Perfect Squares and Patterns

Perfect Squares and Patterns

If you have been following my last blog, you may find this new post helpful for you in solving questions of Pythagorean Theorem.

Firstly, I am going to talk about perfect squares.
The following are some examples and non-examples of perfect squares from the online resource:

 http://www.mathwarehouse.com/arithmetic/numbers/what-is-a-perfect-square.php.

Examples of perfect squares

• 9
  • 9 is a perfect square because it can be expressed as 3 * 3 (the product of two equal integers)
• 16
  • 16 is a perfect square because it can be expressed as 4 * 4 (the product of two equal integers)
• 25
  • 25 is a perfect square because it can be expressed as 5 * 5 (the product of two equal integers)

Non examples of perfect squares

• 8
  • 8 is a not perfect square because you cannot express it as the product of two equal integers
• 5
  • 5 is a not perfect square because it cannot be expressed as the product of two equal integers
• 7
• 7 is a not perfect square because you cannot express it as the product of two equal 
• integers

Hopefully, these above examples has helped you understand more about perfect squares.

Secondly, I would like to introduce some easier and faster ways for students to calculate their questions as I did in my last post.

Usually, students would just know the square of one digit number which is from 1² up to 9². However, it would be even better if students can memorize up to 19². They could either memorize them just by memorizing or they could memorize them using a pattern.

1) Perfect Squares - Pattern 1

#’s    Squares             Difference between squares            Difference increases by

11²   = 121                 [+23]                                                        +2

12²   = 144                 [+25]                                                        +2

13²   = 169                 [+27]                                                        +2

14²   = 196                 [+29]                                                        +2

15²   = 225                 [+31]                                                        +2

16²   = 256                 [+33]                                                        +2

17²   = 289                 [+35]                                                        +2

18²   = 324                 [+37]                                                        +2

19²   = 361                

Pattern: the difference between 11² and 12² is +23, and then the difference increases by +2 as the number increases by 1.

2) Perfect Squares - Pattern 2

#’s    Squares             Difference between squares            Difference increases by

152    = 225                   +400                                                      +200

252    = 625                   +600                                                      +200

352    = 1225                 +800                                                      +200

452    = 2025                 +1000                                                    +200

552    = 3025                 +1200                                                    +200

652    = 4225                 +1400                                                    +200

752    = 5625                 +1600                                                    +200

852    = 7225                 +1800                                                    +200

952    = 9025          

Pattern: the difference between 15² and 25² is +400, and then the difference increases by +200 as the number increases by 10.

By knowing and memorizing these patterns, students can calculate squares faster. 

Hope this post helps you in calculating squares as well.

Pythagorean Theorem

Pythagorean Theorem

A right triangle has a 90^。angle. The longest side of the triangle, c is opposite to the right angle and it is called the hypotenuse.  The other two sides, a and b, are called legs.

Pythagorean Theorem:


For a right triangle ABC with sides a, b and c, and ∠C = 90^。

c² = a² + b²                                      

The Pythagorean Theorem is the most famous and most important result in geometry. It allows us to compute the unknown length in a right-angled triangle, if given the other two sides.

Example: Calculate the length of the unknown side.

c² = 3² + 4²

    = 9 + 16 = 25

c = √25 = 5 m

Example: Calculate the length of the unknown side.

10² = 8² + b²

100 = 64 + b²

100-64 = 64 + b²-64

36 = b²

B =√36 = 6 cm

Practice:
Calculate the length of the unknown side, to one decimal place.

          15² = 13² + r²






Pythagorean Theorem Extension

After introducing Pythagorean Theorem, I really want to talk about one convenient way to calculate the length of the three sides in a right angle triangle. 

If you have done a lot of calculations on Pythagorean Theorem, you would have noticed that there is a ratio for the three sides that is always true. 

The ratio is a : b : c = 3 : 4 : 5, with 5 being the c (hypotenuse) side.

For example:

The length of side a (adjacent) and b (opposite) are 6cm and 8cm. Find the hypotenuse. 

