Sample for self learning and self evaluating Program HCF&LCM

Sample for self learning and self evaluating Program

*This program is designed for recapitulation of your concepts of Prime and composite numbers which you have already learned through this episode “Playing with Numbers.” **After completing this program you will be able to : 1. Differentiate between PRIMES and COMPOSITES. 2. Factorise Composite numbers. 3. Use the principal of prime factorization in finding the H.C.F and L.C.M of numbers *** You will feel that some of the steps are very simple, but do not omit any since all are essential. ****Before working on the next frame check your answer by comparing it with the answer given in right column of the next frame. **** If you note that your answer is correct then go to the next frame otherwise go back to the previous frame(s) Go to the next frame:
Multiplication is one of the four fundamental operations of Arithmetic which assign a third number to a given pair of numbers: 5 x 7 = 35 6 x 8 = 48 6 x 9 = -----
If you said 54, you are correct. Try more: 2 x 27 = -------	54
Did you get 54 again? Both 6 x 9 and 2 x 27 are equal to 54. 54 is called the ------------ of 6 and 9. It is also the -------------- of 2 and 27.	54
We know that 2 x 27 = 54 is a multiplication fact in which 54 is called the -------------. 2 and 27 are called the ------------- of 54. In the same way in the multiplication fact: 6 x 9 = 54, ---, ----- are the ----------- of 54 and 54 is the -------------- of 6 and 9.	Product Product
By now we may note that ------,-------, ------- and ------ Are all factors of the number 54. Note that the number 54 is exactly divisible by each of the numbers -------------------and ----	Product Factors 6,9 factors product
This in fact the definition of a FACTOR. A factor of a number is any number that will divide into it ----------.	6,9,2 27 6,9 ,2 27
3 and 18 will also divide in to 54 exactly. Therefor3 and 18 are also the ---------- of 54.	Exactly or evenly.
Can you think of still 2 more factors of 54? They are -------- and ------	Factors
Recall that 1 and number itself is always a factor of a number. Note that the list of all possible factors of the 54 is ------------------------------------------. List all the factors of 36 -------------------------------------------------------	1 54 1, 2, 3, 6, 9, 18, 27 and 54.
For practice list all the factors of: 5 ---------------- 12 -------------- 17 ----------- 98 --------------------- 172 ------------------	1, 2 ,3, 4, 6, 9,12, 18, 36
Again notice that ------- and ---------- are the factors of each of the above numbers. Can you identify a unique number which is an exception? ---- is a positive number which exactly one factor. Hence, we can say that every whole number other than ------------ has at least 2 factors.	1,5 1,2,3,4.6.12 1,17 1,2,7,14,49,98 1,2,4,43,86,172
There are some numbers which have exactly 2 distinct factors: ‘1 and number itself’. E.g. numbers 5 and 17 in the above frame have exactly two distinct factors. List all the numbers less than or equal to 20 which have exactly 2 distinct factors. -----------------------------------------------------	1 Number itself 1 1
Numbers which have exactly two factors are called __________ numbers. Which of the following are prime numbers: 13,18,25,31,36,89,111,321,509,385	2,3,5,7,11,13,17,19
Again recall that a prime number has only two factors i.e. ----- and ------------------------.	Prime numbers 13,31,89,509
List all primes which are less than or equal to 30. ----------------------------------------------------------	1 and number itself
Till now we have learned about two types of numbers. One is unity and others are ----------. Unity has exactly ------- factor and prime numbers have --------------------------.	2,3,5,7,11,13,17,19,23, 29
Therefore, a number which is neither unity nor prime must have at least one more factor other than 1 and number itself. Such numbers are called ‘COMPOSITE NUMBERS’	Primes One Exactly two
Numbers with more than two distinct factors are ------------- Number 6 has the factors ------------------. Therefore, 6 is a ----------------- Number
List all the numbers less than 1000 and greater than 99 which have more than two factors: -------------------------------------------------------------------------------------------------------------------	Composite Numbers. 1,2,3,4,5,6 Prime/Composite
To classify the numbers of a given set in the categories of ‘Primes’ and composites we must be able to discover the factors of the number. For example to identify the number 63 as prime or composite, we must list all possible ------------ of 63.	100,102,104,105,106,108-------
Since possible factors of 63 are ----------------- Therefore, 63 is a ----------------- List all possible factors of the following composite factors: 25: --------------------- 77: --------------------- 38: ---------------------- 49: -----------------------	Factors
