Mathematics: It Started With Numbers

I’ve made a harrowing discovery. There is a gaping hole in my understanding and knowledge of mathematics. For most, my statement doesn’t seem concerning. Mathematics isn’t a popular subject. But for a future math teacher, like myself, it’s a confession of a big, terrible shortcoming.

My revelation came when my class was asked a very simple question, “What is mathematics?,” and I couldn’t think of a satisfactory answer. The confidence I had in the very subject I was preparing to teach was gone. How can someone teach a subject, when they can’t even define it in words?

Before that question, I believed I knew what mathematics was. I’ve been doing it for years - adding, subtracting, multiplication, matrices, algebra, geometry, and proofs! But those words are either what people do in mathematics or a specific area within mathematics. Not one of them encapsulates what mathematics is, and listing everything in mathematics does not describe what it is either. So, what do all those words have in common to help define mathematics in a clear, precise way?

At first, I thought numbers. Recalling my elementary years, learning math was learning adding, subtracting, and other operations. But before I could do or be taught math, I needed to know numbers. To learn numbers, I counted physical objects before and during kindergarten.

Others in the class shared my view, stating that math is about describing the physical world with numerical values. Some expanded on this concept and said putting numerical values in an order is math. Others mentioned measurement, explaining how trade spurred the use of mathematics, while others mentioned astronomy and how mathematics was used to find the length of a year and day. Finding patterns was also mentioned, and a particular person said, “mathematics is the science of patterns.”

“But is that enough?,” our professor asked and continued to ask with each new claim of what math is. Is having a pattern enough for something to be labeled math? How about counting? Are people doing math when they count? How about labeling a quantity with a specific value, like 4 or 9? Does knowing you have 9 cows make it math? Or could any conceptual form of measurement make it mathematical? A student used an example of a stick as an example of the first rudimentary means of measurement - walk 4 stick lengths and then turn left. How about time? Does knowing the position of the sun in the sky or time of the year make it mathematical? Does assigning that position with a numerical value like 1:00 pm make it mathematical? Just because it has a number or value or quantity, does it make it math?

Since the class period, I’ve been thinking a lot about these questions the professor posed to the class, and I’ve come to the tentative conclusion that yes, just because something has a number or could be described or substituted with a number, it is math. To do the simplest math, like adding, numbers needed to exist or at the very least some form of measurement had to be in place to describe and order our world. To figure out time, the concept of a start and end needed to be established, which could then be described more specifically with numbers. Then operations could have been conceptualized from there. Imagine two people looking at a sundial. One says to the other, “Meet me in two sun-lengths time from now.” The other person sees the shadow on the sundial, and then imagines the position of the shadow in two sun-lengths time. That person has just done addition. Mathematics started with numbers.

Lastly, I have another confession to make. My knowledge of the history of mathematics is very and utterly limited to pop culture. Another embarrassing fact about the math teacher to-be. When asked to think of five top moments or milestones in mathematics, I could barely think of five at all. The first one I thought of was Albert Einstein’s E=mc2. My knowledge of that equation is very lacking. Do not ask me to explain it or tell you what it is for. My embarrassment is bad enough.

The second milestone was the discovery of the number Zero, since it took civilizations a long time to create a number to describe nothing. A student in my group had also mentioned this when we were discussing the “What is mathematics?” question, and I remembered hearing that fact in high school when I learned about how certain civilizations influenced others when they traded or were conquered in war. Again, don’t ask me which civilization came up with the concept of Zero first.

The third milestone was Sir Isaac Newton discovering or creating Calculus. The fourth was Alan Turing’s use of mathematical theory (I’m assuming from the movie) to create a computer during World War II - thank you very much, The Imitation Game! The fifth…didn’t the first woman receive the Nobel Prize in mathematics for something last year? I remember reading an article about it. The math portion was completely over my head.  

Luckily, someone acknowledging they have a problem is the first step towards a solution, and my remedy to fix my lack of knowledge is to read Clifford Pickover’s The Math Book. According to Amazon.com, the book gives brief explanations to many historical moments in mathematics. Just the book I need.

