What Does a Median Mean?

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No, not that kind of median.

Yesterday I gave the same test to all four of my math classes. Each class had received the same lessons, done the same classwork, the same homework, received the same pre-test review and study guide. Three of the four classes did as well as I expected them to. The median grade for those three classes ranged from 70% to 77%, which is a C, which is average.

I was astounded and disappointed by the outcome for the fourth class. The median grade for that class was a 51%. I have been analyzing their answers and wracking my brain since trying to determine why the 20 point difference.

All four of my classes are at about the same academic level. Based on their classwork and quiz grades, I expected each to perform about the same on the test. I’ve looked hard at the errors the low scoring class made and cannot understand why so many in that class made so many simple mistakes, many of them in basic arithmetic.

As you may have read in my earlier posts, the students have been learning to add and subtract integers. There is a process they learned for how to do such. Many times the work of the low scoring class showed they knew what to do, but then they would come up with the wrong answer when they added or subtracted positive numbers from each other, or they would subtract two numbers that had a plus sign between them and vice versa.

Today I reviewed with them every single problem on the test, had them make the corrections on their papers, and informed them that in the not too distant future they would be taking a retest on this objective. Unfortunately, this will put them a day behind the other three classes but it can’t be helped. We cannot move forward to the next objective until I feel confident they have gained some skill with the current one.

I know 73% of my students came to me performing below grade level and the work they are doing for me is especially challenging for them. I’ve suggested to the students and their parents many different and free online tutorial and education websites they can utilize to help the student improve their math skills. Even though more than 2/3 of my students are eligible for free lunch, nearly all of them have some access to the internet at or near home. Some have already begun to take advantage and the students love to come in and tell me how much they've accomplished on the sites.

There just isn’t time during the school day to catch them up on what they missed in prior years, not if I’m going to teach them everything they’re supposed to learn this year. I will just have to incorporate those catch-up lessons into what we’re doing now.

As always, I remain,

The Exhausted Educator

A Week of Adding and Subtracting and Subtracting by Adding

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Integers are wonderful things. To add and subtract Integers all you need are three simple rules. The first two rules, which I discussed in my previous post, Same Sign Sum and Different Sign Difference, are used for both Integer Addition and Integer Subtraction. However, for Integer Subtraction, on additional rule needs to be applied first.

Theoretically, when it comes to Integers, there is no such thing as Subtraction. In order to subtract one Integer from another, one instead adds the opposite of the Integer being subtracted to the Integer being subtracted from. To help students remember to do this, I use the acronym KCC, which means Keep, Change, Change.

Keep, Change, Change reminds the student to Keep the first Integer in the expression the same, change the minus operation sign to a plus sign, and change the second integer to its opposite. For example:

        -5 – (-7) when KCC is applied becomes -5 + 7.

The student then applies the appropriate Integer Addition rule, in this case, DSD because two Integers with different signs are being added. The difference between the Absolute Values of (-5) and 7 is 2. [|7| - |-5| = 7 – 5 = 2] {Note: The bars on either side of the Integer are known as Absolute Value Bars.} Since the Absolute Value of the Positive Integer is greater than the Absolute Value of the Negative Integer the sum, 2, will be Positive.

Let’s look at another example.

        4 – 9

In this example we are attempting to subtract a larger number from a smaller number. In the set of Whole Numbers this would not be possible. Since Integers include all the Whole Numbers and their opposites (the Negative Numbers) this subtraction can be done using Integers.

4 – 9 = 4 + (-9) = (-5) because |-9| - |4| = 9 – 4 = 5 and since the Absolute Value of (-9) is greater than the Absolute Value of 4, the sum, 5, will be Negative, or (-5).

Both of these examples result in using the rule Different Sign Difference to determine the answer. Now we’ll look at an example that uses Same Sign Sum.

-18 – 14. {Note: The 14 being subtracted is Positive.} Use Keep, Change, Change to rewrite this expression as -18 + (-14). Since both Integers are Negative, we take the sum of their Absolute Values |-18| + |-14| = 18 + 14 = 32, and we give the sum the same sign as the original pair of Integers, (-32).

Students have more difficulty with Integer Subtraction than Integer Addition mostly due to them wanting to change both terms in the expression as well as the operation. To counter this, we practice Integer Subtraction more than Integer Addition. Of course, by practicing Integer Subtraction we are getting additional practice in Integer Addition. But don’t tell my students.

As always, I remain,

The Exhausted Educator

Adding Integers – There Are Two Simple Rules

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This past week my math classes learned how to add Integers. Adding Integers is the next logical step after learning about Absolute Value since understanding Absolute Value is essential to understanding how to add Integers.

