Hi, I'm Alex, curriculum director at Math Academy.
As Justin mentioned, there are several criteria that we must meet in our high-school pathway that aren't needed for studying higher-level (e.g., undergraduate) math, or they can be postponed. We decided to remove some of these in the Foundations series.
The idea behind the foundations series is to provide adult learners with the most efficient path possible to get onto the higher-level material.
Examples of topics that were removed from the high-school series to create the foundations series include some of the following:
* Various Geometry topics: All of the _essential_ geometry is covered. However, we removed topics on inscribed angles, Thales' Theorem, Triangle congruence, and similarity criteria (apart from the AA, which is the only one that seems to come up in practice), midpoint and triangle proportionality theorems, a fair amount of solid geometry, except what's fairly standard for calculus (volumes and surface areas of spheres, volumes of cones), lots of stuff on different types of quadrilaterals.
* Conic sections: The essentials are covered in both pathways. But in the high-school path, we go into a little more detail about foci, directrices, eccentricity, and utilizing their geometric definitions (e.g., focus-directrix properties).
* Trig identities and Equations: Covered in both pathways, but the high-school versions go into more detail and consider more cases.
* Some word problem/modeling topics.
* Other arbitrary Prealgebra topics: Divisibility rules, going into more detail about ratios in contextual settings, scientific notation, and some basic data representation topics that one would normally meet in Prealgebra.
* Slope fields. This will be covered in our upcoming differential equations course.
* Some analytical applications of differentiation that are quite specific to the BC Calculus exam: Identifying and removing point, jump, and infinite discontinuities and analyzing graphs of first and second derivatives.
* There are also fewer topics on related rates and optimization, though these topics are still covered.
* Some contextual applications of integration, like volumes of revolution and volumes of known cross-sections.
* Convergence tests for infinite series. When we get to that, these will be covered in real analysis, but other than infinite geometric series (which _is_ covered in Foundations), these tests don't show up too often anywhere else.
* Some ODE models, such as exponential and logistic growth and decay. We cover ODE basics in the foundations course, but particular models will be covered in the differential equations course.
* Taylor series. Again, this can be covered in the differential equations course for anyone wishing to take that course when it's ready.
Happy to answer any further questions you may have.
Removing Taylor series was a tough call. It's one of my favorite calculus topics topics. Something had to give. However, those topics will still serve as prerequisite material for courses that explicitly need them.
As Justin mentioned, there are several criteria that we must meet in our high-school pathway that aren't needed for studying higher-level (e.g., undergraduate) math, or they can be postponed. We decided to remove some of these in the Foundations series.
The idea behind the foundations series is to provide adult learners with the most efficient path possible to get onto the higher-level material.
Examples of topics that were removed from the high-school series to create the foundations series include some of the following:
* Various Geometry topics: All of the _essential_ geometry is covered. However, we removed topics on inscribed angles, Thales' Theorem, Triangle congruence, and similarity criteria (apart from the AA, which is the only one that seems to come up in practice), midpoint and triangle proportionality theorems, a fair amount of solid geometry, except what's fairly standard for calculus (volumes and surface areas of spheres, volumes of cones), lots of stuff on different types of quadrilaterals.
* Conic sections: The essentials are covered in both pathways. But in the high-school path, we go into a little more detail about foci, directrices, eccentricity, and utilizing their geometric definitions (e.g., focus-directrix properties).
* Trig identities and Equations: Covered in both pathways, but the high-school versions go into more detail and consider more cases.
* Some word problem/modeling topics.
* Other arbitrary Prealgebra topics: Divisibility rules, going into more detail about ratios in contextual settings, scientific notation, and some basic data representation topics that one would normally meet in Prealgebra.
* Slope fields. This will be covered in our upcoming differential equations course.
* Some analytical applications of differentiation that are quite specific to the BC Calculus exam: Identifying and removing point, jump, and infinite discontinuities and analyzing graphs of first and second derivatives.
* There are also fewer topics on related rates and optimization, though these topics are still covered.
* Some contextual applications of integration, like volumes of revolution and volumes of known cross-sections.
* Convergence tests for infinite series. When we get to that, these will be covered in real analysis, but other than infinite geometric series (which _is_ covered in Foundations), these tests don't show up too often anywhere else.
* Some ODE models, such as exponential and logistic growth and decay. We cover ODE basics in the foundations course, but particular models will be covered in the differential equations course.
* Taylor series. Again, this can be covered in the differential equations course for anyone wishing to take that course when it's ready.
Happy to answer any further questions you may have.