Teaching Students with LD and ADHD

# Math Manipulatives

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Posted Mar 14, 2001 at 12:00:01 AM
Subject: Math Manipulatives

I would like feedback about different math manipulatives? What do people like best? I am thinking about using the math-u-see program, but wondered about manipulatives such as cuisenaire rods, base 10 blocks, unifix, and others. Is it confusing to students to use different manipulatives? Are manipulatives color coded and would this be confusing to students used to one type of manipulative? If they are color coded could anyone tell me what color each number is to be able to compare similarities? Does anyone have a web address for the math-it program? Please let me know your opinions of this program.Thanks!

Posted:Mar 14, 2001 12:00:01 AM

: I would like feedback about different math manipulatives? What do
: people like best? I am thinking about using the math-u-see
: program, but wondered about manipulatives such as cuisenaire rods,
: base 10 blocks, unifix, and others. Is it confusing to students to
: use different manipulatives? Are manipulatives color coded and
: would this be confusing to students used to one type of
: manipulative? If they are color coded could anyone tell me what
: color each number is to be able to compare similarities? Does
: anyone have a web address for the math-it program? Please let me
: know your opinions of this program.: Thanks!Good bunch of questions. Here are a couple of answers, not complete.I used to use a nice abacus, available in many toy storesm, that had ten rows of ten simple wooden beads. It was good for counting, simple addition, why the sums over ten work out as they do (eg 7 + 8 = one row of ten and 5 ones, or 15), base 10 up to 100, carrying in two-digit addition up to 100 (and with a second abacus, we could have gone further), and excellent for borrowing. I recommend this as a nice basic tool.I like base 10 blocks because they are simple and have very few distractors. The ones I have seen are wooden and nice and simple. Also the weight is a nice clue to the comparative size of the numbers. These are excellent for doing extended base ten, grade 2 and up. Especially good for carrying and borrowing, and if you can get a big enough set, good for multiplication and division.I have never used unifix, but the one teacher I did see using it was making a complete hash of things. Whether that was because she was just a confused and disorganized teacher or whether the fault was in unifix I don't know. The kids were just snapping the blocks together and apart at random, doing no math.Cuisenaire rods are colour-coded. The one system I saw in any detail was a 1960's system, and there is probably :-) a newer edition out by now. In the older system, the colour-coding was stressed far too much, and the kids were supposed to learn that cerise + green = orange ( not sure of the colours any more, but that was the gist of it). They wrote cute things on the board that looked like algebra, c + g = o etcetera. In this (older!) system, number values were not even supposed to be mentioned until the kids had learned how to do everything by colour. In practice, the teachers were not using the blocks by the book probably bbecause the plan was so far-out, and tended to fall back on the familiar worksheets while the blocks languished on the shelf.Cuisenaire rods could be a valuable tool if you used them to supplement almost any paper system; I prefer a &quot;transparent&quot; method, where you tell the child exactly what is going on and why, and start by finding counting equivalencies for the rods. I would even write numbers or tape number labels on them.If the child finds changing colour codes to be confusing, that is a sign that he/she has been leaning a little too hard on the colours as a crutch. I wouldn't pressure a change too hard, but I would stress numbers rather than colours, and I would deliberately switch or combine systems to encourage looking at the essentials rather than distractors.Hope you get answers to the rest of these.

Posted:Mar 14, 2001 12:00:01 AM

We have Math-U-See. The color-coding is green for units, blue for tens rods, and red for 100's squares. 2 through 9 are each a different color. At the very beginning the child is encouraged to learn which color is which number, but once they learn the numbers it's not important -- except maybe to help grab the correct piece when laying out a problem.MUS pieces don't snap together. To demonstrate subtraction you're supposed to turn the pieces over so the empty side is up, in which case the smaller pieces do stay on top of the bigger pieces (not snapping together like lego, but they do fit together).At least with MUS, it would be possible to use differently colored pieces than they provide. I think it would be confusing to use multiple-colored rods for the same number -- for example, tens rods in both blue and red. You'd want to keep all the tens one color, all the hundreds one color, etc.We haven't gotten into the fraction overlays yet, so I don't know how they would coordinate with other than MUS rods.Mary: I would like feedback about different math manipulatives? What do
: people like best? I am thinking about using the math-u-see
: program, but wondered about manipulatives such as cuisenaire rods,
: base 10 blocks, unifix, and others. Is it confusing to students to
: use different manipulatives? Are manipulatives color coded and
: would this be confusing to students used to one type of
: manipulative? If they are color coded could anyone tell me what
: color each number is to be able to compare similarities? Does
: anyone have a web address for the math-it program? Please let me
: know your opinions of this program.: Thanks!

