This sample offers just a little taste of what we have to offer.

To get the full solution, your first exstemsion problem, and all of Apples' supporting tools, use the button below to start your free 14-day trial. Your free trial will also give you full access to your second out-of-the-box challenge.

The first key thinking skill at work in this problem is the idea that there are structures and patterns in math, and that once you know them (or discover them), they can be used to your advantage. Here, using a formula to find area, or even just thinking through how to make it easier to find an area rather than just counting squares, is an example of making use of a pattern. Once a solver has a sense of how something works, at its most basic, it’s important to try to make the calculation more efficient, and there is nothing more efficient than using a pattern or a formula! In this problem, the area formula even has a serious mathematical application, in addition to the way it is typically presented and used in school (like finding the area of an irregularly shaped carpet). It’s important to interact with formulas in cases where they really have a value in making it easier to solve a problem (as a true efficiency), rather than just for the sake of using and memorizing.

In this problem, the solver builds a pattern and discovers a structure, a truth about how numbers behave, something that they will quite likely find pretty useful. In order to get to the culminating equation for this problem, the solver had to discern a visual pattern, turn it into a mathematical pattern, and THEN turn it into an equation. That’s a pretty solid example of actually seeing a structure or pattern, and then using it to come to a conclusion. It’s also a really good example of another key thinking skill, which is generalization! This is likely not the first time you’ve seen this idea, because the ability to go from the concrete and specific, to the abstract and general, is a really important thinking skill for a learner to develop. The ability to generalize helps the solver in all subjects, not just math, to draw out the ‘big ideas’ from what they are learning.

A third big skill that is developed by working through this problem is the ability to prove something through pictures. It’s a skill to be able to draw a diagram, and to include all of the relevant information that is needed for someone else to look at the diagram and understand that what you’re suggesting is absolutely true. This is not only a thinking skill, but a vital visual communication skill. We live in a visual world, where being able to express thinking via images is increasingly necessary. Being able to create a mathematical proof visually, as in this problem, helps your student to understand the components needed to ‘prove’ something, while also building their ability to clearly communicate their thinking.