You are here: Home → Blue Series → Fractions 1 & 2

Math Mammoth Fractions 1 & 2 workbooks

102 pages
(of which 66 are lesson pages)

---

Prices & ordering

Download: $4.50 at Kagi
Black & white printed copy:
$9.20 at Lulu

---

Sample pages (PDF)
Contents and Introduction
Understanding Fractions
Part of a Whole Group
Adding Mixed Numbers 1
Adding Unlike Fractions 1
Equivalent Fractions
Subtracting Mixed Numbers 2: Renaming
Comparing Fractions 1

---

98 pages
(of which 68 are lesson pages)

---

Prices & ordering

Download: $5.00 at Kagi
Black & white printed copy: $10.20 at Lulu

---

Sample pages (PDF)
Contents and Introduction
Simplifying Fractions
Multiplying Fractions by Whole Numbers 1
Multiplication and Area
Divide Fractions by Whole Numbers
How Many Times Does It Fit?
Fractions to Decimals 1
Fractions and Decimals in Measuring Units

---

Your order is secure at Kagi
Shopper help at Kagi

Math Mammoth Fractions 1 and Math Mammoth Fractions 2 are self- teaching worktexts dealing with all fraction operations visually.

Both books come as PDF ebooks that you download after completing the purchase, and then print as you need.

Fractions 1 teaches the concepts of fraction and mixed number, the concept of "part of a whole", addition and subtraction of like fractions, addition and subtraction of mixed numbers, equivalent fractions, addition and subtraction of unlike fractions, and comparing fractions.

Fractions 2 teaches comparing, simplifying, multiplying and dividing fractions and mixed numbers. This book will be revised in February 2009.

The books not only show fraction pictures one time at top of page but let the child work with pictures extensively, and to draw them herself in the various exercises.

This helps ensure that different concepts are not just rules without the understanding!

Also, in some lessons the child is asked to try to discover the rule or concept studied, encouraging independent thinking, experimenting, and discovery.

This way fraction studies will not be just one meaningless rule after another but a bunch of interconnected principles that the student understands.

Fractions 1 & 2 workbooks are most suitable for 4th, 5th, or 6th grade, and for remedial studies. However, they are not tied to any grade level and do not mention anything about grade levels, and so the books can be used in various settings. They are very good for remedial teaching or for the learning disabled because they are so visual.

The video below explains the visual method used to explain equivalent fractions in the Fractions 1 book:

Please read below some feedback I've received about the fraction books:

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

Introduction - Fractions 1 book

Math Mammoth Fractions 1 is the first book of two that cover all aspects of fraction arithmetic. This book at hand covers the concepts of fraction and mixed numbers, equivalent fractions, adding and subtracting like and unlike fractions, adding and subtracting mixed numbers, and comparing fractions. The book Fractions 2 covers simplifying fractions and multiplication and division of fractions.

Studying fractions involves lots of rules, and many students learn them only mechanically, not really understanding the underlying concepts and principles. Then they end up making lots of mistakes because they confuse the different rules and either apply the wrong one or apply the right rule but don't remember it quite right. All this can make students even fear fraction math.

To avoid that, this book uses the visual model of a pie divided into slices all the way through the book. It is a very natural model, because it uses a circle that can be divided into any number of circle sectors (slices). When students work with this model from lesson to lesson, they will eventually be able to "see" these pies in their mind. This, in turn, gives them the ability to do many of the easier fraction calculations mentally. It also enables students to really UNDERSTAND these concepts, and not just learn mechanical rules.

You are welcome to use manipulatives alongside the book; however the visual pie model is probably sufficient for most students in 5th grade level. I have also included (in the appendix) printable cutouts for fractions from halves to twelfths. You can use them to make your own fraction manipulatives. To make the manipulatives sturdier, glue the printed pages on cardboard, and cut the parts only after gluing. The whole circle is there to illustrate "one whole" - needed when studying mixed numbers. You will probably need to print at least two copies of each cutout page. You can use the white cutout fractions if you need to save on ink and let children color them. Just use consistent colors so that thirds are always the same color, fourths are the same color, etc.

