Playing at Math: Multiplication and Fractions

Multiplication and Fractions: Math Games for Tough Topics

(Math You Can Play)

by Denise Gaskins

I have to interrupt my little written narrations of reading over the past year or so to tell you about this great book of math games. I've written about a few of Denise Gaskins's books before: Let's Play Math and the combo book Counting and Number Bonds and Addition and Subtraction. This is another games book, but for older students.

My kids almost all struggled more with math as soon as we encountered multiplication and fractions. For the most part, they understood the concepts, but remembering all the multiplication facts was difficult. For fractions, it was recognizing the ones that were the same when they were simplified or had different denominators.

For my younger kids, I scheduled a games day when we'd just play the next game in this book. Some games were more fun than others, but with a loop (going back to the beginning when we finished the book), we got lots of practice with different facts and they knew their favorite games would come around again. I appreciated how the games depended almost entirely on things we had at home (like regular decks of playing cards) or free materials Denise Gaskins offers on her site if you purchase the book. There are also variations for most of the games to make them easier or more difficult.

The information she provides before the games and when introducing each one is really helpful for parents who are trying to understand how their children are thinking and guide the children to consider how all the multiplication problems and fractions are inter-related. The games give lots of practice with the facts themselves, which helps kids remember them, but they are designed to also encourage kids to explore how numbers are related to each other.

Some kids really thrive with games, and many children simply need a solid dose of math facts practice every day. This book was one tool in our homeschool to ensure we were getting some practice in.

I have received nothing in exchange for this post. Links to Bookshop and Amazon are affiliate links.

Pondering the Marvels of Creation: Great Catholic Scientists

Great Catholic Scientists: Discoveries and Lives of Faith

Dr. Guy Consolmagno, SJ and Dr. Michelle Francl-Donnay

This is a wonderful little audiobook, almost like a collection of podcasts. Dr. Consolmagno and Dr. Francl-Donnay chat together about science and religion through the stories of Catholic scientists from all over the world and throughout history. They are collected in twelve chapters of 20-25 minutes each, focused on areas like medicine or mathematics, but freely wandering in the discussions.

Some of the featured scientists are saints. Some are famous. All of them are examples of real people doing science - showing up at work and persevering through mundane day-to-day activities.

Dr. Consolmango and Dr. Francl-Donnay talk often of the characteristics that make a good Catholic scientist (spoiler alert - they are the same ones that make a good scientist), but they also talk about some of the characteristics of a Catholic that provide a foundation of education, learning, and curiosity which allows a scientist to explore God's creation with joy. There are instances where their faith allowed a scientist to ask questions and be comfortable in areas of uncertainty.

I have assigned these lectures to my daughter, currently in eighth grade, at the beginning of her year, as a kind of introduction to science and how to think like a scientist. It's a lovely sweep through the history of science that touches on a wide variety of areas. So many secular books on science begin with an assumption that a person of faith cannot also be a person of science, an erroneous but pervasive assumption. This audiobook provides one more layer of protection for a student who can then more easily discard mistaken ideas like that as he or she is reading.

My daughter is listening to them straight through, but I think any individual lecture could be enjoyed without listening to all of them, and there's no real reason they must be in this particular order. Though I think the authors tried to follow a chronological order, there is a lot of overlap on the lectures. I've made a little list below in case someone wants to choose just one or two or wants to schedule them within a particular study.

Chapter 1 (4 min intro) and Chapter 2: What Makes a Scientist Catholic?
(Hildegard of Bingen, Georges Lemaître)
This is probably the lecture that most directly addresses the apparent disconnect between faith and reason. Instead, it suggests our faith leads directly to exploring the world and the universe, God's creation. Science as a discipline only makes sense if the universe is not random. Dr. Consolmango and Dr. Francl-Donnay find great joy and delight in their scientific studies.

Chapter 3: Overthrowing Aristotle
(Hildegard of Bingen, Albert the Great, José de Acosta, Athanasius Kircher)
In this chapter, they explain how changes in the world (stability, improvements in communications, large enough populations so some people can study just for the sake of knowledge, etc.) led to the sudden growth of "science." Catholic monasteries and universities were an integral part of this culture.

Chapter 4: Into the Depths of the Atom
(Henri Becquerel, George de Hevesy, Roger Boscovich, Amedeo Avogadro)
As Catholics, we already believe in the unseen and do not fear the mysterious. We also have faith that our experiences tell us something meaningful about the world. This chapter talks about mysteries, and how the most interesting answers and discoveries are the ones that lead to more questions.

Chapter 5: What Is Life?
(René Haüy, Louis Pasteur, Gregor Mendel, Marthe Gautier and Jérôme Lejeune)
This chapter begins with a discussion about a philosophical (and scientific) definition of life. It touches on the humanness of scientists. They have the same trials, controversies, and need to make a living as anyone else.

Chapter 6: Materia Medica
(Hildegard of Bingen, Georg Joseph Kamel, Pierre Joseph Pelletier, John Clark Sheehan, René Laennec, Ethelbert Blatter)
In this chapter, they discuss how a fallen world leads to illness but that God's creation and his gift to us of reason allow us to discover and develop methods of healing. Over the centuries, the Jesuits in particular have had the opportunities to explore the world, learning from other cultures, and regular communications with others to share that knowledge.

Chapter 7: Merciful Science
(Laudato Si', Mary Poonen Lukose, Bernardo Alberto Houssay, Carl Ferdinand Cori and Gerty Theresa Cori)
Pope St. John Paul II said that science and theology work together, like two wings that lift us to the contemplation of truth. We have obligations beyond just doing the science, always asking ourselves "What are the implications for the most vulnerable?"

Chapter 8: God’s Language
(Maria Gaetana Agnesi, Baron Augustin-Louis Cauchy, Pope Sylvester II)
This chapter focuses on mathematics and the question "how do you describe the world using mathematical equations?" For some, math is a kind of prayer, a focus on truth as the greatest earthly joy. “Mathematics forces you to confront the infinite.”

Chapter 9: Explorers of a New Space
(Sisters of the Holy Child Mary: Sisters Emilia Ponzoni, Regina Colombo, Concetta Finardi and Luigia Panceri; Mary Kenneth Keller, Mary Celine Fasenmyer, Francesco Faà di Bruno, Roberto Busa)
This chapter describes how an ambitious project to create a catalog of every known star begun in 1887 required the creation of new strategies and ways of thinking that, over time, evolved into digital humanities. Religious life and the support of the Church allow space and freedom for work to be done that doesn't fall into a three year grant cycle.

Chapter 10: Taking the Temperature of the World
(Jean Leurechon, Nicolas Steno, André-Marie Ampère, Alessandro Volta)
Communicating about science and what you've learned about the world depends on a system of measurement that is consistent from place to place and person to person. This chapter shares about some Catholics who devised reliable methods of quantification, thereby making more scientific discoveries possible.