Note that primes are appearing as the factors of composite numbers. In fact, it is possible to write all composites as product of primes, as we may write 36 = 2 x 18 2x18 = 2x2x3x3 3x12 3x12 = 3x2x2x3 4x9 but 4x9 =2x2x3x3 6x6 6x6 = 2x3x2x3 This shows that a composite number can be expressed as product of prime numbers in a unique form. However order of writing primes may vary’ This process of expressing a composite number as a product of primes is called Prime Factorisation.	1,3,7,9,21 and 63 Composite number. 1,5,25 1,7,11,77 1,2,19,38 1,7,49
Prime factorization of 40 is: 40 = ----------------------------------------
While expressing a number in prime factorization form, we must use the divisibility rules. For example, for testing the divisibility of a number by 2, we are to look only at its digit at the ones place. If it is even or zero, then it is divisible by 2.	2 x 2 x2 x 5
Test the divisibility of the following numbers by two: 8,22,39,756,1030,11067 Numbers divisible by 2 are: ----------------------------------------------
Another number to divide by is’5’. For example: The numbers 10,25,40,5675,6000,9875 are all divisible by 5, since, digit at their ones place is ------- or --------. Thus a number is divisible by 5 if its ones digit is --------- or --------.	8,22,756,1030
Similarly a number is divisible by 10 if its ones digit is ---------. Using the divisibility rule identify the numbers which are divisible by : (a) 2 (b) 5 (c) 10 32578, 9735,6520, 33789, 2480, 78421 (a) ------------------------- (b) ------------------------- (c) -------------------------	0 5 0 5
Division by 3 also has a simple rule. Following are the numbers divisible by 3: 12, 8361, 222, 987, 1002. Which of the following are divisible by 3: 105,211,40569, 60021,3046	0 (a)32578,6520,2480, (b)9735, 6520,2480 © 6520, 2480
Did you spot the rule for division by 3? It is: If the sum of the digits of a number is divisible by 3, then the number is also divisible by 3. Using the rule, which of the following are divisible by 3: 32001,85416,3727,79602,5428	105,40569,60021
Some other rules for testing the divisibility of the numbers by 4,6,8,9 and 11 are already known to you, and you may record these rules in the following 5 frames:	32001,85416, 79602
Rule for divisibility by 4:
Rule for divisibility by 6:	A number is divisible by 4 if the number formed by the last two digits i.e. ones and tens digit is divisible by 4.
Rule for divisibility by 8:	A number which is divisible by 2 and 3 both is divisible by 6
Rule for divisibility by 9:	A number is divisible by 8, if the number formed by the last 3 digits i.e. ones tens and hundreds is divisible by 8
Rule for divisibility by 11	A number is divisible by 9, sum of its digits is divisible by 9
Note that each of the following 3 digit numbers is divisible by 11: 396,495,253,176,462 Without dividing or using the divisibility test stated above, can you say that 891 is/is not divisible by 11? --------------------------------------------------	A number is divisible by 11, if the difference of the sum of its digits at odd places and at the even places (starting from the ones place) is either zero or a multiple of 11
Did you spot some special characteristic of 3-digit numbers divisible by 11? A 3 – digit number is divisible by 11, if its middle digit is equal to the sum of the other two digits. e.g. 374 is divisible by 11 since its middle digit 7 = sum of its other two digits i.e. 3 and 4.	Yes, 891 is divisible by 11.
Further note that 946 is divisible by 11(since 946 = 11 x 86) even though 6 +9 does not equal to 4. Can you examine the divisibility of 627 using test stated above for divisibility of 3 digit numbers by 11? -----------	374 = 11 x 34
The above two examples do not fit the rule, for the sum of two outside digits is greater tha9. Hence the correct rule for testing the divisibility by 11 is: ----------------------------------------------------------------	No
Now for practice: Prime factors of a. 296 are ------------------------------ b. 1001 are ---------------------------- c. 342 are ----------------------------	A number is divisible by 11, if the difference of the sum of its digits at odd places and at the even places (starting from the ones place) is
A number that is neither a unit nor a prime must have ------------------------------ factors.	a. 2 and 37. b. 7, 11 and 13. c. 2, 3 and 19.
All of the following numbers are divisible by -------- and hence are ---------------- numbers. 396,495,276,462.	more than two factors.
For practice express the following in prime factorization form: 296 = ------------------- 1001 = ------------------ 342 = ------------------	3 composite
