The Operations

In this post, I’m going to give a brief description of how I introduce each arithmetic operation. Due to my focus on mental models, I give a specific definition for each operation and then give the children a lot of time to explore them. This exploration is done in the context of place value.

I introduce the operations in the following order: addition, subtraction, multiplication, division. I introduce each new operation as soon as the definition and symbol of the previous operation have been internalized. I do not wait for math facts to be memorized or for secondary meanings to be learned. The following are my definitions:

Addition: I’ve talked about this operation briefly in my series about counting on. I introduce addition as “putting together.” In my experience, most children have some experience with this operation from daily life and can think about questions such as “What do we get if put 2 cookies and another 2 cookies together?”

Subtraction: I introduce subtraction purely “taking away.” This means that I do not treat it as obvious that subtraction also tells us how much bigger one number is than another. An expression like 5﹣3 means no more and no less than “what we get if we take 3 away from 5.”

The benefit of this definition is that this provides a robust mental model that makes it clear that subtraction is not commutative: that is, that 3﹣5 is not the same thing as 5﹣3. I’ve met many children without a clear mental model who do not distinguish between the two.

Multiplication: I introduce multiplication as “taking copies.” Specifically, I say that the first number tells us how many copies to take, and the second number tells us what number we’re taking copies of. For example, 3⨉5 is defined as 3 copies of 5 and 5⨉6 is defined as 5 copies of 6.

I do not treat it as given that multiplication is commutative from the beginning. Therefore, for my students 3⨉5 means 3 copies of 5 and does not mean 5 copies of 3, which would be written as 5⨉3.

Division: I introduce division as “splitting into a certain number of groups,” where the first number tells us how many things we are splitting, and the second number tells us how many groups we are splitting into. For example, 12 ÷ 3 is defined as the answer to “What do we get if we split 12 things between 3 people?”

As usual, I do not assume other meanings of the symbol from the beginning. For example, I do not treat it as given that 12 ÷ 3 also tells us the number of groups of 3 that we could split 12 into. In my experience, the fact that the answer to those two questions is the same is not at all obvious to most kids.

Some of these definitions require minor tweaking as we move into fractions, but overall, these mental models stay constant for my students throughout elementary math. This allows them to gain comfort with using them and results in excellent intuition about the operations and the relationships between them.

As I walk you through our lessons, you will see that we eventually learn all the standard properties of the operations that kids are often taught to take for granted. However, this does not happens early on as part of the definition! It happens organically as they gain fluency and as their mental models get more robust.

This is a post in a series about our basic approach. Here’s the introduction to the series. Here’s are my first and second posts about place value. I will soon also add posts about equality and variables, as well as a couple of detailed examples of our daily lessons.

Math Puzzle Monday!

Time for a slightly belated Math Puzzle Monday! This is another “shape” puzzle, in which every copy of the same shape needs to contain the same number. These puzzles are an excellent way. to work on a mental model of variables!

This one only uses addition but is in fact rather tricky:

Click on the image to get a copy of the PDF file for easier printing! And here’s the solution:

Math Puzzle Monday!

It’s time for our weekly puzzle! My apologies for not posting last week﹣we were on the road.

This week’s puzzle, like the hexagon puzzle I posted a while back, can easily be done via counting on, but this time, there’s a twist: consecutive squares on the path need to be connected by the move of a chess knight instead of being straightforwardly adjacent! This makes for an appealingly tricky puzzle apart from the arithmetic.

Click on the puzzle to open a copy of the puzzle as a pdf.

And here’s the solution to the puzzle as a pdf, if you want to check your student’s work:

Math Puzzle Monday!

This week’s puzzle is a “shape puzzle.” To fill in the shape puzzle, you need to fill in every blank shape. Moreover, the same shape in the puzzle must always contain the same number: so, for example, if one of the squares contains a 3, so must every other square.

Shape puzzles are a staple in my teaching, because they both gently introduce children to algebraic concepts and allow kids to use backwards instead of forwards reasoning. Backwards reasoning always requires more fluency and understanding, and as a result, it’s very worth practicing once an initial mental model is in place.

You can click on the picture above to open a pdf of the puzzle.

Here’s the solution to the puzzle:

Please let me know if you have requests for specific puzzles (types of puzzles, operations used in puzzles, numbers used in puzzles, etc.) in the comments!

Math Puzzle Monday!

Time for a new math puzzle! This week, we have a true or false equation grid: each square contains an equation, and the job of the puzzle-solver is to spot the wrong equations and then to color them in. If done correctly, the wrong equations form a pattern! For my classes, this pattern is often a letter or a number, although that’s not the case in the puzzle below.

These grids are an excellent way to make sure that a student has a robust, relational view of the equals sign. They are also a good spot check for whether they understand the difference between expressions like 2﹣3 and 3﹣2.

Staying on theme with my recent posts, this week’s grid only uses addition and subtraction:

Click on the image to open a pdf of the same puzzle.

And as usual, here’s the solution if you need. In this solution, the green squares are correct and the red squares are incorrect:

Math Puzzle Monday!

Today’s puzzle is a magic square! These are really fun: the sum of every column, row, and diagonal must be the same.

In the square below, it’s possible to figure out this common sum without too much trouble. However, these can get much trickier. The trickiest ones require some algebra!

And here’s a solution to the puzzle:

Click on the picture to open a pdf of the puzzle for printing.

Here’s the solution to the magic square with the missing numbers filled in in red:

If you have requests for a kind of puzzle (or for a given puzzle difficulty), please let me know in the comments! These are easy to generate.

Math Puzzle Monday!

I thought it’d be fun to start a tradition of posting a math puzzle every Monday! I’ve been coding puzzle generators for a while, which means I should be able to create an unlimited supply of math puzzles for anyone interested. I’ll try to make puzzles that are relevant to recent blog posts. Correspondingly, the puzzle below is good practice for counting on, provided that a child is fluent with counting on without concrete objects and can rote count to 40:

Click on the puzzle to open a copy of the puzzle as a pdf.

And here’s the solution to the puzzle as a pdf, if you want to check your student’s work!