Chapter 11: Not Where but What
(Angelo Secchi, James Macelwane, Eduard Heis, Agnes Mary Clerke)
This chapter looks at scientists who look beyond the earth's atmosphere or deep into the earth, beyond what we can measure directly. Many scientists never become famous, working in hidden lives at building a foundation of slow painstaking work for greater understanding. “Wherever your niche is, there’s something for you to do.”

Chapter 12: Our Place in the Universe
(Maximilian Hell, Johann Georg Hagen, Georges Lemaître)
This chapter shares the stories of three notable priest astronomers that gave us a sense of where we are in the universe: the scale of the solar system, the position of stars, the motions beneath our feet, and the origin of the universe itself.

Chapter 13: There is always more to know
(Marie Lavoisier, Blaise Pascal, René Descartes)
This final chapter summarizes many of the themes regarding faith and science that surface, submerge, and resurface in earlier chapters. Faith and science are not big books of facts that might agree or disagree with each other. Faith and science both allow us to "grapple with mysteries, the mysteries of how and why we came to be." It includes a litany of all the men and women of God (and science) included in the audiobook.

I have received nothing in exchange for this post. I purchased this audiobook from Audible. Links to Amazon and Bookshop are affiliate links.

Connecting Math: Discovery Journal

Discovery Journal, with quotations

In August 2018, Denise Gaskins (of the fabulous Let's Play Math and Math You Can Play books) offered a review copy of her new Discovery Journal. I requested one for First Daughter, who loves all things quotable. The quotations all appear at the tops or bottoms of pages in the same color as the dots. They don't get in the way. First Daughter did enjoy them. She read through all the quotes as soon as we got the book.

This is an excellent dot journal. The left-hand page has alternating dots to make easy triangles while the right-hand page keeps the dots lined up vertically and horizontally for a perfect grid. Ms. Gaskins shared some ideas for sketching with the journals on her blog and another one in an email to me. I tried to encourage First Daughter to try some of these in her journal. I also encouraged her to use our compass to draw circles and then made designs inside of them. But First Daughter really just wanted to use it for her math assignments. Every once in a while, she would doodle or use the dots to sketch out a triangle or graph for one of her problem sets, but she wasn't very interested in exploring all the fun ways you can connect dots.

The journal itself held up really well. First Daughter carried it everywhere and often left it lying around, but not a single page fell out and the both covers are still attached. The weight of the pages is good, too. They are thicker than the pages you would find in the dotted composition books in the school supplies aisle. Pen does not go through it, though our markers did. The colored pencils work well on them, too.

The post contains my honest opinion. I received a free copy of the Discovery Journal from the author. The link to Amazon is an affiliate link.

Becoming More Fully Human: Beauty for Truth's Sake

Beauty for Truth's Sake: On the Re-enchantment of Education

by Stratford Caldecott

This book was my meaty read for summer 2018. I didn't finish it before becoming swamped by high school planning, so it carried over into 2019. For many years, we have struggled in our homeschool to avoid viewing math as drudgery. I think we are fairly good at creating an environment in which math is often fun with Life of Fred books as our math texts and plenty of math games from books and our shelves.

Yet this environment is only partially meeting my goal. I have always thought the children should also learn to find the beauty and truth in mathematics, that it should somehow connect them to the natural world,  even though this beauty is something I only vaguely understand myself.

I hoped this book would show me how to reveal the beauty and truth of mathematics to my children in our homeschool.

The Forward is by Ken Myers (of Mars Hill Audio Journal):

Since the Logos is love, and since all things are created through him and for him and are held together in him, we should expect the logic, the rationality, the intelligibility of the world to usher in the delight that beauty bestows. 

A substantial part of the book focused on arguments explaining why the study of science and mathematics is enhanced and fulfilled through explicit relationships to the humanities and liberal arts. Among the many voices Caldecott gathers together in his reasoning are those of James S. Taylor in Poetic Knowledge, Bl. John Henry Newman in The Idea of a University, and Josef Pieper in Leisure the Basis of Culture. That last one is on my wish list.

An integrated curriculum must teach subjects, and it must teach the right subjects, but it should do so by incorporating each subject, even mathematics and the hard sciences, within the history of ideas, which is the history of our culture. Every subject has a history, a drama, and by imaginatively engaging with these stories we become part of the tradition.

Most of these ideas are not new to me and frankly, I was convinced of this much before I started the book, but Caldecott drew connections throughout history from ancient Greece to modern times that I found helpful. His prose is as elegant as you might hope based on the gorgeous cover of this book.

The purpose of an education is not merely to communicate information, let alone current scientific opinion, nor to train future workers and managers. It is to teach the ability to think, discriminate, speak, and write, and, along with this, the ability to perceive the inner, connecting principles, the intrinsic relations, the logoi, of creation, which the ancient Christian Pythagorean tradition (right through the medieval period) understood in terms of number and cosmic harmony.

Homeschooling with Charlotte Mason's philosophy means this relationship of ideas is already integral to our curriculum. We are reading history and science and geography together, allowing the story of humanity to be woven by the student from these different threads. Or rather, allowing the opportunity for these relationships to be developed; each student does his or her own hard work.

Moreover, though we have every intention of our children going to college or trade school and learning how to earn a salary so they can care for a family, either in a domestic church or in the Church, our educational goals are focused on providing the wonder and wisdom for our children to become the people God wants them to be. A job is only a small part of their lives.

The principle remains the same: knowledge is its own end--"worth possessing for what it is, and not merely for what it does." It is not to be valued for the power it gives us over nature, or even for the moral improvement it may bring about in us (even if these things may flow from it). It is to be valued for its beauty. "There is a physical beauty and a moral: there is a beauty of person, there is a beauty of our moral being, which is natural virtue; and in like manner there is a beauty, there is a perfection, of the intellect."

The quotes are Newman's from The Idea of a University.

After these basic arguments, Caldecott begins exploring numbers, shapes, and supernatural relationships. For example, he examines the "irrational beauty" of the golden ratio, phi, and the Fibonacci sequence. Supernatural relationships, like that between Father, Son, and Holy Spirit, can be represented geometrically, revealing a greater depth to the relationship.

Then [pi] could be read as describing the relationship between the Persons, a relationship that is infinitely fruitful and never ending. Thus the endlessly flowing numbers of [pi] suggest the super-abundance of God's mercy, the infinite quality of his love, and the unlimited space opened up within the Trinity for the act of creation.