In the first lesson, Understanding Fractions, the student learns to draw pie models of certain common fractions. This skill is needed later on in various exercises through the book.

The lesson Mixed Numbers teaches the concept with pictures. The child also writes mixed numbers as fractions.

The next lessons, titled Part of a Whole Group 1, 2, and 3 have to do with finding a part of a certain number of objects, and of course has lots of practical applications. It ties the concept of a fraction with division of whole numbers.

Next we study adding and subtracting like fractions, which is an easy topic. Next lesson reviews mixed numbers and further practices changing mixed numbers to fractions and vice versa. Next we add and subtract mixed numbers with like fractional parts.

Then it is time to study equivalent fractions, as a prerequisite for adding unlike fractions. Presented equivalent fractions as parts that have been split further. The rule is to multiply both the numerator and the denominator by the same number, but try to emphasize the terminology of "splitting the existing parts into so-and-so many pieces" or something similar. That should help students to understand the concept instead of memorizing a mechanical rule.

Add Unlike Fractions 1 is an introductory lesson in the sense that the student is not yet introduced to the rule for finding the common denominator. In this lesson, the common denominator is either given, or the student figures it out using pictures.

Add Unlike Fractions 2 emphasizes the idea that we need to find a common denominator, and then convert the fractions to like fractions before adding. Many textbooks introduce here the concept of Least Common Denominator, LCD, which is the best common denominator to use since it is the smallest. That concept tends to be only memorized and poorly understood, so the lesson here does not overtly emphasize that one always needs to find the LCD. Your student will encounter the concept of LCD again in 6th and 7th grade.

Finally we also add and subtract mixed numbers with unlike fractional parts, and add & subtract several unlike fractions.

The last new concept in this book is that of comparing fractions. Once the student has mastered converting two fractions to equivalent, like fractions, this should be fairly easy.

Answers are in the end of the book.

Introduction - Fractions 2

Math Mammoth Fractions 2 continues the study of fraction topics after Math Mammoth Fractions 1. The book at hand is meant for 5th or 6th 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.

The topics of simplification, multiplication, and division of fractions are, again, first studied with the help of visual models. Students also complete various exercises using these visual models, before embarking on the more abstract exercises. This helps ensure that the student understands the underlying concepts, instead of merely following a rule.

The study of fractions involves a lot of "rules" or shortcuts. Shortcuts in themselves are not bad; it certainly is easier and quicker to multiply or divide fractions using them. Therefore, it is needful to study and know the shortcuts, but students should not learn them only. With each topic, the students study the visual models and mental math problems first, before practicing the mechanical application of the shortcuts.

The topic of simplifying fractions is presented as joining or merging slices together. This is the opposite process of splitting the pieces (equivalent fractions in Math Mammoth Fractions 1). Working with the visual model leads us to the "shortcut" of dividing the numerator and the denominator by some same number.

Multiplication of fractions is studied in several steps. First, we study multiplying fractions by a whole number, which can be solved with repeated addition. The next lesson studies the connection between (fraction) × (whole number) and (whole number) × (fraction) (for example, 1/4 × 20 and 20 × 1/4), and explains how (a fraction) times (a whole number) means a fractional part of that number. This is all leading towards the full understanding of fraction multiplication being a fractional part of another fraction, and using the shortcut to solve such problems.

The mental idea of finding 2/3 of 4/5, say, is developed through several steps. First, students learn to find a fractional part of another fraction that in the form 1/n (n being a whole number). For example, 1/3 of 1/5 is 1/15 (this is justified visually). Then they learn that 2/3 of 1/5 is just double the answer to 1/3 × 1/5, so it is 2/15. Then, the shortcut (the rule) for fraction multiplication is introduced, and it is justified by a "roundabout way" of solving fraction multiplications, which just means figuring it out in several steps. Students also study simplifying before multiplying, multiplication and area, multiplying mixed numbers, and compare fraction vs. decimal multiplication.

