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Mental Math

What is mental math?• Consider the time when you first learned to

drive. Think about how much attention you had to pay to each of the discrete skills that make up driving. You had to deliberately check your rear view and side mirrors. You had to remind yourself, consciously, to move your foot from the gas to brake pedal, and so on. Today, as accomplished drivers, each of those skills happens for you at an almost subconscious level. You are freed, while driving, to have a conversation, pass a snack to your child in the back seat, etc.

Why is mental math important?

• Mental math is a lot like the discrete skills that make up driving. Mental math is the calculations that are done in a student’s head without the guidance of a pencil and paper, calculators, or other aids. Math facts can be a part of mental math but so are strategies to help add, subtract, multiply or divide larger numbers.

How mental math is developed?• Like the discrete skills that make up

driving, mental math strategies need to be, initially, focused on deliberately. After a while, they become fairly automatic, allowing us to use them to accomplish more comprehensive tasks and solve more rigorous problems.Therefore, mental math is important to help students tackle multi-step and challenging problems. These are the types of problems that constitute a bulk of their math education.

A few strategies to support mental math

include:• - decomposing numbers: breaking numbers down into smaller and easier-to-use numbers (when multiplying 13 by 7, recognizing that 13 can be decomposed into 10 and 3 making multiplying 10 by 7 and 3 by 7 and adding the products easier than the original problem)

• - friendly tens: manipulating numbers to work with easier or “friendly” tens (when adding 73 and 59, recognizing that breaking 73 into 70 and 3 and breaking 59 into 50 and 9 makes adding easier because of the friendly tens ... this is a specific way to decompose numbers)

Subitizing-the 1st step in Mental Math

• Students in Kindergarten and first grade learn to automatically recognize the amount of objects and then connect it to a number.

• Students use this mental math strategy instead of constantly recounting amounts that they know.

• This translates into learning and under-standing number bonds, addition, and subtraction.

Ten Frame Practice

Directions - Look at the Ten Frame. Without counting, say the number of dots that are shown. See how fast you can get!! Automaticity is the goal!

Number Bonds• Number bonds are mental pictures that

illustrate the relationship between a number and the parts that combine to make it.

• It helps children to understand that a whole is made of parts.

• When you have two parts add to make a whole

• When you have one part and a whole subtract to find the missing part7

512

7

Number Bonds• Number bonds help students see

that numbers can be broken into parts which will make computation easier. Students learn to recognize the relationship between numbers through a written model that shows how the numbers relate.

• Strengthens a student’s ability to do mental computation.

Through a first grade lens…

This example shows one way a 1st Grade child might determine the sum of 8 and 5 using the skills learned in Kindergarten. The student knows that 2 more needs to be put with the 8 to make 10. So the child decomposes 5 into 2 and 3.  So now he is thinking of 10 + 3, or 13.

In the eyes of a 2nd grader…Students are expected to "add

and subtract within 100 using strategies based on place value, properties of operations, and/or the relationship between addition and subtraction".  Now our student is faced with finding the sum of 37 and 25, which would normally be thought of as a "regrouping" problem.  This student is still looking to make 10s, so she knows she needs 3 more ones to make 37 into 40.  And notice that she is really still splitting 5, even though now it's actually 25.  After splitting, she's got an easy mental problem in 40 + 22.

Elapsed time maybe? Doesn't this actually mirror what we do when we calculate

elapsed time mentally?

Time

Measurement

Notice that all of these examples follow the same format for recording the “splitting" process, and that it all

starts in Kindergarten!

Fractions

In grades 3 and above, students must

"decompose a fraction into a sum of

fractions with the same denominator in more than one way"

and "add and subtract mixed numbers with like denominators".

Compensation• This is a technique we use all the time in math

without necessarily thinking about it. We use it to convert a problem to a more manageable one in order to calculate the answer more easily.

• The key idea with this strategy is balance.

AdditionWhatever you do to one side, you have to do the opposite to the other addend using the same value.

Example:

99 + 34 = ?

99 + 34 = ?+1100 + 34 = 134

134 - 1 = 133

SubtractionIf you add or subtract any value from both numbers you will get the same answer. It is called a constant difference.

Example:

87 - 19 = ?

202 - 187 = ?

87 - 38 = ? +287 - 40 = 47

47 + 2 = 49