These explorations were exactly the kind of material I sought. Much of it is understandable without knowing too much higher level math, but the combination of mathematics and philosophy and theology made many of the discourses difficult to follow. Thales (before Pythagoras) showed how

the perpendicular line drawn from a right angle touching the circumference back to the hypotenuse will always equal the mean proportional between the segments into which it divides the diameter[.]

There's a diagram in the book for this one (and many others) that helps a little, but I still often found myself reading sections a second or third time to try to understand exactly what Caldecott meant. I'm certain I could glean even more from the book if I read it again.

In the end, though, the important idea is that these sorts of explorations reveal an inherent perfection of the universe which point us always to the Creator and his relationship with Creation.

Speculations like those I have mentioned in this chapter will appear forced to many. Yet we must return to the central idea that God's archetypal forms or Ideas are inevitably found within nature at every level, reflected with greater or lesser degrees of accuracy. That is not pantheism but Christian Platonism, perfectly compatible with the insights of theology and revelations of scripture.

Discussions of frequency, harmonics, and Chladni patterns allows Caldecott to connect a celestial harmony with liturgy, worship, and prayer. He quotes C. S. Lewis (Planet Narnia: The Seven Harmonies in the Imagination of C. S. Lewis by Micheal Ward).

[Celestial harmony] is the only sound which has never for one split second ceased in any part of the universe; with this positive we have no negative to contrast. Presumably if (per impossibile) it ever did stop, then with terror and dismay, with a dislocation of our whole auditory life, we should feel that the bottom had dropped out of our lives. But it never does. The music which is too familiar to be heard enfolds us day and night and in all ages.

All of these subjects must come together in our education. According to Caldecott, integrating science with poetry, art, music, and the humanities allows students, all of us, to understand the universe in a more complete way, one which will at the same time, allow for greater understanding in scientific and mathematical disciplines.

Music, architecture, astronomy, and physics--the physical arts and their applications--demonstrate the fundamental intuition behind the Liberal Arts tradition of education, which is that the world is an ordered whole, a "cosmos," whose beauty becomes more apparent the more carefully and deeply we study it. By preparing ourselves in this way to contemplate the higher mysteries of philosophy and theology, we become more alive, more fully human.

After reading this book, I have a greater appreciation myself for the beauty of mathematical thought and how the underlying principles of mathematics can reveal universal truths. It is not, however, a book I can simply read to my children or even realistically assign to a high school student. While it's been many years since I was in a college classroom, I have a far greater knowledge base than most high schoolers, and certainly a greater intrinsic interest, and I often struggled while reading the book.

So what I need know is for someone to take the next step. Use Caldecott's philosophy to write a mathematics curriculum or supplement or something I can share with my children.

I have received nothing in exchange for this post; all opinions are my own. I purchased this book at a local bookstore. Links to Amazon are affiliate links. 

Mathematics and Mathematicians in Real Life: The Lady Tasting Tea

The Lady Tasting Tea: How Statistics Revolutionized Science in the Twentieth Century

by David Salsburg

Dr. Salsburg's remarkable book focuses not on statistics, but on the mathematicians who shaped the statistical world in which science and social science now functions. It's clear throughout the book how much the author admires the brilliant and the generous pioneers of this field, though he also includes stories of arrogance and dismissive attitudes. He often writes eloquently about his encounters with exemplary texts from mathematicians of the past. His far-ranging anecdotes and references (even including a mathematical poem from the 1500s) introduce real men and women and share stories of their relationships with each other.

There is no record of anyone getting angry at Samuel W. Wilks. He approached everyone he dealt with, whether a new graduate student or a four-star general of the army, with the same informal air. He was nothing but an old Texas farm boy, he would imply, and he knew he had a lot to learn, but he wondered if...What followed this would be a carefully reasoned analysis of the problem at hand.

There is also a chapter devoted to a few of the many noteworthy women in mathematics and statistics.

Early in the book, he talks about the difficulties in conducting scientific experiments. For example, classroom students may be asked to calculate the gravity constant but find their calculations vary widely from the published 9.8 meters per seconds squared. Teachers may say it's merely mis-measurement or sloppy techniques, but in fact, that's exactly what happens in every experiment. Scientists repeat an experiment hundreds of times and their measurements are never exact. Plotted all together, they show a distribution. (Even so, let's hope the scientists have less mis-measurement and sloppy techniques in general than the high school students.)

The results of individual experiments are random, in the sense that they are unpredictable. The statistical models of distributions, however, enable us to describe the mathematical nature of that randomness.

The style is often humorous.

The reader may recall those terrible moments in high school algebra when the book shifted into word problems...Imagine a word problem where nobody knows how to turn it into a formula, where some of the information is redundant and should not be used, where crucial information is often missing, and where there is no similar example worked out earlier in the textbook. This is what happens when one tries to apply statistical models to real-life problems.

There are lots of descriptions of the real-life work of mathematicians which may be of interest to a student considering studying math in college or as a career.

It is better to do mathematics on a chalkboard than on a piece of paper because chalk is easier to erase, and mathematical research is always filled with mistakes. Very few mathematicians can work alone. If you are a mathematician, you need to talk about what you are doing. You need to expose your new ideas to the criticism of others. It is so easy to make mistakes or to include hidden assumptions that you do not see, but that are obvious to them.

There are also many examples from real-life, seemingly pulled from our newspaper headlines of economics, sociology, psychology, and health and nutrition. This book provides a useful background for those of us who must sift through all the information thrown at us, giving a basic explanation of what studies may actually show and where they may fall short.

Despite his high praise, he raises interesting questions about the future of statistical practices. I found this discussion particularly interesting as I was also reading Small Is Still Beautiful with its critique of our economic models and indices, many based on the pioneering world of statisticians highlighted in The Lady Tasting Tea. At the end of the book, he asks three questions and answers them in surprising ways:

1. Can statistical models be used to make decisions?
2. What is the meaning of probability when applied to real life?
3. Do people really understand probability?

He invites the reader to consider the assumptions of our society, immersed as it is in statistics, and to ask whether those assumptions provide a firm foundation.

In logic, there is a clear difference between a proposition that is true and one that is false. But probability introduces the idea that some propositions are probably or almost true. That little bit of resulting unsureness blocks our ability to apply the cold exactness of material implication in dealing with cause and effect.

I'm considering assigning this book to my high schoolers in their senior year. I do think it's probably best suited to students who are focused on math, science, or are considering research as a career. Even with few formulas and math explained in words, there are some complicated ideas in the text and lack-luster students may find it more troublesome than enjoyable. For those who are truly intrigued, there is an extensive annotated bibliography, some delightful footnotes, and a helpful timeline.

I have received nothing in exchange for this review. I checked this book out from our library. All opinions are my own. Links to Amazon are affiliate links. 

