Material:
• Box of assorted colored bead bars; white and gray cards from 1 to 9 the same size as decimal cards; several slips marked +, -, x and =.
• Set of small decimal cards 1 to 3,000 (colored cards); 6 or 8 coin envelops, brackets, little slips of paper (2 cm sq.); felt mats to work on.
Introduction: These activities use knowledge of multiplication in a game like fashion conveying the idea that it can be fun to do. This series brings to the child’s consciousness some properties of the number system – in particular
Commutative Law of Addition
a + b = b + a
Addition is commutative. This means that the order of adding any two numbers does not affect the result.
Associative Law of Addition
(a + b)+ c = a + (b + c)
Addition is also associative. This means that the order of grouping numbers together does not affect the result in addition.
Commutative Law of Multiplication
a x b = b x a
The commutative law of multiplication states that the order of the multiplicands does not affect the answer, just as is true in addition:
3x5=15 or 5x3=15
The numbers can change places just as you can when you commute to school on the bus.
Distributive Law of Multiplication over Addition
a (b + c) = ab + ac
Exercise One: Number x a Number (Commutative Law)
1. Hold up a 5 bar. Place on the mat horizontally.
2. State, “I’m going to do a multiplication. I’m going to take this 5 bar 3 times.” Place a gray 3 card on the mat to the right of the 5 bar.
3. “5 taken one time, two times, three times. What is 5 taken 3 times worth?” Place 3 5 bars under the 5 bar and the answer below vertically using a 10 bar and a 5 bar.
4. “Now let’s take 3 5 times. 3 taken once, twice…what is 3 taken 5 times worth?”
5. Place a 3 bar and gray 5 card to the right of the 5 bar and 3 card; place 5 3 bars below the 3 bar and the answer, 15, in beads below the 5 card.
6. Note that 5 x 3 = 15 and 3 x 5 = 15. State: “I wonder if it would work with 2 other numbers. Would you like to try?”
7. Later give the definition of the commutative law of multiplication.
Exercise Two: A Sum Multiplied by a Number (1 + 1) x 1
1. State, “We’re going to do more multiplication. Look what’s inside this envelope. We’re going to take this 3 times.”
2. Place a 2 bar and 5 bar horizontally in brackets; place a 3 gray card to the right.
3. “Because the 2 and 5 are together, we’ll put brackets around them. We’ll multiply this by 3.”
4. Place 3 2 bars below the 2 bar then 3 5s below the 5 bar. State, “2 taken one time, 2 times, 3 times. And now 5 taken one times, two times, 3 times. How much is this all together?”
5. Place the answer (21) vertically below in bead bars.
6. “Now we’ll multiply what’s in this envelope by 3.” Lay out a 3 bar to the right of the first lay out. Place a 2 and 5 gray card in brackets to its right.
7. “What’s our answer? Look, we got the same answer! I wonder if this will work with other numbers. Would you like to try?”
8. Place answer below second lay out.
Purpose: To bring to the child’s attention the concept of multiple terms in the multiplicand or multiplier, and the use of brackets.
3 5
Excercise 1
Excercise 2
Exercise Three: Multiplication of a Sum by a Sum (1+1)x(1+1)
1. State, “This time we’ll multiply what’s in this envelop by what’s in this envelop. We’ll set this up like this”
2. “We’ll turn this card over(5)and begin by multiplying by 3. 2 taken 3 times is ... 5 taken 3 times ...” Place 3 2 bars below the 2 bar and 3 5 bars below the 5.
3. “Now we’ll multiply by 5 (turn both cards over). 2 taken 5 times .... 5 taken 5 times....” Place 5 2 bars below the 3 card and 5 5 bars below the 5 card.
4. “Let’s find our answer. Here we have 6 (place a 6 bar below the 3 2 bars). What do we have here? (15) an here?
(10) here? (25)” Place answers below each quantity.
5. “We can add them all together for our final answer. (56) 6. Place in bead bars of to the right of the sums.
Purpose: For a further awareness of the distributive principle;
stress you must multiply each term of the multiplier for each term of the multiplicand. This is indirect preparation for binomial multiplication.
Exercise Four: Multiplication of a Sum by a Sum with Signs & Cards Part A – Introduction of Operational Signs
Repeat multiplying a sum by a sum (4 and 2 bead bars in one envelop, 3 and 5 gray card in another)adding operational signs where appropriate by talking through what you’ve been saying and how signs will express this. Read as (4 + 2) x (3 + 5) =
Work the problem as usual; use decimal cards for answers combining and exchanging for the final answer.
Part B – Introduction of Cards
Lay out as above; state that you will write what you’ll do before using the beads. Lay the problem out in expanded form using white number cards. Place beneath the problem in order:
Part C – Using Cards Only
Work out completely without beads using cards; the child can be shown horizontal addition.
Part D – Completing on Paper
The child works the problem out on paper. This can be checked by adding the numerals in each bracket and multiplying them.
