Magically Squaring Numbers

My college math students lack confidence (I classify them as mathphobics.); so, I like to show them math "tricks" which they can use to impress their peers.  I encourage them to know their squares through 25. (Yes, I know they can use a calculator, but the mind is so much quicker!)  When we get to solving equations using the Pythagorean Theorem, I introduce this trick. Please note: For the trick to work, it must be a two digit number that ends in 5.
Suppose we have 35².  (This means will be making a square.)

• First, look at the number in the hundred’s place. In this case, it is the “3”. 
•  Next think of the number that comes directly after 3. That would be “4”. 
•  Now, in your head, multiply 3 × 4. The answer is 12. 
• Finally, multiply 5 × 5 which is 25. 
•  Place 12 in front of 25 to get the answer. Thirty-five squared is 1,255.
• 3 × 4 = 12      5 × 5 = 25       
• The answer is 1,225.

This means that we can build a square that is 35 by 35, and it will contain 1,225 squares or have an area of 1225 squares.

Now let's try 65². 

• One more than 6 is 7; so, 6 x 7 is 42. 
• Place 42 in front of 25 (5 x 5) and so 65 squared is 4,225.
• 6 × 7 = 42      5 × 5 = 25      
• The answer is 4,225.

How about finding the square root? We begin by looking at the numbers in the thousands and hundreds place. In the answer of 1,225, we would use the 12. Think of the factors of 12 that are consecutive numbers. In this case, they would be 3 and 4. Use the smaller of the two which, in this case, is 3. Now place a five after it. You now know the square root of 1,225 is 35.

Thirty-five represents the length of one of the sides of a square that contains 1,225 squares.

Now, try some numbers on your own. When you get comfortable with the "trick", try it with your students. They will find out that math can be magical!

The Best Laid Plans. . .Writing Lesson Plans

Lesson plans have always been an Achilles heel for me.  I have taught for so-o-o long, that how to teach the lesson as well as knowing the content is not an issue.  I always have a Plan B, C, and D ready - just in case.  I now teach on the college level where no one checks my plans; however, I still write an outline for the day so I know that I have covered the important points. 

My first job, when I retired from our local school system, was teaching math at a private school.  Mind you, I had been teaching math for over twenty years; yet, the administrator wanted me to do detailed plans which had to be turned in every Friday. I grudgingly did them, but would add little comments in the comment section. That space became my way of quietly venting; so, I would write such things as:  "So many lesson plans; so little time. Writing detailed plans is not time well spent.  To plan or to grade, that is the question.  I am aging quickly; so, I need to make succinct plans."

My supervisor finally relented and allowed me to do an outline form of plans. However, he visited often to observe my teaching, which I didn't mind.  At least he knew what was happening in my classroom.  I have learned from teaching and observing student teachers that anyone can come up with dynamite plans, but the question is: "Do the plans match what the teacher is doing in the classroom?"  Remember Madelyn Hunter?  Oh, how my student teachers hated her lesson plan design, but they did learn how to make a good plan. To this day, I still do many of the items such as a focus activity and a lesson reflection at the end.


My Husband's Lesson Plans for a Week in October

As many of you know, my husband is a middle school science teacher. He is the "Sci" part of my name. Anyway, he is in his 43rd year of teaching, and he still does lesson plans - not the detailed ones we did our first couple of years of teaching, but plans he does have. He divides one of his white boards into sections using colored electrical tape as seen in the illustration on the left.  He then writes what each class is doing for the week in a designated square. In this way, the principal, parents, and students know the content that will be covered. Even the substitute (he is rarely sick) has a general idea of the day's activities. If plans change, he simply erases and makes the necessary corrections.

So what kind of plans are you required to do?  Maybe there are no requirements for you, but do you still write plans?  Are they in outline form or just brief notes to yourself?  I am interested in knowing what you do; so, please participate in the this conversation by leaving a comment.



