you do to both sides of the equation”. In the
in classes. The original goal of the activity was to have primary students using all four operations
Teachable Moment system these are known as the Principle of Zero and the Principle of Balance.
(+, -, x, ÷) in a context of meaning that was openended.
Replacing 31 - 5 = 26 is the Principle of Equivalence. Successful students use the Principle of One (2 ÷ 2 = 1) as the last step in the problem and divide both sides by 2.
The criteria has grown exponentially; some of the most recent additions demonstrate:
2x + 5 = 31 -5 -5 2x = 26 2x = 26 2 2
a sense of tenness
(9 + 4 + 1 + 7 + 6 = 27)
a sense of hundredness
x = 13
(81 + 54 + 19 + 46 + 37 = 237)
The principles of balance, equivalence,
using doubling or halving
(4 x 5 = 20) (8 x 2.5 = 20) (16 x 1.25 = 20)
one and zero are all that most students will ever need to solve equations. There is no need for cross-multiplying, canceling, inverting and multiplying, moving the term to the other side, or
The “hundredness” criterion was added after watching Colleen Howard’s class do a “Weekly
any number of other tricks for solving equations. The beauty of these four principles is that they
Graph” and a “Making Change” activity and realizing that students who had already developed
do not have many exceptions (balance does not work for inequalities or if you square both sides).
these skills would be better at multiplication facts and adding and subtracting if they could integrate
The other three always work and will allow students to solve most equations they encounter in life.
the “hundredness” ideas into a new context such as the “How Many Ways” activity.
The important thing is that these primary students are using the principle of equivalence every time they do a “How Many Ways” activity and most
Two years ago the ideas of fancy zeros and fancy ones were added after observing in Marla’s class. Later, the implication of the importance of using these ideas became obvious. These two principles are inherent in algebra. When a grade eight student solves the equation 2x + 5 = 31, she uses the principle of one and zero all the time.
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Power of Ten
students are also making fancy zeros. Only a few make fancy ones but this probably reflects the fact that we think division is hard so we do not teach it very much. The Japanese education system teaches division very early because they have noted that children divide much more in their preschool life (share) than they add. Division has great meaning to preschoolers.
How Many Ways
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How Many Ways
The goal of the “How Many Ways”
Note:
activity is to encourage students to explore all the operations in a safe, creative environment.
Teachers have recommended that students use sa blank sheet of paper without the line down the
When this happens many students create their own meaning and understanding of numbers and
middle. The middle line prevents students from writing long complicated questions like those written
how numbers work together in complex patterns. Brackets are introduced as a way to clarify one’s
by Rhys and Callum used previously as examples.
thinking. Other new ideas come from students and often these new ideas exceed the requirements of the curriculum. Many teachers who teach “How Many Ways” using these ideas and tools report that their students are excited about math and ask if this activity can be done daily. Who could ask for more?
How many different ways can you make __________ ?
+
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Power of Ten
-
x
÷
How Many Ways
page 39
How Many Ways
Notes and Sug gestions Sugg
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Power of Ten
How Many Ways
page 40
How Many Ways - Teaching Number Sense and Place Value
MAKING
CONNECTIONS
How Many Wa ys
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How Many Ways - Teaching Number Sense and Place Value
Sustainable Strategies for Teaching Mathematics •
Sustainable strategies is a phrase borrowed from Colleen Politano that I have adapted to mathematics teaching to mean: • Strategies that make teacher preparation more efficient. • Strategies that become part of the teacher’s daily practice. • Strategies that provide familiar routines for the students so that they can take control and manage their own learning. • Strategies that are learner focused.
•
•
One of the main ideas in creating the Power of Ten and Teachable Moment systems was to model teaching strategies that paralleled those used successfully in teaching Language Arts. Some of the successful strategies that I observed for teaching reading include: • Using meaning as a major motivator so children want to learn (board messages, variety of books available, writing journals). • Giving students choice (variety of books available, children choose topics for writing). • Creating a climate of trust by focusing on process, treating errors as jumping off points, teaching strategies, and emphasizing that we learn through hard work and practice over time. • Valuing diversity by asking children what comes next in the story and accepting a wide variety of answers, and by asking children how they arrived at their ideas and how they know what they know.
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Power of Ten
Using time as a tool rather than a restraint by not focusing on speed in reading over meaning (there are very few timed tests in reading and writing). Using sustainable teaching strategies that create a familiar feeling (choosing books, working in variable groups, reading at the same time, reading with buddies, daily board message, and so on). Providing time for meaningful review. Most primary teachers cover almost the whole reading curriculum every week so there are forty reviews over the year.
