Strategies for multiplication
- Counting by ones
- Skip counting
- Repeated addition
- Unitize units - separates units into groups. Can simultaneously consider the number of objects and groups as two subgroups of objects for the number of groups. (Unitize 5X 8 by regrouping five groups of eight - into five groups of five and five groups three (5X5) + (5X3)).
- Distributive property (5X5) + (5X3) = 40
- Commutative property
- Associative property
- Array, area model
- Trees, hierarchical patterns
- Use of tens
9 * 30 (used factors) 9 * 3 * 10 = 27 * 10 = 270
15 * 18 = 15 * 2 * 9 = 30 * 9 = 270
10 * 18 + 5 * 18, (think of 5 * 18 as 1/2 of 180) = (180 + 90) = 270
Can use a multiplication fact table to reduce by finding the common factors. Students explained procedurally find number in table and then look to see what numbers are left and up and use them for common multiples or factors to reduce. Ask students to explain if it will always work and how they know that they are sure. Multiplication table is a list of multiples and factors for 1-10 or whatever. If you can find numbers in the same column or row, then they have a common factor. E.g. 21/28 both are in the row or column with 7 (3*7 , 4*7). If you take two numbers that you want to find a common multiple and find one on the side and the other at the top, you will a common multiple at the intersection. When you multiple two numbers the product will always be a common multiple of the factors, but not necessarily the least common multiple.
Ideas for teaching multiplication of decimals
Connect multiplication of decimals to fractions. 1/10 of 2 or 2 x 1/10... Doing several so students see that a whole number multiplied by tenths is tenths. Then the same with 1/100 of 2 and 2 x 1/100... Then 1/10 of 1/10 and 2/10 of 2/10... Then tenths of hundredths...
Can also relate to money 2 dimes is .2 of a dollar, 3 pennies is .3 of a dime and .03 of a dollar
Do the same kinds of problems with graph paper that is 10x10 or 10x100, 100x100 if want to get to 10,000ths
Can also use Cuisenaire rods, squares, and cubes. Use each of the following, in turn, (white cube, orange rod, orange square, and orange cube) to represent one unit and have the students determine what the others. Have them let one vary for each and tell what different collections are. E.g. if an Orange cube is one what is an orange square? an orange rod, A white cube?, Then ask what .1, .01, or .001 of each would be and what if multiply?
Also have had them use calculators to check and see if the calculator agrees with their reasoning.
I have found that students who know number values and can read and write numerals well (see reading, writing, and saying numbers lesson in the math and intro. to instruction notebook) will know what the answer is and place the decimal point without counting (for decimal numbers in tenths, hundredths, and thousandths). And if they know that, then they can always figure the rule or procedure for larger numbers, if they need to.
Many ideas for multiplication and division were adapted from Fosnot, Catherine Twomey; & Dolk, Marten. (2001). Young Mathematicians at Work: Constructing Multiplication and Division. Portsmouth, NH: Heinemann.
Quotative, Measurement Sample problem If I want to put 28 donuts on platters and I want 7 donuts put on each platter, how many platters are needed? How many groups of seven can be made with twenty-eight? How many groups of __ can be made with ___ (28 / 7). The size of the groups is specified and the number of groups must be determined. (Measurement how many units of 7 in 28)
Partitive, Distribution, or Dealing
If I have twenty-one beads and want to use all the beads to make three bracelets, how many beads can I put on each bracelet? (21 / 3)
This involves finding how many in the group when the number of groups are known (3). The size of the groups is not known. Because children begin with a building up strategy, they focus on the number in the group, rather than on the group and the whole simultaneously, this makes partitive problems more difficult. While counting can solve quotative problems, partitive problems are initially attempted with trial and error,
Partitive problems are difficult because to understand distributing or dealing out to a given number of groups produces equal or fair sharing requires that children know that one to one correspondence for dealing-out provides a fair share to each group. This is harder than it sounds. Children frequently see adults dealing cards. However, they don't know it will result in all players getting an equal number of cards. Many times they will count to see. Further they must consider the number of groups, the number in the groups, and the whole all simultaneously. The part whole relationship makes this a big idea.
