Math Mammoth Fractions 1 & 2 workbooks
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 concepts of fraction and mixed number, equivalent fractions, plus addition and subtraction of fractions and mixed numbers.
Fractions 2 teaches comparing, simplifying, multiplying and dividing fractions and mixed numbers.
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.
Please read below some feedback I've received about the fraction books:
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
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 doing lots of mistakes while they confuse the different rules and apply the wrong one. That can even make them to fear fractions or at least not like them too well.
To avoid that, the student needs to UNDERSTAND these things, not just learn mechanical rules.
Fractions don't have to become enemies - they can be our friends! Really, we use them all the time in everyday life and they are very needful.
To achieve this goal, we use visual models all the way through this book. I encourage you use
some fraction MANIPULATIVES. The basic model in the book is a 'pie' divided into equal parts.
In the first lesson, Understanding fractions (see sample) the student also learns to draw pie models of certain often used fractions.
Encourage your student to draw the fractions for the problems in the following lessons, too! That will help her to VISUALIZE the fractions, and then she won't make the stupid mistakes of
blindly applying a rule and getting silly results because of making a mistake. She will be able to spot her mistakes.
The second lesson Part of a Whole Group (see sample) talks about part of a certain number of objects, and taking a part of that. It ties fraction concept in with division of whole numbers. For example, to find 1/4 of a number of objects, you would divide by four. To find 2/5 of a number, first find 1/5 by dividing by 5, then take double that.
This subject is 'revisited' in the end of the ebook, as it may be a little hard for some
kids to grasp quickly. Don't worry if that is the case with yours - this topic is not crucial in the development of the next subjects, so you can always come back to the topic later.
The lesson about Mixed numbers (see sample) teaches the concept of mixed numbers with pictures. The child also writes the mixed numbers as fractions.
Next lesson, Mixed Numbers to Fractions (see sample), starts out without mentioning the rule. Instead the child first gets to practice the conversion with drawing pictures. It is almost always BETTER if the child learns to do this conversion picturing the fractions in his mind. That assures understanding.
The rule is good too but is so easily forgotten.
And the other lesson Fractions to Mixed Numbers - and Fraction/Division
Connection (see sample) talks about an important point: the connection between a fraction and division (7/3 is the answer to 7 ÷ 3). This also gives us a way to convert fractions to mixed numbers.
Adding and Subtracting Like Fractions (see sample) is the easy topic. Do some problems with the manipulatives first to let the student see the concept. The student needs to fix in her mind how easy it is to add like fractions so that she doesn't later 'apply a rule' of adding the numerators AND the denominators.
Equivalent Fractions (see sample) are presented here as parts that have been split further. The rule is of course 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 the understanding of the concept instead of a memorization of a mechanical rule.
Adding and Subtracting Unlike Fractions (see sample) requires that the fractions are first converted into like fractions, and for that one needs to find a common denominator.
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 also to be only memorized and poorly understood, so the lesson here first lets the student to add unlike fractions without discussing the LCD. This is done to emphasize the main idea, which is that you need to convert the fractions so they become like fractions, and then add.
Some students can intuitively see what kind of parts will work, and in that case you can omit the discussion on how to find the common denominator for now. Your student will encounter that concept in 6th or 7th grade math.
The lesson gives in the end the rule for a common denominator that it has to be in both multiplication tables of the denominators. That is the same as saying it has to be a common multiple, but you can choose which terminology you use at this point. Also, ANY common multiple works - it doesn't have to be the least!
The lesson on Adding Mixed Numbers lets the student first develop 'a rule' on her own. You as the teacher should discuss that rule and correct it if there are any mistakes. The real 'rules' are found in the end of the lesson.
The same applies to Subtracting Mixed Numbers (see sample). The student can first try to do the subtractions on her own. There are a couple of different cases: the fractional parts can be like fractions or unlike fractions,
and one may be able to subtract the fractional parts separately or not (in which case you need to rename
one whole into a fraction or subtract in parts). This topic is divided into three lessons in the book.
Part of a Whole Group Revisited (
see sample) goes back to taking a part of a certain number of objects, or taking a fractional part of a whole number.
The book also has a lesson about measuring in inches, word problem lesson, and a review lesson.
Introduction - Fractions 2
See complete contents here.
Comparing Fractions (see sample) starts with a visual model to prompt the student to think about converting the fractions to
have the same denominator. Also included comparing fractions with number
lines.
Simplifying fractions (see sample) is presented as
joining or merging slices together, in other words as the opposite process of
splitting the pieces. Visual models are studied extensively to ensure
understanding instead of mere memorization.
Multiplying Fractions by a Whole Number (sample) discusses multiplying by whole number as repeated addition.
Another topic is how fraction times a whole number means
fraction of that number. For example, 1/4 of 20 is
expressed mathematically as 1/4 x 20. The lesson shows the
interesting connection between (fraction) x (whole number)
and (whole number) x (fraction).
Multiplying Fractions by Fractions (sample page) really
is not nearly so difficult a concept as many people think; it is based on the
simple notion that for example 1/2 x 1/4 means half OF one fourth. So if
you have one fourth of a pie left, and you take half OF it, you end up with
one-eighth part. We will present this notion and try to work towards
finding the actual easy rule. Also included a discussion why the rule
works.
Multiplication and area (see sample) interprets the
process (fraction) x (fraction) as an area calculation.
Pictures are at the core of understanding this.
Multiplying Mixed Numbers lesson contains lots of word problems.
Simplify Before Multiplying (see sample)
The lesson justifies the rule in a couple of different ways, starting from the
notion that if you first multiply and then divide by the same number, you have
in essence done nothing.
Dividing Fractions by a Whole Number (see sample)
is thought of as dividing part of pie between certain number of people.
Then is presented the connection to multiplying by the reciprocal number.
Dividing Fractions by Fractions (sample page) first discusses mental math problems where you think "how many
times does this go into this?" The student is then led to notice how
the first easy divisions in the lesson are the same again as multiplying by the
reciprocal number. The rule for dividing fractions by fractions is
presented.
Dividing Mixed Numbers (sample page) practices
fraction division further, and includes a discussion why the division rule
works.
There are three Review lessons (see sample).
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