The parallel structures in the Power of Ten / Teachable Moment system are:
How Many Ways
•
Using meaning as a major motivator so children want to learn (Weekly Graph, student written problems, use of projects such as sports, pets, nutrition, height).
•
Giving students choice (student written problems, children choose the topic for the Weekly Graph, children choose the order in which they do All the Facts sheets and they create the equations in “How Many Ways”).
•
Creating a climate of trust by focusing on process, treating errors as jumping off points, teaching strategies, and by emphasizing that we learn through hard work and practice over time.
•
Valuing diversity by asking children how they solved the problem and then asking if they or anyone else can do the question in a different way.
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How Many Ways - Teaching Number Sense and Place Value
•
• •
•
Using predictable routines that create a sense of control (Weekly Graph, Calendar Time (How Many Ways), All the Facts sheets, Weekly Games day, and Problem Solving days) on a regular basis. Using time as a tool rather than a restraint by not focusing on speed. Focusing on teaching the whole curriculum several times over the year. In traditional math classes many topics are only visited once or twice during the year, so topics like division, geometry and even subtraction get less time even though they are wonderful sources of personal meaning,. Focusing on all operations all year. In traditional programs, subtraction gets left until the middle of the year and division is often hardly taught at all. I estimate that some students finish their primary years having spent 75% of their time adding, 20% subtracting, 4.9% multiplying and 0.1% dividing. Guess what most students can not do well!
The web on page 44 of this document shows a heuristic for teachers to begin to teach strateg-ically and holistically. The great advantage of teaching this way is that the structure is repeated over and over through the year but always with new material that is often student driven. This reduces preparation for the teacher and increases interest and meaning for both the learners and the teacher. When the teacher says, “We are going to study how to add 9 today because of the problem Carla created yesterday” the students feel a sense of ownership. The web on page 44 is focused on the Weekly Graph but several of the main topics (Weekly Graph, Calendar Time, Problem Solving, Projects, and Games) could be put in the center position. In the second web (page 45), Problem Solving is shown as the keystone. The key is that teaching this way focuses the teacher on strategies, continuous assessment and personalizing the curriculum by responding to the students’ interests, errors and creations.
In summary, teachers work hard at personalizing instruction in language arts by focusing on strategies and meaning, while in mathematics teachers are often forced through the structure of the day and the resources to deliver curriculum in a linear fashion. Textbooks and workbooks in all of the new programs are intended to be used selectively as resource materials but it is easy to fall into the old familiar trap of following the book. This often prevents a focus on strategies and personalization.
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Power of Ten
How Many Ways
page 43
How Many Ways - Teaching Number Sense and Place Value
Making Memories Give Choices
Value Diversity
Personalize Meaning
Teach & Assess Over Time
Create a Climate of Respect
All the FFacts acts Problem Solving • Word Wall • Student Written Problems •
(Problem Posing) Ten Frames
Games
+ 9s + 8s + 5s Even Numbers Doubles Multiples of 2, 5, 10 Friendly Numbers Specific Differences such as subtractions that = 2, or = 1
Weekly Graph
• Power of 9, 8, 5 • Friendly Concentration • Set up cooperative groups so
Projects 1) Calendar Time 2) Height 3) Nutrition 4) Whole School Graph
there is time for assessment
Assessment 1) Keep Weekly Record of Best Result on All Facts
Technology Students enter data into spreadsheets and have the computer create a graph
2) Use Criterion Referenced Assessment for Problems 3) Set criteria each term to assess progress
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Power of Ten
How Many Ways
page 44
How Many Ways - Teaching Number Sense and Place Value
Games Assessment Games W eekly Graph
Problem Solving (section 7, page 8)
How Many Ways (section 3, pages 20-21)
All theGames FFacts acts
Make a graph of the countries’ medal standing. (section 6, page 11)
Adding – start with all facts that equal 10 (5 + 5, 6 + 4, 7 + 3, 8 + 2, 9 + 1) (section 9, page 48)
PROBLEM SOLVING Games Games
Check the sum with a calculator.
Pro jects Projects Games
Friendly Concentration (section 10, pages 6-7) Solitaire 10 (section 10, page 10)
Games TTechnology echnology
HowGames Many Wa y s
Olympics (section 6, page 11)
• tenness • breaking up numbers For example, if a student wrote the following problem in grade 3: “In the Summer Olympics, France got 13 gold medals, the USA got 97 and Canada got 3. How many medals did these countries get in all? 13 + 3 + 97 10 + 3 + 3 + 90 + 7 10 + 3 + 90 + 10 = 113
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Power of Ten
How Many Ways
page 45
How Many Ways - Teaching Number Sense and Place Value
Using the Weekly Graph as a Focus Teachers with access to the Teachable Moment should take the time to read Section Five: Weekly Graph (this section is also available for free on the website: www.poweroften.ca).
