Math

Exponent Work for Homeschool Math

Posted by The Beauty of Play on

Playing with Doubles in Cellular Biology

In our last block, cellular biology, we ran across something in our reading that talked about how some bacteria are able to double every 20 minutes. I thought to myself, “That would be some fun math for the day.” I scrapped my previous math plans, and wrote a few things on the board:

“Some Bacteria can double every 20 minutes. 

How many bacteria are there at 1 hour, 2 hours, 3 hours, 10 hours?

What’s the formula?

Graph it.

What time would we have 256 bacterium?”

In this Post You’ll Find:

Definitions

Before I begin, let’s do a short math vocab review, so you know what I’m talking about.

Sequence: set of numbers in a particular order. There are at least 3 different types of sequences: arithmetic, geometric (our sequence,) and triangular numbers.

Term: a location of a number is a sequence. The terms in our work are the number of 20 minutes that have passed. 

Constant: a number that has a known value and often stands alone (and stays the same.)

Resources

You can find out more details about sequences, definitions, and the different types that they are HERE. If you are going to go over this with your child, I’d not share this information up front. It ruins all the fun of the mathematical inquiry and nourishing logic skills and thinking. It may be helpful to you though in guiding a child through this process. It’s also fun to explore without knowing the information. 

Our Process

This type of sequencing we are doing here is called a geometric sequence. The increase (or decrease) is determined by multiplying(or dividing) the previous term with a known constant. In this case, that constant is 2. Because we are doubling, we are multiplying the previous term by 2 each time. 

Initiating the Questions

In the beginning, I just present this questions. Here’s what we know about the bacteria, how can we start finding some of the answers to some of our questions. We are talking about what is happening, and if she understands the information and questions being asked.

From here, she starts to find some of the information by doubling numbers. We do not have the chart at this time, she is just writing down some doubles in a rather messy way. It soon becomes apparent that we are going to have a lot of doubles, and that we don’t remember which double goes with which 20 minutes. I suggest a process change. “I think we maybe we need a system to keep up with our doubles.” 

The Chart

At this point, we draw the long chart that you see here. It has a place for our 20 minute increments and it has a place for the number of bacteria. It’s much easier with the chart in place to see where we are in time and the number of bacteria we have at each of those time intervals. Eventually we add the terms, but not until later.

From filling out a few places in the chart, she’s quickly able to find the answers to the first three spaces in our chart – the number of bacteria in 1 hour, 2 hours, and 3 hours. This ease is intentional on my part. I want her to be able to answers these questions once she gets an understanding of the information and questions. This will keep her engaged and interested, and also give her some confidence that she can, indeed, figure this out. 

From here we jump to 10 hours. This is beyond the ease of simply doubling. She could do it, and was almost willing to, but it is an increasingly greater amount of work to make it happen, so noticing the patterns and coming up with a formula is far easier. 

Noticing a Pattern

We step back and take a look at what is happening. If you’ve been following me for a while this is our ‘notice and wonder.’ You can find our process for that on my post: “Notice and Wonder” with Cuisenaire Rods for Homeschool Math.” 

This particular pattern isn’t difficult to spot. We’ve already been told within the information what we are doing. We are doubling, and that means multiplying by 2 each term. Because we have done this before, she understands what term is and we now begin to write down the number of 20-minute increments that we are using. (Side note: If you are planning to do something similar with your child, I would start with bacteria doubling each hour. This makes the term the same as the hour and much simpler to understand. Here there is a two-step process of not only the doubling of bacteria, but understanding that we have 3 terms within a given hour. It makes the inquiry slightly more complex.)

Writing the Pattern Down

At this point, we begin to write down what is happening. We ARE NOT calculating. There is a difference here that makes seeing the pattern and finding the formula a little easier. We already have our calculations within our chart. The first term is 1, so we aren’t writing anything down quite yet for that. The second term is 1 x 2. This ends up being our base. 

This is where it gets a little more interesting. What are we doing here? We are now taking our 1 x 2 and doubling that, so multiplying by 2, so our mathematical expression of what we are doing is 1 x 2 x 2. This continues. Our last term now is ‘1 x 2 x 2’ and we are doubling that so adding an additional ‘x 2’ making it ‘1 x 2 x 2 x 2’. This pattern continues with adding another ‘x 2’ with every term.

Introducing Exponents

You can see at this point, we are going to be writing ‘x 2’ A LOT. That’s exhausting right. It turns out there is a notation that makes this easier. It’s called the exponent. I can write the number that I am multiplying here as the base, and then add a number as superscripts (super = above, script = write) to tell me the number of times we are multiplying that number. 

After reviewing exponent notation, we switch to use exponent in our mathematical expressions. In addition, we backup and fill in the spaces with the same notation. This is where we notice our ‘1 x 2’ is 21 and our ‘1’ is 20. This can be insightful if you do several of these explorations with different bases, because it becomes apparent that any base with an exponent of ‘0’ will be ‘1’.

Finding the Formula

At this point, it’s pretty easy to see what the formula is: 2n with n being our term. Here I go head and introduce a scientific calculator, so we aren’t doing doubles forever. She is able to determine that 10 hours will be our term 30, because we have 3 increments of 20 minutes in one hour. She plugs in the formula and quickly figures out what the number of bacteria will be.

Graphing the Formula

From here, we went and graphed our formula. We started by labeling the x-axis by the hour. We then discussed how high our y-axis should go, which was over 4,000 because of our highest number in our chart. From that point we decided that our increments on our y-axis would be in the thousands. We charted our bacteria at 1, 2, 3, and 4 hours, but then decided that we needed more dots to see a clearer graph. We went back and placed in our 20 minutes intervals and graph those dots. It was fun to see our graph climb so quickly.

Expanding

This could also be a good work towards logarithms or finding the exponent (term in this case). I ask one such question in our inquiry which is “At what time will we have 256 bacteria?” My question is easily answered in the from our chart. However, if you want to do more work on logarithms, it would be helpful to have questions with numbers that require much more of a challenge and learning a new concept to make it easier.

Video

Let's Chat