Fractions

You need to do something to make the darn things meaningful to her.Right now you are giving her three hieroglyphic symbols which might as well be &*^ for all the information they carry to her. Then you tell her to look up the symbol ^ on a big chart full of symbols, find, not the symbol & which isn’t in that row, but the symbol # which is the nearest before it, then write down the # and subtract (whatever that means to her) # from & and write the result as %*^. She has no idea what she is doing or why, and the fact that she hasn’t started throwing things through the windows shows what a decent and patient person she really is.I posted on this board a few weeks ago some general suggestions on fractions and how to teach them concretely. I’ll try to find it and re-post it for you. At least start with number lines, the fact that 1/4 means divide the space between 0 and 1 into 4 equal parts, then take 1 of them; 3/4 same division, but take 3 of them; etc. Also do other denominators until the concept of divide the unit by the bottom and take top number of parts is settled (draw a lot, or cut a lot of paper.)Then 9/4 actually means divide up the unit spaces into 4 equal parts, and take 9 of them. Obviously if 4 parts is one unit, you’ll need to go above 1- divide up a couple more units. Now, 4 parts = one unit, 8 parts = 2 units, so 9 parts = two whole units and one fourth part left over, or 2 1/4. Yes, it takes longer to actually teach the meaning than to say divide here and subtract here and shove this number over there, but it goes a lot further in both making sense and in being retained.I’ll try to re-post the detailed note later.
: I am teaching a young lady in a Special Ed environment (learning
: disability).: She struggles with math. She does not yet know her times tables, and
: I shall continue trying to help her to learn them. (She seems to
: have a memory problem, in other words, her memory is part of her
: disability). However, I feel the best thing is to move on to
: fractions, for it seems that she is capable of the concepts
: involved. She is 15 and a half.: She does this, if we are doing changing an improper fraction to a
: mixed numeral: I let her look at the times table chart. Say we are
: changing 9/4 (nine fourths) to a mixed numeral. She has difficulty
: grasping that 4 goes into 9 two times, in other words, picking up
: quickly that 9 divided by 4 is 2 with a remainder. I thought it
: might help her to build up a foundation by coming down a notch to
: improper fractions that divide evenly: 8/4, 9/3, 10/5, etcetera
: and then go on to the fractions that yield a mixed numeral: 10/3,
: 11/5, etcetera. She understands quite well the end of the process,
: that is, after doing the division, the result is the whole number
: (9/4= 2 remainder 1; 2 is the whole number in the mixed numeral,
: and the remainder becomes the numerator, keeping the same
: denominator): I do not know if you get many questions from Special Educators…we
: have to give these children a fighting chance in this world. The
: media does not talk much about them but there are millions of them
: out there in the schools and streets.: Thanks very much!: John