Repeating Decimals Repeated

In the previous post, I discussed the connection decimals have with fractions, and specifically went over when fractions have repeating decimals. I explained how we can tell when a fraction is a repeating decimal, and that concept is so important and fundamental to mathematics, I decided to bring up repeating decimals again to talk about it more deeply. And that concept is finding patterns. We know when we have a repeating decimal, and what those repeating decimals are, by finding repetition in the remainder during long division.

One of the best fractions to illustrate this point of finding and using patterns in math are the ones with 7 as the denominator, x/7. For a homework assignment, I explored repeating decimals and fractions. So I started with fractions with 3 in the denominator, then 6, then 7, and then 11. The fractions with 3 or 6 in the denominator had a simple pattern with one or two repeating decimals. With 7 as the denominator, the pattern was more complex with 6 repeating decimals. But an even more interesting pattern came to light. Usually when the numerator changes, so do the numbers in the fraction's decimal form. That wasn't the case when the numerator changed with 7 as the denominator. Instead, the numbers were the same, they were just in a different order that could be determined by the sequence or pattern of the remainders, as shown in the figure below.

To show how the fraction x/7 is a special case to students, I would start with 1 in the numerator, or x=1. In the previous post about repeating decimals, I showed the long division process using both decimals and whole numbers. But in the this post, I only used whole number because I wanted to keep the number of figures down. But during a class, I would do both. Now, the number 7 can't go into the number 1. So we add a zero, shown in red, to get 10. 7 goes into 10 once, with a remainder of 3. 7 can't go into 3, so I drop down a zero, shown in purple, to get 30. 7 goes into 30 four times, with a remainder of 2. I need to drop down a zero again, shown in turquoise, to get 20. Then 7 goes into 20 twice, with a remainder of 6.

At this point, a student might say that this fraction doesn't have a repeating decimal. This is another good reason to explore this example with students. It shows that a repeating decimal can have any number of repeating digits in it's answer, not just 1 or 2. So, I show the students we need to continue by dropping down another 0, shown in green, to get 60. 7 goes into 60 eight times, with a remainder of 4. Drop down another zero, shown in orange, to get 40. 7 goes into 40 five times, with a remainder of 5. Drop down another 0, shown in pink, to get 50. 7 goes into 50 seven times, with a remainder of 1, which is the number we started out with in the numerator. And thus, we finally have the repeating reminder that indicates which numbers are repeating in the decimal form of this fraction. This should help illustrate that fractions will either have a finite number of digits or an infinite number of repeating digits, whether it's 1 or 12, in their decimal form.

Next, I would ask the students to change the numerator and divide by 7 to see if they notice any patterns when they do so. As shown in the examples below, I have compared 1/7 to the other fractions that have 7 as the denominator. I've color coordinated the drop-down-zeros to show how the order of the remainders, and consequently the order of the decimal numbers, changes when the numerator changes. I have also labeled the remainder and decimal order based on 1/7 to help illustrate this pattern even more. 

In 1/7, there are 7 numbers, 0-1-4-2-8-5-7, before the remainder 1 repeats itself again in the long division process. This indicates that there are 6 repeating digits in the decimal form: 1-4-2-8-5-7, as indicated by the long bar over these 6 numbers.  As we can see, whenever the numerator changes, that numerator corresponds to one of the remainders when we divided 1 by 7. For example, the numerator 2 corresponds with remainder #3 from 1/7, and the numerator 3 corresponds with remainder #2. Consequently, this shifts the order of the remainders, which shifts the order of the digits in the decimal. So, for 2/7, the remainder and decimal order is 3, 4, 5, 6, 1, 2. For 3/7, the remainder and decimal order is 2, 3, 4, 5, 6, 1 as shown in the figure below. 

 For 4/7, the remainder and decimal order is 5, 6, 1, 2, 3, 4. And for 5/7, the remainder and the decimal order is 6, 1, 2, 3, 4, 5, as shown in the figure below.

Lastly, we have 6/7. And the remainder and decimal order for this fraction is 4, 5, 6, 1, 2, 3, as shown in the figure below.

Hopefully, the students will notice a pattern at the very least, and the class can share their reasons for why the pattern happens the way it does. The figures above should help them along in their reasoning and understanding. By the end of this example, I hope they achieve a better understanding of repetition patterns with remainders to determine a repeating decimal. Also, I hope they gain a better ability to notice patterns in math in general, since patterns greatly help in understanding math concepts and solving problems. 

