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Math Mammoth Fractions 2

119 pages

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Math Mammoth Fractions 2 continues the study of fraction topics after Math Mammoth Fractions 1. I sincerely recommend that the student study the Fractions 1 book prior to studying this book, if he has not already done so.

I have made a set of videos to match many of the lessons in this book. You can access them at
http://www.mathmammoth.com/videos/fractions_2.php

This book is meant for fifth or sixth grade, and deals in-depth with the following topics:

• simplifying; including simplifying before multiplying
• multiplication of fractions (and of mixed numbers);
• division of fractions (and of mixed numbers);
• converting fractions to decimals.

We start out by simplifying fractions. Since this process is the opposite of making equivalent fractions, studied in Math Mammoth Fractions 1, it should be relatively simple for students to understand. We also use the same visual model, just backwards: This time the pie pieces are joined together instead of split apart.

Next comes multiplying a fraction by a whole number. Since this can be solved by repeated addition, it is not a difficult concept at all.

Multiplying a fraction by a fraction is first explained as taking a certain part of a fraction, in order to teach the concept. After that, students are shown the usual shortcut for the multiplication of fractions.

Then, we find the area of a rectangle with fractional side lengths, and show that the area is the same as it would be found by multiplying the side lengths. Students multiply fractional side lengths to find areas of rectangles, and represent fraction products as rectangular areas.

Simplifying before multiplying is a process that is not absolutely necessary for fifth graders. I have included it here because it prepares students for the same process in future algebra studies and because it makes fraction multiplication easier. I have also tried to include explanations of why we are allowed to simplify before multiplying. These explanations are actually proofs. I feel it is a great advantage for students to get used to mathematical reasoning and proof methods well before they start high school geometry.

Students also multiply mixed numbers, and study how multiplication can be seen as resizing or scaling. This means, for example, that the multiplication (2/3) × 18 km can be thought of as finding two-thirds of 18 km.

Next, we study division of fractions in special cases. The first one is seeing fractions as divisions; in other words recognizing that 5/3 is the same as 5 ÷ 3. This of course gives us a means of dividing whole numbers and getting fractional answers (for example, 20 ÷ 6 = 3 2/6).

Then students encounter sharing divisions with fractions. For example, if two people share equally 4/5 of a pizza, how much will each person get? This is represented by the division (4/5) ÷ 2 = 2/5. Another case we study is dividing unit fractions by whole numbers (such as (1/2) ÷ 4). We also divide whole numbers by unit fractions, such as 6 ÷ (1/3).  Students will solve these thinking how many times the divisor "fits into" the dividend.

After these types of divisions, students learn the “shortcut” for fraction division, that is, the usual rule for dividing any fraction by any fraction (the rule of “invert and multiply”). We also study dividing mixed numbers.

The lesson on introduction to ratios is optional. Ratios will be studied a lot in 6th and 7th grades, especially in connection with proportions. We are laying the groundwork for that.

The last major topic is converting fractions to decimals. Problems accompanied by a small picture of a calculator are meant to be solved with the help of a calculator. Otherwise, a calculator should not be allowed.

The book Math Mammoth Fractions 1 is a precursor to this book. It teaches the concepts of fractions, mixed numbers, equivalent fractions, addition and subtraction of like fractions and of unlike fractions, addition and subtraction of mixed numbers, and comparing fractions.

The two videos below show how division of fractions is taught in the Fractions 2 book. First, we study divisions that can be done "mentally", without the rule. Then, we study reciprocal numbers and the regular rule or shortcut for fraction division.

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Customer reviews and testimonials

Your [Fractions 1, Fractions 2] books have been such a help to us! My son has attended a private school since kindergarten, but we recently found out the school hasn't been teaching to grade-level standards in math. So, while he has wonderful grades, he is behind his peers from public school. He will be attending public school this fall, so we are using your worktexts to get him caught up this summer. I am not much of a "math person", so we are both benefiting from your series!

Thanks so much,
Robbie
July 2010

I began homeschooling my daughters back in May of 2008. Math Mammoth was one of the first websites that I came across on the internet. I had looked over Maria's site. I began targeting my math homeschool after what she provided as far as course knowledge. I began throwing long division and fractions at my oldest daughter. She totally freaked out. She kept telling me that she was not taught many of these subjects. I began teaching her what I knew about these particular subjects. As time moved forward, I realized that I need to check and see where she was at in her math.

I emailed Maria on this. She suggested many websites that I could test my daughter. I found out that she was a year and half behind. I thank God that I found Maria Miller's website because I have purchased the fraction book to help my daughter. She now has fractions down pat. She also has the long division down as well.

Maria takes these problems down to the basics and explains in excellent detail how to solve a fraction or long division math problem. Without the tools that she has given me, I would not have been able to bring my daughter forward and current on her math skills. Maria teaches it as I was taught many many years ago. She provides the tools to help our children succeed in mathematics. I just wish that there were more websites like hers in the other subjects.

I am currently using the Geometry module for my youngest daughter. She is enjoying that module immensely. It explains it perfectly for her.

Thank Maria for having an incredibly useful website and books. I thank the heavens above for someone like you.

Sincerely,
Amy K. Burt
November 30, 2008

I recently bought two of your books [Fractions 1 and 2]. I am a public school 8th grade math teacher. I use your materials to go back and fill in the "gaps" for children who have missed important concepts in their past education. Your products have worked wonders!!!!! I can't get over how complete and rigorous they are.

I plan to homeschool my daughter when she is old enough and I can't wait to use these materials with her...

thanks!!!!!
Jill Gaspard
The workbooks are WONDERFUL and I love how fractions are explained so simply and attractively, my 13 yo daughter is sure to understand fractions now. Thank you for a great teaching product!

Shannon
The students and myself liked the pictures and the step by step breakdown in all of the lessons.

Tia Kline
These books fill the gap between using manipulatives (concrete level) and abstract (symbolic) level. Too often students are expected to make that leap from the concrete level without spending time at the representational level.

Paula Listzwan
I like the clear explanations. The fractions 2 book had the best explanation of why dividing by a fraction works like it does of any I found anywhere.

Jennifer Wheelock
I am amazed at how effective these lessons are. I think what makes them interesting to my student is that the problems are often interesting to solve, with an interesting word problem or just the mathematical reasoning interests her in a way that I've never seen with her.

Cyndi Kane

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