Start by asking the students, “What questions could we ask?” about the data. Teachers of primary students will have to model these questions at first and may even want to have some typical questions written on chart paper. Eventually all questions should be created by students.
Start by asking children what they would like to learn about their class, school or community and then begin to collect the data. Notice how this parallels the process of science (trial, error, investigation) and therefore reinforces one of the major life skills.
How many boys vo ted all together? voted ted all together? voted How many girls vo
For example, the class might decide to investigate what kind of fruit is the most popular (this might be followed by a survey of what kind of fruit they actually bring in their lunches). This investigation could be part of a collection of data for the Nutrition Project in the Teachable Moment manual (see Projects section, pages 20-33).
If a new student ca me into the class came class,, how would you pr ed ict the student would vo te? pred edict vote? Why?
First collect the data in raw form (this can be done with tallies then converted into a table).
Favourite Fruit
Boys
Which was the most popular for the boys? Which was the most popular for the girls?
What if the student was a boy? What if the student was a girl? This series of questions lays the foundation for probabilistic thinking which is essential for scientific enquiry in the modern world.
Girls
apples oranges grapes other
How many mor moree boys than girls chose “other”? Repeat for other categories.
Favourite Fruit
Boys
Girls
apples
3
3
oranges
2
3
grapes
2
1
other
5
3
Total
Reorganize the data into a bar graph* (this is very easy to organize if the students vote with magnetic cards with their names on the front of the magnet – a model of this is in the Mathematics as a Teachable Moment – Weekly Graph section, page 3).
Total
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Power of Ten
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page 46
How Many Ways - Teaching Number Sense and Place Value
Favourite Fruit
1
2
3
4
5
6
apples
B
B
B
G
G
G
oranges
B
B
G
G
G
grapes
B
B
G
other
B
B
B
B
B
Keep investigating questions, write them as word problems then investigate the answers. A.
How many boys voted all together?
3+2+2+5= B.
How many girls voted all together?
3+3+1+3= C.
How many students voted all together?
6+5+3+8= D.
How many students voted for “other” or “oranges”?
8+5=
Power of Ten
8
G
G
9
10
There are many possible questions. However, looking at these four, what are the possible instructional interventions that the teacher could use to help the students solve the equations? Let students work on the problems. In Kindergarten / Grade One, have egg carton ten-frames available with Power of Ten tenframe cards posted. In Grade Two / Three have the Power of Ten Place Value cards available and Power of Ten cards posted. Note that Power of Ten cards are posted when adding but are not on student desks because students who “count on” the cards lose the cards’ major visual value for adding 5, 8 and 9. Once students learn to use the Power of Ten cards as counters it is difficult to use them as a visual tool for breaking up numbers (which is the major strategy for adding and subtracting in groups). Number lines and rulers have the same problem – once they become habituated as counters, students can not use them for breaking up numbers and adding in groups.
*
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G
7
How Many Ways
In the new Western Canadian curriculum, the students are not required to create graphs from data. This does not prevent teachers from creating graphs. The collection of the data is important because it gives meaning (life) to the numbers and then allows the teacher and students to make connections.
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How Many Ways - Teaching Number Sense and Place Value
All the Facts These forms are available in the Appendix of the Power of Ten Adding and Subtracting manual or in the Assessment and Evaluation section of the Teachable Moment manual.
Do not place egg cartons, Power of Ten cards, rulers or number lines on students’ desks (they can be available at stations or on the wall, but encourage students to visualize the tool in their “mind’s” eye). In Grade One, use the same sheet over several days (one day the teacher may focus on adding 5, the next day on doubles, doubles plus 1, adding nines
When using the All the Facts sheets: Focus on a single strategy (you may review a previous strategy). Have students circle or highlight questions where that strategy will be useful. This helps students learn how
friendly facts, differences involving 5, 8 or 9, differences of 1 or differences of 2, using adding to subtract, using tens when adding to subtract). In Grades Two and Three, use a different sheet each time. Have students mark their own sheet while the teacher reads the answers. For Grades Two and Three, record the number left to learn each time a sheet is done. Sheets will probably be given out two or three times a week until most students know all their adding and subtracting facts.
to scan and builds confidence. Instruct students to do any other questions they can do without counting. This teaches scanning and builds confidence. Do not time the students (do not give more than 10 minutes in early primary and 8 minutes in late primary).
Connecting the equations from the graph to the All the Facts sheets: A=3+2+2+5
B=3+3+1+3
C=6+5+3+8
D=8+5
Group to get fives
Doubles 3+3
Adding 5s or 8s or doubles plus 1. Focus on one strategy only and if possible review a strategy.
