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.
Sample pages (PDF) Contents and Introduction Multiplying Fractions by Whole Numbers 1 Multiplying Fractions by Fractions Multiplication and Area Dividing Fractions 2: Fitting the Divisor
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 contains 119 pages, which includes the answers.
This book is enabled for annotation, which means you can fill it in on a computer (with Adobe Reader 9 or higher) or on a tablet using a PDF app with annotation tools.