After this, we turn our study to division. Again, there are several lessons that start out with easy, mental division problems, and gradually work towards the general case and the shortcut or rule for fraction division.

The last major topic is converting fractions to decimals. Students learn to use first long division, and then the calculator for the conversion. They learn about repeating decimals. In this context, we also study converting measurements in the customary system into a decimal form, such as writing 2.34 miles as 2 miles 1,795 feet, or changing 4 ft 5 in to 5.42 ft.

The last lesson in the book compares ratios and fractions, and shows how both concepts can be used in solving word problems.

The answers are in the end of the book. After the answers, I have included printable fraction cutouts, with which you can make your own fraction manipulatives. Most children probably don't need the physical manipulatives anymore, if they have a solid foundation with the pie pictures from earlier fraction studies, but they are included here for the sake of completeness.

Customer reviews and testimonials

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

Better Yet - Package Deals!
Blue Series package - $80 (download) or $85 (CD)
Blue Series for grades 1-3 - $40 (download)
Golden/Green Series package - $65 (download) or $70 (CD)
Light Blue Series package - $99 (download) or $104 (CD)
Blue/Golden/Green Series package - $120 (download)
All Inclusive package - $158 (download) or $163 (CD)

Learn more about the package deals!

*BONUS*: Buy any package listed above, and get the fully upgraded Soft-Pak software with more features and options for FREE (includes 4 math, 1 language arts, 2 testing programs).

*BONUS* - SOFT-PAK MATH SOFTWARE

I bet you'd have to agree: this is a steal! You can get Soft-Pak math programs (list price $24.95) for just a small $3.50 s&h fee - just by ordering ANY of my books at Kagi or CurrClick!

* Math Maker, for printable math activities with answer keys
* Math Master, for on screen practice of various math topics
* Math User, for answering the 900 multi-level word problems
* True/False Math Challenger for mastering facts 1-12


NOTE: if you purchase one of the package deals at Kagi store, you will get Soft-Pak in the download or included on the CD for FREE.

Soft-Pak may also be purchased at great 75% discount without a Math Mammoth purchase - click here for details.

How and where to order

You can buy Math Mammoth books at:

• Kagi store - all of the downloads and CDs. I maintain five different order pages at Kagi:
  • One for the Blue series,
  • another for the Golden and Green Series, and
  • third for the Light Blue complete curriculum products, and
  • one more for the Packages (CDs or downloads).
  • And yet an order page for Make It Real Learning workbooks.

  Kagi accepts all major credit cards, Paypal (USD), checks, cash, money orders, purchase orders, wire transfers, TrialPay, and even cash. Read more about the payment methods here. Please see Kagi Shopper Help for any other questions about the ordering process.

  After completing the order at Kagi, you will see the download links on the receipt page. You will also receive an email with download links for each book you bought, or for a zip file for the packages. You click on the links and download the books to your computer's hard drive.

  In case of any problems with the download, you can always email me, and I can email the books to you directly. My contact info will be in the email you receive.

  
   
• CurrClick.com carries all Blue, Golden, and Lightblue series books as downloads. They accept credit cards and Paypal. You will be able to download the products immediately upon the purchase, and also return to your account at CurrClick to redownload.
• Lulu has printed copies for the Blue, Golden, and Light Blue series books. For each book at MathMammoth.com, you can find a link to the printed copy sold at Lulu on each individual book's page (right sidebar).

---

By purchasing any of the books, permission IS granted for the teacher (or parent) to reproduce this material to be used with his/her students in a teaching situation; not for commercial resale. However, you are not permitted to share the material with another teacher.

In other words, you are permitted to make copies for the students/children you are teaching, but not for other teachers' usage.