Then you can find out the hypotenuse by:

       a : b : c = 3 : 4: 5 

=>  a : b : c = 6 : 8 : 10

Since 3 : 4 = 3 * 2 : 4 * 2 = 6 : 8

Therefore 5 * 2 = 10 cm.

a : b : c = 6 : 8 : 10

Ans: Hypotenuse is 10 cm.

This way of calculation is NOT applicable to Isosceles Right Angle Triangle.

Calculator Concerns

I have had many people, students and parents alike, ask me the point of learning how to do mental math when we have calculators.  I mean, it seems like students as young as elementary school students have personal devices these days, so they always have a calculator on them.  I can't argue that.

However, we cannot forget that calculators, like other pieces of technology, are tools to help us.  They do not replace our brains.  We still have to tell the calculator what we want to do, and it will give us an answer we may or may not reject (this is where number sense and mental math comes in!).

Calculators help us become more efficient, but here are some challenges around calculators that we must keep in mind:

1. A calculator is a piece of technology, and it must be explicitly taught.
Now that electronics are part of our everyday lives, it’s easy to forget that we don’t just learn how to do things by osmosis.  Back when I was in elementary school, we had computer classes, where we learned to type through software such as All The Right Type.  Students these days are expected to “just know how” and guess what?  They type with two index fingers at about ten words per minute.  Some skills just need to be taught and practised, and using a calculator is one of those skills.  Human error and syntax are two problems that can be minimized with practice.

2. We need to use the same calculator all the time.
Remember the previous point about how using a calculator is a skill that must be practised?  If a student uses one calculator at school and another calculator at home, how ready will that student be for a test at school, when they’ve studied for it at home?

The most basic of calculators have numbers, the four operations, and an equal sign.  These calculators do not have brackets, so imagine what happens when a middle school student tries to type in an order of operations question without the brackets.  The answer would be completely off, unless of course they were taught the appropriate way to use a calculator (i.e., only typing in the part of the equation that is being solved).  I suppose this particular problem – human error – wouldn’t exist if #1 was done well.

Another type of calculator is one where syntax matters.  Syntax is the order of the input of your numbers (e.g., square root button first and then number, or number first and then square root button?).  Calculators made today don’t really have this problem.  However, older calculators as well as those on devices (like the iPhone calculator) do care about syntax.  I have personally witnessed students using their device calculators (probably for the first time in their lives) on a test, and seeing the frustration on their faces because this supposed tool to help them actually messed them up.  Again, using a calculator is a skill that needs to be developed and refined.

For the purposes of proving that all calculators need to be practised, I will describe one more type of calculator that is usually free from human error and syntax problems, but we should still be cautious about.  These are the amazing handwriting-input calculator apps on devices.  An example of this is the MyScript Calculator (available on both iOS and Android).  These calculators recognize handwriting input and immediately solves it on screen, so it minimizes the chances of human error, since you can see exactly what is in the equation.  They also don’t care about syntax, because you can easily add or delete numbers and operations with the stroke of a finger.  What is a problem, however, is that handwriting input isn’t perfect at the moment.  

In my opinion, all calculators have potential for error (i.e., human error, syntax error, or other).  The only way we can minimize these errors is to make sure that we as teachers reinforce the fact that students need to always be using the same calculator in class, at home, and on assessments.

Prioritizing "Real-Life" Mathematics

As you can probably infer from the photo above, I went to the shopping mall the other day and walked into a store with lots of savings.  I overheard the following conversation between a middle-school-aged boy and (presumably) his mother:

Boy:  I like this shirt.
Mom:  How much is it?
Boy:  Um... $37.99 with 30% discount.
Mom:  So how much is it?
Boy:  I don't know, this chart doesn't have it.
Mom:  Use the calculator on your phone to figure it out.
Boy:  How do I do that?

To be fair, I'm not exactly sure how old this boy is, but it did make me rethink our Ontario math curriculum.  Correct me if I'm wrong, but the learning of percentages (e.g., how they relate to fractions and decimals) begins in Grade 6, but it's not until Grade 8 that students actually learn to calculate percentage discounts and sales tax.  In my opinion, it's far too late.