Referring to the divisibility test for divisibility by 6, can we say that a number which is divisible by 2 and 4 both will always be divisible by 8? If no give an example to justify your answer. ------------------------------------------------------	296 = 2 x 2 x 2x 37 1001 = 7 x 11 x 13 342 = 2 x 3 x3 x 19
In fact ‘A number which is divisible by two co prime numbers then it is divisible by their product also’. By Co Prime we mean ‘Pair of numbers which do not have any common factor other than 1’. In this case numbers 2 and 4 are not ---------------. Hence 20 which is divisible by 2 and 4 both is not divisible by --------.	No, it is not true as number 20 is divisible by 2 and 4 both but not by 8.
Similarly following are the two more important facts pertaining to divisibility rules: 1. If a number is divisible by another number then it is divisible by each of the factors of that number. 2. If two given numbers are divisible by a number, then their sum and difference are also divisible by the same number. For example: *Number 60 is divisible by 15 and 3, 5 are the two factors of 15. We may note that ----- and ------- both are the factors of 60 also. ** Numbers 65 and 105 are both divisible by 5. We may note that 105 + 65 = ----------- And 105 – 65 = ------------ are also divisible by ---------------.	Co Prime 4
Do you think that every composite number will always contain the only primes 2, 3, 5 and 11? Justify your answer by giving an example.	3 5 170 40 5
For expressing 91 in prime factorisation form we use hit and trial method. Under this technique we can determine that 13 is a prime with only ------ and --------- as factors. We may further see that we need not try with a prime greater than 13 since in this case the other factor 7 is also prime.	No. for example prime factorization of 91 is 7 x 13
By testing with primes smaller than 37, we can tell that 37 is/is not a prime number.	1 13
Oh! Good, you have determined that 37 is prime. For doing so, did you try with each prime number smaller than 37? For example did you try with 31?	Is
Of course you did not have to test whether 31 is a factor 37. It is obviously too ----------.	No
How about 19? It is/is not necessary to try 19 as a prime factor of 37 because it is too large. Which of the following are too large to bother testing? 5 17 13 23 29 7	Large
To see why this is so, remember that if a prime is a factor of a number it is one of a pair of factors of the number. For example, if 7 is a factor of 21 then ----- is also a factor.	Is not 17, 13, 23, 29
If we would have tried 3 as a factor of 21, we would have known that ------------ was a factor also. So it was not necessary to try 7 as a factor.	3
Since factors occur in pairs, it is easier to test the ----------- of the two factors first.	7
Remember that since both factors must be whole numbers, we can rule out any pair of factors in which one of them is not a --------------------.	Smaller
Now let us try a number for factor pairs. Complete the following table for possible factors: Possible prime factor Other factor 2 x No whole number = 49 3 x = 49 5 x = 49 7 x = 49 11 x = 49 13 x = 49 17 x = 49	Whole Number
Note that as one prime factor gets larger the other factor gets -------------- and vice versa. Hence for testing primes as possible factors of a number, the only once we need to test are smaller than or equal to the ----------------------- of the factors. For example for testing 53 as prime or composite we need to consider is -----------	No whole number No whole number 7 No whole number No whole number No whole number
To see if 293 is prime, the largest prime we need consider is: -------------.	smaller Repeating factors 7
State in which of the following number statements prime factorization has been done; a. 32 = 2 x 2 x 8 b. 68 = 2 x 2 x 17 c.90 = 2 x 3 x 3 x 5 d. 45 = 3 x 15	17 (since 17 x 17 = 289 is very near to 293)
Express the smallest of 4 digit numbers in prime factorization form. -----------------------------------------------	b and c
Determine if 52110 is divisible by 15? Give reasons in support of your answer.	1000 = 2 x 2 x 2 x 5 x 5 x 5
Use divisibility test to decide if the sum of two consecutive odd numbers is divisible by 6. Justify your answer with the help of an example.	Yes, since it is divisible by 3 and 5 both and 3 and 5 are co prime.
Find the common factors of 12 and 35. Can we say that numbers 12, 35 are co prime numbers?	No. For example: Two consecutive odd numbers are 3 and 5. Their sum 8 is not divisible by 6.
Write the first 3 common multiples of 12 and 15. What pattern do you observe?	Common factors of 12 and 35 are: 1 only. Yes 12 and 35 are Co primes.
If you did this correctly, you have succeeded in achieving the objectives stated at the start of the program. Congratulations! Proceed further with full confidence.	First three common multiples of 12 and 15 are: 60, 120, 180 Each of the next common multiple appears in the table of first common multiple i.e. 60.

