MTH 231 Section 3.4 Mental Arithmetic and Estimation

Overview Mental arithmetic and estimation are essential part of a student’s development. Students must become proficient in one-digit facts for multiplication. They must also recognize and be able to apply the properties of whole numbers.

Easy Combinations This method involves regrouping to find easier sums or products. Regrouping to find multiples of 10 is a common strategy.

Adjustment At the beginning of a calculation, numbers are modified to minimize mental effort. Generally, the same number is added to one number and subtracted from the other.

Examples

Working From Left to Right Utilizes expanded notation. For adding and multiplication, this is the reverse of the traditional algorithms. Examples: – x 235

Estimation Skill in estimation allows a student to determine whether his or her answer is reasonable. The goal of estimation is to be able to see, without doing much computation, how large or how small an answer should be or what it should be close to. NCTM: “Students in grades 3 – 5 will need to be encouraged to routinely reflect on the size of an anticipated solution.”

Examples Will 7 x 18 be larger or smaller than 100? If 3/8 of a cup of sugar is needed for a recipe and the recipe is doubled, will more or less than a cup of sugar be needed? There is a 2-mile long traffic jam on the highway. How would you decide how many cars are in the traffic jam?

Front-End Estimation Start at the left and (pretty much) ignore the remaining digits. Be careful: this method will sometimes cause you to significantly underestimate your result: = = 900 (actual is 999) – 224 = 700 – 200 = 500 (actual is 515) If a more accurate estimate is needed, consider another method.

Rounding A way to determine which of two given values is my number closer to? The level of estimation is determined by place value. For earlier grades, focus more on “closer to” than “place value” Introduce place value rounding in later grades, or once students have learned their place values.

Rounding Example Round to the nearest: 1.Ten thousand 2.Thousand 3.Hundred 4.Ten Don’t forget: zeros to the decimal point!

Approximation by Rounding Round each of the numbers to the leftmost one or two digits. Use the rounded numbers to make the calculation = = 1000 (rounded to the nearest hundred) = = 1000 (rounded to the nearest ten) – 224 = 740 – 220 = 520 (rounded to the nearest ten)