Thinking Strategies for the Basic Facts EDN 322

Behaviorist Theory Helpful Procedures Repetition is important Sequence the content in a series of small steps Learner fatigue decreases the effectiveness of drill Responses must be elicited from the learner so that correct responses can be rewarded and incorrect ones can be modified

Strategies for Learning the Addition Facts Commutativity Adding 0, 1 Adding Doubles or Near Doubles Counting on Adding to 10 and beyond Commutativity – The task of learning the basic facts is simplified because of the commutative property. Changing the order of the addends does not affect the sum. 2 + 3 is the same as 3 + 2 Adding one increases the number by one; adding 0 does not change the number Most children learn doubles quickly. Adding a near double is easier. For example: 7 + 8 = 14 + 1 or 15 2 + 6 – start with 6 and count on…6, 7, 8 is answer 8 + 5 = Add to 10 …8 + (2 +3) …8 + 2 = 10 + 3

Strategies for Learning the Subtraction Facts Using 0 and 1 Doubles Counting Back Counting On 15 – 0 doesn’t change the number; 15-1 changes the number by 1 Doubles – 16-8… 8 +8 = 16 therefore 16 –8 must be 8 Counting Back - 9 –3; Think 9…8, 7, 6 is the answer Counting On - 8 – 6; Think 6…7, 8 or 2

Strategies for Learning the Multiplication Facts Commutativity Skip Counting Repeated Addition Splitting the Product into Known Parts Using 0 and 1 Pattterns 4 x 6 = 24 and 6 x 4 = 24 Hundreds board is great for this. 4 x 5 – Skip count the 5’s four times – 5, 10, 15, 20 or by 4’s 5 times – 4, 8, 12, 16, 20 – Skip counting around the clock face is a good way to reinforce multiples of 5 Can be used more efficiently when one of the factors is less than 5. 3 x 6 = 3 6’s added together or 6 + 6 + 6 Can be done by thinking of ONE MORE SET or TWISE AS MUCH AS A KNOWN FACT. An example of ONE MORE SET… if 2 x 5 is known, one can figure out 3 x 5 by multiplying 2 x 5 to get 10 and adding 5 more = 15 More difficult when the addition requires renaming. If students are trying to figure out 7 x 8 and they know 7 x 7, then 7 x 7 = 49 + 8 = 56 TWICE AS MUCH AS A KNOWN FACT – Given the problem 6 x 8, if students know 3 x 8, then can double the product = 3 x 8 = 24 doubled = 48, then 6 x 8 = 48. Children can eliminate facts if they know that multiplying by 1 does not change the other number and that multiplying by 0 results in a product of 0 Finding patterns can be helpful. For example the nines. The sum of the digits is 9. Finger multiplication. Hold down 1st finger – 1 x 9 = 9; hold down 2nd finger – 2 x 9 = 18

Strategies for Learning the Division Facts Think Multiplication Repeated Subtraction Division is the inverse of multiplication. For the division fact 54/9 = ?, students can think what x 9 = 54 ..oh, that’s 6. 12/ 3 = 12 – 3 – 3 – 3 – 3 ; I had to take away 4 3’s; the answer is 4 Thinking strategies for division are far more difficult for children to learn than are the strategies for other operations. The child must remember more and regrouping is often necessary. The primary burden falls on the child’s facility with the multiplication facts. Being able to recall those facts quickly is critical!

Division – Why is it so difficult? Computation begins at the left, rather than at the right as for the other operations. Involves not only basic division facts, but also subtraction and multiplication. There are a number of steps, but their pattern moves from one spot to another. Trial quotients must be used and may not be successful at the first or second attempt. Teachers struggle to teach division, and children struggle to learn it.

Four Computational Tools Mental computation Estimation Written computation Calculators As teachers we need to think about which computations are needed and important enough to learn in school, and how instructional time should be directed. Research indicates that more than 80 percent of all math computations in daily life involve mental computation and estimation of numerical quantities rather than written computation. Research also shows that 70 to 90 percent of the instructional time in elementary school math directed toward computation is focused on written computation procedures. Mental computation – Children need to build facility with mental computations. 1) More than ¾ of the calculations done by adults are done mentally. Point out Math Strategies Series – Each week begins with Mental Math (CP p. 187 for example) 2) Many problems are easier to solve mentally and 3) proficiency in mental math contributes to increased skill in estimation. Estimation – Estimation is the process of producing an answer that is close enough to allow for good decisions without making elaborate or exact computations. Front-End Estimation – Using the first or front-end digit to make an estimate Adjusting or Compensating – 4 items at $0.78 . One person might think 4 x $1 but less than $4. Another might think: 4 x .75 is $3.00, a little more than $3.00 Compatible Numbers – Numbers that go together naturally. 6 and 4 are compatible with 10 Problem – 348 + 427 + 555 + 7 + 675 + 198 = Use front end and then 98 is about 100. 75 and 27 is about 100. And 55 and 48 is about 100.

NCTM Principles and Standards Children should select appropriate computational methods from among “mental computation, estimation, calculators, and paper and pencil according to the context and nature of the computation and use the selected method or tool” (NCTM, 2000, p. 148)