	Let us gain more knowledge of Divisibility Tests Recall that we used: the divisibility tests for 2 and 3 to check divisibility by 6. We cannot, however, use the test for 2 twice to check for divisibility by 4. Using 2 and 3 works because they are Co Primes, that is, the only factor they have in common is 1. The test for divisibility by 2 can be modified for testing divisibility by 4 and 8. Here's how it works. Since 4 does not divide 10, but 4 does divide 100, rewrite the number, such as the five-digit number 24512, into two parts: 24512 = 24500 + 12; = 245 x 100 + 12; = 245 x 4 x 25 + 12; Since first part of the number 25412 i.e. 245 x 4 x 25 is a multiple of 4, therefore it is divisible by 4. Hence for testing the divisibility of 24512 by 4, we are to simply test the divisibility of the second part i.e. 12 of the number under consideration. Note that second part of the number is nothing but the number formed by the last two digits 1 and 2 of the number. Hence we may generalise that: A number is divisible by 4 if the number formed by its tens and ones digit is divisible by 4 For example, to check whether the number 23456 is divisible by 4, rewrite the number as: 23,400 + 56. We know that 4 divides 23400. Since 4 also divides 56, then 4 divides 23456. The test for divisibility by 8 continues this pattern. Because 8 does not evenly divide either 10 or 100, but it does divide 1,000, separate the number -- for example, separate 35478 into two parts: 35478 = 35000 + 478; = 35 x 1000 + 478; = 35 x 8 x 125 + 478; This clearly shows that the first part 3500 of the number 35478 is a multiple of 8 and hence is divisible by 8. Thus for testing the divisibility of the number under consideration by 8, we need to simply check the divisibility of the second part 478 of the number. Further note that second part of the number is nothing but the number formed by the last three digits 4, 7 and 8 of the number. Hence we may genralise that: A number is divisible by 8 if the number formed by its hundreds, tens and ones digit is divisible by 8 For example for testing the divisibility of the number 23456 by 8, we write the number 23456 =23,000 + 456. Since 8 divides 23,000 and 8 divides 456, Therefore 8 evenly divide 23,456. Now try these tests with the numbers in the chart below: Row Numbers 1 1 11 21 2 2 12 22 32 152 2462 10562 3 3 13 23 33 153 2463 10563 4 4 14 24 34 154 2464 10564 5 5 15 25 35 155 2465 10565 6 6 16 26 36 156 2466 10566 7 7 17 27 37 157 2467 10567 8 8 18 28 38 158 2468 10568 9 9 19 29 39 159 2469 10569 10 10 20 30 40 160 2470 10570 Blue: Divisible by 4, but not 8 Red: Divisible by 4 and 8
Problem Use the divisibility test to determine whether the following numbers are divisible by 4 and 8: a. 32464 b. 82,426