Adding 5s or adding 8s
Use Power of Ten cards to show how to add five
Reorganize as 6 + 4 and circle the “friendly numbers”
Use Power of Ten cards to show how to add 5 or 8
Use Power of Ten cards to show how to add 5 or 8
Circle all the 5s on one of the All the Facts forms
Circle all doubles on one of the All the Facts forms
Circle all doubles plus 1, or circle 5s or 8s on one of the forms
Circle 5s or 8s on one of the forms
All the strategies above could be used on different days in the same week. This ensures connections to the Weekly Graph and gives meaning to the numbers.
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Power of Ten
How Many Ways
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How Many Ways - Teaching Number Sense and Place Value
Connecting the Equations from the graph to “Calendar Time” and “How Many Ways” A=3+2+2+5
B=3+3+1+3
C=6+5+3+8
D=8+5
Group to get fives 5+2+5
Show using groups 3x3+1
Make into doubles 6+8+8=8+8+6
Rewrite as 5+8
Reorganize as 5+5+2
Reorganize as 6+4
Reorganize as 2x8+6
Reorganize as 5 + (5 + 3)
Reorganize as 3 + (2 + 8)
Write as “groups of” 2 x 5 +2
Students who do a “How Many Ways to Make _____” activity regularly should be aware of the criteria for marking (see Section 9, page 65).
Connecting the Equations from the graph to “Games” A=3+2+2+5
B=3+3+1+3
C=6+5+3+8
D=8+5
Play Power of Five which is an adaptation of Power of Ten
Reorganize as 6+4 and play Friendly Concentration
Play Power of Five or Power of Eight
Play Power of Five or Power of Eight
Play Friendly Concentration because 5 + 5 = 10
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Power of Ten
How Many Ways
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How Many Ways - Teaching Number Sense and Place Value
Using the Power of
Ten Place Value and Problem Solving Cards with the Weekly Graph equations generated previously
Equation A Step one:
Equation C
A=3+2+2+5
Step one:
yields 5 + 2 + 5 yields 10 + 2 trade in 3 + 2 for 5 trade in 5 + 5 for 10 put together to make 12
yields 6 + 8 + 8 yields 6 + 10 + 6 yields 10 + 2 + 10 yields 20 + 2 = 22 trade in 5 + 3 for 8 trade in 8 + 8 for 10 + 6 trade in 6 + 6 for 10 + 2 to make 12 trade in 10 + 10 for 20
Connection to All the Facts sheets: teach doubles or friendly numbers Connection to games: Friendly Concentration
Connection to All the Facts sheets: teach doubles Connection to games: Doubles Challenge
Equation B Step one:
C=6+5+3+8
B=3+3+1+3
Equation D
yields 6 + 4 yields 10 trade in 3 + 3 for 6 trade in 6 + 4 for 10
Step one:
Connection to All the Facts sheets: teach friendly numbers Connection to games: Friendly Concentration
D=8+5 yields 8 + 2 + 3 yields 10 + 3 trade in 3 + 2 for 5 put together to make 13
Connection to All the Facts sheets: teach adding 5 or adding 8 Connection to games: Power of 5 or Power of 8
For more examples of how to use the Place Value Cards see Numeracy, section 4 – pages 34-39 and Special Days and Seasons Project, section 8 – pages 38-43 of Mathematics as a Teachable Moment.
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Power of Ten
How Many Ways
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How Many Ways - Teaching Number Sense and Place Value
All the Facts Whenever a new word is used it should be added to the mathematics word wall. When students write or speak problems bonus marks (non-recorded) should be given for using a mathematical word.
A=3+2+2+5
B=3+3+1+3
C=6+5+3+8
D=8+5
Add the word “altogether” to the word wall.
Add the word “altogether” to the word wall. Add the word “doubles”.
Add the word “altogether” to the word wall.
Add the word “altogether” to the word wall.
Encourage students to write problems
Encourage students to write problems
Encourage students to write problems
Encourage students to write problems
Make connections for the Family of Facts for 8 + 5*
8 + 5 = 13 5 + 8 = 13 13 - 5 = 8 13 - 8 = 5
* Cover-Up Method for Teaching Families Write
8 + 5 = 13
Cover up 5, then 8 with your hand and say and write
5
8
+
5
=
Cover up 13, then 5 and say and write
8 +
5
8
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Power of Ten
+
13
13 - 5 = 8
= 13
Cover up 13, then 8 and say and write
8
8 + 5 = 13
13 - 8 = 5
5 = 13
How Many Ways
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How Many Ways - Teaching Number Sense and Place Value
Teaching Subtraction Teachers need to be mindful that subtraction is often unconsciously avoided. This may be because of the way teachers were taught. Subtraction always follows addition in textbooks and workbooks. This implies subtraction should not be taught early in the year. Try to focus on subtraction every week. Students relate to subtraction in their personal life because they often lose things or their E. siblings “borrow” things such as Halloween candy which leaves them with less than they had when they started. Using the previous data it is possible to create subtraction problems. F.
