The farther we go in math, the more division disappears. It ceases to exist as a separate concept.
Instead, we learn to see division as:
Each of these perspectives offers us a new way to think about and make sense of our calculations.
The methods in last week’s Advanced Multiplication post only work for certain numbers, but we have another, more powerful multiplication tool: We can always use a ratio table to make sense of any multiplication.
Ratios are the beginning of proportional thinking. We can systematically alter the numbers in a ratio to reach any quantity required by our problem.
Students begin working with ratios in story problems that help them visualize and make sense of a proportional relationship.
Continue reading Mental Math: Advanced Multiplication, Part 2
Mental math is the key to algebra because the same principles underlie them both.
As our children learn to do calculations in their heads, they make sense of how numbers work together and build a strong foundation of understanding.
Remember that while mental math is always done WITH the mind, reasoning our way to the answer, it doesn’t have to be only IN the mind. Make sure your students have scratch paper or a whiteboard handy to jot down intermediate steps as needed.
Besides, math is always more fun when kids get to use colorful markers on a whiteboard.
Continue reading Mental Math: Advanced Multiplication, Part 1
Over the course of this series, we’ve seen how mental math relies on a child’s own creative ways of thinking. In mental math, children develop understanding of how numbers interact with each other in many ways.
In this way, they learn the true 3R’s of math: to Recognize and Reason about the Relationships between numbers.
And the principles that underlie mental calculation are also fundamental to algebra, so that flexibility and confidence in mental math is one of the best predictors of success in high school math and beyond.
But as we went through the various example problems, did you find the written-out calculations hard to follow?
Don’t force your children to write down their mental math. It looks dreary when I write the calculations out step by step, but that’s not how it works in a child’s mind. With regular practice, this sort of thinking becomes second nature.
As our children grow and develop their math skills, the mental math strategies grow with them.
The basics of mental math don’t change:
But we have new ways to help children do math in their heads as the numbers get bigger and the problems more challenging.
For example, how might kids figure out a multi-digit subtraction like 67 − 38?
First, we need to adjust our mindset…
Mental math is doing calculations with our minds, and perhaps with the aid of scratch paper or a whiteboard to jot down notes along the way.
But we cannot simply transfer the standard pencil-and-paper calculations to a mental chalkboard. That’s far too complicated.
We still want to follow our basic strategies of using friendly numbers, estimating, and adjusting the answer. So how can we help children do math in their heads as the numbers get bigger and the problems more challenging?
How might kids figure out a multi-digit addition like 87 + 39?
Here are three useful strategies…
Mental math is doing calculations with our minds, though we can use scratch paper or whiteboards to make notes as we work.
Doing mental math, children use the basic principles of arithmetic to simplify problems so they can think about number relationships, mastering the basic structures of how numbers work, the same structures that underlie algebraic reasoning.
As always, we rely on two key mental-math strategies.
Division is the mirror image of multiplication, the inverse operation that undoes multiplication, which means we are scaling numbers down into smaller parts. Important friendly numbers include halves, thirds, and tenths, plus the square numbers and any multiplication facts the student happens to remember.
Children learn best through interaction with others, and mental math prompts can lead to fascinating conversations, listening as our kids apply their creativity to the many ways numbers interact.
With mental math, students master the true 3R’s of math: to Recognize and Reason about the Relationships between numbers.
And these 3Rs are the foundation of algebra, which explains why flexibility and confidence in mental math is one of the best predictors of success in high school math and beyond.
Multiplication involves scaling one number by another, making it grow twice as big, or three times as much, or eightfold the size. Multiplication by a fraction scales the opposite direction, shrinking to half or a third or five-ninths the original amount.
The key friendly numbers for multiplication and division are the doubles and the square numbers. As with addition and subtraction, students can estimate the answer using any math facts they know and then adjust as needed.
How many ways might children think their way through the most-missed multiplication fact, 8 × 7?
By doing mental math, we help our children use the basic principles of arithmetic to simplify problems so they can think about number relationships, mastering the basic structures of how numbers work.
And the more our children practice these structures in mental math, the better prepared they will be to recognize the same principles in algebra.
The basic idea of subtraction is finding the difference between two quantities: comparing a larger amount to a smaller one, figuring out what’s left when you remove a part, or finding the distance between two measurements (or two points on the number line).
When you work with young children learning subtraction, remember our two key mental-math strategies.
For early subtraction with numbers less than 20, the most important friendly numbers are 5 and 10, the pairs of numbers that make 10, and the doubles.
When children apply their creative minds to reasoning about math, they can use friendly numbers to get close to an answer, and then tweak the result as needed.
From the very beginning of a child’s experience with math, we want to focus on reasoning, making sense of numbers, thinking about how they relate to each other and how we can use these relationships to solve problems.
The basic idea of addition is putting like things together: combining parts to make a whole thing, putting together sets to make a collection, or starting with an original amount and adding the increase as it grows. Connecting two numbers in relationship with a third number we call the sum.
When you work with young children learning addition, remember the two key mental-math strategies I mentioned in the previous post.
For early single-digit addition, the most important friendly numbers are 5 and 10, the pairs of numbers that make 10, and the doubles.
When children apply their creative minds to reasoning about math, they can use friendly numbers to get close to an answer, and then tweak the result as needed.
Denise Gaskins' Let's Play Math 