Another change is that we need to be able to summarize data and extract meaning, and express that meaning in a way other people can understand. And we need to be able to look at a situation and make decisions about which things are most relevant to the problem, so we can ignore the parts that are less relevant. Keith Devlin has been talking about these changes for years, especially the need to switch from a focus on executing calculations to a focus on thinking about how things relate and interact within a problem. That's not a new type of thinking, of course, since it's what the Problems Without Figures book was trying to develop. Both types of math have always been important, but schools used to emphasize the former, and now we need to emphasize the latter. Here are some of my favorite Devlin posts, from earliest to most recent: Telling stories with numbers What is mathematical thinking? Most Math Problems Do Not Have a Unique Right Answer Your Father’s Mathematics Teaching No Longer Works Hard fun – video games creep into the math classroom

Card games make for fun practice, and you don't have to spend money: Free Multiplication Bingo Game Times Tac Toe Math Facts: 5 Minutes a Day Contig Game: Master Your Math Facts The Game That Is Worth 1,000 Worksheets

Passive voice and linking verbs have a blurred line between them, as you have discovered. An answer book that is too rigid does the student a disservice. I would let the student's answer stand, as long as he or she can explain why it makes sense. One thing that may help in making the distinction, is that a linking verb in some sense serves as an equal sign. But that's most helpful when the verb links two nouns. In a case like "We were not impressed", the equal sign is wobbly---are we talking about our state of being, or are we talking about a (metaphorical) action that was (not) done to us? Incidentally, if we are talking about a non-metaphorical action, then it's definitely a passive verb. "Jason and Gwen dodged the stompers. Although they lost Jason's gun, they themselves were not impressed." (For fans of Galaxy Quest.)

He does state in the intro that he put trick questions in: A few “catch problems†are put in to entrap the unwary. To stumble occasionally into a pitfall makes a pupil more watchful of his steps and gives invigorating exercise in regaining his footing. The groove runner thus learns to use his wits and see the difference between a legitimate problem and an absurdity.

I've posted a lot of math games on my blog over the years, so I'm sure you'll find something of interest there: Math Games Blog PostsI have organized the best of my games into books, roughly by grade level, though I'm not sure they can stand in for a full curriculum. If you're curious, you can follow the link in my signature to download free excerpts.

Lulu used to offer samples, but I don't see them now. Hmm. Well, maybe you can decide whether you like his teaching style by checking out his G'Day Math free courses, or some of his older YouTube videos.

How about "Problems Without Figures"? A whole book of word problems without numbers, to stimulate thinking and discussion. http://www.schoolinfosystem.org/pdf/2008/10/problemswithoutfigures.pdfI keep meaning to transcribe and modernize this public-domain gem, but haven't gotten around to it yet. You can take a similar approach to word problems from any source, just leaving out the numbers. Another good approach is to leave the numbers in, but omit the question -- let kids tell you what questions *they* would ask. Or combine the two: Give *part* of the problem setup, and ask what the kids notice or wonder. Then add the rest of the problem setup (but not the problem question), and see which of the kids' questions they can answer. What additional information would they need to answer the others?

I've posted quite a few math games on my blog. They aren't played online -- mostly they are card games, but some use dice or other easy-to-get supplies. Low stress, fun, and plenty of learning potential.

Are you set on doing a curriculum? Or perhaps I should ask: How comfortable are you with math yourself? If you are comfortable talking about numbers, shapes, and patterns with your kids, there are all sorts of early-elementary math things you can do without having to use workbook pages. If you do want to stick with a curriculum, I suggest the following adaptations: Do as much as possible orally. Use the workbooks as a coach's manual for you, but just talk about the numbers and let your kids think about them. If there *must* be written work, consider being your child's secretary and taking dictation. Use a whiteboard and colorful markers, instead of paper and pencil. Let your kids use manipulatives if they want, even if the workbook page doesn't mention them. (And in my mind, fingers count as manipulatives, because they help kids think their way through math problems.) Play math games. (Link goes to the game posts on my blog.) Games are worth much more than worksheets. Use lots of games.

