With permission of the list owner, I've dug out some of those posts--mostly for my own benefit, as Crayons is doing the same math now that Ponytails was a few years ago. I've edited them to use our blog nicknames and to take out some repetitive details. I've also added links to things I posted here after we started the blog.
The sequence of the six Miquon Math workbooks is Orange, Red, Blue, Green, Yellow and Purple. The average speed on them for most homeschoolers is about two workbooks a year.
The teacher's manuals for the program are the Lab Sheet Annotations, Notes to Teachers, and the First Grade Diary. There is no "Lesson 1", "Lesson 2" and so on; each topic is introduced generally in the Annotations, and then notes are given for the worksheets. Often there are activities suggested to go along with the sheets, such as chalkboard games. Many Miquon users just have their children work right through the books, especially if they are independent learners; but we had more success mixing up the topics and not depending too heavily on the worksheets.
August 1999: Introducing Myself
This is our fourth year using Miquon and I never imagined there was a loop just for us. I've been reading through the posts and I already see I'm in good company. Our Apprentice just started the yellow book today. She loves the rods, but doing operations with them doesn't make sense to her. She loves the books, likes adding and multiplying (HATES subtracting), but would rather use a hundred chart, break-apart cubes, two rulers one above the other (anybody tried that? it's a "magic adding machine"), turning a question into a word problem, just about anything except putting two rods together and figuring out how long they are.
Answering a question about using two rulers as an adding machine:
I can't take credit for this idea--I got it from a library book but I can't remember which one! You need two rulers, metric works best because there are more numbers. I use two identical transparent ones. This idea doesn't work well with very young children, but school age children find it fun.
To add 3+12, set the rulers against one another so that the 3 on the upper one is above the 0 on the lower one. Find the 12 on the lower ruler, and whatever number is above the 12 is the answer! To subtract 10-7, place the 10 on the upper ruler over the 7 on the lower one. Find the 0 on the lower ruler, and whatever number is above the 0 is the difference.
We started the Yellow Book this fall and for some reason we just can't get into it; my daughter groans "oh no, Miquon" when I get the book out. We are playing a lot of math games, doing questions with a muffin tin and macaroni (if you put 10 pieces in each compartment, you can do addition and subtraction up to 120) and with coloured cubes and pieces of coloured paper for the units, tens and hundreds (if I give you 5 cubes for the blue tens "column" and 4 for the red units "column", what number do you have? and if I give you 3 more for the blue and 7 for the red?--regrouping is necessary here)....
I never saw anyone as sure yesterday that 48-30 equalled 2 (on D-21). She took away the three tens and then insisted that you had to do something with two (I think she was thinking 10-8). I had to get the muffin tin out and prove the answer was 18, and I still don't think she believed me.
I read a very interesting article by Ruth Beechick (I think it was in her Question and Answer Book). To summarize...She describes a class where the children solved problems with some kind of flat counters. The teacher would say "make three piles of two" or "eight piles of ten" and so on. The kids got comfortable with the idea of not counting every single one. As I remember it, she eventually suggested to some of them that they could write out the solutions so that they wouldn't have to count out all those piles every time; she showed them how they could take two piles of two, three piles of two, four piles of two etc. and write the answers going across the page; and then they caught on and started writing out the threes, fours etc. The idea spread around the classroom and soon many of the kids were writing out their own tables for reference.
Later she suggested that if they learned some of those facts they wouldn't have to refer to the tables all the time, and again the idea caught on; the kids saw the usefulness of carrying the facts around in their heads instead of on their papers. I don't know how well they all learned their times tables, but I'm sure that was a happier classroom than the one in "Hans Christian Andersen!" (If you never saw the movie, the schoolchildren are droning "two and two are four, four and four are eight...")
December 1999 again
In spite of my complaints [about the Yellow book], we have found one way to make some of the addition and subtraction a little clearer (for example, page E-53, which is addition and subtraction with a sum or difference of 100, 200, 1000, etc.) We get out our pile of pennies, dimes and Canadian dollar coins, and do the problems as if they were money.
We have a couple of favourite scenarios we use with this. One is "you had this much money in your piggy bank this morning. Your sister came along and "borrowed" a few coins. When you came back later, you found you only had so much money left. How much exactly did she take?"
The opposite is, "you had this much money; grandpa came for a visit and slipped a few coins in; when you counted all your money you found you had this much; so how much money did he put in?"