Posted:Mar 14, 2001 12:00:01 AM

PASSWORD>aa4um5Lp2CxdUCuisennaire rods and Math-u-see rods are color coded. The cuiesennaire rods are smooth, so until you get used to the colors, you can't really tell the difference between a 6 and a 7 rod without lining it up next to a row of unit cubes. I think the math-u-see rods are marked at 1 cm. intervals, so you could just count the segments on a rod to figure out it's length.Switching back and forth between math-u-see and cuisennaire might be a bit confusing because the colors are different Unifix cubes are simply individual cubes that you can snap together. They come in a pack of 100 cubes, 10 of each color, but they aren't color coded like cuisenaire rods (1= white, 2= red, etc.)I like using unifix cubes with my boys (we're homeschooling) but we aren't doing anything fancy with them- Just 2 +3 =5, 3 x 4= 12 sort of thing. We could do the same thing with M &amp; M's, but there are limits to how much candy I want them to eat during math! I agree with Victoria that a set of base 10 cubes is great for borrowing, carrying, etc. In general, I'd say that different types of manipulatives are best suited for different tasks. You don't need to spend a fortune on different manipulatives, but don't feel you have to stick with the same thing ALL the time.Jean: I would like feedback about different math manipulatives? What do
: people like best? I am thinking about using the math-u-see
: program, but wondered about manipulatives such as cuisenaire rods,
: base 10 blocks, unifix, and others. Is it confusing to students to
: use different manipulatives? Are manipulatives color coded and
: would this be confusing to students used to one type of
: manipulative? If they are color coded could anyone tell me what
: color each number is to be able to compare similarities? Does
: anyone have a web address for the math-it program? Please let me
: know your opinions of this program.: Thanks!

Posted:Mar 14, 2001 12:00:01 AM

PASSWORD>aaI221mi7wL3IVictoria,My 11 year old daughter has been doing hands on equations which teaches LD and gifted kids about algebra using manipulatives. Today her teacher approached me and said, &quot;how do you teach subtracting negative numbers?&quot; The way that hands on algebra does it is different and we understand what they are doing but it is confusing to teach it to my daughter. I told her to stop with the program until I got some advice. I also know my daughter needs to understand more about fractions and other pre-algebra concepts. Hands on Equations has been great for her self esteem and she has just been sailing through until we hit this glitch with negative numbers and subtracting them. Two negatives become a positive but WHY the child asks.... How do you teach the concepts of subtracting negative numbers? I memorized rules but I don't understand the rationale behind the concept enought to explain it to a child.Thanks!

Posted:Mar 14, 2001 12:00:01 AM

: Victoria,: My 11 year old daughter has been doing hands on equations which
: Today her teacher approached me and said, &quot;how do you teach
: subtracting negative numbers?&quot; The way that hands on algebra
: does it is different and we understand what they are doing but it
: is confusing to teach it to my daughter. I told her to stop with
: the program until I got some advice. I also know my daughter needs
: to understand more about fractions and other pre-algebra concepts.
: Hands on Equations has been great for her self esteem and she has
: just been sailing through until we hit this glitch with negative
: numbers and subtracting them. Two negatives become a positive but
: WHY the child asks.... How do you teach the concepts of
: subtracting negative numbers? I memorized rules but I don't
: understand the rationale behind the concept enought to explain it
: to a child.: Thanks!Pattim,I'm a tutor currently working with a 7th grade student on pre-algebra skills like integers and rationals. The way the text explains it with manipulative is like this:Take the problem 5-(-3). You have 5 positive markers +++++ and you want to take away 3 negatives, but you don't have any negative markers. So what you need to do is add what are called neutral pairs, (a positive and a negative marker) which don't change the value of the original problem.+++++ (+-)(+-)(+-)Now you have 3 negative that you can take away which leaves you with 3 more positives than when you started. +++++ (+)(+)(+) or 8.Another way I've seen it explained is to think of a mailman bringing bills and checks to you. If he brings a check for \$5 and takes away a bill for \$3 then you are up by \$8.I&quot;d also like to know if anyone has some better ways of presenting the concepts.Lila

Posted:Mar 14, 2001 12:00:01 AM

Posted:Mar 14, 2001 12:00:01 AM

One more way to present &quot;negative of a negative is a positive&quot; if the abstract concept of &quot;negative means opposite&quot; isn't working, is with the VCR: If you're watching a movie that is already backwards, and you rewind, it goes forward.Another more concrete presentation of the number line is an elevator in a building where the ground floor is zero, but I don't like that because nobody numbers their floors like that (and what about that thirteenth floor?)Sue (who is *hoping* her website returns in the 24-48 hours promised, and looking at the thermometer hovering at the zero mark...)

