Stringing in Zero

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Objective

SWBAT to develop an intuitive sense of the laws of exponents through the use of pattern strings. This lesson gets students ready to work on a project to create a book about exponents.

Big Idea

Students can develop their intuition around the laws of exponents by using the patterns formed in strings.

Start Up

15 minutes

This is one of the first lessons on exponents and is aimed at developing understanding relevant to evaluating x^a where a <= 0. I start with a problem that seems tough: 

Start Up Equation

As students enter the room, I give them a chance to observe the problem and give them a minute or two to solve it. You are looking for students thinking around the most fundamental law of exponents, where x0 = 1, if x≠0. This property is the often the most misunderstood and can lead to major issues as we work toward tougher problems.

I circulate and encourage students in their work and ask them to take on this challenge. I remind them that mathematics is about solving problems precisely and effectively. Its about finding truth in the universe. Here we are trying going to use the language of math to understand why this expression can be simplified so easily. Specifically, we are going to focus on patterns and strings to understand this expression.

I try to start conversations that focus on specific components of expression. I will ask things like:

  • "What term seems overwhelming?"
  • “What operations are happening in this equation?”
  • “Is there one number or operation that might be more helpful than the others?”
  • “What do you think the little 4 applies to?”
  • “What does the little 4 mean?”
  • “What does the zero apply to?”
  • “What do you think the zero power means?”

These questions are designed to give students an opportunity to reflect on the components of the expression, to remember what they already know about exponents, and to try and apply them to the zero power. Essentially we are having causal conversations about their mathematical observations.

As we continue, I expect the most obvious misconception will be that a number to the 0 power is 0. The Start Up Expression is designed to help students with this misconception and still encourage them on this problem. For example, I will circulate in the first minute or so and help students who think the answer is -100, since 0 – 100 = -100. Instead of saying wrong, I can say, “Whoa, you are only 1 off!” This will get other students curious, they will wonder, “How did they do it so fast?” 

Investigation

25 minutes

Now that my students have named the exponents in the expression and shared their basic understanding, I will focus on using strings to understand the basic laws of exponents. I will play the Law and Order theme music with slight adjustments in the graphic and intro dialogue. This will get them excited and help them laugh as they prepare for this exciting project.

I'll say, “We will return to the Start Up Expression at the end of class and discuss the most useful terms in the expression. I want everyone in the room to have an intuitive understanding of the answer to the Start Up. To do that I will show you some strings.”

A string is a pattern that evokes a property, idea or law in mathematics.

I'll say to the class, “Earlier some of you told me that 24  = 16. Why is this?”

This gets the conversation started on our first string. We set up a sequence of equations to elicit the structure leading to the definition of 20. I would show the string step-by-step and involve the class as we build each line:

24 =  2 x 2 x 2 x 2  = 16

23  = 2 x 2 x 2        = 8

22 = 2 x 2              = 4

21 = 2                   = 2

2=

2-1 =

I will give the class a minute to write out their solutions to 20 and 2-1 and to write a sentence or two explaining why they think they are correct. I stress that mathematics is a tool invented to explain the universe and that their voice can contribute to that process. I want them to imagine that they are the first person ever writing out this pattern and to think about the choices they have. Any choice is fine, but they must explain why they think they are correct. I would circulate and quote some interesting remarks to share with the class. For example, some students will write that 20 is 0 since it represents “no 2’s” and that 2-1 = -2 since it is the “opposite of 21”. These are both logical inferences, but they will lead to problems in other areas of mathematics. Although these answers lead to incorrect conclusions, they get us thinking about the right areas. 20 does mean “no twos,” but there must be something else going on if the answer isn’t zero. 21 and 2-1 are opposite, but not in the same way that 2 and -2 are opposite. They aren’t opposite numbers, they are opposite operations. The goal is to help students reach this level of understanding.