Games of Math: Counting and Number Bonds and Addition and Subtraction

Math You Can Play Combo: Number Games for Young Learners

by Denise Gaskins 

I reviewed Let's Play Math by Denise Gaskins in February 2016. At the time, I think I'd already been using these two books for half a school year, but I haven't take the time to review them. On Facebook and in person, I highly recommend Let's Play Math to just about any homeschooling family struggling with math. I also think it's just about the best thing a new homeschooling mother or father could read. No matter what curriculum you choose for math, Let's Play Math can help you understand it and implement it for your children and help you create an environment of mathematical curiosity that can benefit any student. I know my family would have been much better off if I'd read it earlier!

So now that you know I want you to read that first, let's talk about Math You Can Play Combo. This book includes two books: Counting and Number Bonds and Addition and Subtraction.

This book (or rather, these two together) constitute the spine of our math curriculum for prekindergarten, kindergarten, and first grade. I begin with Counting and Number Bonds and we just work through the books playing one game a week (sometimes multiple times) through the whole book. If I had started really young, I may have repeated Counting and Number Bonds, but with the two who started this way, I moved right on to Addition and Subtraction.

In addition, sometimes I read math stories out loud to the whole family as a read-aloud. Let's Play Math has a large list of such books as does the author's website. We also play games from our well-stocked game shelves during math time. I specifically schedule those games through first grade, so Second Son is currently the only one who gets to pick a game once a week. We don't start "formal" math until second grade with the Life of Fred books and Khan Academy. I do let Second Son use Khan Academy, because he felt left out, but only for a short time a few times a week.

I almost can't say enough fantastic things about the Let's Play Math game books. My youngest two children, Second Daughter and Second Son, are the ones who benefited from them. Of course, we changed up a lot between the older two and these two because that's when we switched away from Saxon over to Life of Fred, so it's hard to say if any particular change made all the difference, but our whole attitude toward math is dramatically improved with the younger two.

They love playing these games. In fact, it's not uncommon for Second Daughter to insist she's done with her independent work so she can play with Second Son during his game time. Or she'll ask to play some of the games outside of school time. (Snugglenumber is a particular favorite; she's even taught a friend to play it with her.) The great benefit here is that interest in the game encourages repetition of the math facts and skills. More practice with the basic counting, grouping, addition, and subtraction in these games leads to consistent success and, eventually, the ability to focus instead on more complex math.

The games almost all use cards you already own. (There are a few boards you can make and lots of printables included with purchase of the book if you don't want to make your own.) For the most part, I open the book and we play the game without any preparation. Along with the games are brief explanations of the math behind the games and strategies for guiding children through the math involved.

Some of the games seem like they'll be too challenging for the child, but we always give them a try. Over and over again, I see them succeed when I might have neglected to even give them the opportunity.

Though First Son and First Daughter seemed to do well with the mental math exercises when we were using Saxon, both Second Daughter and Second Son seem to do mental gymnastics without even realizing it's a skill. Sometimes when Second Son tries to walk me through his mental process to come up with a math answer, I can hardly follow him, but it works!

I have the Kindle version of the game books and they work great. The pictures are clear. The text is well-formatted and easy to navigate from the contents. Still, if I could go back, I'd invest a little more money to get the paperback versions. It's a hassle to pull out my Kindle during school time and then switching around within the two ebooks to find where I am with different children as we work through them. I also think the kids might play some of the games on their own if they had access to the paperback. (I don't let them use my Kindle.) If you use a e-reader more regularly during school time, have only one child, or combine children for math games, that's not as much of an issue.

Because we had such a good experience with these books, I purchased the Kindle version of Multiplication and Fractions. First Daughter (age 11, in 5th grade) and Second Daughter (age 9, in 3rd grade) each play with me once a week.

I purchased this books separately for the Kindle (rather than the combo book) and received a discounted price. I can't remember if they were discounted for everyone or if I received a discount from the author. Either way, this post gives my honest opinion. Links above to the books are Amazon affiliate links.

Homeschool Review: Time

Time from Teacher Created Resources

Second Daughter was struggling a bit with learning how to read a clock. It's not that she couldn't do it; she just didn't want to make the effort. I picked up this book at a going out of business sale and it was just enough to get her going. She liked lining up the stickers on the reward chart in the front as she completed exercises in the book.

Each page had just a few problems, so she could practice without getting overwhelmed. They increased in difficulty so while in the beginning she was choosing from multiple choice options of what a clock said, later she was drawing in the hands herself for a specified time. Exercises at the end of the book included time word problems and comparing clocks to determine the change in time. There were a few pages of calendar problems at the end of the book as well.

This was a nice little inexpensive book on practicing telling time.

Measuring a Tiger

I found this post in my drafts. These are pictures from a year ago, when Second Daughter was in first grade. I asked her to measure her ridiculous and enormous white tiger stuff animal for a math lesson. She happily measure bits and pieces of him.

Safe to say, this was one of her favorite math lessons of all time.

Patterns and Japanese Dolls

As part of our Reading Around the World in Asia picture book study this year, we read both Yumi and Kimonos by Aneelore Parot. When I selected these books, I had my seven-year-old daughter in mind, but my five-year-old son was just as interested (as well as the nine-year-old daughter). Neither of these stories has what I'd call a plot. Instead, each spread has a little activity for children to do with the little dolls. The second book, Kimonos, was especially interactive. Second Son loved searching for the ladybugs, choosing the correctly patterned kimono, and opening all the flaps.

There are Japanese words (including the calligraphy) but no pronunciation guide. I knew a few words and guessed at the others if the children asked. (Most of them are probably easy to find online, if you happen to have an easy way to search quickly for them.) I'm not sure how much they added to our knowledge of Japan, but they were a lot of fun.

It occurred to me as we were reading these books together, that searching for the matching patterns and thinking about what the dolls look like from the back (to choose the right one on the page) engage mathematical skills for pattern recognition and spatial manipulation. These books are a delightful way to introduce these skills, especially if you have a little girl who loves tiny dolls.

Physics with Fred

Life of Fred: Elementary Physics

It is now called Pre-Algebra 0 with Physics but the content has remained the same. I believe I purchased my copy from CBD. You can find a placement test there.

I wrote last year about how we started using Life of Fred and how First Son managed with Fractions and Decimals and Percents. He finished Decimals and Percents at the beginning of this year without problems, working steadily.

The Physics book contains forty chapters. At the end of each chapter are a few questions under Your Turn to Play. The answers appear on the following page (unlike the previous and ensuing books in which the answers appear immediately after the questions. While a student could avoid looking at the answers, I find it difficult myself to avoid accidentally seeing them.) Every six chapters or so, there's a "Bridge" of ten questions including recent and review material. The Bridge answers are in the back of the book. It's recommended students can answer at least nine questions correctly before moving on to the next chapter. With that in mind, there are five Bridges provided each time so the student has multiple chances to pass. In the beginning of the book, the author writes that some students complete multiple Bridges merely for additional practice and we did that a few times as well. If First Son seemed just a little shaky on something even if he had passed, I encouraged him to do a second Bridge and he usually did.