Posted:Mar 14, 2001 12:00:01 AM

: Today her teacher approached me and said, &quot;how do you teach
: subtracting negative numbers?&quot; The way that hands on algebra
: does it is different and we understand what they are doing but it
: is confusing to teach it to my daughter. I told her to stop with
: the program until I got some advice. I also know my daughter needs
: to understand more about fractions and other pre-algebra concepts.
: Hands on Equations has been great for her self esteem and she has
: just been sailing through until we hit this glitch with negative
: numbers and subtracting them. Two negatives become a positive but
: WHY the child asks.... How do you teach the concepts of
: subtracting negative numbers? I memorized rules but I don't
: understand the rationale behind the concept enought to explain it
: to a child.: Thanks!

Posted:Mar 14, 2001 12:00:01 AM

PASSWORD>aaI221mi7wL3IWhat a bunch of incredibly smart women I can count on and consider friends! Thank you so much!! I will be reading and devouring your counsel..inbetween my neurology and child psyche classes this morning. I don't live in the midwest so I can't use a thermometer outside. Where I live it rarely gets below 30 degrees, I am in southern California about 2 miles from the ocean. But I am just amazed at the answers, the more I know the less I know!!

Posted:Mar 14, 2001 12:00:01 AM

Thanks for all the information! Does anyone know of any web sites that discuss different ways to use manipulatives?

Posted:Mar 14, 2001 12:00:01 AM

EMAILNOTICES>noHey, Victoria! How'd you like to become my math tutor? =)Kathy G.

Posted:Mar 14, 2001 12:00:01 AM

: Hey, Victoria! How'd you like to become my math tutor? =): Kathy G.I'll teach math to anyone if I can hold them down long enough.Seriously, this is my profession; short questions answered for free, longer help available as distance learning on a basis of you pay for photocopies, S&amp;H, and a little extra for my time.

Posted:Mar 14, 2001 12:00:01 AM

: Personally, I like to keep things as simple as possible for my kid
: who can get very confused over wordy or complex explanations. By
: all means, spend lots of time on the concept of what a negative
: number means. Adding negative numbers is relatively simple to
: visualize. Stress the concept that the subtraction operation is
: the same as adding a negative number 10-5 = 10 + (-5): If you think of the subtraction operation as reversing addition then
: you can use that operation like a kind of &quot;opposite
: switch&quot;.: if a problem reads 10 - (-10) look at the opposite problem first 10 +
: (-10): It is easy to see using markers that the solution is 0 since the
: negative and positive 10 cancel each other out. In order to solve
: the problem you had to subtract 10 from 10.: When you change the operation from addition to subtraction, you have
: flipped the &quot;opposite switch&quot;. To get the correct
: answer, you must do the opposite of what you did before so instead
: of subtracting 10 from 10 you must add the two 10s together.: As you get into multiplication and division of negative numbers the
: process gets more and more difficult to visualize and think about
: logically in terms of real life situations. I have seen attempts
: by college proffessors to come up with real life models to show
: why the rules work. It is very amusing and not very helpful. The
: very best explanation is the simplest to my mind.: Our mathematical system is invented. We could have done it another
: way. The only thing critical in a mathematical system is that the
: rules MUST be consistant. They must follow a pattern and maintain
: that pattern throughout the system or the system will not work.
: You can show the patterns by making a chart if you want to.: Subtraction is the opposite of addition. A negative number is the
: opposite of its positive. Therefore, changing the operation *OR*
: the sign of a number causes you to do the opposite thing. If you
: change both, then you have changed nothing because you have
: flipped the opposite switch TWICE.: Once that is understood, memorize the rules and use them.: When you get to complex algebraic equations, you won't be trying to
: visualize what is happening when you solve an equation. The
: understanding is required when writing an equation to solve a real
: world problem, but the actual solving is just &quot;following the
: rules&quot;. That's why algebra was invented. To simplify a
: complex problem down to it's barest elements so you can eliminate
: confusion and use the &quot;rules&quot; to get a final solution.
: Mathemeticians found eliquent shortcuts by manipulating equations.
: They call these &quot;proofs&quot;. These proofs created new rules
: to follow that allowed eqautions to be solved faster and easier.: For a kid who is not going to be a mathematition, keep the
: explanation simple enough for them to follow. They may or may not
: internalize that explanation, but at least you have shown them
: that there is a reason for what they are doing. Then let them use
: the rules. Those who are not initially able to internalize the
: explanation *may* find a deeper understanding later after much
: practice applying those rules. If they don't, at least they can
: solve the problems. That's what the rules were created for in the
: first place.: Personally I LOVE understanding what I am doing thoroughly. In fact,
: it drives me nuts when I don't. I have finally come to the
: conclusion that some folks aren't built to have that kind of
: understanding. My daughter and my husband will never
: &quot;see&quot; math like I do but that doesn't mean they can't
: &quot;do&quot; math. My husband is at least as good (if not
: better) at calculating than I am. He is very good at applying
: those rules he has learned. He gets along in life just fine.: My poor daughter is going to school at a time when we pretend that
: &quot;understanding&quot; is THE most important thing. It doesn't
: matter nearly as much whether or not you get the correct answer
: (try telling that to the IRS!) They are presenting complex
: mathematical concepts before she is ready to understand them,
: glossing over the &quot;doing&quot; and then moving onto the next
: topic with lightning speed. They have spiraled through math topics
: in this way for her entire 7 year school career. This means that
: not only does she not understand the math concepts, but she can't
: &quot;do the math&quot; either. She is the type of learner that
: has to DO before she can UNDERSTAND. I can't seem to get this