Re: Fractions Posted By: victoria Date: Monday, 29 January 2001, at 4:07 a.m.In Response To: Fractions (Ida Krill): I’m doing my student teaching in an LD middle school classroom. I
: need to start a fractions unit with very low level student’s any
: ideas?Yes, be concrete, be real, be applied.But please — Beware the food approach: Allergies, sometimes deadly Diabetes, possibly deadly Religious deitary rules Cultural dietary rules Diets to reduce hyperactivity VegetarianismThere will be at least one kid, and usually two or three, in every class, for whom the food “treat” is big trouble and stress and possible illness and a situation of conflict between parental rules and teacher rules. Just not worth it. Yes, this applies to every food. I and my daughter cannot have chocolate or M&M’s or ice cream or cream-filled cake (four big teacher favourites) due to serious milk allergies. I can’t have pizza either (celiac disease). Peanut candies, and even food with peanut oil including maybe that pizza, could kill my landlord’s kid in five minutes. And my diabetic five-year-old student cannot anything at all not on her dietary balance plan (and she’s too young to figure out exchanges)and the sugary “treats” will send her to the hospital in a coma. And my vegetarian student’s family won’t thank you for any of these junk foods.Also, once you get the class to think that math will be rewarded with treats, be prepared to bribe them with treats forever after.OK, back to fractions. Start with dividing something into equal parts. Make sure they understand the parts have to be (as close as possible) exactly the same size. Pies are OK but get really hard to divide equally for numbers other than 2, 4, and 8. Better to use a bar 12 or 24 inches long (make bars out of construction paper)(or if you can get Cuisenaire rods, use all the parts that add up to 12, 12 ones, 6 twos, etc.) or for students on their paper, make dittos with bars 12 centimeters long. 12 and 24 divide easily. Show that 1/2 means 1 part out of 2 (equal), 1/3 means one part out of 3 (equal) etc. Also show the fractions on a standard measuring cup. Draw diagrams with a bunch of bars the same length and colour in 1/2 of first, 1/3 of second, etc. Compare and see that 1/3 is less than 1/2. Note the pattern that bigger number on the bottom means smaller unit fraction. Spend a couple of days on just this. Then introduce different numerators. Explain that we know the bottom number means how many equal parts in one bar; the top means how many of those parts to take. Draw diagrams of 1/4, 2/4, 3/4, 4/4. Note that 4/4 means 4 out of 4 so 4/4 must equal one whole bar. Do the same with other denominators. Again, a few days just on the meaning concept. Practice measuring fractions of a cup, fractions of a yard, fractions of an inch, just as we do in real life in cooking, sewing, and carpentry. Do fractions of a group: If there are 6 girls in the class and two of them hagve red sweaters, what fraction of the group is wearing red sweaters? (No simplifying yet) Draw dots and colour them in to represent the group anmd parts of the group, and draw loops around equal sub-groups. Introduce mixed numbers and measure them — how much is 2 1/2 cup? 1 2/3 yards? 4 3/8 inches? Draw and diagram and measure everything. Once we have a good idea what fractions are, introduce equivalent fractions meaning EXACTLY THE SAME SIZE. We measure and count up. We saw that 4/4 has to be 1. Now we see that 2/4 is the same as (in the sense of exactly the same size as) 1/2, 3/12 as 1/4, and so on.We can show this by drawing circles around equal-sized groups; we have 12 sections of 1/12 each in one unit; if we take a different colour and draw a box around every set of 3 of these, we get four equal groups, and 3e/12 colours exactly one of these four. NOTE THAT I HAVE NOT MENTIONED A SINGLE COMPUTATION SO FAR. Computation is the enemy of thinking in this area. Give a student a simple, short recipe, tell him to hurry up or he will be punished for not finishing the page, and he will use it in panic forever after whether or not the results are in any way meaningful or sensible. After the students are well aware of what fractions are and how to use them to measure and how two different ways of counting (lots of little pieces versus fewer bigger pieces) can get to the same measurement, *then* you can start introducing some computational rules. The system of multiply or divide top and bottom by the same thing can be made sensible by drawing pictures and splitting up parts or re-grouping, as above. By the way, if you want students to actually learn math, avoid computational shortcuts until the students have earned the right to use them by showing they understand what is going on. The usual system of simplifying by crossing out is a magical mystery trip to most kids, and this is the point where many develop a math phobia — they are told to do things that defy logic and common sense, so of course they resist. You’ll get far better results in the long run if you first have diagrams to show that 2/6 = 1/3, then show computationally that 2/6 divide by 2 BOTH top and bottom (2 divide by 2 over 6 divide by 2) comes out to 1/3. Addition/subtraction is best learned on some form of number line (large ruler, bar model). If you start with same denominator, such as 3/4 - 1/4, and point out that “fourths” is the name of the pieces, then you are just taking one piece away from three pieces, all the same size, so you have two pieces left. (Easy to illustrate this) When you get to different denominators, if you have been diagramming/modelling all along, it is easy to see that thirds and halves are different spieces, and to add them is to add apples and oranges. So you work out an “exchange” of both of them for an equal amount in sixths. Use similar “exchanges” to do improper fractions and mixed numbers; 2 = 8/4 (draw it and look at it; one whole bar is 4 parts so two make 8 parts) so 2 + 3/4 is 8/4 + 3/4 or 11/4. And the same in reverse, 11/4 = 8/4 + 3/4 = 2 + 3/4 = 2 3/4. Be warned: if you introduce the recipe computation of multiply this numner here by that number there and add on this one over here and shove it all on top of this other one down here, knowthat you will speed up your students’ computations for two or three days and then stall them out in confusion for a long time after that. To do this all properly will take months, don’t forget. Be patient with the students and yourself. For multiplication, if you get that far this year, use an area model; a mile by a mile makes a square mile. Cut in two equal parts one way for 1/2 and three equal parts on the other side for 1/3 or 2/3. Then it’s easy tio see that 1/2 of 1/3 is 1/6, and so on. For division, use the question “How many ___s in ___?” How many 1/3 cups in 1 cup? 3. So 1 divide by 1/3 = 3. How many 1/3 cups in 2 cups? 6, because 2 = 6/3. So 2 divide by 1/3 = 6. You can lead up to the “multiply by the reciprocal of the divisor” rule this way, but don’t rush it.I have an excellent old textbook that covers much of this, and will mail you photocopies if you are seriously interested. It costs money and time, so only if you really intend to use it, please; but if you do, just email and ask.Messages in This ThreadFractions Ida Krill — Saturday, 27 January 2001, at 7:41 p.m. Re: Fractions victoria — Monday, 29 January 2001, at 4:07 a.m. Re: Fractions Sara — Sunday, 28 January 2001, at 8:51 a.m. Re: Fractions coridroz — Sunday, 28 January 2001, at 4:24 a.m.