This is the last book for which the student is not supposed to use a calculator, and there's plenty of multiplication and division to practice those skills.

Because I didn't remember much of my high school physics, I read the chapters and answered the questions as well. I included it with my other preparation over the weekend for the upcoming week. He rarely needed help, but I was glad I was ready the few times he did. 

First Son loved placing mathematical problems and concepts within the framework of physics. Rather than replacing the physics he was doing for science, I found this book a nice complement, bringing in additional information and providing greater practice with the concepts introduced in his other books. (The physics books he read this year focused much more on the development of thought in physics and the theorems than practice problems with actual numbers.) Some of the topics covered include: friction, the meter, Mu, measuring force, Hooke's law, energy, work, transfer and storage of energy, the metric system, measuring mass, pressure, density, buoyancy, vacuums, volts, amperes, ohms, Ohm's Law, parallel circuits, and the history of physics.
 
Along the way, there's plenty of math, including review from Fractions and Decimals and Percents.

The Life of Fred books excel at revealing a greater world of mathematics and all of creation:

Fred asked a question that stopped Kitty: "What makes you think that human beings with their three-pound brains should be able to understand everything? There are mysteries in physics. There are mysteries in mathematics. There are mysteries in religion. Only lunatics and God say that they know all the secrets of the universe."

The last few chapters of this book are a condensed history of physics without any math problems to work.

The more we discover, the less we pretend to understand.

First Son spent fifteen weeks, four days a week, working through this book, including repeating a few of the Bridges. He continued with xtramath until he mastered division again and is also completing the sixth grade module on Khan Academy. He's nearly finished with that and then we'll set it aside until next year as well. (I didn't do as well keeping up with him on Khan Academy as I did with the physics!)

He has already eagerly started Pre-Algebra 1 With Biology, though he'll need to set it aside at the end of the year and finish it next school year. We continue to be pleased and encouraged by Fred here on the Range.

February 2016 Book Reports

Herodotus and the Road to History by Jeanne Bendick is suggested as a family read-aloud for Connecting with History volume 1. I read it aloud to the children. They enjoyed it and it does give a glimpse into what Herodotus's life may have been like. (purchased copy, I think from the publisher which has fantastic sales if you follow them on facebook and wait patiently)

Pegeen - by Hilda van Stockum, read aloud to the children. Read my thoughts here. Sorry the sale has ended, but follow Bethlehem Books on facebook or get on the email list to be notified of the next tremendous sale. (ebook purchased for the Kindle, directly from the same publisher as above)

The Black Cauldron and The Castle of Llyr by LLoyd Alexander, are the second and third books in the Prydain chronicles. I enjoyed them and think First Son (12) and First Daughter (9) would as well. (library copies)

The Big Alfie and Annie Rose Storybook by Shirley Hughes. Alfie and Annie Rose books are mentioned on the Pre-K page at Mater Amabilis but we'd never read any of them. The last pages show Alfie turning five and Annie Rose is just a baby. Second Daughter (seven) and Second Son (five) loved these sweet stories of Alfie and Annie Rose just living life. I loved the wedding story that showed a black couple getting married with Alfie as the ring bearer. There's no mention of race in the text, just the illustrations depicting different races celebrating family life together. (library copy)

Don Camillo Takes The Devil by the Tail by Giovanni Guareschi, translated by Frances Frenaye, is the fourth book of Don Camillo available in English. This is a particularly wonderful book for discussions about discerning the ethics of actions in the world, rather than only theoretically (though of course it's fiction). There are plenty of examples of good people (maybe) doing the wrong thing and (maybe) bad people doing the right thing. (library copy)

The Children's Own Longfellow by Henry Wadsworth Longfellow - Read my thoughts here. (library copy)

Let's Play Math: How Families Can Learn Math Together and Enjoy It by Denise Gaskins - Read my thoughts here. Sorry, the giveaway has ended. (PDF version provided by the author)

It Happened In Yellowstone by Erin H. Turner is a young reader book of short stories about real things that happened in the area of Yellowstone National Park, beginning with the "Making of Wonderland" 600,000 years ago. I might read some of this book to my children before a planned trip to Yellowstone later this year, though I intend to skip three of the chapters I thought might make them more nervous than excited - about an Indian raid, a murder, and a deadly earthquake. There's nothing particularly gory or violent in those chapters; I just prefer to focus on the excitement of our trip rather than the dangers of it. The stories vary between animals, historical figures, and events in history. It's a good introduction to Yellowstone. (library copy)

String, Straight-Edge, and Shadow: The Story of Geometry by Julia E. Diggins, illustrated by Corydon Bell - Read my thoughts here. (purchased copy)

George Washington, Spymaster: How the Americans Outspied the British and Won the Revolutionary War by Thomas B. Allen - Read my thoughts here. (purchased copy)

Books in Progress (and date started)

• Norton Anthology of English Literature Vol 2 (sixth edition) (August 2014)
• For the Children's Sake: Foundations of Education for Home and School by Susan Schaeffer Macaulay (January 2015, with my Start Here book club)
• Humility of Heart by Fr. Cajetan Mary da Bergamo (February 2015)
• Towards A Philosophy of Education (Charlotte Mason's Original Homeschooling Series) by Charlotte Mason (May 2015, with my Start Here book club)
• Ten Ways to Destroy the Imagination of Your Child by Anthony Esolen (November 2015)
• Everything Must Change: When the World's Biggest Problems and Jesus' Good News Collide by Brian D. McLaren (December 2015) 
• Father Junipero Serra by Ivy Bolton (January 2016, aloud to the children) 
• City of Saints: A Pilgrimage to John Paul II's Kraków by George Weigel (February 2016)
• Mystery of the Roman Ransom by Henry Winterfield (February 2016, aloud to the children)
• The Story of the Treasure Seekers by E. Nesbit (February 2016, aloud to the children)

The italic print: Links to Amazon are affiliate links. As an affiliate with Amazon, I receive a small commission if you follow one of my links, add something to your cart, and complete the purchase (in that order).

Links to RC History are affiliate links.

Other links (like those to Bethlehem Books) are not affiliate links.

These reports are my honest opinions.

The Beauty of Geometry: String, Straight-Edge, and Shadow

String, Straight-Edge, and Shadow: The Story of Geometry by Julia E. Diggins, illustrated by Corydon Bell

I can't remember where I purchased this book, but I'm linking to RC History because it's presence on the syllabus for Connecting with History Volume 1 is the reason I purchased it and I am entirely grateful. Through the ancient civilizations of Mesopotamia, Egypt, and Greece, it tells a story of discovery, the discovery of the eternal truths of geometry.