Five boys voted for “other”. Three boys voted for “apples”. How many more boys voted for “other” than for “apples”?
5-3 Two boys voted for “grapes”. One girl voted for “grapes”. How many fewer girls voted for “grapes” than boys?
2-1 G.
How many more students voted for “other” than voted for “oranges”?
8-5 H.
Twelve boys voted altogether. Five voted for “oranges”. How many didn’t vote for “oranges”?
12 - 5 I.
Two boys voted for “grapes”. One girl voted for “grapes”. Find the difference between the number of girls and boys who voted for “grapes”.
2-1 J.
Five boys voted for “other”. Three boys voted for “apples”. Compare the number of boys who voted for “other” to the number who voted for “apples”.
12 - 5
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Power of Ten
How Many Ways
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How Many Ways - Teaching Number Sense and Place Value
All the Facts These forms are available in the Appendix of the Power of Ten Adding and Subtracting manual or in the Assessment and Evaluation section of the Teachable Moment manual. Before using the All the Facts sheets refer to the principles outlined earlier in the Adding Section.
Compare two cards. For 9 - 6, show both cards and ask students what would have to be done to make the two cards the same (so they won’t be different). Relate this to the family of facts and adding on.
Using the Power of Ten cards to subtract: Use the cover up method. If the beginning number is greater than 9 (10, 11, 12 ... 18) hold up a ten card and another card to make the desired number (e.g. for 12, hold up a 10 card and a 2 card).
1234567 1234567 1234567 1234567 1234567 1234567 1234567 1234567 1234567 1234567 1234567 1234567 1234567 1234567 1234567 1234567 1234567
Use a blank power of ten cutout for the number being subtracted (this only works well for 5, 8, and 9). For 12 - 5, hold up a 10 and a 2 then take a blank five-strip and cover up 5 squares on the 10 card (the student will now see 7 red squares left so 12 - 5 = 7). This method only works well for subtracting 5, 8 or 9.
Making tens can be useful. For 13 - 7, show a 13 and a 7 and note that one easy way to make 7 into 13 would be to add 3 (make a 10) and add 3 more. 13
7
1234567 1234567 1234567 1234567 1234567 1234567 1234567 1234567 +3 1234567 1234567 1234567 1234567 1234567 1234567 1234567
So 12 - 5 = 7
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Power of Ten
How Many Ways
7
3
So 13 - 7 = 3 + 3 = 6
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How Many Ways - Teaching Number Sense and Place Value
Use family of facts. Ask if students know an addition question that will help them out. For 13 - 7 many students use doubles. They reason that 7 + 7 = 14 so 7 + 6 = 13 and then use fact families to deduce 13 - 7 = 6.
Use a number line. Relate the previous process to a number line. Notice that this is counting on in groups rather than counting on in ones which is very inefficient and open to errors.
13 - 7 = 6
10
7
13 7
13
10
7
14
12 - 5 = 7 5
©
10
12
E=5-3
F=2-1
G=8-5
H = 12 - 5
Notice that the difference is always 2. Find other differences of 2 such as: 6 - 4 7-5 8-6 9-7 10 - 8
Notice that the difference is always 1. Find other differences of 1 such as: 6 - 5 7-6 8-7 9-8 10 - 9
Subtracting 5s is easy on the Power of Ten ten-frame cards.
Subtracting 5s is easy on the Power of Ten ten-frame cards.
Use Power of Ten cards to compare 5 and 3 and the other examples above
Use Power of Ten cards to compare 2 and 1 and the other examples above
Use the cover-up method or the comparison method
Use the cover-up method, tens method, or number line
Circle all the questions that have a difference of 2 on one of the All the Facts sheets
Circle all the questions that have a difference of 1 on one of the All the Facts sheets
Circle all the 5s on one of the forms. Do any other questions you know without counting.
Circle all the 5s on one of the forms. Do any other questions you know without counting.
Power of Ten
How Many Ways
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How Many Ways - Teaching Number Sense and Place Value
How Many Ways E=5-3
F=2-1
G=8-5
H = 12 - 5
The difference is always 2. Extend the pattern of differences of 2 beyond 20. e.g. 20 - 18 = 2
The difference is always 1. Extend the pattern of differences of 1 beyond 20. e.g. 20 - 19 = 1
The difference is always 3. Extend the pattern of differences of 3 beyond 20. e.g. 20 - 17 = 3
Show how to subtract 5 by breaking it into 2 and 3 so
120 - 118 = 2 30 - 28 = 2 and so on.