To get there, we can observe another pattern in the results of each power. Students should notice that the results are 16, 8, 4 and 2. This means that we are dividing by 2 each time. This means we are dividing by 2 each time. So when 21 = 2 we divide this by 2 to get 20 = 1 and 2-1  = ½.  I help the class explain this by drilling them with different questions that help them conclude this is true. I would help them realize that exponents are not simply numbers but operations. Secondly, we need a slight redefinition of exponents to understand this pattern better. The redefinition is stunningly simple, we just redefine exponential expansions by starting with the number 1, the identity element of multiplication and certainly my favorite number:

241 x 2 x 2 x 2 x 2  = 16

23  = 1 x 2 x 2 x 2        = 8

22 = 1 x 2 x 2              = 4

21 = 1 x 2                   = 2

20 =

2-1 =

By starting with the Identity Element of Multiplication, we can see that 24 really means to multiply 1 by 2 four times. And we always start with 1, (since 1 is the identity of multiplication and exponents represent repeated multiplication). These observations help students accept that 20 = 1, since the zero means multiply 1 by 2 zero times. In other words, leave the 1 alone. It also looks great on the string:

241 x 2 x 2 x 2 x 2  = 16

23  = 1 x 2 x 2 x 2        = 8

22 = 1 x 2 x 2              = 4

21 = 1 x 2                   = 2

20 = 1                              = 1

Of course all of this would come out through conversation with the class. I would bring this back to earlier misconceptions that 2is zero because there are no twos. They were write, there are no twos, but we can’t forget that there is still a 1.

The next step is to notice that we always multiply 1 by some number of 2’s when the exponent is positive. This helps us reconsider the meaning of a negative exponents. If a positive exponent means we multiply then a negative exponent means we divide. Multiplication and division are opposite operations, like the opposite numbers they expected at the start.

I would discuss this with the class and then show the next steps of the string and help them see the reciprocal results between a base and its positive and negative exponent. For example 22 = 4 and 2-2 = ¼ . This is essential. A final string might look something like this:

241 x 2 x 2 x 2 x 2  = 16

23 1 x 2 x 2 x 2       = 8

22 = 1 x 2 x 2             = 4

21 1 x 2                 = 2

20 1                           = 1

2-1 = 1 ÷ 2                = ½

2-2 = 1 ÷ 2 ÷ 2          = ¼

2-3 = 1 ÷ 2 ÷ 2          = ⅛

This type of instruction is tough to write out but only takes about 15 minutes in class. I would then offer students a second example with -2 and reconstruct a second string. Again we would show that (-2)0 = 1.

At this point, partners have a choice on how to proceed. They can select a mild, medium or spicy problem to try. They have 15 minutes and can work on multiple problems, but the goal is to construct a string at a level that is beneficial to them.

Example Mild: Construct two strings. One that start with a positive integer and another that starts with a negative integer.

Example Medium: Construct two strings. One that starts with a positive simple fraction and another that starts with a negative simple fraction

Example Spicy: Construct two strings. One that starts with a positive improper fraction and another that starts with a negative improper fraction.

The guidelines for strings are simple: explore the base from the 4th power to the -4th power.

If students encounter large or tedious numbers, I help them to use a graphing calculator to find the values (MP5). As they work with negative values I remind them of the importance of placing parentheses correctly when writing expresssions. I expect that my students will struggle with the idea that (-3)2 ≠-32

Summary

20 minutes

I start the summary by asking students to share the numbers they chose and ask them about their results. As students share I write out the results of their work. The goal is for students to know that every number raised to the 0 power is 1, so I would list out all the different numbers they chose in a column, so that students could see that any number to the 0 power is 1 (except 00, which is a special case).

30 = 1

(-4)0 = 1

½ 0  = 1

-⅚0 = 1

This would allow us to revisit the start up and share why it is so easy:

Start Up Problem Revisited

You can think of it as just 1- 100 = -99. This is an exciting moment and gets students ready for their upcoming Stringing Strings with Stringy Strings project. They now understand on a basic level that the math we are about to review makes things easier. Students are so used to thinking that math is meant to make things tougher that they forget its true meaning: to make sense of the world we live in.

A brief note about the upcoming project:

For their project, we are creating a book of strings. They can use their strings from class, but they have to write or record a video explaining how to solve each equation. They especially have to explain the step involving the zero power and the meaning of the negative exponents and how they connect to the positive exponents. They also have to explain how these patterns help them think of algorithm for simplifying exponents. For example, they might explain that 2-4 can be thought of as 24 and then one can take the reciprocal. This means that 2-4  = 1/16. I would scan student work if it done on paper, but if strings are made online, they can email them to me. They would also need to email me a link to their video (created most likely on iPads with the educreations app) or submit some type of written explanation. I would assemble all the strings and create our first volume of “Law and Order.”

The book will be digital and one could click on any string to watch a student video and/or read a student explanation surrounding the math in their strings. The book will be catalogued as a future study guide and shared with family and posted online.