We began this book last year, when First Son was in fifth grade. I would read the selections to him and then he would narrate it back to me. This year, in sixth grade, I asked him to read it independently and then narrate it. Some of the chapters, ones that deal specifically with theorems more than story, are a bit more challenging, but worth the effort.

In the chapter The Golden Age and the Golden Mean, the author writes of the rise of Athens under Pericles.

High on the hill of the Acropolis rose new marble temples and bronze and painted statues. Crowds thronged the vast new open-air theater nearby, to hear immortal tragedies and comedies by the greatest Greek playwrights. These splendid public works were completed under the direction of the sculptor Phidias and several architects, all of whom knew and used the principles of geometry and optics. "Success in art," they insisted, "is achieved by meticulous accuracy in a multitude of mathematical proportions." And their buildings had a dazzling perfection never seen before--the beauty of calculated geometric harmony.

This is not just a book that teaches geometry. This is not just a book that lists off the names of important men and their achievements. This is a book written by someone who appreciates the intrinsic beauty of geometry and skillfully shares that appreciation with the reader.

But working on what may seem useless has frequently been the task of mathematicians, and such tasks, pursued with care, patience, and persistence, have led to most useful results. A whole book could be written about useful results from useless problems.

Kansas Dad taught a class on theology and mathematics a few years ago and agreed whole-heartedly with the above quote. Over and over, he says, investigations into pure mathematics have revealed insights that have clear and immediate real-world applications.

The book reaches its end and its pinnacle in discussing Euclid's Elements.

It was as though Thales and the Pythagoreans had quarried great marble slabs from nature, and through the centuries that followed many minds had carved and polished each piece until at last the whole was put together by Euclid into a simple and perfect structure as lovely as any Greek temple.

Excellently written. Clearly and attractively illustrated. A fantastic addition to our homeschool and a book I thoroughly enjoyed myself.

Links to RC History are affiliate links. 

Get Ready, Get Set: Let's Play Math (A Review with a Giveaway)

Let's Play Math: How Families Can Learn Math Together and Enjoy It by Denise Gaskins

Mathematics is mental play, the essence of creative problem solving. This is the truth we need to impart to our children, more important than fractions or decimals or even the times tables. Math is a game, playing with ideas.

I discovered the Let's Play Math website a few years ago. I felt like my oldest son was learning his math skills just fine, but he was also learning to hate math with a passion. He dreaded every minute of math and was convinced he was terrible at math, despite my protestations. Ever since, I have been open to ideas to reveal to all my children the beauty and joy of mathematics, even though I've never felt certain I understood it myself. I recently joined the review team for new Let's Play Math books and was excited to read and review the updated version of Let's Play Math.

The book is divided into five main sections. The first one, "How to Understand Math," is one of the most valuable. It describes how mathematicians, people who revel in math, think about math. They play. It's not just about skills and arithmetic; it's about how people learn and the problems that have fascinated them for centuries.

Real mathematics is intriguing and full of wonder—an exploration of patterns and mysterious connections. It rewards us with the joy of the “Aha!” feeling. These characteristics make it easy to stick with real math, even when a particular concept or problem presents a difficult challenge. Workbook math, on the other hand, is several pages of long division by hand followed by a rousing chorus of the fraction song: “Ours is not to reason why, just invert and multiply.”

"Playful Problem-Solving" explores math through stories, manipulatives, and solving interesting and challenging problems that are about more than just arithmetic. It's full of stories you can tell your children and games you can play together, describing interactions that are centered not around a school day but around family life. (All the problems mentioned have solutions listed at the end of the book.) "Math with Living Books" encourages parents to present math history with children, to trace the excitement of mathematical discoveries and advances over the centuries. Also, math concepts can be presented more interestingly than most textbooks in picture books, chapter books, and puzzles.

In "Let's Get Practical," the author describes what math might look like in a homeschool setting, starting with some of the ideas she herself has implemented. Buddy math, for example, involves taking turns solving practice problems and talking through the process. (I was particularly struck by this idea which I've used for years in reading but never considered for math.) Working through problems together is not about getting the right answer at all.

Our discussion is the important thing. The answers are an almost insignificant byproduct. Sometimes we don’t even bother to work out the final calculation because what intrigues us is the web of ideas: how can we think our way through this problem?

She answers frequently asked questions and provides some descriptions of educational offerings, textbooks, and online resources for different ages. These are not homeschool curricula reviews, but rather highlights of materials she has found to be comprehensive, adaptable, or particularly useful for different subjects.

The "Resources and References" section begins with a booklist for parents and teachers and just gets better after that. There are probably hundreds of print and online resources mentioned including worksheets and lessons found online.

A Table-ful of Math

Because I am familiar with some of Denise Gaskins's other work (emails, blog, and other books), I had already modified our homeschool math a bit based on her ideas. Second Daughter and I have been playing math games for a few years and am pleased with her progress and her attitude toward math. For the past few days she has joyously managed a store of boxes in the living room.

Reading this book, however, has made me consider challenging myself even more.

I'm considering adding a puzzle time or math read aloud to our "cultural studies loop."

I'm considering how to incorporate more time with me into math lessons with First Son and First Daughter. The author points out how families, especially homeschooling families, make family read-aloud time a priority; we read to our children and we ask them to read to us. When it's time for math, though, we send them off to another room with a book and then just point out all their mistakes. This accusation struck home. How can I maintain the personal connection with my children as they explore math without spending an additional two or three hours each day in one-on-one lessons?

I'm considering how we might take time every now and then, at natural breaks in our current math curriculum (with which I am still quite happy), to explore some of the intriguing ideas or websites mentioned in Let's Play Math.

I'm considering ways to foster a relationship between my children and math that encourages problem-solving for it's own sake rather than merely to get the answer to a question. I think most math worksheets and lessons inherently reward behavior that just gets it done. Get the answer right and you can move on.

We recognize that children are short-term thinkers, wanting to finish their school work with as little effort as possible so they can get back to the truly important things in life, such as Minecraft. We resist their efforts to turn math into answer-getting and insist on their taking time to explain and justify their conclusions.

I've already decreased the time we spend on math facts and drilling for the younger ones and will continue to consider how to ensure we're memorizing the facts well enough to facilitate higher level math without drilling so much we damage the growing love of math. Let's Play Math acknowledges the necessity of learning math facts and practicing skills, but insists we must consider the larger goal when we decide on our methods.

Mathematical literacy is a worthy challenge. But most of us want more than literacy for our children. We want them to be educated. An educated person is interested in more than merely what is useful. He or she loves to learn, studies for the sake of gaining knowledge, and grows in wisdom.