Posted:Mar 14, 2001 12:00:01 AM

PASSWORD>aa4um5Lp2CxdUI love your advice about focusing on a few major topics for each year and covering them thoroughly, rather than dabbling with every possible mathematical concept. With my older son, not only has this enabled him to gain a solid understanding of math, but this approach will actually allow us to cover MORE material in the elementary years than he would cover if we just went through a typical curriculum as written. Since the amount of &quot;review&quot; material in the text increases each year, and since he learns it the first time around, we have weeks of time available to focus on other things, like problem solving skills, and at the rate he's going, I think we may end up almost a whole year ahead by the time he gets to 6th grade. That gives us a whole year to cover a lot of &quot;extra&quot; stuff before we get into pre-algebra.Jean.

Posted:Mar 14, 2001 12:00:01 AM

: possible -- work out a simple idea of what a negative number is;

Posted:Mar 14, 2001 12:00:01 AM

My kiddo number one was taught in school using that marker method. She had no trouble with it but it drives me nuts. I do wonder what would have happened if we hadn't done that afternoon of negative numbers the summer before her 7th grade year though. It works, but there are so many simpler ways to represent this concept I cannot figure out why on earth they chose that one. I know I have to teach kiddo number two negative numbers BEFORE she is presented with them in school. Thank goodness kiddo number one came first so I always know what's coming for number two!: Pattim,: I'm a tutor currently working with a 7th grade student on pre-algebra
: skills like integers and rationals. The way the text explains it
: with manipulative is like this: Take the problem 5-(-3). You have
: 5 positive markers +++++ and you want to take away 3 negatives,
: but you don't have any negative markers. So what you need to do is
: add what are called neutral pairs, (a positive and a negative
: marker) which don't change the value of the original problem.: +++++ (+-)(+-)(+-): Now you have 3 negative that you can take away which leaves you with
: 3 more positives than when you started. +++++ (+)(+)(+) or 8.: Another way I've seen it explained is to think of a mailman bringing
: bills and checks to you. If he brings a check for \$5 and takes
: away a bill for \$3 then you are up by \$8.: I&quot;d also like to know if anyone has some better ways of
: presenting the concepts.: Lila

Posted:Mar 14, 2001 12:00:01 AM

I am a middle school 8th grade math teacher for students who Learn Differently (LD) and they too have difficulty with negative and positive numbers. Ironically, I also have a &quot;regular&quot; 8th grade homeroom and those students also have the same problem. I have explained negative numbers and the relationship of positive numbers as the ground being zero. Now, when you dig a hole each shovel full of earth is a negative number and as you put the earth on a pile next to the hole you are creating a positive number. So, when you have dug 3 shovel fulls of dirt (-3) and you dig down 3 more shovel fulls + (-3) you get -6. -3 + -3 = -6. Now lets say you have dug 5 shovel fulls (-5) and you decide to fill in your hole with three shovels of dirt from the pile of dirt next to the hole (+3) how many shovels have you dug? -5 + 3 = -2. You could even go outside with your students and dig with shovels to provide a real &quot;hands on&quot; experience!!! Hope this helps.Petra