Figures were created using GeoGebra.

Repeating Decimals

There is a direct relationship between fractions and decimals. All fractions can be rewritten as decimals, and those decimals (but not all) can be rewritten as fractions. However, the relationship between fractions and decimals may not be clear to students. The first exposure students have with using decimal notation is with money, since American money is written like this: $1.43. With the dollar sign written in front of it, people know 1.43 is representing money and people say that value as 1 dollar and 43 cents. However, money doesn't give students the sense that decimals can represent fractions. But without the dollar sign, the money context is removed, and $1.43 becomes 1.43, which is communicated as one and forty-three hundredths. In this context, the relationship between decimals and fractions becomes clearer, since 1 43/100 is communicated in a similar way: one and forty-three one-hundredths.

To illustrate this point further for my students, I would show them how fractions are hidden division problems. For example, 3/4 can be said as three-fourths or 3 divided by 4. The answer to 3 divided by 4, which is 0.75, is pretty straightforward and the answer doesn't have a remainder. The main focus of this blog post is introducing students to the concept of the repeating decimal, which is when a fraction in decimal form doesn't end neatly, but instead the answer has repeating digits that goes on indefinitely.

The first example I would use to illustrate repeating decimals to my students is 5\6, as shown in Figure 1. Now, I would tell my students that dividing by a number that is bigger than the number being divided, has a similar process as having a smaller number dividing a bigger number. So, I would use the same language that the students are familiar with when doing long division. So, 6 goes into 5 zero times. So, I would put a 0 as the first digit. Since 5 is in the ones place, then I  must use the tenth place (I've designed this lesson assuming I've already discussed place values with decimals with my students). So, I need to place a decimal point at the time when I use the tenth place to show where the whole digits stop and the decimal digits begin. And that spot is right after the 0, as shown in Figure 1. Now, when I was being taught decimals using long division, the teacher had used whole numbers when subtracting. But my professor, John Golden, showed my class a way using decimals. So I will illustrate both ways in my blog post and to my students. That way, students who prefer one method over the other can choose for themselves. In this blog post, the left side of the figures will use whole numbers and the right side will use decimals.

Figure 1: Dividing 5/6

To continue, we need to add a zero, as shown in Figure 2. This concept might be easier to grasp when using decimals, since 5 and 5.0 has the same value, while 5 and 50 does not. But either way, the value for tenth place in our answer will be the same. 6 goes into 50, 8 times, since 6 x 8 = 48. Then I subtract and get 2. Now, I will use decimals. 6 goes into 5.0, 0.8 times, since 6 x 0.8 = 4.8. Then I subtract and get 0.2. This also serves as a good example to show students the trick of shifting the decimal point over one place when multiplying with tenths.

Figure 2: Dividing 5/6

Again, I need to add another 0, so 50 becomes 500, and 5.0 becomes 5.00. Then, I can drop down the 0 to get 20 or 0.20, as shown in Figure 3. Either way the number in the hundredth place of the answer will be the same. 6 goes into 20, 3 times, since 6 x 3 =18. Then I subtract to get to 2. For the decimal side, 6 goes into 0.20, 0.03 times, since 6 x 0.03 = 0.018. Then I subtract to get 0.02. As we can see, especially with the whole number example, we got the number 2 again (the remainder) after we subtracted the two previous numbers. So, I would ask the students, "do you think we will get the same number for the thousandths place in the answer?" To check, I would go through the steps of long division one last time.

Figure 3: Dividing 5/6

To begin one more round of long division, I need to add another 0 again. So, 500 become 5,000 and 5.00 becomes 5.000. I drop the zero down, to get 20 and 0.020. Again, 6 goes into 20, 3 times, since 6 x 3 = 18. And 6 goes into 0.020, 0.003 times, since 6 x 0.003 = 0.018. So, 3 goes into the thousandth place, and when I subtract, we will get the number 2 as the remainder again, as shown in Figure 4. This will show the students that the answer will have a repeating decimal that will continue indefinitely. It will also show the students how to recognize a repeating decimal when it happens, and that's by seeing remainders repeating during long division. I would then show them the proper notation for a repeating decimal, and that is placing a bar over the number or numbers that are repeating, as shown in Figure 4.