According to Ctrl-F, the word "real" appears in the mathematics curriculum 50 times (e.g., real world, real-life).  In the section under "Principles Underlying the Ontario Mathematics Curriculum" (page 3), it explicitly states that it will challenge students to make connections between mathematics and the real world.  If that's the case, shouldn't we prioritize "real-life" mathematics in the curriculum?  Perhaps introduce concepts like sales tax much earlier in the curriculum, rather than in Grade 8?  I realize that comes with some complications in the sequential learning of mathematics, but there's really nothing more relevant than calculating sales tax, discounts, and gratuity in real life.

Making Math Fun Is Possible

For decades there has been a common trend of many students not liking Math for various reasons. Some students don’t like it because they don’t understand concepts and find it difficult while other students don’t enjoy math because they find it repetitive and boring.

As a new teacher, I want to change this trend by not only changing instruction so that students can understand various concepts being taught but to also change instruction so that students LOVE math and have fun while learning.

           

From my past experience, I have found that most classes love competition. Therefore incorporating games into math instruction where students can compete against each other is one way to get them excited about math. These math games can be as simple as using math activities from IXL Learning https://ca.ixl.com/math/grade-7 where there are multiple exercises that follow the curriculum for all grades. When using these activities you could split your class into groups and have students compete against each other to solve the questions.

            I also like incorporating math into other subjects that students enjoy-sometimes that might not even realize they are using their math skills as well! For example, including geometry into an art project where students have to create a figure with a certain area and perimeter. Or incorporating math into physical education where students have to complete an ‘Amazing Race’ using their physical abilities as well as their mental skills to complete certain tasks. There are many ways you can incorporate math into other subjects, making fun and cross-curricular activities.

            These are few of many ideas that I would like to apply to my teachings someday as a classroom teacher, follow this blog for more to come! 

Mathemagic

Videos are a great way to get students interested in Math content that is being presented to them.

TED Talks is a great resource for videos that students can learn from and get them excited about Math.

The video posted below is wonderful for talking about squaring large numbers. In this video we have ‘Mathemagic’ where a gentleman can square large numbers in seconds-it is a fun and interesting video! At the end of the video it shows his process of thinking to solve these questions.

https://www.ted.com/playlists/189/math_talks_to_blow_your_mind

There are many more videos on TED Talks that can be applied to a math lesson.  

Math Trick

At the end of a lesson, unit, or week I like to incorporate something fun at the end of class to make students happy and get their minds off of the topic at hand; I feel this is useful because it gives students a mental break. A cool way to do that and look like a math genius at the same time is by following this video. You can use this trick any time and it will get the whole class engaged and they will never figure out how you are doing it! It will be a very fun component added to class whenever you feel it would be useful. Then, when you decide to teach your class this trick, you will have a class full of math geniuses.

Cool Activities

The following website would be very useful to teachers because it provides some cool activities and a variety of lesson plans for many strands. This website would be helpful as well because math is not looked upon as an exciting subject for some students, so these games and lessons would add a bit of excitement to the classroom. Students are familiar with traditional homework checks and the same routine over and over again. Providing different methods of teaching and routines in the classroom will get students excited and will change the mood/tone within the classroom environment.

Common Core Math Standards

This article talks about the debate with common core math, this has been something we as teachers have begun seeing more and more. The argument is students not just being able to memorize math skills but to actually understand why they are doing those skills. Students should understand how multiplication tables work, rather than just memorizing the tables. For example 5x3 = 15, but why? How can we understand this? By showing students that there are 5 groups with 3 sticks in each group, this is how we get the number 15. I really enjoyed this article!

http://www.usnews.com/news/special-reports/articles/2014/02/25/the-common-core-math-standards-content-and-controversy

Math Anxiety

Working the elementary environment you begin to see these small signs of anxiety in some students. When I came across this article I thought, if we can help these issues when they begin at a young age will it have an effect on the student later on?
The article is from the University of Chicago but I believe there are some good points made about this topic of math anxiety and how we as teachers can help our students overcome this problem.

http://news.uchicago.edu/article/2012/09/12/math-anxiety-causes-trouble-students-early-first-grade