Since you are comfortable with math yourself, I think you could do very well without a formal curriculum until your son is ready for Beast Academy. Really, there's not that much to early-elementary math --- just paying attention to the numbers, shapes, and patterns all around you, and leading your son to notice and think about them. With a discussion-based approach, it's easier to be sensitive to your son's true level. Here are some resources that might help you think more creatively about early math: Talking Math With Your Kids: Christopher Danielson helps parents support their children’s mathematical development. Highly recommended! Tell Me a (Math) Story. From my blog, my favorite approach to early math. Moebius Noodles: “Adventurous math for the playground crowd.†Plenty of ideas for sharing rich math experiences with your children. Sooooo much good stuff here! Math by Kids!: A 78 page workbook of original math problems (including solutions) created by homeschooled students aged 4 to 17, edited by Susan Richman. Internet Math Resources. Also from my blog, more of my favorite online goodies.

Do your kids like fantasy fiction? My daughter's second novel is out, written when she was 15yo: Hunted: The Riddled Stone, Book 2 Read an excerpt: the first five chapters of Hunted. Review snippet: “… a captivating fantasy story with a well-thought-out plot … people who like medieval-style fantasies with wraiths, spirits, and even an attacking swamp tree will enjoy the story. I certainly did, and the excitement, adventure, and suspense will easily keep the reader’s attention …†— Wayne S. Walker Home School Book Review And if you order the paperback from CreateSpace, we have a discount code good for 30% off through July 31. (Sorry, but we have no control over discounts at other booksellers.) The Riddled Stone Series paperback books at CreateSpace --- (Use discount code LEUCWCSQ for 30% off the list price.)

Do your kids like fantasy fiction? My daughter's second novel is out, written when she was 15yo: Hunted: The Riddled Stone, Book 2 Read an excerpt: the first five chapters of Hunted. Review snippet: “… a captivating fantasy story with a well-thought-out plot … people who like medieval-style fantasies with wraiths, spirits, and even an attacking swamp tree will enjoy the story. I certainly did, and the excitement, adventure, and suspense will easily keep the reader’s attention …†— Wayne S. Walker Home School Book Review And if you order the paperback from CreateSpace, we have a discount code good for 30% off through July 31. (Sorry, but we have no control over discounts at other booksellers.) The Riddled Stone Series paperback books at CreateSpace --- (Use discount code LEUCWCSQ for 30% off the list price.)

Do your kids like fantasy fiction? My daughter's second novel is out, written when she was 15yo: Hunted: The Riddled Stone, Book 2 Read an excerpt: the first five chapters of Hunted. Review snippet: “… a captivating fantasy story with a well-thought-out plot … people who like medieval-style fantasies with wraiths, spirits, and even an attacking swamp tree will enjoy the story. I certainly did, and the excitement, adventure, and suspense will easily keep the reader’s attention …†— Wayne S. Walker Home School Book Review And if you order the paperback from CreateSpace, we have a discount code good for 30% off through July 31. (Sorry, but we have no control over discounts at other booksellers.) The Riddled Stone Series paperback books at CreateSpace --- (Use discount code LEUCWCSQ for 30% off the list price.)

Are you trying to divide by (x - 2) instead of (x + 2)? That was a typo up above, I think. If x = -2 is a solution, then the factor you need to divide out is x - (-2), which means x+2. That's because if x = -2 is a solution, then the function has to = 0 when x is -2. And that means the function must have a factor that would turn into zero at that point. The factor that becomes zero when x = -2 is (x + 2), since -2 + 2 = 0.

I don't have access to the Thinkwell site, but here are some general resources that might be helpful: PurpleMath http://www.purplemath.com/modules/variatn.htm Algebra Lab http://www.algebralab.org/lessons/lesson.aspx?file=Algebra_LinearEqDirectVariation.xml

The formal, two-column proofs typical of many high school geometry texts are never used again, as far as I know. But the ability to think logically and explain your reasoning to another person is vital. Try encouraging her to write her proofs as paragraphs, like a short essay: "I know _______ is true because ____________. And then that fact helps me see that _____________..."