Excercise 3
Excercise 4
Exercise Five: Multiplying with Numbers Greater than Units
The child must have completed exercises 1 to 4 and understand the process.
Part A
– Using the Golden Bead Material
1. On a longish slip of paper write the problem: 42 x 23. Place in the middle at the top of the felt mat.
2. Lay out in cards, symbols and brackets:
3. Introduce as the multiplier and multiplicand.
4. Turn the 3 card over. State that you’ll multiply 40, 20 times. Lay out 4 ten bars horizontally:
5. Lay out 19 other groups of 4 ten bars underneath to make 4 groups of 20 leaving a space between groups of ten.
6. To the right of this work out 2 taken 20 times using unit beads.
7. Continue with 3 x 40 under the groups of ten bars and 2 x 3 under the units.
8. Talk through and exchange groups of 10 10s for hundreds, and the lines of unit 10s for 10 bars.
9. Discuss what categories you end up with when multiplying different categories together, i.e. 10s times 10s gives you 1000s, etc.
10. Find out the answer by counting and placing out decimal cards for each section; add partial products, exchange as needed and place the answer after the equal sign.
11. The children can repeat with their own problems.
Part B
– Using Cards and Beads
Same procedure; expand the problem in cards but still lay out the beads.
Part C
– Completing on Paper
Complete the problem on paper.
Part D
(
40 + 2)
x(
20 + 3) (
40 + 2)
x(
20 + 3)
(
40 + 2)
x(
20 + 3)
( )
203402+ x + ( )
(40 x 20)
(40 x 3) (2 x 20)
(2 x 3) Excercise 5
Multiples
Material:
• Bead cabinet and tickets
• The box of colored bead bars
• Paper called Multiples of Numbers with numbers 1 to 100 printed in an array
• 3 charts called Calculation of Multiples – tables A, B and C, numbered 1 to 50
Presentation 1: Concept and Language of Multiples
1. Lay out the 5 bead chain and skip counting markers; state, “Let’s count the 5 chain. 1, 2, …5 (place the 5 ticket) 6, 7…10 (place the 10 ticket).” etc.
2. Point to and count the numbers in order.
3. State, “Each number contains 5 with nothing left - 5 contains 5 once with nothing left over. Does 10 have 5 in it? (Yes) How many times?” (10 has 2 five’s.) Is there anything left over?” etc.
4. Explain that 5, 10, 15, 20 and 25 are all multiples of 5 and that all numbers that have five in them are mul-tiples of five.
5. State, “When a number contains another number perfectly, it is called a multiple of that number.
6. Repeat with a second chain (6). Use such words as nothing left over, exactly, this many times.
7. Ask the child to state the multiples of the new chain; combine the tickets from both chains and have the child place these in the correct spots.
8. The child can repeat with 2 other chains.
Presentation 2: Concept and Language of Common Multiples
1. Tell the children, “Let’s do multiples of 2.”
2. Take out the bead box. Lay out a two bar horizontally and ask: “How many 2’s are there in two?”
3. Place a two bar vertically below the two bar.
4. Lay out 2 2 bars to the right of the first 2 bar and state, “Two taken two times is 4” Place a four bar underneath the 2 2 bars.
5. Lay out 3 2 bars and ask: “2’s taken 3 times is six?” Place a 6 bar below.
6. Continue by two’s to twelve.
7. State that these are multiples of 2; review how many 2’s are in different numbers.
8. Lay out the 3’s table in the same fashion underneath; have the children find numbers that are in both and intro-duce as common multiples.
9. Invite the children to do other numbers; 2 digit numbers or 3 numbers rather than just 2 can be used. This
Presentation 1
Presentation 2
etc...
Presentation 3: Investigation of Multiples
1. Introduce the multiples of number paper stating you’ll look at multiples using numbers.
2. Choose a number, write it at the top, and circle all it’s multiples on the paper.
3. Repeat with a second number and sheet of paper; use a different col-ored pencil or draw a different shape around each multiple.
4. Later do 2 on the same paper using different colored pencils; the child can also do 3 etc.
5. Large sheets of graph paper can also be used.
6. Children can make booklets of their sheets.
Presentation 4: Tables A, B and C
1. Table A involves finding multiples of each number (1 – 10) to 50. Bead bars can be used.
2. Table B continues the work of Table A to 100. Counting bead bars may be helpful.
3. On Table C, have the children use tables A and B to fill in the problems that correspond with the numbers on the chart, e.g. for 6, children can write 2x3; 3x2. They can also find prime numbers by asking them to under-line in red numbers whose only multiples are 1 and themselves (1, 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, etc.).
4. Children may also color dots next to multiples on Table C.
Notes:
Multiple (Definition):
When a number contains another number perfectly, it is called a multiple of that number.
‘Multiple’ comes from (Latin) meaning many Prime Numbers:
Numbers that are only a multiple of one and itself.
1 2 3 4 5 6 7 8 9 10
2 x 1 =