Figure 4: Dividing 5/6

In my next example, I will show the students how a series if numbers can repeat. So, I will use the fraction 1/11 to show this. I'm dividing 1 by 11, so 11 goes into 1, zero times. So, 0 goes in the ones place. Again, since I need to use the tenth place to continue, I need to place the decimal point next to the 0 to indicate where the decimal values start, as shown in Figure 5. Again, I will be using whole numbers for long division on the left side of the figure and decimals on the right.

Figure 5: Dividing 1/11

Now, I need to add a 0. So, 1 becomes 10 on the left, and 1 becomes 1.0 on the right, as shown in Figure 6. 11 goes into 10 and 1.0, zero times again. So, 0 goes in the tenth place. Then I subtract, and get 10 on the left side, and 1.0 on the right side, as shown in Figure 6.

Figure 6: Dividing 1/11

Again, I need to add another 0, so 10 becomes 100 and 1.0 becomes 1.00. I drop the 0 down, and get 100 and 1.00 at the bottom. Now, 11 goes into 100, 9 times, since 11 x 9 = 99. I subtract the previous two numbers and get 1 as the remainder. For the right side, 11 goes into 1.00, 0.09 times, since 11 x 0.09 = 0.99 times. I subtract the previous numbers, and I get 0.01 as the remainder. Since I have the number 1 in the remainder again, I know I will have to repeat 0 and 9 again indefinitely. So, 09 is my repeating decimal and I show that by placing the bar above those two numbers in the answer, as shown in Figure 7.

Figure 7: Dividing 1/11

Hopefully by the end of both of these examples my students will have a better understanding of the relationship between fractions and decimals. Furthermore, my students should be able to recognize when they will have a repeating decimal by noticing when they have a repeating remainder. 

The images were created using GeoGebra.

Instrumental and Relational Understanding with Fractions

For my teaching class, I read an article "Relational Understanding and Instrumental Understanding," by Richard R. Skemp from Mathematics Teaching in Middle School, September 2006, Vol. 12 No. 2, pages 88 - 95. It defines instrumental understanding as the ability to follow rules and know when to use them, while relational understanding is defined as the ability to know and explain the concept. When I learned multiplying and dividing fractions, it was just a bunch of rules. For multiplication, just multiply the numerator and denominator across to get the answer, as shown in Figure 1. For division, the rule is flip the numerator with the denominator of the second fraction, and then multiply across, as shown in Figure 1. But learning math shouldn't be memorizing a bunch of rules without knowing how they work. It should be about true understanding; knowing the how behind the rule. So, I've designed a lesson showing how the rules work with visual explanations, which will incorporate student instrumental understanding with relational understanding.

Figure 1: Multiplication and Division of Fractions

 First, I would start with dividing 1 by 2 using one circle as the unit. The unit defines what the whole is. In Figure 2, we have 1 circle divided by 2. Students usually know what a half of something is, since division is associated with splitting an amount into groups. So, instinctively they should say that the answer of 1 divided by 2 is 1/2 because the circle is being split by a green line into two groups of 1/2, as shown in Figure 2. Then I would ask the students: does the division rule of fractions fit in with this example and how? Some might see the connection to the flip rule. But if not, then I would show them how dividing by 2 is the same as multiplying by 1/2, which gives us the division flip rule.

Figure 2: Dividing 1 by 2

 Next, I would take the answer from the last problem and have 1/2 divided by 2, which is 1/4. According to Figure 2 above, the circle is already split into halves. Then, I would split the two halves of the circle by 2 as demonstrated with the red line in Figure 3. This will give us 1/4. To show that the division flip rule still works, I will flip the 2 to make it 1/2 and then multiply 1/2 with 1/2 to get 1/4.

Figure 3: Dividing 1/2 by 2

In the past 2 problems, I've only been dividing one circle repeatedly. Now, I'm going to show how the division flip rule still works when dealing with numbers greater than 1. In Figure 4, there are 11 circles, which are being divided by 2. So, the circles will be split into two equal groups, where 5 circles and a half of one circle go into each group which is 5 1/2, or 11/2. Again, dividing 11 by 2 is the same as 11 being multiplied by the fraction, 1/2.