Teachers often tell students that division is repeated subtraction. This is an incomplete analysis and hinders children's ability to construct an understanding of the part/whole relationships in multiplication and division.
I have $12; if socks cost $3 a pair, how many pairs can I buy? Quotative - measurement -(the total amount and the size of the groups - find how many groups) It can make sense to subtract $3 from $12 because each $3 is matched to a pair of socks.
Socks are on sale at 3 pairs for $12; how much is one pair? Partitive (find the size of the part) distributive (have the total size and total number of groups - find the number in the groups by distributing or dealing equal amounts to the groups) Both require that 12 be divided by 3. However, what would you subtract in the partitive? It doesn't make sense to subtract 3 pairs from $12.
Children solve both problems differently. In the quotative / measurement they lay out 3 and then 3 more until 12. Then they count the piles. They don't subtract, they add.
In partitive they use trial and error and maybe a dealing strategy. They solve both problems very differently. How do they discover that both of these kinds of problems are division?
Students were asked to do two similar problems.
One If a hotel has 240 rooms, 24 per floor, how many floors?
Two if another hotel has 240 rooms and 24 floors how many rooms on a floor?
Students did each problem differently. As part of the discussion they were asked to look and explain why the numbers were the same. An array of squares was drawn to illustrate both hotels 10 x 24 and 24 x 10. Next they talked about multiplication being commutative.
Another group of students was given the following.
The teacher bought 186 pencils and wanted to put them on six tables, how many would s/he put at a table? S/He also said that the pencils came in a box of six, how many boxes did s/he buy?
One way to get from tallies and pictures to unitizing groups.
How many tables would be needed if they invite 81 guests and can sit 6 at each table.
Many students drew tables with 6 and counted on until 81 one student figured 6 * 10 = 60, so 4 more would be needed. 60 + (4 x 6) = 60 + 24. During class discussion the student explained his/her process and why s/he thought it worked. The students wanted to try other problems to see if it would work.
To help students abstract (the use of symbols or distributive property) and make more connections a teacher decided to create a problem with objects that wouldn't be easy to partition.
If you were going to serve each person a glass of lemonade and each pitcher holds seven glasses, how many pitchers are needed for ___ people? The result some did and some didn't.
Remainders can be dealt with four ways: round up, round down, divide equally, or toss away.
From repeated addition to subtraction to using multiplication
A student in a class said that s/he made $328 selling Sega games at $8 a piece. How many games were sold?
Students solved the problems and compared the different methods to see how the numbers matched in each procedure. Most used repeated addition and one used repeated subtraction. Another knew 6 x 8 was 48 so repeatedly added 48. To verify this process Multilink cubes were matched and rubber bands put around 6 groups of 8 to show 48. It is very hard for students to construct the idea that numbers can be used to count elements in the group (8) and the groups (6) simultaneously. So when 48 is used it must be thought of as six groups and six groups of $8 simultaneously.
When the student was asked, why they did not use 8 X 10? S/He didn't know why. Probably because they knew 6 * 8 and place value is harder yet, so 6 x 8 made more sense. But if others did use 8 x 10 it would have been explored and related to the others.
Area models are difficult for students to construct.
- Sometimes students count around one side and down the middle.
- Students struggle to draw straight lines when making a grid. Many times the number of lines is wrong or not to scale.
- Some have trouble drawing the grid with a ruler
- Some have trouble drawing the grid with graph paper.
- Some don't understand how the rows and columns are related.
- Many students want to count the rows and columns twice.
- Takes time to see that each square is in a row and column simultaneously.
- Sometimes students draw 12 lines for 12 squares and end up with 11.
Combinations and permutations models
If 12 people played checkers with 26 people how many games would be played altogether.
Different outfits with 7 shirts and 4 pants.
Common multiplication strategies.