One problem was x + 32 = 100. My dd didn't know how to do this mentally, so I said, "Pretend I'm 30 years old. In how many years will I be 100?" "70," she said. "Now it's two years later; I'm 32 years old. Now how many years will it be until I'm 100?" "About 72" she said. "Really? But two years ago you told me it would only be 70 years. How can it be more now?" She thought about this, and then decided 68 made more sense.
[There is a long gap here because The Apprentice switched to another math program, and I unsubscribed from the Miquon-Key list until our second daughter, Ponytails, was in the first grade and starting Miquon.]
I'm starting Miquon again with renewed enthusiasm for the program, and I'm trying to use it more creatively. One difference from our first time through is that Ponytails is a bit older than The Apprentice was when she started the workbooks.
Another difference this time around is that Ponytails has had lots of experience playing with rods over the last couple of years. She's not quite comfortable yet thinking of them in terms of their actual length; she thinks of them as a blue rod or a green rod, not as a four or a five; but I think actually that's good; it means she can think about them for themselves without worrying too much yet what they represent.
I inadvertently found a way to heighten Ponytails' interest in learning the lengths of Cuisenaire rods. This morning I held up a yellow rod and asked her how many white rods long it would be. She said four. I said "Okay--prove it." She took the bait and said, "Okay then--I WILL." She lined up the white rods and admitted that she needed five. When I held up a light green and asked how many whites that would take, she said, "Three--and I'll PROVE it to you." And she did. We continued like that, and she kept on PROVING stuff to me--and the math got done.
I found a homemade substitute for the wooden rod tubes described in the First Grade Diary (wooden tubes just long enough to hold one 10 cm orange rod, or the equivalent in shorter rods). (They are described in the entry for Sept. 30.)
Cut a piece of index card to a 10 cm length and wrap it fairly tightly around an orange rod, trying to crease the edges as you go (to keep the rectangular shape). Secure each end well with tape, and there you go. I made four in under 10 minutes. They're obviously not made to last forever, but at that price they could be replaced easily when needed.
The purpose of the tubes, if you don't have the Diary, is to play hidden-rod sorts of games. The simplest version would be to insert two rods that add up to ten, show the child one end of the tube, and ask what colour should appear at the other end. A more complicated version would be to use three (or more) rods, show both ends and ask what is hidden in the middle.
September 2003: Centimeter Cubes
We use these for a lot of things (besides giving the toddler something to play with [update: I don't remember what our toddler was doing playing with something so small--I must have been right beside her at the time because obviously cm cubes aren't an appropriate toy for children small enough to eat them]); I used them a lot when my oldest was learning subtraction, because you can't break a piece off a Cuisenaire rod! We also used them later on to demonstrate multiplication/division--twenty cubes snapped together, then broken into four groups of five and so on.
Young children can use them to do patterns with--two reds, a blue, two reds, a blue. You can draw a pattern with markers and have them copy it.
You can build things with them besides just straight trains. Build a 3-D model for the child to copy, something made out of several cubes stuck together. (Easy to do with Duplo or Lego too.)
You can use them to visualize word problems; for instance, you can use a piece of green construction paper to represent a field, and use some white cubes to represent sheep in the field--for counting, adding, subtracting. Or a blue piece with different coloured "fish".
We also use them to play a simple guessing game at about the K-1 level. You snap a few together (5 or 6), hold it behind your back and break a couple off; hold out the remaining piece and have the child guess how many are still behind your back. (You can do the break-off game with Duplo or Lego blocks too. Anything that snaps. Or even with small groups of non-joining objects, such as crayons or coins.)
And I suppose you could use them for actually measuring things
October 2003: After A Month of Grade One
....I have to say, I am more impressed than ever with Miquon and I'm glad I chose it for my first grader (I might have been discouraged after my older one's so-so experience).
The amazing part of this that we are doing a maximum of 20 minutes of math a day, doing a lot of rod games and the sorts of short activities suggested in First Grade Diary, and Ponytails is NEVERTHELESS picking up not only addition concepts but has started subtraction as well and ENJOYS it.
What a great program this is. I keep wondering why I see all these entire unused sets of Miquon at used curriculum sales; it makes me wonder if people just give up too soon.
On the difficulty of problems such as x - 3 = 2:
Ponytails and I played around with some problems like this today using coloured popsicle sticks and a small empty box which had originally contained frozen chicken wings. I put six popsicle sticks in the box without showing her how many, and said "I was all by myself for dinner and I was hungry, so I took out four chicken wings to cook. (I removed four sticks.) The next time I looked in the box (I showed her), there were two left. How many were there before I ate all those chicken wings?" She looked at the two in the box and the four on the table and said "six." We repeated this with some different sets of numbers, and she had no trouble with that.