120 - 117 = 3
120 - 119 = 1 30 - 29 = 1 and so on.
30 - 27 =3 and so on.
12 - 5 = 12 - 2 - 3 = 7 Try it with 21 - 5 = 21 - 1 - 4 or 33 - 5 = 33 - 3 - 2
Make patterns like 15 - 13 and extend
Make patterns like 32 - 31 and extend
Make patterns like 128 - 125 and extend
Make patterns like 22 - 15, 32 - 25 and extend
Use equivalent expressions and brackets (10 ÷ 2) - (2 + 1) = 2
Use equivalent expressions and brackets (10 ÷ 5) - (20 ÷ 20) = 1
Use equivalent expressions and brackets (16 ÷ 2) - (10 ÷ 2) = 3
Use equivalent expressions and brackets (10 + 2) - (10 - 5) = 7
Games Using Power of Ten Cards
©
E=5-3
F=2-1
G=8-5
H = 12 - 5
Play Salute
Play Salute
Play Salute
Play Salute
Play More / Less / Different / Compare* (see Games,
Play More / Less / Different / Compare* (see Games,
Play More / Less / Different / Compare* (see Games,
Play More / Less / Different / Compare* (see Games,
section 10, page 27)
section 10, page 27)
section 10, page 27)
section 10, page 27)
Power of Ten
How Many Ways
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How Many Ways - Teaching Number Sense and Place Value
Problem Solving Whenever a new word is used it should be added to the mathematics word wall. When students write or speak problems bonus marks (non-recorded) should be given for using a mathematical word.
Teachers should also model questions such as:
Mary has 5 candies. Bill has 12 candies. How many does Mary need to have the same number of candies as Bill?
Subtraction is very rich in words. Teachers should use the words more, less, different, compare in different contexts.
Mary had 12 candies. She lost 5 candies. How many candies does Mary have left? Mary has 5 candies. Bill has 12 candies. Compare the amount Bill has to Mary’s. Mary has 5 candies. Bill has 12 candies. Find the difference.
©
E=5-3
F=2-1
G=8-5
H = 12 - 5
Add the words:
Add the words:
Add the words:
Add the words:
more less different compare
more less different compare
more less different compare
more less different compare
to the word wall
to the word wall
to the word wall
to the word wall
Encourage students to write other problems
Encourage students to write other problems
Encourage students to write other problems
Encourage students to write other problems
Power of Ten
How Many Ways
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How Many Ways - Teaching Number Sense and Place Value
Assessment and Evaluation The following is a very brief account of what is available in the Mathematics as a Teachable Moment series. The Manual is 600 pages and includes an extensive discussion regarding why it is important NOT to time students with strict time limits. There is always a limited amount of time – just do not have races – unless, of course, this is the way you teach reading and art!
Evaluating All the Facts
GRADE THREE: All students with few exceptions
Once per week issue a new sheet, give students approximately eight minutes in Grade 3 and ten minutes in Grade 1 to work on as many questions as they can without counting. Record the number left to learn. When students have mastered one of the All the Facts Addition sheets, have them move to Subtraction, then Multiplication and finally Division as they master each operation. KINDERGARTEN: Only exceptional students
would have access to an All the Facts sheet.
will be able to do an Addition sheet and a Subtraction sheet. All will begin work on Multiplication sheets (be sure not to send Multiplication sheets home or parents may start drilling the times tables) and many students will be able to get over 50 correct. Some students will get over 60 correct and a few will get all 66 correct. I have seen classes where most students had all 66 multiplication facts correct. All students will work on Division and some will get it all correct (especially true if Families of Facts are emphasized).
GRADE ONE: Some students would be able to
complete an All the Facts Addition sheet without counting (no time limit). Usually I allow ten minutes but occasionally let students have longer if they are still interested. GRADE TWO: All students, with few
exceptions, will be able to do an Addition sheet and most will do a Subtraction sheet. A few will start work on Multiplication sheets (be sure not to send multiplication sheets home or parents may start drilling the times tables).