Figure 4: Dividing 11 by 2

Then, I would take the answer from the previous problem, and have 5 1/2 divided by 2 to get 2 3/4, or 11/4. 4 circles get divided equally, so both groups get 2 circles. 1 circle gets split into two halves, so each half goes into a group. And the one-half gets split into two one-fourths, so each fourth goes into a group. Add one group up (2+1/2+1/4), and we get 2 3/4 circles, as shown in Figure 5. And the division flip rule still applies, since 11 x 1 = 11, and 2 x 2 = 4, which gives us 11/4, or 2 3/4.

Figure 5: Dividing 5 1/2 by 2

I have shown how the division flip rule works for whole numbers. Now, I'm going to show that the division flip rule is the same when dividing by fractions. So, again we will take the answer from the previous problems and have 2 3/4 divided by 1/2, which will get 11/2 or 5 1/2 as shown in Figure 6. This is the same as multiplying 2 3/4 by 2, which is the opposite of multiplying by 1/2. So, instead of the answer getting smaller, like in the previous problems, the answer will get bigger. Specifically, it will be the number that got multiplied by 1/2 in the previous problem, which is 5 1/2.

Figure 6: Dividing 2 3/4 by 1/2

So far, I've been multiplying and dividing by unit-fractions. Next, I will show the students how the rules for multiplying and dividing fractions hold for any fraction. So, I will give them the problem 2/3 x 3/4 = 1/2. So, we take the circle and divide it into thirds, and then divide each third into fourths, which gives us a circle split into twelfths, as shown in the Figure 7. In the numerator, 2 x 3 = 6, so we count 6 pieces, which will give us 6/12 or 1/2, which is the purple line in Figure 7. The rule for multiplication holds, since 2 x 3 = 6 and 3 x 4 = 12, giving us 6/12 or 1/2. The red lines indicate the thirds, the blue line indicate the fourths, and the purple line indicate the half of the second circle in Figure 7.

Figure 7: Multiplying 2/3 by 3/4

Since division is the opposite of multiplication, I would again show how dividing the previous answer will get the number that was divided before. So, the problem is 6/12 divided by 3/4 = 2/3. So, we have a circle that is split into twelfths. Then, we split each twelfth into thirds, and then we use the division flip rule. This makes the problem 6/12 x 3/4. And since 6 x 4 = 24, we count 24 pieces in the second circle which gives us 2/3, as shown in Figure 8. The red lines in the second circle indicate thirds. Again, the division flip rule still holds, since 6 x 4 = 24 and 12 x 3 = 36, which gets 24/36 or 2/3.

Figure 8: Dividing 6/12 by 3/4

Hopefully, by the end of this lesson, my students will have the visual explanation that will help them understand how these fraction rules work. That way, the students will have the relational understanding to go with the instrumental understanding.

Figures were created using GeoGebra. 

Distribution Dilemma

Students at a middle school I was observing were simplifying equations, and I noticed a consistent issue among many of them; they did not know how to distribute properly. Many would distribute to the first number inside the parenthesis, but not the second. Others would distribute the outside number to both sets of parenthesis if there were more than one.  So, if I were a teacher and saw this issue with my students, how would I fix it? What is it about the concept the students aren't getting?

The underlying issue might be that the students don't understand the function of the parenthesis and how the number outside each parenthesis got there. Since distribution involves factors and multiplies of numbers, teaching that connection could help the students understand how distribution works, and understand the purpose of the parenthesis. As a way of teaching this concept, I will use a great problem solving method I learned from my professor, John Golden, called the work-backwards method. For example, lets take the problem 8x = 48. Even though 8 is being multiplied by x to get 48, 48 needs to be divided by 8 to get the solution for x, which is 6. In other words, we start with the answer and work backwards. And that's how I approached designing this lesson. I'm going to start with an expression in distributed form and then take out the common factor to show how the parenthesis groups together a set of numbers that has a common factor. This will show how distribution is factoring in reverse.