Answer: (a) Divisible by both (b) Neither divisible by 4 nor by 8.

Set A: Easy problems:

1. Indicate whether each number is prime or composite.

a. 2 b. 78 c. 51 d. 19 e. 31

Solution

a. The only factors of 2 are 1 and 2. Therefore 2 is prime.

b. Because 78 is even, it is divisible by 2. Having 2 as an “extra” factor—in addition to 1 and 78—means that 78 is composite.

Do you see why all even numbers, except for 2, are composite?

c. Using the divisibility test for 3, we see that 51 is divisible by 3 because the sum of the digits 5 and 1 is divisible by 3. Because 51 has more than two factors, it is composite.

d. The only factors of 19 are itself and 1. Therefore 19 is prime.

e. Because 31 has no factors other than itself and 1, it is prime.

Finding the Prime Factorization of a Number

2. Fill in the blanks:

a. Every composite number can be written as the product of ------- factors. This product is called its ------------------------ For instance, the prime factorization of 12 is --------------

b. The prime factorization of a whole number is the number written as the product of its ----------------------..

c. There is ---------------prime factorization for any composite number.

d. A good way to find the prime factorization of a number is by making a -------------..

Answers: a. prime prime factorisation

b. prime factors

c. only one

d.

3. Write the prime factorization of 72.

Solution

We start building a factor tree for 72 by dividing 72 by the smallest prime, 2.

Because 72 is 2 · 36, we write both 2 and 36 underneath the 72. Then we circle the 2 because it is prime.

Next we divide 36 by 2, writing both 2 and 18, and circling 2 because it is prime. Below the 18, we write 2 and 9, again circling the 2. Because 9 is not divisible by 2, we divide it by the next smallest prime, 3. We continue this process until all the factors in the bottom row are prime.

The prime factorization of 72 is the product of the --primes.

72 = --2 x 2 x 2 x 3 x 3

4. Express 60 as the product of prime factors.

Solution

The factor tree method for 60 is as shown.

Set B:

A. Computational Skill

1. Use the divisibility test to determine whether the following numbers are divisible by 4 and 8:

a.	32,464
b.	82,426

Mental Skill

2. Complete the following table by entering Yes or No in each grid after checking the divisibility of the number in first column of each row by the number written in each grid of the first row:

	Is divisible by
Number	2	3	4	5	6	8	9	10	11
256
1390
6851
88688
50759
20800
6582

Knowledge – cum – Computational Skill

3. Use divisibility tests to examine which of the following are divisible by:

a. by 2 but not by 6 b. by 3 but not by 6 c. by 4 but not by 8

d. by each of the numbers 2,3,6 e. by each of the numbers 3,6 and 9

(i) 3582 (ii) 76431 (iii) 2268 (iv) 51792 (v) 777771

(vi)181 (vii) 246 (viii) 38001 (ix) 45260 (x) 500000

4. Which of the following are divisible by 2:

2345, 9547, 37321, 732754, 542737, 573256

5. Which of the following are divisible by 3:

181, 38001, 1315, 8470,45260,2743

6. Which of the following are divisible by 5:

1315, 1513, 2743, 7324, 2745, 4527, 8470

Skill of observing patterns – cum application

4. Use Sieve of Eratosthens to write:

a. All even prime numbers less than or equal to 100.

b. 3 consecutive composite numbers.

c. Three prime numbers whose sum is even.

d. 2 consecutive prime numbers

5. Which of the following are Primes?

a. 321 b. 10000 c. 2337 d. 79111

6. Write prime factorization for each of the following:

a. 365 = ----------------- b. 1122 = -----------------

c. 23480 = -------------------- d. 10001= -----------------------

e. 35730 = -------------------- f. 462762 = ---------------------

Let us actually play with numbers:

Number Riddles

See if you can solve the riddles by using the hundreds chart below.