Then we went to our blackboard and I drew an empty square and told her that was for "something." I wrote the problem "Something" - 4 = 2, and reminded her about the "chicken wings." I also showed her again with the appropriate Cuisenaire rods. She caught on to that quickly. I did one more simple one with her, and then she wanted to write one. She wrote "Something" - 5 = 10. I asked her, "Do you know how to do that one?" She wasn't sure. I took a yellow and an orange rod and said, "here's our 5, and here's our 10." She still wasn't sure how much that would be, so I had her figure out how long they would be together in white rods, and she got it.
October 2003: Odd and Even
I notice in the First Grade Diary that the topic of odd and even is suddenly introduced at about this point in the school year, although there's no comment there about how it was first presented (there are some suggestions in the Annotations, though), and odd-and-even worksheets aren't included until the Red Book.
I asked Ponytails if she knew what odd and even numbers were, and she said, "One is odd. Two is even. Three is odd. Four is even, and like that." I asked her if she knew how you know if something is an odd number, and she gave me a pretty hilarious explanation that I can't even try to reproduce here. (Some of our adult explanations must sound just like that to her.)
So obviously she does have some idea that some numbers are odd or even, but doesn't really understand what that means.....One explanation given in the Miquon materials was that you can build an even number with red rods, but an odd number you can't. I also saw an arrangement of cubes (similar to an activity in Family Math) where you take one cube, two cubes, three cubes etc. and show how you can arrange the even ones in pairs but not the odd ones.
I've noticed that some topics are covered VERY briefly in the Miquon worksheets, for instance, skip counting. In the Red Book, page F-15 is a dot to dot based on counting by 3's; F-18 is counting by 4's; F-21 is counting by 5's and 6's. Obviously a one-page puzzle is not going to be enough for a child who doesn't yet know how to count by 3's or whatever. Another topic, similarly, is time; there are a few sheets in the back of the orange book, but if the child doesn't really get telling time to the hour or half hour the first time around, there's no point in going on to the quarter hours right away.
However, this isn't a complaint! I'm just pointing out why I think it doesn't work as well just to go through the books in a rigid sequence. The curriculum obviously expects that the children are going to be learning skip counting, for instance; and if you look in the Annotations and First Grade Diary, there are extra suggestions for activities for the different topics. If I see pages on a topic I think Ponytails has already mastered (like the simple counting dot-to-dots at the beginning of the Orange Book), I use those sheets to see for sure that she knows what she's doing and has no difficulty.
But I guess what I want to say--to make it short--is that as we move into the Red Book, I'm using the topics in the book as a guide to what we should be covering--more addition/subtraction/multiplication, work in fractions, skip counting, doubles, finding tens, inequalities, and eventually division concepts. But I know there's no way she's going to learn those things, like counting by threes, by doing one worksheet. The sheets are there for her to practice on or to nail down something I think she's almost there on; but the bulk of our time will be spent on finding out how numbers work, and the ways we can move them around and do things with them.
February 2004: A Measurement Story
We're going to spend the next two weeks doing the T pages of the orange book, which are about measurement. When I looked at the Labsheet Annotations for this section, there were some classroom experiences described that sounded a bit difficult to replicate with just one student, so I took the description and rewrote it as a story for my daughter. (She always likes to hear what "the Miquon teacher" did with her class according to the First Grade Diary.)
We are doing some of the C (adding) pages in the Red book, and the focus is grouping for tens. I've noticed that it helps if we start off with a no-book rod activity and then move on to the worksheet. That way too, if Ponytails isn't following the concrete activity, I know it's not time yet to move on to the written part.
Here are a couple of quick ones we've done or are doing with those pages. For C-12, which is 10 + 2, 10 + 4, and so on, I made two piles of rods; one pile was just orange rods, the other was all the other ones. We took turns pulling just one rod out of the mixed pile, but as many oranges in the other hand as we wanted, and "guessing" what number the other person had made. (Guessing is the wrong word, figuring out is probably better.) So I took two orange rods and a red one, and my dd said "10, 20, 22. You have 22." She took four orange rods and a blue one, and I said "49." I tried to make mostly numbers in the teens and twenties to make sure she understood those, but she kept taking as many orange rods as she could to try to make it really hard. I think we got up to 129.
Anyway, after that, the worksheet was no problem.
We're going to do C-15 today, which brings in the concept of grouping for 10s. I think I'll take a group of four or five rods (similar to the groupings on the page) and show her how to add them quickly by building pairs that make ten. I think this follows very well from the C-12 activity--instead of pulling an orange rod, you're pulling two rods that make ten.
