Mathematical Reasoning | Real Application

In the post about Making Mathematical Reasoning a Homeschool Priority, I had presented the first part of the lesson where we had changed the value of white. Today, we are going to dive into the second part of the lesson where we change the measuring rod and its value and see real life application.

This added detail of changing the measuring rod adds another level of thinking. When students decide for themselves to use the white rod to measure everything, it can be fun to challenge them to measure by another rod.

Real Life Application

But before we jump into the lesson, let’s see how students could use these skills later down the road in a real-life situation.  I am not going to pick some crazy biology application or advance mathematical application.  Let's work with something that could be a project your kid might want to tackle one day.

If I only had enough material to build a simple bookshelf with 4 shelves with the total height of bookshelf being 48 inches, how should I space each shelf if I wanted each shelf to have equal spacing? Ultimately, the “if then statements” we have been playing with have been just multiplication and the inverse of multiplication is division.

It’s like seeing the 4 shelves as 4 steps in the staircase. So, if the 4^th step is the height of 48 inches, then using the white rod, we can determine that each shelf must have a spacing of 12 inches from the bottom of one shelf to the top of the next shelf.

What if we want varying heights for each shelf?  We can still use the information to determine what combination of heights and number of shelves we can build.  Maybe we need 3/4 the height of red or 3/4 the height of 24 inches for one of the shelves.  What would remain left?  1/4 the height of red plus 1/2 of purple or 30 inches. 

While not everyone is ready to be a DIYer, preparing students to be comfortable with the many relationships that exist between numbers is essential to cultivating confident young adults.  Let's move into part 2 of the lesson on mathematical reasoning.

Lesson 2 on Mathematical Reasoning

Today, we change not only the value of a rod but the measuring rod. We already changed the value of the measuring rod in the last part of the first lesson to orange plus a white, but we did it in subtle way by changing the value of white to 1/11^th. Today, we are going to be clear that we are changing the measuring rod to red.

If red equals 1, then what is the value of white? Remember, that math is a language, and it is always better to begin where the student is most familiar. Children who have gone through Gattegno’s literal work will know that we describe the relationship that exist between white and red in few ways. One of those ways is by describing how white is ½ of red. You could ask the student, “How much of white makes up red?”

If this is familiar to them, give the students time to consider the implications of white’s new value if red equals 1. Once they have concluded for themselves that white equals 1/2, they are ready to move on to filling out the rest of the table.

​Don't forget to add the challenge of equations.  Adding fractions under this framework is easy.  They seem not to get confused by the idea of not adding the denominator.  They also have the table to look back to for help, and it just makes sense how fractions are added.  

Preparation for Advanced Mathematics and Life

Carpentry is an excellent skill to have but it is firmly founded in one’s ability to manipulate numbers and to see how numbers relate to each other. These “if, then” task cards are meant to develop that deep, fluid number sense, and it has real application from finding the right spot to hang a picture to how many shelves you can fit in a certain space.

Manipulating the rods help students to really grasp these relationships and to make use of their understanding. This understanding also preps them for logarithms which help us to condense really large numbers and extend really small numbers to make them manageable.

Sonya from Arithmophobia No More talks about it over HERE, and it has me thinking of a new activity to extend these “if, then statements.”  Wouldn't it be fun to have young kids playing around with logarithms, sequences and series?