MTH 231

Section 3.4Mental 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:1.352 + 6472.739 – 2243.4 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:

1.352 + 647 = 300 + 600 = 900 (actual is 999)2.739 – 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 49854 to the nearest:1.Ten thousand2.Thousand3.Hundred4.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.1. 352 + 647 = 400 + 600 = 1000 (rounded to the

nearest hundred)2. 352 + 647 = 350 + 650 = 1000 (rounded to the

nearest ten)3. 739 – 224 = 740 – 220 = 520 (rounded to the

nearest ten)