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Power of Ten
How Many Ways
All the Addition Facts You Ever Need to Know (A) 8 + 8=
6 + 0=
2 + 9=
4 + 0=
8 + 3=
6 + 9=
7 + 7=
2 + 1=
6 + 8=
6 + 3=
8 + 1=
5 + 5=
3 + 2=
3 + 4=
1 + 4=
7 + 5=
1 + 1=
4 + 9=
4 + 7=
4 + 4=
7 + 9=
2 + 2=
3 + 3=
1 + 3=
9 + 9=
5 + 6=
0 + 1=
1 + 7=
0 + 9=
2 + 5=
6 + 7=
2 + 0=
3 + 9=
4 + 5=
8 + 0=
2 + 8=
0 + 3=
1 + 5=
2 + 4=
7 + 0=
5 + 8=
4 + 6=
3 + 5=
2 + 6=
5 + 0=
6 + 1=
2 + 7=
1 + 9=
4 + 8=
5 + 9=
6 + 6=
8 + 9=
7 + 8=
3 + 7=
Name
Column 1
Column 2
Column 3
left to learn
Column 4
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How Many Ways - Teaching Number Sense and Place Value
Evaluating Problem Solving
Beside each student’s name, record the level and sophistication of the student’s work under one of the heading.
Once per week write three equations on the board. Instruct the students to pick an equation and write a story.
4
In Grade 1 do this orally in the fall, then move toward written stories: 8-3= 10 - 6 = 13 - 8 = In Grade 3: 13 - 8 = 24 - 16 = 124 - 117 =
Exceeds expectations
Uses mathematical language extensively Chooses the most difficult equations The story makes sense The student knows the solution to the problem
3
Meets expectations
Uses some mathematical language Chooses the simple/medium equations
Adjust the level of the questions as the class improves, always having one relatively easy, one difficult and one in between. As your students become more sophisticated, mix up the operations and include groups (multiplication) and sharing (division).
The story makes sense The student knows the solution to the problem
2
Almost meets expectations
Seldom uses mathematical language Chooses the easiest equations
Mark records as follows. Use a class list with the following headings:
The story almost makes sense (little things are missing) The student knows the solution to the problem
Writes a story for adding Writes a story for subtracting Writes a story for multiplying Writes a story for dividing
1
Does not meet expectations
No mathematical language used Chooses equations randomly The story makes no sense The student does not know the solution to the problem
KINDERGARTEN: all work will be done as a group with one or two exceptional students creating their own stories.
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Power of Ten
GRADE ONE: students should be able to write or verbally create an addition story and some able to write or verbally create one for subtraction. Only exceptional students will have multiplication or division stories.
GRADE TWO: students should be able to write or verbally create addition and subtraction stories. Some students will have multiplication or division stories.
How Many Ways
GRADE THREE: students should be able to write or verbally create addition and subtraction stories. Most students will have multiplication or division stories.
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How Many Ways - Teaching Number Sense and Place Value
Evaluating Written Problems and Solutions Review the evaluation criteria below for writing and solving problems. Start collecting student-written problems and use these as starting points in your class. Discuss the criteria for a good problem with the class; and eventually show them how to display their work when solving a problem.
Writes an addition problem
Name Alice
4, 4, 4
Solves an addition problem
3
Start an evaluation sheet for problem solving and enter data as you collect a written problem or a solution. The blank spaces indicate those students who have not yet contributed. When a student contributes more than once, enter the data as shown on the following sample.
Writes a subtraction problem
Solves a subtraction problem
Writes or solves multiplication problem
3
Bethany Charles
2
Dana
1
Ellison
3, 4
2
Francine
Mark 4
• • • •
3
• • • •
2
• • •
1
• • •
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Power of Ten
Student Solution to a Problem
Student Written Problem
(may be verbal in K/1/2 or with some exceptional students)
The student demonstrates excellent use of mathematical language. The student chooses a difficult question. The problem makes sense. The student knows the solution to the problem.
• • •
The steps to reach a solution are shown. The solution route is easy to follow. The question is answered in a complete sentence that reflects understanding.
The student demonstrates some use of mathematical language. The student chooses a simple/medium difficult question. The problem makes sense. The student knows the solution to the problem.
• • •
The steps to reach a solution are shown. The solution route may be difficult to follow. The question is answered but the sentence is not complete or may not show understanding.
The student demonstrates no use of mathematical language. The problem almost makes sense (little things are missing). The student knows the solution to the problem.
• • •
The steps that may reach a solution are shown. The solution route is difficult to follow. The answer to the question shows some understanding but not complete understanding (the answer may be incorrect).
The student demonstrates no use of mathematical language. The problem does not make sense. The student does not know the solution to the problem.
• • • •
No steps to reach a solution are shown. The solution route is impossible to follow. If there is an answer, it makes no sense. There may be no work to show.