Posted:Mar 14, 2001 12:00:01 AM

: I have good stuff on multiplication and division too -- when you're

Posted:Mar 14, 2001 12:00:01 AM

Posted:Mar 14, 2001 12:00:01 AM

That was an excellent explanation. But, I always was told that if you had an even number of negatives, you'd have a positive result; an odd number of negatives, you'd have a negative result. Does that hold up?: OK, two ways that are logical and not excessively complicated:
: Verbal/graphic/financial people: A teacher, Miss June Smith, likes
: to go on educational tours in Europe. Every year she puts aside
: \$200 per month from September to June in a special savings
: account.: Then she goes on a six-week tour in the summer and spends \$300 per
: week.: Three basic ideas: time forward, in the future, is positive; looking
: back to the past is negative. Deposits are positive, withdrawals
: are negative. Increase or upwards is positive, decrease or
: downwards is negative. (All ideas which should be made clear in
: introductory work on signed numbers): Let's sit down with Miss Smith on December 31 as she balances up her
: finances, checks over back records, and makes plans.: (a) In six more months, will she have more money or less money?
: Obvious answer, more money, as she keeps putting it in the
: account. How much more, and how do we calculate it and why? We
: multiply when we have equal groups. 6 months ahead (pos) X \$200
: each month deposit (pos) = \$1200 more in the account (Pos,
: increase) So (+6) x (+200) = (+1200) and pos X pos = pos (Stress
: pos TIMES pos; never say pos &quot;and&quot; pos; &quot;and&quot;
: either means addition or is too vague to be meaningful) We can
: graph this with a bar graph -- write months of the year each one
: square wide, then a bar \$200 for Jan, \$400 for Feb, \$600 for Mar,
: etc. The bars climb up like steps.: (b) She checks back for errors in the past. Did she have more money
: or less money three months ago, at the end of September? Answer is
: clearly less; she hadn't put it all in yet. How much less and how
: calculated? 3 months ago (neg) x \$200 deposits (pos) = LESS by
: \$600 (neg) so (-3) X (+200) = (-600) and neg X pos = neg (That's
: neg TIMES pos): Graphical: re-copy the graph from (a) above, putting January in the
: middle so you have room to work back. Then look what we have to do
: to keep the stair-steps going leftwards. December must be at zero
: (this is where we started counting, end of December, so we started
: at nothing). November must be at -200 (200 less than Dec., 400
: less than Jan, etc.) Oct. must be at -400, Sept. at -600. Draw it
: and look at it -- the patern makes sense.: (c) OK, now we visit Miss Smith on July 31, as she views the ruins of
: Rome. She's halfway through her vacation, and she wants to check
: her finances. She is spending \$300 per week now. In three more
: weeks, will she have more or less? Answer: clearly less, because
: she's spending away and not putting in. How much less, and how to
: calculate? 3 weeks ahead (pos) X \$300 withdrawal (neg) = \$900 less
: (neg) (+3) X (-300) = (-900) so pos X neg = neg ((Please say
: TIMES, not &quot;and&quot;)): Graph as above; start at zero end of July, and make bars getting
: longer and longer *downwards* as the weeks go on.: (d) Now the punchline. Still on vacation, July 31, she looks back at
: her spending so far. Two weeks ago, did she have more or less
: it all yet! How much more, and how do we calculate it? 2 weeks ago
: (neg) X 300 withdrawal (neg) = \$600 **MORE** **pos** (-2) X (-300)
: = (+600) so neg X neg = **pos** (negative TIMES negative): Graphical: re-copy the graph from (c) above with July 31 in the
: middle, and extend it leftwards. Since the bars get longer and
: longer downwards as you go forward in time and you keep stepping
: down each week, they must go upwards as you look back. To the
: right step down; to the left, step up.: Summary: pos X pos = pos (please, please say TIMES) neg X pos = neg
: pos X neg = neg neg X neg = **pos**: ***************************************: For division: sign rules come out to be exactly the same as for
: multiplication. You can do this several ways: (i) invent a story
: like the above if you wish. (ii) Look at division as
: multiplication by reciprocal: EX: 20 divided by (-4) is the same
: as 20 X (-1/4) = -5 so pos divided by neg = neg, etcetera, same as
: multiplication.: (This assumes reciprocal of a negative is negative, but that makes
: sense) (iii) Look at division as &quot;undoing&quot;
: multiplication (a most productive way to do it) EX: (-5) X (-4) =
: +20 so just read backwards (+20) / (-4) = (-5) (using fraction
: slash / for division): pos X pos = pos so pos/pos = pos neg X pos = neg so neg/pos = neg pos
: X neg = neg so neg/neg = pos ** neg X neg = pos so pos/neg = neg: *********************************** So there you have it. Now the
: hard part; get your algebra book and do a couple of pages of mixed
: practice (all four operations) and work on keeping them straight.
: Enjoy!

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