Photo by Cog Dog

Previously, we looked at the big idea behind inductive learning. Let’s build on this idea within the context of math.

### Math Example: Squaring

A typical lesson on squaring numbers might look like this:

- State the rule: “To square a number, multiply it times itself.”
- Offer some examples.
- Students practice with teacher
- Students practice together
- Students practice independently

### Squaring Inductively

Now, let’s approach the same topic using an inductive approach:

#### Give Unorganized Examples

Ask students to group up (or work solo). Give them these examples:

- 5
^{2}= 25 - 4
^{2}= 16 - 3
^{2}= 9

Ask them to determine the job of that *incredible flying two*.

**Scaffolding Note** I purposefully avoided 1^{2} and 2^{2} right now because they are unusual examples that set up false patterns.

- 2
^{2}= 4 is identical to 2 × 2 - 1
^{2}= 1 is confusing for several reasons

Even though we want students to work towards the meaning on their own, we still set the stage for success. *Enhance* the pattern by choosing careful examples.

#### Formalize

Eventually, students will recognize the pattern:

Some Student: I know! It’s the number

times itself!

Others: Wait.. OOOHHHH!

Now you can offer a carefully worded definition:

You: YES! The flying two means we multiply the big number times itself.

And then formalize their understanding by defining the terms:

You: That flying two has a name. It’s called an

exponent. And the big number it’s flying over is called thebase.

Note that students have a *reason to care* about the academic vocabulary, because they’ve been trying to communicate these very ideas in their groups.

#### Test The Pattern

Now we’ll test the pattern. Give students some sample problems, this time without the answers. They should solve on whiteboards so you can see gauge their understanding. Some kids are going to struggle. Identify them for small group practice.

Try: 6^{2}, 7^{2}, and 8^{2}.

Celebrate mightily when you see correct answers.

Notice you’ve practically tricked them into doing repetitive exercises because they’re *proving their own patterns.*

#### A Monkey Wrench

Now unload the two weirdos on them, and throw in zero, since it’s always interesting:

- 2
^{2}= ??? - 1
^{2}= ??? - 0
^{2}= ???

Again, note kids who struggle and provide support. Everyone else can move on to some independent practice or an extension.

#### Extension

For an extension (you know someone’s going to need it!), toss a couple examples in that force students to *extend their pattern:*

- 2
^{3}= 8 - 4
^{3}= 64 - 3
^{3}= ??? - 3
^{4}= ??? - 3
^{5}= ???

### Multiply Fractions

Now let’s look at teaching students to multiply fractions by allowing them to uncover patterns and formulate the rule on their own.

#### Give the Examples

Again, I’ve carefully chosen examples to enhance the pattern. For example, none of these have a simplification step at the end, making the pattern more obvious.

^{1}/_{2}×^{3}/_{5}=^{3}/_{10}^{2}/_{3}×^{5}/_{7}=^{10}/_{21}^{3}/_{5}×^{4}/_{7}=^{12}/_{35}

#### Students Identify Patterns

Eventually, someone will figure it out:

Student: Oh! I know! You multiply the numerators and then multiply the denominators!

This gives you the chance to formalize their definition, and make sure they’re using the correct math vocabulary.

#### Test the Pattern

Now, toss them some more interesting examples as practice (and as a check for understanding). This is a great time to enforce that simplification rule.

^{1}/_{2}×^{2}/_{3}= ???^{4}/_{3}×^{3}/_{4}= ???^{99}/_{100}×^{0}/_{2}= ???

### In Closing

With inductive learning, we still define terms, explain rules, and practice, but the order is different. We’re harnessing students’ natural abilities to enhance our lessons. As you try this, play to students’ ability to recognize patterns and determine the rules, but also keep close to the struggling math students who may flounder in this setting.

For more inductive math ideas, please read these articles:

Next, we’ll look at some awesome math games that build on students’ inductive abilities.

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