1. I am a two digit number. I am in the forties. I am two more than forty three. A. What number am I? B. Am I even or not? C. Am I prime or not? 2. I am in the middle of a row. My row is in the middle of the hundreds chart. I am an odd number. My tens digit is a five. A. What am I? B. Am I divisible by 3? C. Am I a multiple of 9? 3. I am a one digit number. I'm odd. I'm on the first row. I'm less than eight but I'm more than five. A. What number am I? B. Am I the H.C.F. of the numbers 35 and 42? C. Am I the L.C.M of 35 and 42? 4. I am on the fifth row. My first digit is a four. My last digit is even. A. What numbers can I be? B. What can be my highest value? C. Can my highest value be evenly divided by 5?

4. I am a number on the fifth row. I'm an even number. I'm greater than 46 and less than 49. A. What number am I? B. Can I be a prime number? C. Can my successor be composite? 6. I am a two digit number. I am on the first row. My sum is one. A. What number am I? B. Am I divisible by 4? C. Am I a factor of 250? 7. I am on the ninth row. I have a nine in the tens place. I am less than 91. A. What number am I? B. Am I a multiple of 5? C. Am I divisible by 3? 8. I am a two digit number. I am also even. I'm on the second row. My sum is nine. A. What number am I? B. Am I the L.C.M. of 3 and 6? C. Am I the multiple of 9?

9. I am in the ninth row. I have a 0 in the ones place. A. What number am I? B. Am I co prime with my successor? C. What number is my predecessor? 10. I am a two digit number. My number is even. I am on the last row. A. What numbers can I be? B. Is the sum of my predecessor and successor a multiple of 11? C. What can be my least value? 11. I'm on the 4th row. Five is in my ones place. A. What number am I? B. Which are the first three numbers which are my multiples? C. What are my possible factors?

Answers:

1. A. 45 B. Not C. Not

2. A.55 B. No C. No

3. A. 7 B. Yes C. No

4. A. 42, 44, 46, 48 B. 48 C. No

5. A.48 B. No C. Yes

6. A. 10 B. No C. No

7. A. 90 B. Yes C. Yes

8. A. 18 B. No C. Yes

9. A. 90 B. Yes C. 89

10. A. 92, 94, 96, 98 B. May or may not be, e.g. 91 + 92 = 183 is not a multiple of 11 whereas 93 + 94 = 187 = 11 x 17 is a multiple of 11.

c. 97

11. A. 35 B. 35, 70 and 105 C. 1, 5, 7, 15

Suggested Class Room Activities:

Even and Odd -Teacher Activities

Even and Odd Game Show

Host a game show in your classroom. Write different numbers on index cards and tape them to a large board. Then have three student volunteers come up at a time. One student can select a card and you can turn the card over and read the number aloud. The first student to raise their hand and identify the number as odd or even wins the point.

Twenty Questions

Think of a number and have students try to guess the number by asking questions. You can give them clues to the number, such as “It is a even number between zero and fifty.” Have student volunteers guess and keep track of the number of questions that have been asked. After twenty questions, give an obvious hint so students can guess correctly.

As an extension, have students split up into small groups and repeat the activity with each other. Encourage students to think of different clues and questions to ask using even and odd numbers.

Even Pairs

How many students are in your class? Is this an even or odd number? Discuss with the class and have students make predictions. Then divide the students into pairs. Is there a person left over? Can the class be divided into pairs without remainders? Can the class be divided into two equal groups without remainders? Explain to students that even numbers can be divided into two equal groups with no remainders and odd numbers cannot be divided evenly.

As an extension, have students place counters, paper clips, or other small items into a box. Have them take a handful of objects out and count them to see if there is an odd or even number. Instruct them how to divide the items into pairs. Have students write the numbers in their notebooks and identify them as even or odd.