Cat and Mouse Game (comments)
Last Friday I planned to have Ponytails do the Red Book, page C-25, the dice game where the cat tries to beat the mouse to the mouse hole. The way it turned out, The Apprentice (seventh grader) had some spare time that morning so I asked her to play the game with her sister instead. I stuck around to watch.
What I quickly realized is that for any child who regularly plays board games, this game is so simple that it quickly becomes boring. (The corresponding subtraction game, D-9, has the same problem.) After one quick round, I suggested that maybe they could spice up the game a bit by designating one or two of the numbers as "go back that many spaces." They decided to make three of the numbers "go forward" and three "go back."
The inevitable happened: with three forward and three back, they ended up going back as much as they went forward! I didn't want to interfere, so I let them play until they'd both been sent back to start several times; then I suggested again that maybe just one of the numbers should be "go back." They decided on 6, and played that way until one of them won.
Just wanted to share about this--that pages like this that may seem too simple can still be made interesting if you add an extra twist!
We are doing 20 minutes of math a day but it's not all from the books; we spend about half the time doing activities with the rods or other manipulatives, practicing skip counting and counting backwards, measuring things and so on. We got through the Orange Book, most of it except for some bits like clock arithmetic, in the first half of the school year, and are now mostly doing the Red Book. At this rate, she will be done it by June, no problem.
Today Ponytails begged me to teach her "the dividing thing, you know, there's a line with a dot on each side." So I did. :-)
Ponytails is almost through the Red book, but we didn't do every page in the books, and we didn't do them in order. What we did do, especially with the Orange book, was use the First Grade Diary as a model. Not exactly every day as described (that would be impossible), but as kind of a plan for what typical math lessons would look like--often incorporating two or three seemingly unrelated activities, and having only one of those (at the most) be a worksheet. We did pages from the Red book (such as the Odd/Even section) before finishing the Orange book. We've worked on concepts like telling time and geometry all through the year, not just for the few pages on that that are included in the Orange book.
I've seen a strong thread of "tens awareness" building through the first two books, even though "place value" as a topic doesn't come up until a later book. If you keep that in mind, Miquon seems to make more sense. You do many activities based around the orange rod--finding combinations of rods that make ten (this covers both adding and subtracting), and learning to group for tens when given a string of numbers to add.
There's also a lot around the concept of equations. You start with very simple equations, like 3 + 3 = 6; later you get into equations where you have to figure something out on each side, like 3 + 3 = 2 + something. (There's a Miquon game called "How much wood" or "Lumberyard" which introduces this concept.) Then inequalities, and greater-and-less-than.
So what I'm saying is that if you do a lot of hands-on activities--using the blackboard, a hundred chart, a number line, the rods, small objects for oral word problems--involving tens and equations, you'll help your child get in touch with some of these basic concepts. Then what you'll find is that the worksheets are kind of like the follow-up activities to the concepts that he's already learning, and you'll find it easier to pick which sheets he should be doing next.
May 2004: Numberlines
The Lab Sheet Annotations is not the best laid-out teacher’s manual....I’ve added “improvements” to my copy, including colour-coded notes at the top of the pages to show which of them relate to which workbook. It does leave a lot up to the teacher to decide. I originally had the impression that Miquon students visited the math lab and were just turned loose with balances and blocks. I think that was partly true (according to the manuals), but there was also a sense of progression and growth in understanding that could only come through a certain amount of planning and guidance by the teacher.
I responded to questions by a list member about Section N.
Thank you for your questions about this, because it led to a really interesting afternoon reading carefully through that section, and I got some good ideas for number line activities. You mention a problem with Louis’ answer in the Grasshopper game on page 227, and you’re right–it’s a typo where he says the grasshopper goes back 3 jumps and it should say units instead. This game description is almost identical to one in the First Grade Diary, but a few of the numbers were changed, and maybe the word got changed accidentally in the process. (There’s also a typo at the bottom of page 229, top of 230; the section heading and a whole paragraph are repeated.)
About the Number Line topic in general...I hated number line exercises in grade school, and I’ve read that some math educators don’t believe that children understand them. Miquon takes the opposite approach, making them concrete and integral to the program; work on the number lines relates very closely to the work in adding and subtracting, and later, multiplication and fractions. It’s just too bad that they’re hidden in this little section and that the N pages are the ones you’re probably going to skip–what are you supposed to do with a bunch of blank number lines?
