Multiplication+Combinations


 * __//Fluency Benchmarks for Learning Combinations Through the Grades//__**
 * Addition:** fluent by end of Grade 2, with review and practice in Grade 3
 * Subtraction:** fluent by end of Grade 3, with review and practice in Grade 4
 * Multiplication:** fluent with multiplication combinations with products to 50 by the end of Grade 3; up to 12X12 by the middle of Grade 4, with continued review and practice
 * Division:** fluent by end of Grade 5


 * Download this chart* [|MultiplicationCombinations.doc]


 * A. Students Who Need Review and Practice of Combinations to 50**

Students who have difficulty learning the multiplication combinations often view this task as overwhelming - an endless mass of combinations with no order and reason. Bringing order and reason to students' learning of these combinations in a way that lets them have control over their progress is essential. Traditionally, students leaned one "table" at a time (e.g., first the x2 combinations, then the x3 combinations, the x4 combinations, and so on). However, the multiplication combinations can be grouped in other ways to support learning related combinations.

I want to share with you one possible sequence for learning the combinations. This method has been used in Shauna Roshone's classroom with a great degree of success!!! +++++++++++++++++++++++++++++

First, make sure that students know all multiplication combinations that involve x1, x2, x5, and x10 fluently. Please note that skip counting by 2s, 5s, and 10s does NOT equate to fluency! The fluent student does not need to skip count to determine the product of these combinations.

Once students have mastered these basic combinations, we can provide small groups of combinations that students can relate to what they already know.

1. **//Learning the x11 combinations.//** Once students know the x1 combinations, it is a simple matter of recognizing the double digit pattern of the x11 combinations. Help students use the x10 rule of adding a "0" to solve 11x10=110. For 11x11 and 11x12, show students the nifty strategy of splitting the digits of the factor, and then adding them. So, for 11x12, we split the "1" and the "2," and then add them together for the center digit. The product becomes 132. Likewise, 11x11 becomes 121, split the "1"s and then add them together.

2. **//Learning the x4 and x8 combinations//** Help students see that by doubling the x2 combinations, they can deduce the x4 combinations they don't know. So, 4x6 is (2x6)+(2x6), or 4x6=2x(2x6). Students may verbalize this idea as "4 times 6 is 2 times 6 and another 2 times 6," or "to get 4 times 6, I double 2x6." Facility with doubling will prepare students for some of the more difficult combinations, as well as later work in multiplication and division strategies for multi digit numbers.

Have students use the same strategy of doubling for their x8 combinations. Double their 4s! So, 4x8 can be thought of as 2x(2x8).


 * //2. Learning the square numbers.//** Next, students learn and review the combinations that produce square numbers. The multiplication combinations chart is very useful for this step. Students will immediately recognize the diagonal pattern of squares. If they need to, they can now also use some doubling strategies to help them.

//**3. Learning x3, x6, and x12 Combinations**//Students can now focus on their x3 combinations. Once these are mastered, move them to x6 - now they are prepared to double their x3 combos when necessary! And the once uber-challenging x12 combinations are a simple matter of doubling the x6 combos!!!


 * Practice, Practice, Practice**

Even when they possess the conceptual understanding this sequence attempts to teach, students will need to practice if they are to become fluent. We encourage the use of array cards and games, flash cards, computer games, etc.Students will be internalizing these combinations alongside the good conceptual teaching you are providing through //Investigations//.

In Sauna's class, we practiced twice a week for 15 minutes each session. Kids took a quiz on Tuesday and Thursday, tracked themselves on their own progress monitoring instrument (actually, they officially recorded their score after Thursday's quiz only), and then practiced those combos that the quiz showed they hadn't mastered yet.




 * Download this Tracker* [|combinationsprogressmonitoring.xls]