Even and Odd Family Activities

Shopping Spree

Together with your child, go shopping at a grocery store, drug store, or a 99¢ store. Look at different items and their prices. Have your child identify if the number is odd or even. Remind your child to look at the ones place to see if a number is even or odd. Then ask your child to look for an item with a price that is even or odd. As an extension, you can add a dollar or subtract a dollar from a price and have your child identify if the new price is even or odd.

Odd and Even Cubes

Roll two number cubes to make a two-digit number. Have your child identify if the number is even or odd. If he or she has difficulties, have your child use beans, coins, or other small objects to divide the number into two groups. If the counters can be divided into two equal groups, the number is even. If the counters cannot be divided into two equal groups, the number is odd. Have him or her record all the numbers in a notebook and use the page as a reference in the future.

Even and Odd Items

Have your child look for items in your house that are an odd or even number. For example, have your child count the number of pickles in a jar, the number of coins in a dish, or the number of eggs left in a carton. Is the number odd or even? How does he or she know? A good method to see if a set if item is odd or even is to divide the items into two equal groups. Encourage your child to use this method when possible.

Exploring Pascal's Triangle

Coloring Even and Odd Numbers

The Mathematics in the Patterns

1. Students practice the concept of even and odd numbers by coloring odd-numbered balloons red. They should see that the triangle is outlined in red balloons because the number 1 is an odd number.

2. The Commutative Property of Addition
This property, together with the structure of Pascal's Triangle, explains the symmetry that can be observed in the colors of each row of balloons. It is not necessary for students to know or understand this property to be able to appreciate the symmetry in the coloring: if the triangle is folded through its center, balloons of the same color will fall on top of each other.

3. The sum of two odd numbers is an even number. When two red balloons are next to each other, the balloon below will be white.

4. The sum of two even numbers is an even number. When two white balloons are next to each other, the balloon below will be white.

5. The sum of an odd number and an even number is an odd number. When a red balloon is next to a white balloon, the balloon below will be red.

---

Extensions

Use one of the numbered Pascal student worksheets to repeat the activity. Six identically colored triangles can be joined to form a hexagon. These constructions make great classroom or hall decorations. Looking at the center point gives the optical illusion of a cube in three dimensions.

Let Us Play with Numbers:

A.

1. The rule is given. Some of the frames are empty. Fill in the blank frames.

Solution: Write 4, 12, 28, and 36 in the blank frames.

2. Looking at the pattern, identify the rul:

Solution: The rule is subtract 4 or minus 4 or -4.

3. Some of the frames are empty. The rule is missing. Find the rule and fill in the empty frames.

B.

Number grids have many wonderful features that help children with pattern recognition and place value. Their original use, however, was simply to solve the problem of number lines being unmanageably long. Number grids may be considered number lines that fit nicely on a page or a classroom poster.

The number grid lends itself to many activities that reinforce understanding of numeration and place value. By exploring the patterns in rows and columns, for example, children discover that any number on the number grid is:

(i)1 more than the number to its left

(ii)1 less than the number to its right

(iii) 10 more than the number above it

(iv) 10 less than the number below it.

In other words, as you move from left to right, the ones digit increases by 1 and the tens digit remains unchanged. As you move down, the tens digit increases by 1 and the ones digit remains unchanged.

Children can be asked to solve puzzles based on the number grid. These puzzles can be pieces of a number grid in which some, but not all, of the numbers are missing. For example, in the puzzle below, the missing numbers are 356 and 358.

Number grids can be used to explore number patterns. For example, children can color boxes as they count by 2s. Starting from 0 and counting by 2s, they will color the even numbers.

Using the number grids children may be asked to identify the prime numbers (highlighted below) using the Sieve of Eratosthenes. Named after the ancient Greek mathematician and scientist, Eratosthenes, it is an algorithm for identifying all prime numbers up to a given number n (in this case, up to 100).