Conversion Factors: How Old Are You in Nanoseconds?

Posted on by Denise Gaskins
birthday cupcake with sparkler

Homeschool Memories

[Based on a problem I made up for my co-op students, once upon a time…]

Conversion factors are special fractions that contain problem-solving information. Why are they called conversion factors?

  • “Conversion” means change, and conversion factors help you change the numbers and units in your problem.
  • “Factors” are things you multiply with. So to use a conversion factor, you will multiply it by something.

For instance, if I am driving an average of 60 mph on the highway, I can use that rate as a conversion factor. I may use the fraction:

Or I may flip it over to make:

It all depends on what problem I want to solve.

After driving two hours, how far have I gone?

But if I am planning to go 240 more miles, how much longer will it take?

Any rate can be used as a conversion factor. You can recognize them by their form: this per that. Miles per hour, dollars per gallon, cm per meter, and many, many more.

Of course, you will need to use the rate that is relevant to the problem you are trying to solve. If I were trying to figure out how far a tank of gas would take me, it wouldn’t be any help to know that an M1A1 Abrams tank would get about 1/3 mile per gallon. I won’t be driving one of those.

Continue reading Conversion Factors: How Old Are You in Nanoseconds?

Math Prompt: True-False-True

Posted on by Denise Gaskins
girl writing in a notebook, sitting on couch with her corgi

Book, Charlotte Mason's Living MathOne of the stretch goals for my Charlotte Mason’s Living Math Kickstarter campaign is to add a math journaling prompt to the end of each chapter. So, I’ve been playing around with ideas to get readers writing.

Since the book’s all about how to build mathematical reasoning, I’m looking for ways to prompt creative thinking and flexibility in math calculations.

Check Out the Kickstarter

I found some fun ideas in Guy Gattegno and Martin Hoffman’s Handbook of Activities for the Teaching of Mathematics (which you can download here), including the following riff off a puzzle created by Lewis Carroll.

Continue reading Math Prompt: True-False-True

Mental Math: Advanced Division

Posted on by Denise Gaskins
Father and daughter working mental math

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:

  • an inverse multiplication
  • a fraction (ratio)
  • a proportional relationship

Each of these perspectives offers us a new way to think about and make sense of our calculations.

Continue reading Mental Math: Advanced Division

Mental Math: Advanced Multiplication, Part 2

Posted on by Denise Gaskins
Father and son celebrate a mental math answer

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: Advanced Multiplication, Part 1

Posted on by Denise Gaskins
Mother and daughter working mental math together

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

Happy Pythagorean Triple Day!

Posted on by Denise Gaskins
Pythagorean Theorem demonstrated with tangrams

Thursday is Pythagorean Triple Day, one of the rarest math holidays.

The numbers of Thursday’s date: 7/24/25 or 24/7/25, fit the pattern of the Pythagorean Theorem: 7 squared + 24 squared = 25 squared.

Any three numbers that fit the a2 + b2 = c2 pattern form a Pythagorean Triple.

Continue reading Happy Pythagorean Triple Day!

Monday is Square Root Day

Posted on by Denise Gaskins
square tree with roots

On May 5, we celebrate one of the rarest math holidays: Square Root Day, 5/5/25.

Here are a few ideas for playing math with squares and roots.

What is a Square Root?

Five is the square root of twenty-five, which means it is the number we can “square” (multiply times itself) to get 25.

The root is the base number from which the square grows. In physical terms, it is the side of the square.

Imagine a straight segment of length 5, perhaps a stick or a piece of chalk. Now lay that segment down and slide it sideways for a distance equal to its length. Drag the stick across sand, or pull the chalk across paper or a slate.

Notice how this sideways motion transforms the one-dimensional length into a two-dimensional shape, a square.

The area of this shape is the square of its root: 5 × 5 = 25.

What do you think would happen if you could drag the square through a third dimension, or drag that resulting shape through a fourth dimension?
How many shapes do you suppose might grow from that original root of 5?

Continue reading Monday is Square Root Day

If Not Methods: Dividing Fractions

Posted on by Denise Gaskins
Mother and daughter working together on math homewrok

As I said in an earlier post, we don’t want to give our children a method because that acts as a crutch to keep them from making sense of math.

But what if our children get stumped on a tough fraction calculation like 1 1/2 ÷ 3/8?

Continue reading If Not Methods: Dividing Fractions

Middle School Math Proof

Posted on by Denise Gaskins

Homeschool Memories…

Kitten (my daughter) and I sat on the couch sharing a whiteboard, passing it back and forth as we took turns working through our prealgebra book together.

The chapter on number theory began with some puzzles about multiples and divisibility rules.

Continue reading Middle School Math Proof