Math facts and skill practice are important but not sufficient. Mathematicians don't just add, subtract, multiply, and divide: they solve puzzles.

Instead, we need to introduce our students to the thrill of tackling tough, challenging puzzles. We need to give children a taste of the joy that comes from figuring things out, the “Aha!” factor. We need to adopt the mathematician’s view of math as mental play. Learning to think a problem through can be hard work—and that is exactly what makes it fun.

Changing how we address math can be frightening, especially for parents that learned math in the very kind of environment or with the kind of attitude we're attempting to avoid. My oldest is in sixth grade and I feel like I'm just beginning to find our way in approaching math, even though we've been learning math all along. I believe I can establish a few small changes, a little at a time, and we'll see results in the end.

Don’t worry about taking a less formal approach to math in the elementary school years. If you are always doing something—reading library books, telling each other stories, enjoying math crafts, drawing geometric pictures, playing with calendar numbers, and so on—then your children will pick up an amazing amount of knowledge. As Julie Brennan explains: “Early exposure to real mathematics in natural settings, without requiring mastery of arithmetic on a set timetable—this has been a key to the ease with which my kids attain mastery when the time is right for them.”

Here on the Range, I'm determined to establish an environment where math is not just numbers and answers. I firmly believe my children can learn all the math they want, when they're ready, as long as they don't convince themselves they can't learn it, they don't like it, or that it's too hard. To reach this goal, math must be a regular part of our lives in a way that encourages conversation and exploration.

Our kids can only see the short term. If we adults hope to help them learn math, our primary challenge is to guard against viewing the mastery of facts and procedures as an end in itself. We must never fall into thinking that the point of studying something is just to get the right answers. We understand this in other school subjects. Nobody imagines that the point of reading is to answer comprehension questions. We know that there is more to learning history than winning a game of Trivial Pursuit. But when it comes to math, too many parents (and far too many politicians) act as though the goal of our children’s education is to produce high scores on a standardized test.

Let's Play Math could be the very introduction a young family needs as they contemplate the first few years of homeschooling. First Son's early years may have been completely different if I had read this book when he was five. It could be a fantastic book for a family with a child that's struggling (in homeschool or otherwise) with math. A few years ago, when First Son first showed signs of a potentially life-long hatred of all things numerical, reading this book may have helped me adapt the curriculum we were then using to meet his needs and enrich him. (We ended up switching and I'm happy with that, but I could have avoided quite a bit of angst.) This book would be perfect for a parent who has always struggled with inadequacies in math or for someone like me, who always did just fine in math but never understood the claims of math's beauty or fascination. I find myself excited to explore some of the resources the author has gathered together for my own growth and new challenges.

In the few years we have our children at home, we cannot possibly teach them everything they will need to know as adults. At best, we can give them the tools for learning and the ability to reason, so they can continue their own education. And one of the most important tools for learning is a solid understanding of real mathematics—math taught the mathematician’s way, as mental play.

THE GIVEAWAY - Win Your Own Copy of Let's Play Math

Denise Gaskins has graciously offered her book as a giveaway hosted right here at Our Home on the Range! She is providing an autographed paperback copy of Let's Play Math for the grand-prize winner AND an electronic version (winner's choice of Kindle, PDF, etc.) for the runner-up.

I'm trying something new here on the Range. Use Rafflecopter below to enter the giveaway.

• One comment and you're entered. Tell me your favorite math resource or, if you don't have one, that you need this book to discover one.
• Contest opens on Thursday, February 25th, at 7:00 am and ends at midnight on Monday, February 29th. (Central time - it's Kansas folks!)
• I'll email the winners on Monday or Tuesday and will choose new ones if I don't hear back within 48 hours - so be sure to watch your email!

Just to be clear, the author offered to provide prizes for this giveaway without knowing what kind of a review I would write. No purchase is necessary for the giveaway, and no benefit is gained by making any kind of a purchase.

a Rafflecopter giveaway

I received a free copy of a PDF version of this book in advance of its publication as part of Denise Gaskins's review program. This post is my honest opinion. The link above is an Amazon affiliate link. I receive a small commission if you click the link, add something (anything) to your cart at Amazon, and make a purchase in whatever time frame Amazon currently employs.

Just Playing Games: Math Games Our Family Enjoys

Second Daughter has always loved games. I love the idea of playing games with my children, but I don't really love playing games. Incorporating them into our school day meant I felt more justified in letting the dishes or the laundry sit a few minutes (or an hour) longer. My checklist (play a math game with Second Daughter) made me accountable for devoting some of our lesson time to something she would enjoy as much as I enjoy reading aloud to her.

This year, I purchased Counting and Number Bonds: Math Games for Early Learners (Math You Can Play Book 1) and Addition and Subtraction: Math Games for Elementary Students (Math You Can Play Book 2) which I'm using with both Second Daughter and Second Son, but I've also continued our habit of playing a "math" game from our shelves once a week. I thought I'd share a few of our favorites. Once I started paying attention, I realized math concepts are hidden in many games. Second Daughter has also successfully argued for many of her favorite games based on counting and comparing skills. (Munchkin* comes to mind.)


Count Your Chickens - Second Son received this game as a gift and I love it. It's beautifully made, a cooperative game (everyone wins!), and is all about counting. I highly recommend it for the very young. If you can avoid losing the little chicks, this game would last through lots of young children learning to count.

Sum Swamp - I purchased this game when I first decided Second Daughter's kindergarten year would include math games. I wanted one on addition and this one had great reviews. Personally, I found it a bit repetitive, but there is no doubt that Second Daughter loved it. She and a friend played it about once a week for the entire school year. I think the swampy animals were a big attraction.

Blink -  I also purchased this card game. It's one of my favorites, but I almost always win. We started coming up with handicaps so it would be more of a contest.


Rat-A-Tat-Cat - We've enjoyed this game so much, I've given it as gifts to others. Comparing numbers, addition, memory, and strategy. This game is easier for younger players to win than Blink, too. My little ones love the silly illustrations on the cards.

Battleship - strategy and grids. First Son received this game as a gift years ago and we are all still playing it. At first, Second Daughter couldn't get through a whole game, but she's gained endurance over the past year or so.

Qwirkle - shape recognition, sets, and strategy. Second Daughter likes this game best if we work together to make as many qwirkles as possible.


Skippity - I received this game as a gift a few years ago. Mostly I like all the bright colors, but I'm certain there's math in there somewhere. I also like how it balances the players well, so children have a shot at winning even if an adult doesn't ignore the best moves.

Blokus - This game is probably my favorite of our "math" games. The bright colors and shapes are so inviting, the game pieces usually end up being used in individual play after the game is over. The last time we played, Second Daughter swept the game off the table and returned an hour or so later with this perfect square.