How Many Ways
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How Many Ways - Teaching Number Sense and Place Value
How Many Ways Sheets At least once per week have students record all the sentences that they can for the date. Share sentences with the class and mark the class using the rubric on pages 20-21 of this section. The criteria are:
Students may get more than one mark for a single equation. For example, the following equation written by a student would get 11 marks because it meets the following criteria: (20 x 50 - 1000) + (159 ÷ 159) - 1 + (
1) Give one mark for one of the following (if a student has two equations that meet the same criteria, she still receives only one mark). The student gets one mark for a sentence containing or showing:
• • • • • • • •
• • • • • • • •
©
The addition operation The subtraction operation The multiplication operation The division operation More than two terms (e.g. 4 + 6 + 20 = 30) Two or more operations (e.g. 5 x 5 + 5 = 30) Numbers that substantially exceed the goal number (e.g. 159 - 129 = 30) Evidence of a pattern (3 or more equations that can be linked by a pattern such as: 27 + 3 = 30 28 + 2 = 30 29 + 1 = 30 Knowledge of the Power of Zero (e.g. 30 + 100 - 100 = 30) Knowledge of the Power of One (e.g. 29 + 100 ÷ 100 = 30) Knowledge of the Commutative Principle (e.g. 20 + 10 and 10 + 20) Knowledge of the Standard Form of the number (20 + 4 = 24 or 2 x 10 + 4 = 24) Knowledge of the use of brackets (5 x 20) - 70 = 30 Uses “tenness” Uses “hundredness” Any proper use of exponents, square roots, or fractions
Power of Ten
•
• •
+ ) + 29 = 30
The addition operation The subtraction operation The multiplication operation The division operation More than two terms (e.g. 4 + 6 + 20 = 30) Two or more operations (e.g. 5 x 5 + 5 = 30) Numbers that substantially exceed the goal number (e.g. 159 - 129 = 30) Evidence of a pattern (3 or more equations that can be linked by a pattern such as: 27 + 3 = 30 28 + 2 = 30 29 + 1 = 30 Knowledge of the Power of Zero (e.g. 30 + 100 - 100 = 30) Knowledge of the Power of One (e.g. 29 + 100 ÷ 100 = 30) Knowledge of the Commutative Principle (e.g. 20 + 10 and 10 + 20) Knowledge of the Standard Form of the number (20 + 4 = 24 or 2 x 10 + 4 = 24) Knowledge of the use of brackets (5 x 20) - 70 = 30 Any proper use of exponents, square roots, or fractions
The above equation is not totally unusual. Every part it was taken from a How Many Ways sheet written in Marla Margett’s Grade 1 class.*
How Many Ways
*
To obtain a copy of this video, e-mail
[email protected]. The video is 20 minutes in length and illustrates what can be achieved by using Calendar Time creatively and allowing students to “fly”.
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How Many Ways - Teaching Number Sense and Place Value
The main idea is not to get a lot of equations but rather a few equations of high quality that show a lot of mathematical knowledge. A sheet with the following equations would not show a lot of knowledge but would show a lot of work. 29 + 1
28 + 2
27 + 3
26 + 4
25 + 5
24 + 6
23 + 7
22 + 8
21 + 9
20 + 10
19 + 11
28 + 12
Even though there are 12 equations, this student would only receive two marks for meeting the following criteria: • • • •
The addition operation The subtraction operation The multiplication operation The division operation More than two terms (e.g. 4 + 6 + 20 = 30) • Two or more operations (e.g. 5 x 5 + 5 = 30) • Numbers that substantially exceed the goal number (e.g. 159 - 129 = 30) Evidence of a pattern (3 or more equations that can be linked by a pattern such as: 27 + 3 = 30 28 + 2 = 30 29 + 1 = 30 • Knowledge of the Power of Zero (e.g. 30 + 100 - 100 = 30) • Knowledge of the Power of One (e.g. 29 + 100 ÷ 100 = 30) • Knowledge of the Commutative Principle (e.g. 20 + 10 and 10 + 20) • Knowledge of the Standard Form of the number (20 + 4 = 24 or 2 x 10 + 4 = 24) • Knowledge of the use of brackets (5 x 20) - 70 = 30 • Any proper use of exponents, square roots, or fractions
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Power of Ten
Once per month, assign a goal number (it could be the date, a student’s birth date, or a random choice), then give the students ten minutes and have them work on their own, writing as many equations as they can on a blank sheet. Mark each student’s sheet using the rubric below with a 4-3-2-1 criterion reference system similar to the one for problem solving. The minimum guidelines for each grade are included in the table below.
(1) Below expectations
(2) Just meeting expectations
(3) Meeting expectations
(4) Exceeding expectations
N/A
N/A
2 of the criteria
4 of the criteria
Gr. 1
4 of the criteria
Gr. 2
6 of the criteria
Gr. 3
7 of the criteria
K*
* only exceptional students try
These criteria have been applied consistently over two years across the Pembina Hills School District in Alberta and appear to have reliability and validity.
How Many Ways
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How Many Ways - Teaching Number Sense and Place Value
TEA CHING NOTES EACHING
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Power of Ten
How Many Ways
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