The number lines and other activities in this section are interesting because they don’t depend on the usual plus-minus operation signs. For that reason, I think they could be useful and non-threatening for kids who get scared by minus signs. The activities described in the Annotations go from very simple (the basic description of a number line) to quite complex; they should be done throughout the year and the program (there are “N” sheets used in three different workbooks), getting more complex as you go on.
Think of all the different kinds of number lines there are to use. “Real” number lines (on paper or chalkboard) can vary by beginning at points other than 0 or 1, can be marked with fractional units, can be marked in multiples of 2, 3, 5 etc. But there are other number lines that children may find easier to understand: thermometers, rulers, measuring tapes, board games (the kind where you roll the dice and move along a numbered track–there are a couple of simple ones in the workbooks), even life-size number lines drawn with chalk on the driveway. A hundred chart (we have a poster-size one on the wall) is great for a lot of number-line-type activities (Ruth Beechick’s booklet on math gives ideas for using one). We’ve also used concrete things like a tower of blocks to illustrate concepts of increasing and decreasing–going up and down our “skyscraper” in an imaginary elevator, stopping at different floors and sometimes pretending that the elevator goes to several levels of basement (negative numbers).
My take on this topic? I think it’s like many of the other sections, like addition: you start simple, then add little twists in (if the children don’t ask about them themselves), and keep adding on to the possibilities. But slowly, a little at a time, as they’re ready. Talking about functions in primary-level math seems way beyond what most of us expect (we’d like to stick safely with our basic adding and subtracting!). But the interesting part of it...for me...is that playing with numbers in this way is going to incorporate a great amount of adding, subtracting and more, and in a painless way. I also like its practicality as children learn to measure with rulers and apply what they’re learning in one area to the other. And I like its possibilities for bigger questions: where do you end up when you are on the third floor and the elevator goes down five floors? What happens if you want to put two jumping rules together? If we had a number line that went on out the door, where would one million be?
May 2004 again
A list member wrote, "Regarding the Functions section, I'm not even sure I would know how to approach or explain this. Any suggestion would be so helpful."
Sheets N2-N4 can be used any way you want; there are suggestions on page 231; you might also look at the problems on page 225.
I like the game presentation of functions here. This is one place where I think the Annotations does do a good job taking you through a series of exercises about number lines and functions, especially at the Green and Yellow levels. It's suggested that you stick to the terminology of the number line games, using words such as: "try my rule," "make up a rule," "2 goes to 4," "start," "beginning number," "land," "landing number," "undoing rules" (inverses), "jail rules," "standstill points," "anywhere goes to anywhere plus four" or "box goes to box plus four." Functions, as far as this book goes, are just more and more complex rules for number line games.
At the Orange level, number lines stay simple, for instance with the basic Gus and Happy game. At the higher levels, the rules of the game become more involved; students experiment with combining more than one rule and with a simple form of graphing the functions (something we spent a lot of time on in high school Functions and Relations).
At one point it's suggested that you try using a number line labelled with letters of the alphabet instead of numbers. You could play that game out loud, too, without written letters: if the rule of the game is to tell me the letter that is my letter plus two more, what is the letter if my letter is B? (That's how substitution ciphers work...also known as secret codes.) This even makes me think of the notes on a piano: if I play C and we make up a rule that says play the note that is two notes lower, you play A. Of course this would only work for adding/subtracting functions, not multiplying/dividing ones.
I wouldn't even worry about the more complex "rules of the game" until your student is very comfortable doing the more basic number line activities. If you notice, there are relatively few worksheets in this section for a topic that can become quite complex; my feeling is (based on the notes) that they originally did a lot of chalkboard work with it. To be honest, I can't even remember going through these upper-level sheets with my oldest daughter, but I know she "did" the Green book. Shows you how much we missed the first time through...
We did K-7 this morning, which is one of two pages dealing with factors of 24 (before the topic of factors is gone into as such). I decided to keep the book closed this time until we had done some concrete work with the rods.
I had Ponytails build a train of 24 (two oranges and a purple) and then asked her to find all the one-colour trains she could of that length. I had intended to wait until she had found them all before doing any recording, but as soon as she lined up six purples, she said "Six fours! Can we write that down?" So we went on like that until she had found three pairs of multiplication facts equalling 24. (After she found twelve twos, I asked her to imagine she had some "twelve rods" and asked how many of those would fit into 24.)
At this point, doing the multiplication part of K-7 would be optional since you have really just done it anyway, but I had Ponytails fill in the blanks anyway. There are division facts (the inverses of the multiplication facts) going down the right hand side of the page, and those are something we're still just getting into, but she didn't have a lot of trouble with them if I phrased it as "how many 4's in 24?"