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By the way, do you need a lesson plan that is easy to use, and yet is acceptable to turn into your supervisor or principal?   Check out my three lesson plan templates. One is a generic lesson plan; whereas, the other two are specifically designed for mathematics (elementary or secondary) and reading.  Checklists are featured on all three plans; hence, there is little writing for you to do. These lists include Bloom’s Taxonomy, multiple intelligences, lesson types, objectives, and cooperative learning structures. Just click under the resource cover.

Is Extra Credit a Worthwhile Option?

Among teachers, extra credit work has its supporters and its critics, and there are a large number of "undecideds" as well. (Sounds like a political poll!) The range of viewpoints is understandable because the whats, when, whys and hows of extra-credit assignments really matter. Many instructors can't determine whether extra credit is a benefit or a liability, whether it is a point of contention or a headache. In other words, often it is a controversial practice.

When considering extra credit, think about these questions.

1) Does extra credit urge the students to spend less effort on their main assignments?

2) Are extra credit assignments meaningful or mere busy work?

3) Will extra credit encourage student behaviors that will not serve them well in the real world?

4) Should extra credit opportunities be extended to every student or be offered only to certain students on a case-by-case basis?

5) Can extra credit work contribute to grade inflation?

Teaching on the college level, I find that particular instructors never offer extra credit under any circumstances. (That’s me!) Others embrace it as a way to help students learn the course material or improve an unacceptable test score. A small minority, if pushed, will confess they only offer it when students wear them down until they finally give in to it. Most instructors understand that if there are too many opportunities for extra credit, it could possibly outweigh the required course assignments to the point where a student could pass the class without meeting all the standards. (YIKES!!)

I have always been anti-extra credit, the central reason being that it can inflate grades and allow students to receive grades that truly do not reflect their abilities or understanding of a subject. (Remember, I teach math.)  This is the way I view it.

• Extra credit reinforces students’ beliefs that they don’t need to work hard because whatever they miss or choose not to do, they can make up with extra credit. 

• Often, students who ask for extra credit tend to be those who aren’t succeeding or those who hope they won’t have to work hard because some easy extra credit opportunities will be available to them. 

• It is an unintended chance to make up for low scores on earlier exams or missed assignments. (I would NEVER create extra credit assignments at the end of a grading period for students who needed a boost in their grades.) 

• Time spent on extra credit means less time spent on regular assignments. 

• Extra credit (especially if it is easy) lowers academic standards for everyone in the class. 

• It is basically unfair to students who work hard and get it done the first time or turned in when it is due. 

• Extra credit means more work for me in that it has to be graded! 

So after all of my rambling about extra credit work, my question to you is:

"What are your thoughts (pros and cons) about extra credit?" 
Leave a comment to participate in the discussion.

Reducing Fractions with Pattern Sticks!

When working with fractions, many of my students seem confident in performing the different operations, but a few are still unsure of how to reduce fractions. 

Although I have stressed learning the Divisibility Rules for 2, 5, 10, and the digital root for 3, 6, 9,  some still have difficulty since they do not know their multiplication tables. As a mathematics tool, I have the students make Pattern Sticks, a visual and kinesthetic aid, similar to a multiplication chart like the one on the left. Notice that an extra column (blue) has been added to the chart. (In this space, a hole is punched so that a 1" ring can be inserted to store all of the sticks in one place.)

On the right are the directions for making the Pattern Sticks using a multiplication chart. 

(Side note: My students cut out individual Pattern Sticks which I prefer over cutting a multiplication chart apart.)

I then give the students fractions such as 9/36 to reduce. Using the Pattern Sticks, they search for a column where a 9 and a 36 are lined up in the same column. They easily find it on the 1 strip and the 4 strip. They then take the two strips and line them up so that the 9 is over the 36. (see illustration above) By moving to the left, they discover that 9/36 is the same as 1/4. This is 9/36 in its lowest terms. Also notice that all the fractions in the illustration are equivalent fractions - fractions that have the same value. The Pattern Sticks can also be used to determine what number to divide by and to change improper fractions to mixed numbers.

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If you are interested in learning more about Pattern Sticks and how to use them in your classroom, check out the resource entitled Pattern Sticks: A Math Tool for Skip Counting & Reducing Fractions at Teachers Pay Teachers.