Doubling 2 * 6 * 12 = 12 * 12
Halving and doubling 4 x 3 = 2 x 6
Using the distributive property 7 x 8 = (5 x 8) + (2 x 8) or 7 x 8 = (8 x 8) - 1
Using the distributive property with ten 9 x 8 = (10 x 8) - 8
Using the commutative property 5 x 8 = 8 x 5
If students memorize the facts they are many times robbed of the opportunity to learn these important mathematical relationships.
Algorithms and number sense.
Multiply 76 x 89 then get a piece of graph paper, draw a 76 X 89 grid, and show where the numbers in the traditional algorithm came from. If this is hard for you then the way you were taught has worked against your own conceptual understanding of mathematics. Don't cheat students by having them memorize an algorithm without an understanding of multiplication. This is critical for anyone to understand algebra E.g. (x + 1) (x + 1)
When you multiplied 76 * 89 did you use any of the following?
- 76x89 = (100 x 76) - (11 x 76) = 7,600 - 760 - 76 = 6,764
- or take 50 x 89 = 4450 then half of it for 25 = 2225 then add 4450 + 2225 + 89 =
- or 76 is close to 75 or 3/4 of 100, 3/4 of 89 = 3 x 22 1/4 = 66 3/4 (then since divided by 100 multiply by 100 and add 89) 6,675 + 89 =
- or 80 x 90 = 7200 - (4 x 90) = 76 x 90 = 6,840 - 76 =
All are easier than the algorithm.
Student created algorithms for two-digit multiplication
Students usually work from left to right starting with tens first and then use the distributive property (backward from the traditional algorithm).
One strategy to transition student's from invented strategy to a more traditional algorithm is with the use of arrays.
Make a 2 x 3 array. Then ask what a 2 x 30 would look like, long short How does it compare to the 2 x 3 array? How many 2 x 3 arrays are in the 2 x 30 array? How do you know? What would it look like? Can you draw it? What about 4 x 4 and 4 x 40?
What about 4 x 39? Decompose and use distributive property. (4 x 30) + (4 x 9)
Then extend it into three dimensions.
Egyptians multiplied by doubling
28 x 12
1 x 28 = 28, 2 x 28 = 56, 4 x 28 = 112, 8 x 28 = 224
4 + 8 = 12,
Therefore 112 + 224 = 336 or 28 x 12
Russian peasants used a halving and doubling algorithm.
28 x 12
They would take half of 28 and double 12, 14, 24
Half and double again 7, 48 remember all odd doubles
If odd half and forget remainder 3, 96 remember all odd doubles
When get to 1 add all the odd doubles 1, 192 48 + 96 + 192 = 336
Muhammad ibn Musa al-Khwarizmi invented the present day algorithm in the ninth century. In Latin his name was Algorismus - hence algorithm. Before he invented the algorithm the merchants and intelligentsia used the abacus. The new system with the dust board democratized computation. In the Renaissance in Europe those who knew how to multiply and divide with algorithms were guaranteed a professional career. But today it's different. In 1621 the first slide rule was made. The first mechanical calculator by Pascal in 1642, and the first handheld calculator 1967. Constance Kamii's research has led her to insist that teaching children algorithms is harmful to their mathematical development. Some reasons follow.
- Children want to do multiplication from left to right - largest to smallest (algorithm is opposite).
- In division they want to go from smallest to largest, right to left (algorithm is opposite).
- Children have to give up their sense making in order to perform the algorithm.
- Doing so hinders children's ability to construct understanding of the distributive property of multiplication.
- Also doesn't encourage understanding of place value.
- It makes them dependent on the spatial arrangement of digits on paper.
- Worst of all it causes children to see themselves as proficient users of someone else's mathematics, not as mathematicians.
Most of the strategies students create are based on some form of the distributive property and on breaking the number into place value components. Therefore, are cumbersome and inefficient. Guidance is needed to help students create efficient algorithms. Mathematics in the City has looked at how to develop in students efficient computation strategies that are based on a deep understanding of number sense and operation that honor children's own constructions. Sample mini-lesson with strings of problems Display one problem at a time and discuss. Sample problems. 5 x 6 = Sample discussion: I knew 6 x 6 was 36 and I subtracted 6 and got 30
I knew 10 x 6 was 60 and I halved it and got 30.
