How do you connect fundamental number concepts to algebraic and geometric ideas?

For many of us, math was experienced as a series of course topics.  In our formative years, a great deal of time was dedicated to the study of arithmetic (or basic math).  Following arithmetic, most of us took algebra and geometry courses — usually in high school.  We continued with advanced algebra topics while being introduced to trigonometry and analytic geometry. And for the most daring, their high school experiences concluded with studies in calculus. 

Unfortunately, it wasn’t until these upper level/advanced courses that most of us began to understand the discipline of mathematics as a whole: how it all integrated together, how the arithmetic we learned connected with algebra, and how geometry built on algebra and led to trigonometry and calculus.  Each area often seemed discrete and we didn’t understand the beauty of mathematics and how so much of what we learned — including even the very elementary arithmetic concepts — all nested together.  We missed the opportunity to ever realize these connections, make sense of the fragmented information, and come to feel intuitive about math by operating from a broad and integrated picture of the discipline.
IntegraMath and "the number lab" project work to change this problem for our children.  

The IntegraMath philosophy is that early exposure to the breadth of the concepts across the discipline of mathematics ensures that children formulate an integrated understanding of the BIG IDEAS of mathematics.  We want our students to recognize the interconnections between major ideas in arithmetic, algebra, and geometry so that they leverage these connections as the math curriculum expands.  Thus, our mission is to refresh and deepen key mathematical concepts introduced during the school year, while also working to connect fundamental number concepts to algebraic and geometric ideas not typically introduced to children in these grade levels. 
How do we do this?  We have long since realized that mathematics is much the same as other disciplines in the way in which skills are acquired and knowledge grown: the topics themselves are only important as they offer a means to elevating and/or refining of skills and knowledge.  Topics are not our focus; BIG IDEAS, or Transferrable Ideas, are the focus. Using the IntegraMatrix, we follow a progression of powerful big ideas that help students develop an integrated understanding of mathematics.
Additionally, when students' skills of observation, communication, questioning, and reasoning are honed while presented mathematical ideas in very deliberate, meaningful ways, they are able to explore various topics of mathematics much earlier and throughout their schooling experience.  While there is indeed a progression, i.e. logical sequencing to extending ideas, topics can be explored at random, because it is the BIG IDEAS that link the discipline.  This allows the teaching and learning of mathematics to happen through an integrated approach.  
Thus, as we refresh key arithmetic concepts during lessons, we also explore algebraic and geometric ideas that involve a similar base of knowledge.  With an integrated approach, students are able to extend their knowledge across topics through their recognition of the threading ideas or BIG IDEAS.  The end result is that our kids are much more excited and intrigued by mathematics, and begin to feel they have a more intuitive understanding.  That’s because they realize once they grasp a BIG IDEA, they find it again and again in topics from arithmetic, to algebra, to geometry.  This approach, combined with direct teaching on what it means to be a mathematician, leads to a fundamentally different mathematics experience and empowers students to realize their potential.