Multiplication and Division

5. Equal Groups: Whole Unknown (Multiplication) Mark has 4 bags of apples. There are 6 apples in each bag. How man apples does Mark have altogether. If apples cost 7 cents each, how much did Jill have too pay for 5 apples? (rate) Peter walked for 3 hours at 4 miles per hour. How far did he walk? (rate) Equal Groups: Size of Groups Unknown (Partition Division) Mark has 24 apples. He wants to share them equally among his 4 friends. How many apples will each friend receive? Jill paid 35 cents for 5 apples. What was the cost of 1 apple? (rate) Peter walked 12 miles in 3 hours. How many miles per hour (how fast) did he walk? (rate) 4 x 6 = 24 5 x 7 = 35 3 x 4 = 12 24 ÷ 6 = 4 35 ÷ 7 = 5 12 ÷ 4 = 3

7. Combination: Product Unknown Sam bought 4 pairs of pants and 3 jackets, and they all can be worn together. How many different outfits consisting of a pair of pants and a jacket does Sam have? You want to make a set of attribute pieces that have 3 colors and 6 different shapes. If you want your set to have exactly one piece for every possible combination of shapes and colors, how many pieces will you need? Combinations: Size of a Set Unknown Sam bought some new pants and jackets. He has a total of 12 different outfits. If he bought 4 pairs of pants, how many jackets did Sam buy? Your set of attribute materials consists of 18 different pieces. There are 3 colors, and each color comes in different shapes. How many different shapes are in the set? In the experiment of missing a coin and rolling a die, there are 12 possible outcomes. Of course, the coin has two sides. How many sides does the die have? 3 x 4 = 12 3 x 6 = 18 12 ÷ 4 = 3 18 ÷ 3 = 6 12 ÷ 2 = 6

11. Doubles Facts that have 2 as a factor are equivalent to the addition doubles and should already be known by students who know their addition facts. The major problem is to realize that not only is 2 x 7 double 7, bus so is 7 x 2. Try word problems where 2 is the number of sets. Later use problems where 2 is the size of the sets.

12. Five Facts This group consists of all facts with 5 as the first or second factor as shown here. Practice counting by fives to at least 45. Connect counting by fives with rows of 5 dots. Point out that six rows is a model for 6 x 5, eight rows is 8 x 5, and so on.

13. Zeros and Ones Thirty-six facts have at least one factor that is either 0 or 1. These facts, though apparently easy, tend to get confused with “rules” that some children learned for addition. The fact 6 + 0 stays the same, but 6 x 0 is always zero. The 1 + 4 fact is one more idea, but 1 x 4 stays the same. The concepts behind that facts can be developed best through story problems. Above all else, avoid rules that sound arbitrary and without reason such as “any number multiplied by zero is zero.

14. Nifty Nines Two patterns that are useful for mastering the nines are: 1. The tens digit of the product is always one less that the “other” factor (the one other than 9), and 2. The sum of the two digits in the product is always. These two ideas can be used together to get any nine fact quickly. For 7 x 9, 1 less than 7 is 6, 6 and 3 make 9, so the answer is 63.

16. Helping Facts The chart shows the remaining 25 multiplication facts. These 25 facts can be learned by relating each to an already known fact or helping fact. For example, 3 x 8 is connected to 2 x 8 (double 8 and 8 more). The 6 x 7 fact be related to either 5 x 7 (5 sevens and 7 more) or to 3 x 7 (double 3 x 7). The helping fact must be known and the ability to do the mental addition must also be there.

18. Pinczer, E. (1995). A remainder of one . Boston: Houghton Mifflin.