While Practicing Exponents, Place Value, Division, Multiplication and Critical Thinking
We had a very short block of one week on Number Bases. Number bases, in my opinion, is not a crucial concept in math. However, it is a fun one and an opportunity to practice exponent, multiples, and division work, and to practice a bit of critical thinking. We needed a break in our history blocks, and this math block provided just what we needed.
In this post you’ll find:
Number Bases
We began the unit with Cuisenaire rods of course. We are taking a bit out of Book 5 of Caleb Gattegno’s textbooks on number systems. We first started with review of our own number system and how place value works. We looked at different magnitudes of 10 using the orange rods and first crosses and then L’s. With the tens place showing one orange rod, the hundred’s represented with two orange crossed showing 10 x 10, the thousands showing a tower of three orange rods representing 10 x 10 x 10 or 103. We took some time here to notice the pattern of 10’s and the exponents of this base as how it showed the place value.
Crosses, Towers, and L’s
If you are not familiar with these terms, crosses, towers, and L’s are used with the Cuisenaire rods to show multiplication and exponents. The crosses are derived from area-model multiplication where we can see we have one length of a rod on one side of the rectangle and another length of a rod on the other. We can use these two rods stacked on atop one another to represent this thus showing one number multiplied by the other.
From here we can move into towers where we can stack different colored rods atop one another to show what we are multiplying. For instance, a 2 x 4 x 5 would be a red rod, a purple rod, and a yellow rods stacked in a tower. When we have a tower of a solid color rod, one number that we are multiplying, we are representing exponents. Eventually our towers become unstable with the height of them, and we can move into L’s to show the base and how high our tower is.
Binary
The second activity we did to explore bases was to use the white and red blocks to talked about what a number system would look like with just digits – the one and the zero. This would be the binary system. In this system, two in our system, would be our ten equivalent. Then we explored what our hundred would be which is 10 x 10 in our system, so 2 x 2 here which is 4. We continued with our thousands and ten thousands being 23 which is 8 and 24 which is 16.
From here, we set out a stair case of the rods and found the equivalent in the binary system. We started with the white rod, which was just 1. Then we moved onto the red rod. We already know from the previous work that red is our 10. Then we moved on the light green rod and looked to see if we could use our red and white to come up with a number. We had one red and one white which gave us a ten and a one, which is 11. Then we moved to the purple rod. Here we were able to get our first 22 or 4 in a number. This was our 100.
From here we looked at our yellow rod, 5 in our decimal system. We were able to get our two squared (four) and then an addition white rod, giving us 101. We went through each of the rods in this way starting at the highest power possible for 2 and working our way through the powers. When we had finished with the binary system, we worked on the base 3 system in the same way. All of this work was done with rods and a white board for the day.
Journal Work
We spaced the material by waiting to place the work in the notebook on following day. This gave a us a chance to review everything we had done the day before. Then we moved into to working with base 5 and placing that work in the notebook as well. Base 5 was explored in the same way the other bases were.
The following two days, we worked on a table of bases. We started with filling in the information that we had done before. This counting in different bases gave us a good sense of how the base worked. This allowed us to fill in the bases for six, and because I wanted to be sure to cover bases over ten, we also did one for eleven. Bases over ten require additional symbols to be created to represent the digits ten and over.
Conversions, Adding, and Multiplying
After we had counted our bases, we worked on conversions for 50, 100, and 150. Choosing these numbers gave us the opportunity to covert using division for finding the equivalent of 50, the double this number, essentially multiplying to get 100 in a different system, and then adding in a different base to find the equivalent to 150.
The conversion boxes that we are using are from Jamie York’s Making Math Meaningful for the Middle Years. They are a simple tool to show the place value and to help convert between systems. These little boxes made it easy for us to find out the place value of each space and then covert the number box-by-box.
Videos
Blackboard Drawing
I also did a blackboard drawing for this block which was a series of nesting polygons. I chose this from our quality of numbers work way back in grade one where different polygons represented different numbers. In the video I describe how I constructed the drawing and also why I chose it.