There are lots of other games with hidden (or obvious) math like Double Shutter, Connect 4, Monopoly Junior, and Trouble (which we have in multiple versions and which Second Daughter insists on calling "Pop the Bubble"), but the ones above are the ones we play and enjoy the most. Do you have any favorites to recommend? (A friend of mine has already suggested Perfection.)


* Please note Munchkin is not for all families. We've covered up the most egregious words on cards, but some families would probably rather just skip it altogether.

Links to Amazon are affiliate links. If you click on one, put something in your cart, and purchase it within Amazon's specified time frame, I receive a small commission. Our family appreciates every little bit. Every link in the post is for something I purchased or something we received as a birthday or Christmas gift.

An Atypical Perfect Homeschool Moment

I glanced around the table and realized I was living one of those elusive moments - a boy working diligently on his math with math-rich games flourishing on either side of him. It was so unusual in its perfection, I had to take a picture.

This, ladies and gentlemen of the blog, is an ideal homeschool moment, so rare I almost tingled when I realized what was happening.

It degenerated into chaos shortly after this picture was taken.

The children pictured claim no memory of such an event.

Math Talk: Life of Fred and Hoping My Kids Don't Hate Math

I devoted time last summer considering math. First Son had just finished 4th grade, during which he completed Saxon 5/4 math. Though successful with the math itself (with some leeway for recalculations in his long division), he hated math. Lamentations surrounded our daily math work. No one liked being around him when it was time for math, but I was even more concerned that he would give up on ever enjoying math before he had a chance to learn much of anything interesting.

I've never felt like I could see the beauty in mathematics, but I believe it is there. I want my children to see it and though I'm not sure how to reveal it to them, I do know that dreading math every day would hide that beauty, possibly forever.

So I considered. A friend struggling with the same sorts of issues decided to invest in the elementary series of Life of Fred. These books are the most likely ones to be recommended when homeschooling moms ask about kids who hate math. They tell a story centered around a child math professor at an imaginary Kansas university (Yay Kansas!) who encounters problems in real life that can be solved with math. Because they are presented in a series rather than by grade level, students can work through them at their own pace without focusing on whether they are ahead or behind. In the elementary series, every short chapter (meant to be read independently by the student) ends with a Your Turn to Play section in which there are a few practice problems.

The intermediate series (starting with Fractions), also has problems at the end of every chapter. In addition, there are "Bridges" which are mastery quizzes every three to five chapters. Those have ten questions each and it is recommended students answer at least nine problems correct. If not, four additional Bridges are provided.

The Life of Fred story hints at exciting higher level math all through the books. The reasons for math appear in the text (like a story problem, but more interesting) and then are addressed with math, compared to most traditional math books in which the concept is introduced and then made more complicated with story problems. Instead of being useful, math is just hard.

I bought a few of the books. First Son started the school year (fifth grade) working through Fractions. His attitude flipped completely. Math quickly became his favorite subject. He often read the elementary series during his free time.

He managed the first few Bridges, but soon started having problems. He was struggling with the long division problems, making small mistakes. His weak multiplication skills were showing. We took a few weeks off, focusing on mastering the multiplication and division facts (using xtramath.org, Multipication Wrap-ups, and Speed on the iPad) then returned to Life of Fred.

I think some children would work well through the problem sets and  comparing their answers with the correct ones. First Son, however, too quickly gave up on trying and would just look at the answer. It would become apparent to me when he couldn't pass the Bridges. To compensate, I would grade his Bridge, then recommend any areas with which he struggled on Khan Academy. I have my reservations about Khan Academy, but the problem sets are designed to reward mastery. Usually, you continue to work through a skill using hints and videos until you can answer five questions in a row correctly. Eventually, I asked First Son to do one chapter of Life of Fred each day in addition to two problem sets on Khan Academy. (He also continued with Xtramath each day until he had mastered division.) The combination of the lively story in Life of Fred and the focused practice on Khan Academy seemed to work well for us. By the end of the year, he had worked his way through about half of the second intermediate book, Decimals and Percents.

The elementary series starts with Apples and progresses through books starting with subsequent letters of the alphabet. All elementary students should start with Apples and can work more quickly through the first few books if appropriate. If the student finishes all ten books before being ready for Fractions (around 5th grade, after all the multiplication and division facts have been mastered), there are three more books you can use (Kidneys, Liver, and Mineshaft). After those, if the student is still not ready for Fractions, the publishers recommends you start back with Apples as there are many higher level skills that can be developed by a second reading.

Last year, First Daughter (in second grade) started with Apples and worked independently through the elementary series. She finished the first seven books, working about four days a week. At first, she was upset to move away from Saxon math (mostly because she was proud of being a grade "ahead" in her math), but after a few weeks, she admitted the new books were much more enjoyable. Every few days I would watch as she answered the problems to see whether she understood the concepts. By the end of the year, she had finished Goldfish. First Daughter continues to work on Xtramath.org and at Khan Academy as well.

I read a chapter or two aloud a week to Second Daughter who was six but only in kindergarten. She enjoyed the story but didn't not understand all the concepts completely. The author recommends not starting before first grade and I would concur. We slowly worked through Apples and the second book, Butterflies, before the end of the school year. Next year, when she is in first grade, we're going to start the year with something else and add in Life of Fred when she is ready to read and work independently through the third book. Given Second's Daughter's progression through the books, there will be plenty of time to read them even if we wait until second grade.

Overall, I'm pleased with how Life of Fred has changed the math attitude in our house. My children can learn math for the rest of their lives, as long as they don't learn to hate it first. An added benefit is the significant decrease in the amount of time I spent presenting Saxon K-3. I wish I had started using it earlier.

One of the reasons I hesitated was a perception that they were expensive. Ten books! $16 each! I did a calculation. For the first child, paying regular price, the Life of Fred books (ten elementary books and three additional books) cost $208. Saxon K-3 is $418 (that's not counting fourth grade, which is probably included in the Life of Fred books). Plus, each additional child is another $121 in consumables. Life of Fred isn't the cheapest math program out there, but it's a lot less expensive than Saxon. From what I can tell, the books should sell very well after you are done with them, too. They've held up to multiple readings already at our house.

All of the links above are affiliate links to Amazon, but that might not be the best place to buy Life of Fred, if you're interested. Check out the Life of Fred website (not an affiliate link; this is an "official" seller website and is not maintained by the publisher) for more information. You can also find Life of Fred at CBD (not an affiliate link) and at Adoremus Books (not an affiliate link). I bought a couple used off Cathswap, most of them at CBD (where free shipping coupon codes are plentiful) and the additional elementary books at Adoremus during their Easter sale.