To review: Integers are the set of whole numbers and their opposites. Absolute Value is the distance a number is from zero on the Number Line.

When we began the lesson on adding Integers, I introduced the students to two acronyms I’ve used for many years. I’ve found these acronyms are simple ways to help students remember the rules for Integer addition.

The first rule is SSS. SSS stands for Same Sign Sum. When you add two Integers with the Same Sign, either both positive or both negative, you take the sum of their Absolute Values and then give the Sum the Sign the two Integers had in common. I often joke with my students that math is easier to do than explain at times. This may be one of those cases. Perhaps an example is called for.

Take the expression 2 + 3. Both the 2 and the 3 are positive. The Absolute Value of 2 is 2 and the Absolute Value of 3 is 3. When you take the sum of 2 and 3 you get 5. Since both addends, the 2 and the 3, are positive, the sum of 5 is also positive.

All this may sound unduly complicated, very New Math or Common Core, but it makes sense to the students. It makes even more sense when you add two negative integers.

Say you want to add (-4) + (-6). The Absolute Value of (-4) is 4 since (-4) is 4 units from zero on the Number Line. Similarly, the Absolute Value of (-6) is 6. When you add the Absolute Values of (-4) and (-6) the sum is 10. Because both of the original addends were negative, the sum will also be negative. Thus, (-4) + (-6) = (-10).

In class we show this using Integer Counters. Positive Integers are represented by yellow chips. Negative Integers are represented by red chips. This helps the students visualize the math. Once they grasp the concept using the chips, we move to the Number Line. By the time they are asked to work out a few exercises on their own, most are ready to do so with mastery and enthusiasm because they really understand the concept.

The second acronym, DSD, stands for Different Sign Difference, and is used for adding Integers with Different Signs. The long version is as follows: When adding two Integers with different signs, you subtract the Absolute Value of the Integer with the lesser Absolute Value from the Absolute Value of the Integer with the greater Absolute Value. The difference between the Absolute Values is the sum of the two Integers, and the sign of the sum is the sign of the Integer with the greatest Absolute Value becomes the sign of the sum.

If you’re having trouble making sense of all that, here is an example.

Find the sum of (-12) + 6. The Absolute Value of (-12) is 12 and the Absolute Value of 6 is 6. Now subtract the Absolute Values. 12-6=6. Once you have calculated the difference, the answer is given the same Absolute Value as the original addend with the greater Absolute Value, in this case (-12). Therefore, the answer is (-6).

Change the expression a bit and the answer changes a bit.

Find the sum of 12 + (-6). The Absolute Value of 12 is 12 and the Absolute Value of (-6) is 6. Now subtract the Absolute Values. 12 – 6 = 6. Once you have calculated the difference, the answer is given the same Absolute Value as the original addend with the greater Absolute Value, in this case, positive 12. Therefore, the answer is 6.

Standing alone, each of these exercises may not seem relevant. However, understanding each will become vitally important when the students start solving equations and have to isolate variables. But that’s for a future post.

As always, I remain,

The Exhausted Educator

Labor Day – A Day to Celebrate Labor

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Does anyone find it odd that here in these United States we celebrate Memorial Day, Independence Day, and Labor Day in nearly the exact same fashion?

Memorial Day is celebrated with cook-outs, camping trips, and visits to the beach. Independence Day is celebrated with cook-outs, camping trips, and visits the beach with the added spectacular of fireworks. Labor Day is celebrated with cook-outs, camping trips, and visits to the beach.

Memorial Day is considered the “unofficial” start of Summer though it precedes the summer solstice by 3 weeks or so. Labor Day is considered the “unofficial” end of Summer though it precedes the autumnal equinox by nearly 3 weeks.

Labor Day and Memorial Day also coincide, roughly, with the beginning and ending of the new school year. In our district, Memorial Day comes about 2 weeks before the students get out of school and Labor Day comes the first Monday after they start back to school.

When I was almost a teenager, I remember asking my mother why we had a Labor Day holiday. I wasn’t familiar at the time with labor unions and all they’d done to improve the lives of workers in this country in the early part of the 20^th Century. My mother, with her unusual sense of humor, told me that on Labor Day we celebrated all the hours of labor that mothers went through in giving birth to their children. If my father hadn’t overheard and started laughing, laughter that earned him a harsh glare from my mother, I might have believed her for more than a moment.

My father, never a fan of unions himself, explained to me the theory behind Labor Day, but never mentioned unions. He simply explained that on Labor Day the working men and women of America get a day off in honor of the hard work they do all year.

