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Have students choose a card and, in their math journal, record the correct comparison and explain which strategy they used to make the comparison. You can download the cards here. One thing I feel the need to mention as I close this series. Many of us were not taught to deeply understand fractions. It is our responsibility to pass on something better than that to our students.

If you are still using tricks in your classroom, set a personal goal to develop yourself in this area. I need advice on teaching fractions to above high students who only want to use computation. Their parents have pretaught and I find they are unable to use strategies you have mentioned.

That is a tough road!

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Once kids have learned by memorization, they are usually very reluctant to entertain other strategies. It can be done, though by educating parents about the need for developing conceptual understanding as well as setting an expectation for mathematical communication and understanding in the classroom. It truly is changing the culture of how we teach math, and that can be like turning the Titanic. Keep fighting the good fight.

I am a Texas teacher, teaching 3. How far would you go with third graders? Thank you! So glad you like it! In Texas, 3rd graders are only required to compare fractions with either like numerators or like denominators. I love this series of posts, and they are just in time for me and my students. Your blog has made me fall in love with teaching kids to use number sense rather than tricks. Thank you so much for your thoughts and generous freebies!! Has this been a problem for others?

Would you know of how to fix this? I just checked the link, Julie, and it works for me. It takes me right to the Google doc. Thanks for a great post! Finally, with this last post, my comparing fractions strategies are made whole!! I am one of those students who was taught tricks and procedures without conceptual understanding. I want better for myself and the precious kiddos that I teach! When the gears begin rotating, we can determine how many rotations the first gear must complete to realign the line segment by making use of LCM m , n. Suppose there are three planets revolving around a star which take l , m and n units of time respectively to complete their orbits.

Assume that l , m and n are integers. Assuming the planets started moving around the star after an initial linear alignment, all the planets attain a linear alignment again after LCM l , m , n units of time. The following formula reduces the problem of computing the least common multiple to the problem of computing the greatest common divisor GCD , also known as the greatest common factor:. There are fast algorithms for computing the GCD that do not require the numbers to be factored , such as the Euclidean algorithm.

To return to the example above,. Because gcd a , b is a divisor of both a and b , it is more efficient to compute the LCM by dividing before multiplying:. Because gcd a , b is a divisor of both a and b , the division is guaranteed to yield an integer, so the intermediate result can be stored in an integer. Done this way, the previous example becomes:. The unique factorization theorem says that every positive integer greater than 1 can be written in only one way as a product of prime numbers. The prime numbers can be considered as the atomic elements which, when combined together, make up a composite number.

Here we have the composite number 90 made up of one atom of the prime number 2, two atoms of the prime number 3 and one atom of the prime number 5. First, factor each number and express it as a product of prime number powers.

Common Place (Common Denominator #3) by Richard David Bach

The lcm will be the product of multiplying the highest power of each prime number together. The highest power of the three prime numbers 2, 3, and 7 is 2 3 , 3 2 , and 7 1 , respectively. This method is not as efficient as reducing to the greatest common divisor, since there is no known general efficient algorithm for integer factorization , but is useful for illustrating concepts. This method can be illustrated using a Venn diagram as follows.


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Find the prime factorization of each of the two numbers. Put the prime factors into a Venn diagram with one circle for each of the two numbers, and all factors they share in common in the intersection. To find the LCM, just multiply all of the prime numbers in the diagram. This also works for the greatest common divisor GCD , except that instead of multiplying all of the numbers in the Venn diagram, one multiplies only the prime factors that are in the intersection. Thanks Bob. Lori: Thank you. What a kind comment!

Please know how much I appreciate that. Passion is a definite must, as is a business sense, of which I had none in the beginning.


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I definitely know people who are passionately broke. Some are so because what they love does not have enough value for others in the way they deliver it and some are just passionate about spending like they are already billionaires. The key is balancing passion, value and impact or reach. And definitely knowing when the horse will never run. Doug: All great points. Great lessons all the way around. Bob, Amar Bose sounds like an earlier Steve Jobs….

Finding the Least Common Denominator - Civil Service Exam Review

Passion alone will likely never translate into huge financial rewards. I have recently added a third component — relevance. Our greatest value comes from the combination of our competencies skills and our passions drivers result in something with relevance. Bill: Yes, indeed; Edison, Rockefeller, and many others.

Always something much bigger than just themselves, and improved the quality of life for the masses. Lene: Thank you. And, indeed, the comments add so much value to the original post. As Dondi Scumaci always tells me, we have the greatest commenters!! Thank you for your kind words. I will try to be short in my description of my struggle with keeping my passion.

I worked in IT at a fortune 50 corporation for I really enjoyed my job. I loved helping people and creating solutions. Some of those solutions were created with code. In , I was forced out of that job. The economic downturn forced the company to lay some people off. Then eventually release them.

Comparing Fractions: Find a Common Numerator or Denominator

I was one of those people. This happened less than a month after I turned I was on course to finish my degree at a nearby university when all of this went down. I kept attending college online until I could no longer pay the bills on it. So my education is unfinished. In , I got certified as a John Maxwell coach to train and coach others in leadership principles. I do have a wealth of knowledge behind me on this, I have been listening and implementing success, leadership, and personal improvement principles for over 20 years.

So, to pay the bills now, I need an income. I have no unemployment, that ran out long ago. I need focus and passion. These are the things that will fuel me to get on my business and work it every day. That is the long and short of it. So yes, you are right. Pretty severe setbacks at that. Thanks for your input. My friends will all attest to the fact that I am not depressed, but rather more upset at working so hard for a goal to have it pulled out from under me. I am far more motivated today than I was 4 years ago.

Day 1: Passionate and motivated. I get working on the business, get mired down in details. Get little to nothing done. Day 2: Still motivated. Going to get back to the business today. I get distracted or busy with something. Day 3: May repeat day two, or I go into being upset about getting essentially nothing done the previous days. At times, someone will say something that will remind me of a failure. That will cause a downward spiral. Believe me, I have talked to my pastor and other ministers about this.

I just want to be back to my normal self again. What has worked best for me is to have a success coach CD playing everyday. That has not become a solid habit for me. I just wanted to help you understand this a bit better.