Next problem 30 x 6 Sample discussion: I knew 30 + 30 = 60, then 60 + 60 + 60 = 180 Next 35 x 6 Sample discussion to try to get students to see as 30 * 6 + 5 * 6. If they do good give more, if not try another like 42 x 7, if they still have trouble, make it two problems 40 x 7 and 42 x 7. Strings like 5 * 6, 30 * 6, 35 * 6 are good to use to encourage students to use the distributive property. 35 x 6 = (30 x 6) + (5 x 6). If students understand, then draw a rectangle model and chart students' answers. 6
30 180 35 = 210 5 3
Suggests more problems and have students put examples in their journals, (25 * 9, 26 * 9, 46 * 5) use the strategy, draw arrays, and share with partner. Eventually students will invent a rule. "Split one factor into tens and ones and multiply each part by the other factor." Before students can use the array as a model they must construct it. Can use Cuisenaire rods and squares, tiles, grid paper, Multilink cubes, and eventually it should become a mental model. Extending the distributive property 49 x 7 = (50 x 7) - 7 48 x 7 51 * 7 98 x 32 = (100 x 32) - (64) Create a model, name the strategy. E.g. "friends more or less" 37 x 84 = (30 x 80) + (7 x 80) + (30 x 4) + (7 x 4) Create a model, name the strategy. E.g. "The ugly one " Using the associative property (30 x 6) 3 x 10 x 6, I added 10 six times = 60, then 60 three times = 180, Create a model 10 x 6= 3 x 6 x 10, 3 x 6 = 18 and added zero, Really add?, No I put it on 4 x 7 10 x 7 40 x 7 4 x (7 x 10) associative property 60 x 8 Suggest students create more problems and put examples in their journals. Doubling and halving Directly associated with the associative property, but because it is so important, highlight it separately. 2 x 24 4 x 24 8 x 24 8 x 12 4 x 12
50 x 42, (100 x 42 = 4200), 4200/2 = 2100 or 100 * 21 = 2100 3 1/2 x 14, 7 x 7 2 x 24, any of the following? (48 * 1), (24 * 2), (12 * 4), (6 * 8), (3 * 16), (1 * 48) 4 x 24, 2 x 48, 8 x 12, 16 x 6, 32 x 3
Generalize to reciprocal
18 x 3 1/3, take 1/3 of 18 and 3 x 18, then 6 + 54 = 60
or 18 x 3 1/3, divide and multiply by 3 to get 6 x 10 = 60
3 x 3 1/3
.8 x 30, multiply and divide by 10 to get, 8 x 3, 24 20 x 9 = 20 x 10 - 20 (add one group to multiply and subtract it later) Money 25 X 9 = 8 * 25 + 1 * 25 (8 quarters is 200 and one more = 225), or (ten quarters is 250 and one less is 225) How many 25's does it take to make 300? 4, 6, 8, 10, 12 What is the value of a 4 x 4 array of quarters? 16 x 25, 4 x 4 x 25, 4 x 100 Using fractions After students have constructed a very good understanding of fractions 1/2, 1/4, 3/4 try: 75 x 80 = 3/4 x 80 = 60 x 100 = 6,000 1/4 x 80 .25 x 80 25 x 80 1/2 x 60, .5 x 60 50 x 60 Illustrate with models 3 1/2 x 14, double and half or multiply and divide by two to get 7 x 7 Using the open array with division Teaching division as goes into is insufficient. Teaching division by isolating numbers is confusing and often makes little sense to children. Children choose to build up to the whole rather than subtract from the whole. The open array can represent repeated addition and subtraction from the whole and the relationship of division to multiplication. Use an open array with reducing (Cuisenaire rods) 24/6 48/2 (an array would have 4 columns and 12 rows, each cell would be 1/12, each column would be 1, and the whole array would be 48/12, and each row would be 4/12 of 1/3) 48/6 96/12