So lets start with a simple expression as shown in Figure 1. One common factor between 40 is 8. The number 4 is also correct, as well as 2. So, 8 is taken or factored out, as indicated by the arrows, and after dividing 40 and 72 by 8, 5 and 9 are left. The next step is crucial, which is explaining the parenthesis. Since I took out the 8, I need a way to show that the 5+9 was originally 40+72. So, I place the parenthesis around 5+9, right after the 8, which will indicate that the 8 gets distributed and gets multiplied by 5 and 9 to get 40+72. This setup tells you that the number outside the parenthesis gets distributed to the group of numbers inside that parenthesis only. In other words, the parenthesis groups different sets of numbers together respective to their distributor.

Figure 1: Factoring out an 8

The next expression will have 3 terms instead of 2. That way the students will see that more than 2 numbers can be within a parenthesis. As shown in Figure 2, we start with 20+45+55. To help the students visually with the parenthesis, I put them in at the first step. Then the 5 gets factored out, and 4+9+11 is left in the parenthesis.

Figure 2: Factoring out a 5

The next expression will show that not all numbers in the expression can be factored. As shown in Figure 3, a 3 gets factored out of 27 + - 60 - 9, but not 11. So the parenthesis group the left over 9 -20 -3 right after the factored out 3.

Figure 3: Factoring out a 3, but not 11

The next expression has 4 terms, which are factored out by 2, which is placed outside the parenthesis, and the left over 4+6+25+5 is inside the parenthesis.

Figure 4: Factoring out a 2

As discussed before some of these can be factored out in multiple ways. So, I repeated the expression in Figure 4, to show how the same expression can be re-written by re-grouping different sets of numbers by their different common factors. 4 can be factored out of 8 and 12, resulting in a 2+3 in one grouping (as indicated by the parenthesis), while 10 can be factored out of 50 and 10, which results in another grouping of 5 +1 (as indicated by the parenthesis).

Figure 5: Factoring out a 4 and a 10

As an evaluation after this mini-lesson on distribution and factors, I would give the students an expression with the variable x, like the one in Figure 6, for them to distribute and simplify. Hopefully, as Figure 6 indicates, they would distribute correctly. The simplified answer to Figure 6 is 1 - 13x. A good indication of how well this lesson worked at increasing student understanding would be seeing less errors with distribution.

Figure 6: Distribute 5 and 2

The figures in this blog post were created using GeoGebra.

Revealio! The Hidden Negative Number

Subtraction is a difficult concept to grasp when introducing negative numbers. I had difficulty coming up with real-world story problems that had negative numbers either as an answer or part of the expression. Thinking in terms of negative numbers is so difficult because the first math concept we learn as young kids is that numbers exist to count physical, tangible objects. Count the number of oranges. Ten. What's left if all of them are taken away? Zero. Take away one more and what do you have? ....But how can you take away something that's gone? Placing a numeric value on something that can't be seen is like the magic trick where the magician pulls a rabbit out of an empty hat. Something is there that has value, but you can't see it!

But there is an old math trick that I was taught in middle school that reveals the hidden negative number, and I'll refer to it as the change-the-sign trick. Whenever there is a minus sign for subtraction, change it to a plus sign for addition, and then change the sign of the number after it. So a positive number becomes a negative, and a negative number becomes a positive. It's a great way to show students that they've been computing negative numbers all along; it has just been hiding! But how do I, as a future math teacher, show how this trick works, so my students don't rely on memorization to do math, but instead rely on their understanding?

In the teaching class I'm taking, our professor created a huge number line on the floor of our classroom, and we had to come up with ways to demonstrate how we add and subtract integers. The way that seemed less like following a bunch of rules was the one that was used and shown in the reading, The Intersection of Language and Mathematics, by Patricia E. Swanson from Mathematics Teaching in the Middle School, Vol 15, No. 9, May 2010, pg. 516-523. The first number is the person's starting point, the sign of the second number determines the direction they are facing - if it's a negative, face the negative numbers; if it's a positive, face the positive numbers - and the number tells how much they have to move to get to the answer. Lastly, the operation tells the person which way they move - if it's addition, the person moves forward; if it's subtraction, they move backward.