Oh--and something funny only Miquon users would appreciate. The other day we were in the car and pulled up to a red light beside a sport vehicle. Ponytails pointed: "Look, Mom! Four fours!" I looked, and she was right: on the side, the truck said 4x4.
November 2004: Blue Book
Ponytails is in her second year now with Miquon and....I find we're using the rods a fair amount, especially for two-digit addition and subtraction,and also a hundred chart. (We made a big one--big is good!-- that has cardboard number disks attached with sticky Velcro.) We use the hundred chart almost every day for oral adding and subtracting problems, and she's getting very good at moving the right number of spaces DOWN for tens and ACROSS for units. (This is great place value stuff.) (Oh, a note about doing hundreds with the rods: we just cut some hundred-sized squares of orange paper.)
We've been playing around with assigning different values to the rods than the usual white = one . This was helpful, for instance, on the "diagnostic page" E-46 which had questions like 60 + 40 + 70 + 30 . I showed Ponytails that if each white rod = 10, she could use the rods easily to add those numbers together (in addition to grouping for tens).
We did E-44 (arranging three given numbers into number sentences) using rods and "symbol cards" (half-pieces of index cards marked with signs for plus, minus, equals etc.). (Haven't done E-45 yet where they are to choose their own three numbers.) After she arranged the rods and cards to make true number sentences, she copied the sentences onto the worksheet.
For E-48 and E-49 I wrote all the questions (10+19, 23-13 etc.) on half-index cards and did them orally using the hundred chart (she did not write the answers). I wrote the addition questions on one side in one colour, and the subtraction ones on the reverse in another colour). This way (since it wasn't done on a worksheet) we can use a few of the same questions as review on another day. Yesterday I picked out three of the addition questions, had her do them with the hundred chart, and then asked her to do the reverse sides as well.
We've also been spending some time on doubles, using pages F-23 and F-25. We started by working down the "1" column on F-25, doing doubles up to 256 (my dd figured that one out with rods). Because she seemed really interested in this, I re-read her a story we have in a Childcraft math book about why there is no mathematical possibility of vampires (because if you double the number of vampires every week, everybody in the world would be a vampire in less than a year :-)); it's just a variation of the story about the servant who asks for one grain of rice, then two, then four etc. until he owns the whole kingdom. I also showed her the doubling function on a calculator (2 x = = = = etc.)
All this is to say that if you have children who are active/easily distracted and who are "global"...I think the word is global...enough to enjoy challenging questions and big numbers even though they may sometimes get stuck on easy ones...I think Miquon continues to be a good choice even past the first grade stage, if you can keep the activities concrete and work within their attention span.
April 2005: Number Lines
We are in the middle of the green book, having just completed the section on multi-digit addition and now working on number line games.
We also found a game today in Family Math that is good for place value and addition. If you have the book, it's called Dollar Digit and it's on page 112. You need a pile of dimes and a pile of pennies; one die; and two sheets of paper each marked in two columns (one for dimes, one for pennies). You number each paper 1-7 down the left hand side. For each turn, someone rolls the die but all the players use that same number. You can either take that number of pennies or that number of dimes (but not mixed), and place them beside the number 1 on your sheet. Next turn, someone else rolls but all players again use the number from that roll. Any time you get 10 pennies, you have to trade them for a dime. After 7 rolls, you add up what you have, and the closest one to a dollar wins. (You can make a rule that you can't go over, if you want.)
Ponytails is transitioning into the Yellow book, although we haven't finished all the end-of-the-book topics in the Green yet.
Anyway, although Miquon's transition into subtraction is slow, I am really impressed with the reasoning behind it. I told Ponytails that this kind of math--that is, learning to make life easier for yourself by changing questions like 48-13 into 50-15) is "Smart Math." Using-your-brain math, rather than having the teacher tell you what to do and you just doing it. I think she pretty much understood that.
May 2005: Functions
A list member asked about Page N-7: "Why is it written 'box arrow box' then the rule to add or subtract a number. Why not just one box?"
What you're working with here are functions. That's why they're set up as they are...the book is giving you a kind of junior version of math that we did in high school.
The Annotations is really helpful with this section. If you present it as they do, it will probably make more sense. Imagine a series of number line games, where each person involved makes up a different rule for how you're allowed to move. (Kind of like how the different chess pieces all have their own rules for moving. Or each person having their own dance step.) Billy's rule says that you start on "box", and you end up on "box plus three." Whatever "box" is, you add three to "box" to make your move. (In this case, you're jumping by threes then.) You can also say "anywhere" or "starting point" instead of "box" if that makes more sense.