Yep. With beginner algebra problems, it can be much quicker to solve using common sense and a bit of arithmetic. Figuring out how to translate the relationship into algebra is the hard part.

I think a lot of the confusion in fraction division (and in other applications of multiplication and division as we move into middle school, like rates and ratios and conversion factors) is keeping track of what the numbers mean. For instance: 1/2 ÷ 1/3 = 1 1/2 Elementary students are used to thinking of division as splitting something up into smaller parts. But how in the world could you split up 1/2 into smaller parts and end up with a number that's bigger than you started with? It doesn't make any sense at all, as long as you are looking at abstract numbers without thinking about what they mean. You can memorize a rule (flip-and-multiply) that will give you the correct answer. But if you're like most kids, the rule will get jumbled in your head and lead to more confusion. High school students are notorious for committing nonsense with fractions due to half-remembered rules. To make sense of the calculation, you have to go back to the original meaning of division: 1/2 ÷ 1/3 = "How many thirds in a half?" So our answer will be the number of thirds (or of pieces that size): 1/2 ÷ 1/3 = 1 1/2 pieces the size of one-third.

Let's do another example: An atom of zinc has 30 electrons. An atom of oxygen has 2 more than 1/5 as many electrons as an atom of zinc. How many electrons does an atom of oxygen have? What is the writer talking about? Chemistry. What atoms are made of. Electrons in zinc and oxygen. <--(This is the most helpful summary.) What declarative sentences (facts about our story) do we have? An atom of zinc has 30 electrons. An atom of oxygen has 2 more than 1/5 as many electrons as an atom of zinc. Edit to make the sentences simpler. Since we are talking about the number of electrons in zinc and oxygen, we will label them [zinc electrons] and [oxygen electrons]. "An atom of zinc has 30 electrons." becomes: [Zinc electrons] are 30. "An atom of oxygen has 2 more than 1/5 as many electrons as an atom of zinc." becomes: [Oxygen electrons] are 2 more than 1/5 of [zinc electrons]. Look for patterns that let you draw conclusions. [Zinc electrons] are 30. [Oxygen electrons] are 2 more than 1/5 of [zinc electrons]. Conclusion: [Oxygen electrons] are 2 more than 1/5 of 30. Translate the words into symbols. First, the verb, because it's easiest. And we'll put in "e" to stand for the number of oxygen electrons, since we're doing algebra: e = 2 more than 1/5 of 30 "More than" is easy, because direction doesn't matter when you add: e = 2 + 1/5 of 30 Now the hard part, because in math we always save the hardest part for last. What is "1/5 of 30"? Remember, you know how to talk and read. You know what words mean. What does it mean when someone says they have a fifth of something? Well, that means the something has been split into five equal parts, right? And what math operation splits something into equal parts? e = 2 + 1/5 of 30 means: e = 2 + 30 ÷ 5 But in algebra, we write division as fractions. Final translation: e = 2 + 30/5