Celebrating Labor Day by taking a break from our labors does seem a fitting way to spend the weekend. Here in our district, Labor Day Weekend is the last 3-day weekend we’ll have until Veterans’ Day this year. We do not recognize Columbus Day as a holiday here in our school district.

Soon, this holiday weekend will be over and it will be back to school for me and my students. There’s math to be learned, activities to be enjoyed, and exercises to be completed and graded. This shortened week promises to be an interesting one.

As always, I remain,

The Exhausted Educator

Hermine Visits Middle School

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This year the first week of school ended with a washout. None of our students washed out, but all of our students were “washed out” of school early on Friday due to the imminent arrival of Hurricane Hermine.

The good news about a storm inspired Early Dismissal is that the day counts as a School Day and will not have to be made up at a later date. The bad news is we had to weather a storm. Thankfully, the impacts of Hermine on our area were minimal. We experienced a good bit of rain and some wind but no serious damage as far as I am aware.

I was pleased with the way our students handled the news of the Early Dismissal when the announcement came. They were excited about getting out of school early, as one might expect. They were also anxious about what Hermine might do to our area. To allay their concerns, I brought up The Weather Channel on our in-class projector and showed them that we were on the very edge of the impact area.

Awaiting the arrival of Hurricane Hermine is how we ended the school week. During the week we had some planned excitement with our first Fire Drill. Even though we all knew it was coming, the Fire Alarm in our building is so loud and abrupt, we all nearly jumped out of our skins when it sounded. Then there was some confusion as the first teacher on our hall led the students down the wrong sidewalk. Our evacuation route is to the left of the Media Center. The eighth grade uses the sidewalk on the right side of the Media Center. The new teacher on our team led our students down the right side sidewalk.

No real harm was done. I was able to meet the students as they came around the Media Center and bring them back into the fold. I was very proud of my class. They’d gone ahead with one of my other team mates and when I caught up, after finding our lost lambs, they were lined up nicely, quietly, and awaiting me to come and take role.

Despite all the excitement this past week, we were able to get some Math done. Even the Early Dismissal on Friday didn’t stop us. The three classes I did get to see were able to complete 2 of the 3 planned activities. Hopefully, I’ll have all four classes back on the same schedule by the end of next week.

For the next couple of days, though, I intend to relax, do some writing, and enjoy what is left of this Labor Day Weekend.

As always, I remain,

The Exhausted Educator

Year’s First Foray Into Math, And Then Some

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Yesterday during homeroom and first period my students and I spent the whole time going over all those pesky start of the year forms and handbooks each new school year brings. By the time we were done with that there was no time left over to talk math.

In my other 3 classes, since we didn’t have to go over any of that bureaucratic paperwork, we were able to delve into some math. The math we touched on was rather basic, consisting primarily of a review of elementary concepts.

Okay, so the ‘yesterday’ of paragraph one has become two days ago and my classes have not only touched on some basic concept review, we’ve begun the curricular lessons.

Prior to beginning our trip up and down the number line learning about integers tomorrow (Hurricane Hermine permitting) we did several group activities today to renew our familiarization with several mathematical properties.

Our first activity involved the Associative Properties of Addition and Multiplication. For those of you who might not remember, the Associative Properties are the Properties where parentheses are used to group operations that are to be done first when evaluating an expression. For instance, to simplify an expression like (x+3)+5 you would regroup by moving the parentheses to get the expression x+(3+5). Now you can combine the two like terms, 3 and 5 to get 8. The simplified expression would be x+8.

The first activity also included the Commutative Property. Using the Commutative Property, you can change the order of the addends in an addition only expression or the factors in a multiplication only expression. For example, to simplify an expression like (5+y)+7, you would first commute the 5 and the y so the new expression would read (y+5)+7. Then you would apply the Associative Property to get y+(5+7). The simplified expression would be y+12.

In the second activity, students used the Properties of Zero and One to simplify expressions. Both Multiplication and Addition have a Zero Property, though they work quite differently.

In Multiplication, 0 times any number results in a product of 0. In Addition, adding 0 to any number changes the number not at all. This is sometimes called the Additive Identity Property.

Multiplication also has an Identity Property When you multiply any number by 1, the value of the number does not change. This comes in handy when finding equivalent fractions or common denominators.

Tomorrow we will be putting these Properties to work when we begin learning about using the four basic arithmetic operations with Integers. We will if Hurricane Hermine lets us get in a full school day.

As always, I remain,

The Exhausted Educator