User:Vtaylor/XO fractions

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On Wed, Jun 8, 2011 at 02:19, Steve Thomas <> wrote: > On Fri, Jun 3, 2011 at 4:23 PM, Edward Cherlin <> wrote: >> >> I have been wishing for this, particularly the Cuisenaire rods and the >> full tutorial on eToys. I'm in. You have the full cooperation of the >> Sugar Labs Replacing Textbooks project, all both of us. ^_^ Plus >> whomever we can recruit now that we have a specific OER project to >> talk about. > > Thanks Randy, Edward and Peter (by way of Edward volunteering you) for > offering to help.

I was actually thinking of Valerie Taylor, but yes, Peter is our third.

> Rather than a complete curriculum/textbook/OER for each grade (which should > be an end goal) I would like to tackle things one bite at a time, with the > first bite being Fractions.

An excellent choice.

> Why, its a subject kids struggle with and > therefore teachers and parents worry/care about. I do not want to re-invent > the wheel here. Lets use good existing ideas and it is possible to teach > first graders to solve problems like: (1/2 x () + (2/5 X 10) -(1/2 x 8) = ?

After a bit, yes. I was thinking about how to teach basic fraction arithmetic first, using pie slices of the kind in the attached Turtle Art image. There are many decent static explanations of fraction arithmetic, but there are never enough examples. Peter does a good job of providing enough examples of arithmetic and proto-Turtle Art operations.

Our first task is to identify all of the separate ideas that go into fraction arithmetic, including the ideas that children need before they start on it, and work out an activity for each. Then we can decide how to program them, and how to structure the entire process. I think that this is about right, but It can no doubt be improved.

Children need to understand counting and the cutting of cakes, pies, fruits and vegetables, and loaves of bread. I use pie in these steps, but we should mix up our examples. We could also use pieces of eight, actual historical money. They also need to be comfortable with some level of arithmetic with non-negative integers.

  • Fractions derive from dividing something into some number of equal pieces. It is also possible to cut something into unequal pieces, where this kind of fraction does not work.
  • Take some number of equal pieces. Less than the number of slices of the pie, or equal or greater.
  • Fraction names (halves, thirds, quarters, etc.) and notation (1/2, 1/3, 2/3, etc.) come after the base idea. They require practice as a language rather than math lesson. Fraction names abbreviate expressions such as, "Cut a pie into four equal pieces, and take three," down to "three quarters", saving a lot of words and time.
  • How many parts make a pie? As many as you cut it into. Two halves, three thirds, four quarters or fourths, etc.
  • What if you cut the pie in one? In words, we have one whole pie.

There is no word "oneth", so we cannot say "one oneth". We have to say "one whole". In notation, this is 1/1, which requires a bit more explaining. Let children practice with 2/2, 3/3, and so on, where they get two whole pies, three whole pies, etc.

  • Divide one pie into quarters. Each piece is 1/4 of the pie, that is, one divided by four.
  • We can have four quarters, three quarters, two quarters or half, one quarter, and no quarters or zero quarters.
  • Can you divide a pie into 0 pieces? No division by 0.
  • The whole pie is four times 1/4. A one quarter piece goes into a whole pie four times. So dividing 1 by 1/4 (= dividing one pie into quarters) gives 4 pieces. Language lesson, transformational grammar,

between words and symbols. In the UI, we will have a place for naming a fraction in numerals, and ask children for the words. In early practice, we can have one of Peter's number buttons showing words rather than digits. Later we can ask the students to type the names.

  • Add fractions with the same denominator visually, numerically in words, and symbolically.
  • Reduce to lowest terms. GCD.
  • Add fractions with different denominators, where one divides the other. 1/3 + 1/6 = 3/6. Do you recognize that shape?
  • Add fractions with different denominators in general, using obvious LCMs.
  • Lesson on LCMs when they are not obvious, but the factors of the numbers are obvious.
  • Lesson on factoring, perhaps.
  • More addition
  • Subtract fractions, in the same progression.
  • What is half of a third? What do you get if you cut a pie into thirds, and then cut each third into two equal pieces?
  • What is half of two-thirds? What is two-thirds of a half? Cut the half in three, and count.
  • Multiplying fractions generally means dividing pieces into smaller pieces, and counting out however many you need. Let the students discover the rules for multiplying with numbers and symbols.

We need about this many topics again before we are done. We need to convert fractions larger than one to integer and fraction parts, and vice versa. (What used to be called converting between improper fractions and mixed fractions. Nobody talks this way except some schoolteachers, so we should not bother the children with these names.) We need to do all of the operations on fractions larger than one, and on integer + fraction combinations.

Then we can introduce continued fractions and infinite series, using ideas from Don Cohen, The Mathman, and the algebra of the rational numbers, using ideas from Caleb Gattegno. We will have to create similar sequences for introducing negative numbers and decimal notation.

Bryan Berry has talked about a simple, wordless counting program his team in Nepal developed that allows students to practice endlessly if they like. Teachers had found that their students were two and three years behind in arithmetic because of lack of counting skills. Some students spent hours a day for weeks on end with this program, and caught up to grade level in just a few months. Now children can get this program before they enter school.

Our principle is that the student is not finished with a topic until the student turns to something else. Whatever additional understanding the student is wringing out of the work is to be treasured, not criticized. The more time we allow for deep understanding at the beginning, the faster we will be able to go later on.

> (see Lore Rasmussen et al - Lab Sheet Annotations and Mathematics for the > Primary Teacher, pg 140). If we can provide good materials to help kids > understand fractions and get evidence that they work we will create a good > entrance path for the rest of our materials. > So a couple of tasks: > > Identify XO Activities we can use to teach fractions (I spend most of my XO > time in Etoys)

I am particularly fond of Turtle Art, but I don't mind which tools we use for this. We can discuss what is appropriate to do in Python as a standalone activity and what is appropriate for Etoys.


  • Identify virtual manipulatives needed (I have cuisenaire rods and pattern blocks in Etoys, here is a nice one at Illuminations)
  • Identify physical manipulatives that can be used (and id how can be easily and cheaply created by kids/teachers)
  • Identify no computer, concrete activities kids can do (ex: 1) Fold a paper in half and cut, 2) Label as 1 piece of X needed to make a whole sheet; repeat)
  • Create/Identify a lesson, manipulative, blog pos or video each day. Okay ambitious, but do-able (for short sprints). It helps me keep the perfect from being the enemy of the good and gets me focused and moving. I look at it as the equivalent of some writers practice of writing something every day.