When students have trouble with word problems, it is almost always in the translation stage. And once they start thinking "word problems are hard", their brain freezes, which makes it worse. BUT your daughter has been talking and reading and making sense of words for most of her life, right? So that means she CAN DO THIS! She just has to fight away the brain freeze (acknowledge the panic, shut her eyes, take a deep breath, relax) and think about what the words mean. Not what they mean in math, but what they mean as words. Just like reading a story or following the plot of a movie. For example... The age of Sarah's mother is 3 years less than 3 times Sarah's age (s). If Sarah's mother is 42, how old is Sarah? Part of the trouble with language is that it takes so many words to describe a single thing. And we like variety in our language, so writers tend to change the words they use. They don't repeat exactly the same words each time they mention the same thing. In the first sentence, the writer is talking about ages, but then in the second sentence, he switches to using the person's name when he still really means the age. So, as a person who knows how to read, your daughter can look over a paragraph and understand it. Just read it as a story, not as math. In five words or less, what is the writer talking about? People's ages. How many people? Two: Sarah and her mom. Okay, so let's simplify the problem by editing the writer's wordiness and making it more consistent. Focus on what our story is all about, the people and their ages. Ignore all the minor details, like the numbers, for now: [Mom's age] is 3 years less than 3 times [sarah's age]. [Mom's age] is 42, how old is [sarah's age]? Now, every declarative sentence, or every phrase that could stand as a sentence, is a statement about what is true in our story. As a person who knows how to read, your daughter can identify sentences. How many declarative sentences, or phrases that could stand alone, do we have? [Mom's age] is 3 years less than 3 times [sarah's age]. [Mom's age] is 42. And your daughter can draw conclusions. Anyone who reads a book needs to sometimes read between the lines to understand what the writer means. Or what about movies: the camera zooms in on a shadowy hand with a knife, and then zooms out to show a pedestrian walking toward a dark alley. Your mind jumps from that pattern to the obvious conclusion, right? Well, there is a pattern in two of our sentences above that should jump out just as obviously, now that we've edited away the excess words: [Mom's age] is 3 years less than 3 times [sarah's age]. [Mom's age] is 42. Conclusion: 42 is 3 years less than 3 times [sarah's age]. And because ALL of the things we are talking about are age years, we don't really need to keep the word "years." We can edit it out: 42 is 3 less than 3 times [sarah's age]. At this point, you should feel encouraged. This looks so much simpler than what we started with, doesn't it? We have mostly numbers and words that we know relate to things we can do with math. The editing is done, and now it's time to translate. Let's start by translating the easiest part, the verb: 42 = 3 less than 3 times [sarah's age]. The next easiest to translate is the word "times" because when something is three times as big, we know that means it is multiplied by three. 42 = 3 less than 3 X [sarah's age]. And we can put in the "s" for Sarah's age, leaving out the times symbol, as is traditional in algebra: 42 = 3 less than 3s Now the absolute hardest part: the phrase "less than." Translating subtraction is always tricky, because direction makes a difference. This is where someone who is blindly using key words will get confused, but a reader who is looking at the words the way they would read a book---that is, looking at the meaning of the words as language, not just as "math"---can get it right. What does it mean for one number to be three less than another number? Which of these subtractions matches the meaning of the words in our story: 42 = 3 - 3s or 42 = 3s - 3 Our story says 42 is three less than something. So if we have that something and take away three from it, we should get 42, right? Final translated equation: 42 = 3s - 3. If you can read a book or follow the plot of a movie, you can also read and follow the plot of a math word problem. It takes time at first, but like any task, it will gradually come easier: Read the words like a real story, as if they have meaning. Don't think of them as math. Start by getting the big picture. Say what the story is about in five words or less. Edit the sentences of the story to make them simpler. Look for patterns that let you draw conclusions. Keep everything in words, so you can use what you know about stories and plots to help you figure it out. Finally, when you've edited it down as simple as possible, start translating the words into symbols. Begin with whichever parts look easiest. But always remember to focus on meaning---don't get fooled by tricky words like "less than"!

The basic question of division is, "How many of these are in this much?" I will demonstrate how the rectangles work with a simpler example than Fawn used in her blog article, Fraction Division via Rectangles. The trouble with fractions is that they can mean different things depending on what they are a fraction of, so the abstract question "How many thirds in a half?" can land you in a pit of trouble, if you aren't careful to make sure they are both thirds and half of the same amount, the same unit --- in this method, the same rectangle. If you make the units different, say, "How many third cups in a half gallon?" then you have a completely mixed-up mess. So we draw a rectangle that will be the "one" unit for each of our fractions. The easiest way to make sure our rectangles are the same size and can be accurately compared is to draw them with each denominator being one side, so for thirds and halves we would draw a 3x2 rectangle. We make a second copy of the same rectangle. We divide one rectangle into half (or whatever fraction we need) and the other rectangle into thirds (or whatever fraction we need). Now, our basic question "How many of these are in this much?" has a visual meaning: How many of this size piece could we cut out of that size piece? If the division comes out even, that's simple. How many fourths are in a half? Two. Easy-peasy. But if we have extra pieces, like with thirds and halves, then we have to do more thinking. 1/2 ÷ 1/3 = "How many thirds in a half?" When we draw our rectangle, we see that we can cut one "one-third" piece from the half, and then we have a bit extra. We could get PART of another third, but not quite a whole third. What size is the part? How close are we to getting another complete piece? Since our extra bit is one square, it is exactly HALF of a one-third size piece. Therefore, 1/2 ÷ 1/3 = "How many thirds in a half?" = one whole third, plus an extra half a third = 1 1/2. Does that help? How this transitions to the standard flip-and-multiply method: As students draw example after example, focusing on the meaning of each division problem, they will begin to notice a pattern. In particular, using the color code above, "How many blues in the red?", they will see that the number of red pieces they need to cut up is always the same as the red numerator times the blue denominator. And the number of blue pieces (the size they are cutting red into) is always the red denominator times the blue numerator. In other words, flip-and-multiply has a very real, visual meaning in the model. It is a short-cut for exactly the procedure they have been doing. And if they forget the short-cut, they can always go back to the visual model to figure out their answer.