Show the relationship of division to multiplication Ask students for problems and make two lists similar to the following: 48/12 = 4 48/6 = 8 48/3 = 16 48/1.5 = 32 Then multiplication 4 x 12 = 48 8 x 6 = 48 16 x 3 = 48 32 x 1.5 = 48 These are very difficult relationships and will take much time for students to learn. However, it is essential that they do if they are to understand mathematics and to prepare for advanced algebraic reasoning. The division algorithm is dependent on the distributive property of multiplication. To divide 275/25 using the traditional algorithm we take (25 x 10) + (25 x 1) = 25 x 11. When you use the distributive property you don't necessarily have to break the numbers up into place value columns as the long division algorithm does. You could as easily break it into (200 + 75)/25 = (8 x 25 + 3 x 25)/25 = 8 + 3 = 11 Or 300 / 25 = 12: 12 - 1 = 11 Divide with the use of the array and different objects for manipulatives (grid, Cuisenaire rods, Multilink cubes, Meter stick, Money ) 1224/24, Use halving 612/12, 306/6, then 51 Or make it friendly: 1200/24 + 24/24, then halve to get 600/12 and divide to get 50, That makes 50 + 24/24 = 50+1 = 51 Or halving 1224/24, 612/12, 306/6, 153/3, then divided by three and got 51. Story problems to solve, model, chart, and represent with symbols. How many ways can you group ____ How many ways can you make an array ____ How many ears in groups of students (1-10)? There are ten kids on the playground that want to swing. There are two swings how many groups would be swinging? How many leaves on groups of three leaf clovers? (1-10)? Pizza party so that each person gets 4 pieces of pizza. How many slices will we need (1-10)? Three children each want 5 cookies. How many cookies do they want? Ratio table Division Algorithm by piling-up. 1 20 20 12 498 240 258 240 18 12 6
Represent 2*3 by drawing two groups with three objects. Be consistent. Always read as two groups of three. Making the first number always the groups and the second the number of objects in each group. Circles and Stars Game Roll a die. For the first roll draw that many circles and for the second roll put that many stars in each circle, write equation, chart all possibilities on class chart, talk about possible combinations for probability, patterns, factors,
Multiplication How can you remember? 0's - 0 x anything is 0, ends in 0 1's - 1 x anything is anything, 0,1,2,3,4,5,6,7,8,9 2's - doubles - add the number to itself, 0,2,4,6,8 3's - skip counting, end in 0,3,6,9,2,5,8,1,4,7,0, 4's - double doubles, end in 5's - multiples of five, use the clock to count fives, end in 0, 5, 6's - being double three, end in 7's - 8's - 9's - fingers, product of digits sum is nine and first digit is one less than the first factor (4*9 = 4-1 = 3 (first digit)3 + _ = 9 36 10's - add zero, count by ten, end in 0 11's double digits, 0, 11, 22, 33, 44, 55, 66, 77, 88, 99, 110, 120, 12's - inches in feet, 0, 12, 24, 36, 48, 60, 72, 84, 96, 108, First digit is same as first factor (3 x 12) 3 and last is first factor times second (3x6), until 10X12. Use the commutative property to switch some to easier problems (this is made easier if students can generate fact families). Have students create a chart with all the facts 12x12 matrix. Then have them cross out the facts that they really know. Most students will only have 5-15 left. Then convince them that learning the few that are left is a manageable task. The key to learning the basics facts is find a fact that students don't know or missed and discuss different strategies to determine an answer, let students select one that they understand and are willing to try, and have them write it and practice it so they can remember to use it next time. Then provide time periodically for students to review their list of strategies until they are comfortable with all facts. E.g. 6*8=48 can be thought of as 5*8=40 plus 8 more = 48 or 6*8=48 can be thought of as 5*8=40 plus 8 more = 48.