Reasoning about Ratios and Proportions

photo of sailboat on the Nile river

“The value of a problem is not so much coming up with the answer as in the ideas and attempted ideas it forces on the would-be solver.”

—I. N. Herstein

Whenever I give a problem or puzzle in an Alexandria Jones story, I’ll try to post the answer soon afterward.

But don’t peek!

In the old days of my snail mail newsletters, the answers always came in the next issue, so students had two months to let the puzzle ruminate without temptation.

After all, if you only read the answer I give, you miss out on the fun of solving the puzzle. Follow the Math Adventurer’s Rule:

  • Figure it out for yourself!

Give the problem a good workout on your own. See what you can notice about the puzzle. Wonder about how it connects to the larger world of mathematics.

Play with the ideas and find out where they may lead.

Then check the answer just to prove you got it right, and maybe to see whether we did it the same way. There’s always more than one way to approach any math problem, and each method offers its own insights.

Continue reading Reasoning about Ratios and Proportions

Introduction to the Science of Counting

fantasy elf archer

One of my favorite rabbit-trails to explore in middle school math is the science of counting and probability.

It’s a great topic because the calculations are relatively easy, mostly multiplication and division. But there’s a lot of thinking that goes into figuring out just what calculations we need to do.

Counting is not only for little kids! It’s part of discrete mathematics, which means the study of things that are not continuous—things that come in discrete chunks—and it’s important in game design, computer programing, and cryptography.

Prealgebra and Geometry games book

This post is a section of my Prealgebra & Geometry Games book where my daughter and I explore the fundamental counting principle, the key to solving many counting and probability puzzles.

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Conversion Factors: How Old Are You in Nanoseconds?

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.

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

Math Prompt: True-False-True

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

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

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

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!

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!

Algebra Puzzle from Lewis Carroll

The Beaver's Lesson by Henry Holiday, illustration for The Hunting of the Snark by Lewis Carroll, public domain

On a friend’s blog, I found this delightful letter written by Charles Lutwidge Dodgson to a young man named Wilton Cox.

The pseudonym Lewis Carroll comes from an alteration of the Latinized name Carolus (for Charles) and the replacement of Lutwidge with Lewis.

Continue reading Algebra Puzzle from Lewis Carroll