What makes this version the best to me is that the steps are less like rules because they are intuitive and use concepts students already use when they add and subtract whole positive numbers. Facing the direction of the sign of the second number doesn't require memorization - if the sign is negative, it makes sense to face the negative direction on the number line. As the reading discusses, students think of addition as gaining and subtraction as taking away. So, it makes sense for the students to move forward to gain, and move backward to take away. So, I've come up with a lesson using these steps to help my future students understand how to compute negative numbers and how the change-the-sign trick works. Students can either have their own number lines to follow along or these can be done on the blackboard as a whole class. (Figures created using GeoGebra.)

First, I would begin by explaining the steps on the number line and show them which direction the arrow points, or faces, according to the sign of the second number, as shown in the figure below.

Figure 1: Number Line Showing Direction of the Second Number

Next, I would use a simple math equation using subtraction that the students would easily know the answer to, like 5-2=3. I would demonstrate the answer using the steps: start at 5, second number is positive, so the arrow is pointing (facing) in the positive direction, and the operation is subtraction, so we are moving backwards. 

Figure 2: 5-2 = 3

  

Next, I'll show the students the change-the-sign trick. We have a minus sign, so we turn into a plus sign. Consequently, we have to change the sign of the second number from a positive to a negative. Then we'll go through the steps on the number line again to show that the answer is the same: start at 5, but this time the second number is negative, so the arrow is facing the negative direction, and the operation is addition, so we are moving forward. 

Figure 3: 5+-2 = 3

  

Next, I would show them the reflexivity property for adding numbers, and how it still holds true for adding negative numbers. So, first I would show them the property using positive numbers, as shown in the two figures below.

Figure 4: 2+5 = 7

Figure 5: 5+2 = 7

Then, I would use negative numbers, as shown in the two figures below:

Figure 6: -2+-5 = -7

Figure 7: -5+-2 = -7

But what happens when the change-the-sign trick isn't used? To answer this, I would demonstrate what happens using the two figures below. As we can see, the answer changes when we flip the numbers and then subtract, which means that reflexivity is not a property of subtraction.

Figure 8: -5-2 = -7

Figure 9: 2-(-5) = 7

Furthermore, I can show them how subtracting a negative number becomes adding a positive number by using the change-the-sign trick and comparing Figure 9 to Figure 10.

Figure 10: 2+5 = 7

If I have time and if the students seem to get comfortable with negative numbers, I would then show them how the number line, with the negatives on the left of zero and the positives on the right, as shown is Figure 1, is a convention. Technically, we could flip the number line, as shown in Figure 11 and the steps we have been using would still work, as shown in Figure 12 and Figure 13. This helps illustrate how math is consistent and not dependent on certain scenarios.

Figure 11: The Flipped Number Line Showing Direction of the Second Number

Figure 12: -2+-5 = -7 on Flipped Number Line

Figure 13: -2-5 = -7 on Flipped Number Line

Lastly, I would assess my students. In my teaching class, we've learned the steps on how to create a lesson plan by learning what students already know and forming a lesson around what they need to learn. The last step is Assessment, which is when I give them some activity to see how much they've learned and understand.

To assess their understanding of integers, I would show them a series of number lines, like the two below, and ask them to write the two equations that could represent what's shown. The red circle in the figures indicate the first, starting number. If they answer 7 + -15 = -8 and 7 - 15 = -8 for Figure 14, then I know they understand that subtracting a positive number is the same as adding a negative number. If they answer -3 - (-9) = 6 and -3 + 9 = 6 for Figure 15, then I know they understand that subtracting a negative number is the same as adding a positive number. Which is the point of learning the change-the-sign trick!

Figure 14: 7-15 = -8

Figure 15: -3 - (-9) = 6

Inequality Card Tricks

So, yesterday was Thanksgiving, and the family was all gathered. What a perfect time to engage your loved ones in a fun, educational, math game! Unfortunately, the only person who was interested was my father, and I'm sure he was just being nice. But what transpired was great material for a blog post!

The game we played was called "Greater Than," and it was created by Professor Golden. I had played the game before with my mother for my teaching class, and it was very straightforward. It's a card game for 2 players or teams and it's designed to help students practice inequalities. The dealer gives 4 cards to each team/player. Then each team/player selects a card from their 4, which is their starting value, and reveal it at the same time. Then, the non-dealer plays a turn, and the operation and value of that card gets applied to both player's cards. Then, it's the dealer's turn, and so on, until all the cards are played. The team/player who has the greatest value at the end wins! The operations were very simple: adding and multiplying by positive or negative numbers. (To see them in more detail, click the link above.)