Here is, hands down, the best way I've ever seen to help kids think through fraction division: Fraction Division via Rectangles

If the memory work is getting in the way, then you need to move on to numbers she hasn't memorized. For instance, if the page says "13+8" but she knows that one automatically, then try changing it to "43+8." Same make-a-ten concept, but without the mental interference.

One method would be to use wet-erase markers to draw a line down the middle of one or two sides of each rod. Let dry, of course, and keep fingers dry while using. Let the marked sides represent negative numbers, the unmarked are positive. Negatives added together make longer and longer negatives, just like positives do with each other. But putting positives and negatives together, they cancel off parts of each other---like matter and antimatter---and all you have left is part of whichever rod is longest. When you get a positive and negative of the same number together, they obliterate each other entirely, leaving you with zero. Adding is easy to show: adding positive to positive or negative to negative just gives you more of the same. When you add a negative to a positive number, it dissolves away part of the rod and leaves you with a smaller positive number. Or, if the negative is the longer rod, the positive will dissolve away part of it and leave you with a "smaller" negative number. Of course, "smaller" negatives are actually greater (bigger, worth more, closer to zero) than "larger" negative, which is somewhat counter intuitive. Multiplying positive times negative isn't hard. Just put out that many of the negative rods. Subtracting is also fairly easy as long as you stay in all negatives or all positives (so you can subtract by taking away rods). Subtracting a negative number from a positive is tricky to model with manipulatives. You can do subtraction if you first teach that the same-size positive and negative together make zero, and you can always add zero to any number without changing your total amount of something. So here you have your positive rod, and you want to take away a negative, but you have no negative rod to take away. What can you do? Add in a zero, which doesn't change the total amount of your number, but make the zero by using the positive and negative of whatever size you are wanting to subtract. THEN you will have a negative rod to take away. Subtracting a positive number from a negative would use the same "add a zero" trick. The hardest calculation to model is negative times negative. Here is one method, which I think could be adapted to rods: Multiplying Negative Numbers with Rectangles

If he can do the word problems without the models, then I'm not surprised he wouldn't want to draw them. You have several options: (1) Skip it. Don't bother with the bar models unless his intuition starts to fail him. (2) Push him to do harder problems, where his intuition is no longer able to organize and interpret the information. (3) Make "bar models" a separate lesson from problem solving. He doesn't have to *solve* the problem, just draw a model that shows the relationship. This was my solution, since it let my kids feel like they were getting off easy by not having to figure out the answer to the problem. But it made them do the analysis, which is what I felt was most important. At the beginning, I even "took dictation" by doing the drawing myself, and they just told me which labels went with which parts of the diagram. (4) Go directly to algebra. Elementary students can learn to do algebra with words, which can give them a way to think through problems where their intuition fails. After all, the end goal is algebra. The bar diagrams are just a visual model that can help along the way, but some children are more comfortable jumping straight to algebra. If that last option seems strange, you might want to read through the first couple of articles in my Word Problems from Literature blog series, which compare an algebraic approach to the bar model method. By the time I got to the 4th and 5th grade articles, I had switched to using only the bar models --- but the algebra approach can work fine all through elementary school, gradually transitioning from word algebra to the more abstract stuff.