Fractions

PIZZA! I did a lesson with my students making a pretend pizza. We made a pizza from construction paper, using whatever toppings they liked. Then, I had them cut into 2 pieces. (1/2). Then we took the halves and cut them in half (1/4). We saved them for the next lesson on comparing fractions. They had fun and they remembered it.: 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?

: 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?Why not take a page right out of the textbook? Most textbooks teach fractions by talking about a pie or a cake. Instead of talking about math, make it real for these children. Bake a box cake or two. Bring it in. The real thing. Talk about it. How to divide it up among them.Bring in apples. One for every two students.Bring in a candy bar, a really long one.The list would easily go on.Math is too often taught in isolation from reality. The best of math students thinks math just exists on the page of a textbook and you’ll never be able to use it.Show these students that the better math students are wrong. Show them that math is very real and it can solve problems. Like how to take a real cake and divide it into parts called fractions so that everybody can have a share.

: 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.