So if you start with 4 and follow "Billy's rule," you end up at ? (7). And so on.
Then "you try Billy's rule." Same thing. Each time you're just adding three.
Then make up some examples of your own.
If you follow the Annotation suggestions, you'll see some variations that you can use while you're playing around with number lines. For instance, what if somebody has a rule that "box" goes to "box plus three", and someone else's rule says "box" goes to "box times two?" If they have a race, who gets further along, or is there a point where one overtakes the other?
Or you could even play "guess my rule." Show a series of moves and see if your child can guess what rule you were following.
May 2005 again
A list member mentioned that they always skipped the Factor House Game in the O pages.
LOL, and I thought we were the only ones . We just did those pages, and I couldn't see the point in cutting out the little houses (other than the scissors practice).
But the sections after that have been great--square numbers and area are so much fun with rods! We built pyramids out of rod squares as suggested in the Annotations, but we built them off-center (rather than each new layer right in the middle, we lined up the edges on one corner of the bottom layer, if that makes sense), so that you could see the different colours of rods and how many extra were added. The next day, I printed out some cm graph paper and we did the same thing only with paper and crayons. We made up a story that a famly started out with a small house (a red, 2 cm square) and then needed to enlarge their house, so they added a red rod on each side and then filled in the gap with a white rod, making it as large as a green, 3 cm square. Then they had more children, so they needed to enlarge again, and so on. We went through several squares this way, writing out the formula each time. (i.e. 3 squared = 2 squared + 2 + 2 + 1; 4 squared = 3 squared + 3 + 3 + 1.) There's no way all this would have made any sense to Ponytails (and to me) without the rods.
A list member asked for help with worksheet H-47.
There are different ways you can illustrate these problem besides with the number line. You could turn each question into a word problem, like this: "If you have two thirds of a pizza, and I have two thirds of a pizza, how many thirds of a pizza do we have? Is there another way to say that? Or, is that more or less than a whole pizza? How much more? It's okay to say we have four thirds of a pizza, and it's also okay to say we have one and a third pizzas."
When you look at each question, try saying it like this "Two two-thirds." "Four two-thirds." "How many two-thirds are there in six-thirds? What IS six-thirds, let's figure that one out first. (Use whatever manipulatives you want.) Oh, six-thirds is the same as two whole things. So how many two-thirds are there in two whole things? (Use manipulatives to figure it out, if you need to.) Do you want to think it through with pizzas again? Well, if you have two-thirds of a pizza, and I have two-thirds, how much do we have left? Another two-thirds. So there are three two-thirds pieces in two whole pizzas."
The last one: "Some number of thirds is the same as two wholes. How many thirds can we fit into two wholes?" (Use manipulatives.)
The manipulatives could include Cuisenaire rods, a paper "Hershey bar" (divided into sections), paper or plastic "pizzas", or plastic fraction pieces.
It's a good idea to vary the shapes if you're doing this kind of exercise on more than one day; you can get into a rut of always thinking about fractions in terms of round things like cakes and pizzas, so it's good to remind kids that fractions come in other shapes like squares and rectangles too. And even, if they're ready to think about this, in groups of things rather than in pieces of a whole. For that, you can use cereal or beans or cubes (white rods work fine) or whatever small objects you want to divide into groups...these kind of problems are definitely more of a mental challenge, but you can turn them into word problems as well. Stories about classes of children, or teams, or a bag of treats that we have to divide up, all work well.
The rods are good fraction manipulatives in themselves, and that's one thing you miss out on if your kids think of them only as representing the numbers one to ten. It can actually be a lot of fun playing around with these ideas...my third grader liked this part of math. For instance, you take a brown rod. "This is two." (You may get protests.) "Well, today it is a two. Maybe it's a two-dollar bill (or coin if you're Canadian). Okay, let's make it be money. If this is a two-dollar bill, show me a one-dollar bill." (Child shows you a purple.) "Show me half a dollar. Show me a quarter dollar." (Short sidetrack into why quarters are called quarters.) "Okay, let's try something else. (Show a light green rod.) This is one. Show me two. Show me three."
The fun, for Ponytails anyway, was when I asked her to do some for me. I think she has caught on to the idea that what we're doing is about relationships. Rods are about relationships. Fractions are about relationships. Fractions are also about dividing and multiplying.