John, Certainly proceed with teaching fractions and make it as hands on as possible, as suggested. Point out real life situations where fractions apply. However, I just have to say that fractions have been taught to my 19 year old daughter many times, using manipulatives, real life situations, etc. by some really good special ed trained teachers, as well as by us at home. She transiently gets a grasp of them and passes the exam during the time it is taught. However, if fractions appear months later on a placement or general evaluation math exam, invariably she cannot solve fraction problems or apply the concept to situations which arise in every day life. I continue to hope that perhaps when she is 30 and has been taught fractions for the nth time it will finally stick in her memory. Best of luck. Mary

: John, Certainly proceed with teaching fractions and make it as hands
: on as possible, as suggested. Point out real life situations where
: fractions apply. However, I just have to say that fractions have
: been taught to my 19 year old daughter many times, using
: manipulatives, real life situations, etc. by some really good
: special ed trained teachers, as well as by us at home. She
: transiently gets a grasp of them and passes the exam during the
: time it is taught. However, if fractions appear months later on a
: placement or general evaluation math exam, invariably she cannot
: solve fraction problems or apply the concept to situations which
: arise in every day life. I continue to hope that perhaps when she
: is 30 and has been taught fractions for the nth time it will
: finally stick in her memory. Best of luck. MaryJust wondering about your post — two questions —Does your daughter have a good concrete grasp of whole numbers, or is she dealing with them by rote? If the whole numbers are not concrete, the fraction teaching has no foundation to rest on, and that could perhaps explain the slipping out of memory.Does she do any cooking or crafts or sewing or woodworking, where measurement and fractions come into play all the time naturally? These are all good things to do in their own right, also help hand-eye-coordination, and make fractions and measurement immediate and useful.

My daughter does have an excellent grasp of whole numbers. If all of math was done with whole numbers life would be easier. She does cook and has no problem measuring for example 1/2 cup of water. However, if a recipe were for 4 people and she needed to scale up for 6, she would not be able to figure this out without a lot of prompting. She can tell time on an analog clock and knows the meaning of terms such as a quarter of an hour, but cannot do elapsed time calculations unless they are very simple. She has remarkably little difficulty solving algebraic equations as long as they involve whole numbers, but cannot seem to grasp that the process is no different if the equations involve fractions. A simple example is solve for x in the following equations:2x=10 2.5x=10 2 1/2x=10She could solve the first without a calculator, would probably know what to do, but need a calculator to solve the second, and would probably need a lot of prompts to figure out what to do with the third. Although she knows 1/2 is the same as .5, she can never seem to independently think of applying this information when needed.She had a head injury as a child and there are mysterious gaps in what she can do and more importantly what she can retain. Retention and insight into when to apply facts and concepts are the BIG problems. Her teachers always love her hard work and she does well enough for the short term until the test, but what they don’t see (that I do see) is how little is retained. Excessive repetition can sometimes allow knowledge to make that leap from no recollection to storage in long term memory, but its a frustrating and unpredicatable process.

: I am teaching a young lady in a Special Ed environment (learning
: disability).: She struggles with math. She does not yet know her times tables, and
: I shall continue trying to help her to learn them. (She seems to
: have a memory problem, in other words, her memory is part of her
: disability). However, I feel the best thing is to move on to
: fractions, for it seems that she is capable of the concepts
: involved. She is 15 and a half.: She does this, if we are doing changing an improper fraction to a
: mixed numeral: I let her look at the times table chart. Say we are
: changing 9/4 (nine fourths) to a mixed numeral. She has difficulty
: grasping that 4 goes into 9 two times, in other words, picking up
: quickly that 9 divided by 4 is 2 with a remainder. I thought it
: might help her to build up a foundation by coming down a notch to
: improper fractions that divide evenly: 8/4, 9/3, 10/5, etcetera
: and then go on to the fractions that yield a mixed numeral: 10/3,
: 11/5, etcetera. She understands quite well the end of the process,
: that is, after doing the division, the result is the whole number
: (9/4= 2 remainder 1; 2 is the whole number in the mixed numeral,
: and the remainder becomes the numerator, keeping the same
: denominator): I do not know if you get many questions from Special Educators…we
: have to give these children a fighting chance in this world. The
: media does not talk much about them but there are millions of them
: out there in the schools and streets.: Thanks very much!: John