After I explained the rules to my father, we started a game. I dealt 4 cards, and my father and I flipped our first card. We both flipped a positive 10. At this point there was no need to continue the game, since whatever cards that were played afterwards would be applied to both cards, and they would end up being equal. So, I dealt another hand, and we flipped both our cards.... We both flipped -11! At this point, we decided to break one of the rules, and thus the first variation was born! The first of three. Here is how it all went...

Variation 1: When a player plays a card, it only applies to their own card. It doesn't apply to both cards.

I dealt 4 cards to each of us. And when we looked at our hand to select our first card, it took longer than the previous two hands, since we could strategize the order of our cards to figure out the greatest value. Before, strategizing was harder, since the value of your hand was controlled by the cards that the other team/player used. Now, each player had sole control over the value of their hand. My hand consisted of +(10), x(-3), +(-4), x(10). If I played the positives first, I would be left with x(-3), which would turn my value negative. And the more negative the number, the lesser the value. So, I played the +(-4) first, followed by the x(-3), which gave me a positive 12. Then, I played the +(10), which gave me 22, and then I played the x(10), which gave me a positive 220. If I had played x(10) before the +(10), I would have gotten 12x10 = 120 + 10 = 130. And 220 > 130, and I want the greatest value possible. So, the order and combinations is very important to think about! Unfortunately, my father had the better hand with 310 (220 < 310). We played 2 more rounds, and my father won all of them. And then I decided to make the math game more challenging and changed the rules again. So, we moved on to play Variation 2.

Variation 2: Adding division to the mix.

So far, we've been adding and multiplying positive and negative numbers. But division is out of the mix. So I decided to include it, and I let the Face cards (not including the Ace, the Ace is still 11) be the dividing or multiplying-by-a-fraction cards. Kings were 1/2, Queens were 1/3, and Jacks were 1/4. With the new rules in place, I dealt 4 cards to each of us. I got x(1/3), +(-11), x(9), x(-5). The +(-11) had to be played first, so I could then use x(-5) to get +(55). Then I used x(9). This math was a little harder to do in my head. First, I added 55 +55 to get 110. Then I multiplied 110 by 4 to get 440. 2x4=8, and I needed to multiply by 9, so I had one more 55 to add, which gets me 495. Then, I divided by 3 or multiplied by 1/3, which got me the final value of 495/3. This is also equal to 165, which I did by figuring out how many 3's go into 4, drop the next number, and so on. My father this time didn't do so well, this round. He got -11/4. We played 2 more rounds, all of which I won. But then my father wanted to try his variation of the game. So, we moved on to Variation 3.

Variation 3: Diamonds now represent division.
My father wanted a whole entire suit to be division, instead of the Face cards. The rest of the rules would be the same. I pointed out that this would mean that we could only divide by a negative number. My father was fine with that. So, I dealt 4 cards to each of us again, and I got +(-10), +(-7), x(-1/2), +(3). I wanted to add as many negatives as possible, so I added (-10) and (-7) to get (-17). Then I multiplied by (-1/2) to get (17/2). Finally, I added the (9), which I converted to (18/2). 8+7 = 15, so I carried the 1, and 1+1+1=3. So, I got (35/2). My father got (-3/11), so I won this hand. We played 3 more games, which my father won all of them. At this point, Thanksgiving dinner was ready, and the game playing was over!

Reflection: All the versions of the "Greater Than" Game will be useful to teach students about inequalities. Depending on the confidence level of my students, I would have them do the original version first, followed by Variation 1. Variation 1, I believe, will teach them more about strategy and how to think a few steps ahead in order to figure out a problem. In this case, it's which card combination and order will achieve the greatest value. Then, if they are getting really comfortable, I would tell them to move on to Variation 2, and let them use a paper and pencil to help them divide/add/multiply/subtract fractions until they felt comfortable enough to do the math in their heads. I would also allow the students to use paper and pencil to figure out difficult multiplication problems with the original version, especially if I saw them struggling. All in all, the "Greater Than" Game is a great way to get students to practice math, learn about inequalities, and learn how to think steps through ahead of time to solve a problem.