March 30, 2006
The power of the rods is that they can represent so many different concepts. Any one of the rods can represent "one"; then all the others can be identified in relation to that rod. The most usual set of names we give them is to say that the white rod is one, and then red is two and so on; but that is just the beginning of what you can do with them. If your child works with them from the beginning and understands their relationships, then they should have no trouble in using them for more difficult concepts like fractions. (If the orange rod is two, then the yellow rod is one.)
I'm reading a book called The Mathematical Mystery Tour, written for adults--kind of a fictionalized history of mathematics. It got mixed reviews on Amazon as far as its scope and attention to details goes (and whether it was boring or not). The point, though, is that in the first chapter the characters are talking about Pythagoras and early Greek mathematics, which was based only on integers. The mathematician in the chapter points out that the ancient Greeks substituted their geometrical knowledge for the algebra they lacked. In other words, they could draw a diagram of a problem (in the dirt, usually) and figure out how to express the answer as a ratio (since they had only integers to work with). The diagrams in the book look exactly like something you might build with Cuisenaire rods!
When I read that chapter, I realized again what a powerful set of ideas we are giving our children with these manipulatives. Primary-aged children, like the Greeks, do not have any knowledge of algebra; we do not ask them yet to "solve for x." But my third grader is able to *see* the answers, using rods, to problems that she would otherwise not be able to solve.
April 2006: Cookin' with Math
May 2006: Math Stuff
January 2007: Trains Game
A list member asked about number names for rods.
For that reason, I might even suggest that you don't worry about whether the "2" rod is 2 at this point, but stick to the colours and the relationships between the colours. Two reds make a purple, two purples make a brown. How many ways can you make a train as long as an orange rod? You can play trading sorts of games (as if the rods were money)--what will you trade me for a yellow rod? You could even develop that into a sort of playing store--for instance, if a doll is priced at a yellow rod, she could pay for it with five white rods, two reds and a white, etc.
There's a game with variations that's given in the primary-level Idea Book for Cuisenaire Rods, and we've played it many times to help build awareness of the colours without getting hung up on the numbers.
The basic game is that you dump about 40-50 rods on the table or the floor, and then take turns trying to find a rod that the other person can't "match." You hold up one of the rods (any rod except a white one), and the other person has to find two rods that make a train as long as your rod. (Not one, or three--just two.) If they can, then you set those three rods aside and the other person gets a turn to hold up a rod and stump you. You continue the game until somebody holds up a rod that can't be matched with the rods that are still on the table; then that person wins the game.
For instance, you hold up an orange rod; the other person can then match it with two yellows, or a purple and a dark green, or a light green and a black, or whatever. But if it's near the end of the game and you hold up a light green (3 cm) rod, and all the red (2 cm) rods have been used up, then there's no way they can match that, and you win.
April 2007: Yay for Cuisenaire Rods
June 2007: The Second First Year of Miquon Math
August 2007: Math with Lore
September 2007: More Math Lore
December 2007: On Using Balances
Lore Rasmussen did use balance scales to demonstrate equations, and somewhere in the teachers' books, can't remember where now, there are suggestions for simple homemade balances that work more-or-less well enough to get the concept across. (For example, margarine tubs suspended from a coat hanger suspended from a broomstick.)
When our oldest was small, we made a balance with yogurt cups, so now whenever we come across an equation that needs to be simplified on both sides, we say "this one is a yogurt-cup problem."
May 2008: Names of Blocks
Another question about memorizing the colours and number names of Cuisenaire rods.
Yes and no--some of the fascination of the rods is in the fact that they can represent anything. If the white is 1,000, the orange is 10,000; if the orange is 1000, the white is 100; if the orange is 100, the white is 10 and so on.
It is convenient for them to learn, though, that for most of the basic arithmetic they'll be doing, the white does represent 1 and the orange 10. One way I taught the relationships between them was to have the child measure all the rods with a white rod. How many white rods fit into a blue, an orange...Also, the stair-patterns and pyramids children often make with the rods are good for showing those relationships--they will remember that first you take a white, then a red, then a light green.
One other way to help them remember might be to connect particular rods with the ages of children in the family. Make up imaginary names for the rods, if people-names don't confuse them in addition to colour-names and number-names. The light green rod is John, he's three; the yellow rod is Joe, he's five; and you're the dark green, six. Jackie is the orange rod, ten--she's twice as big as Joe. Stand them up as if they were little people and have some fun with them. (If John stands on Joe's shoulders, are they as big as Jackie?)
July 2008: Crayons' Grade Two: Math (planning process and detailed Blue Book notes)