You can create:
- A collection of division from the world around YOU
- Comparisons between different models of division
This activity is similar to our Multiplication expedition. However, we don't have an online gallery yet, so here is the list of some division models for you to use, in text form. It came from a discussion on the Living Math mailing list. I hope people running this activity will add to the model list and also illustrate it with examples from their lives!
Find what each person gets. Natural for dividing by whole numbers ("sharing among people"). Fun examples: pirate booty; pizzas. A hundred pirates looted two thousand gold pieces; what is the share of each?
Extension of Sharing 1 into fractions: a third of a worm gained 2g weight. How much is the gain of each (whole) worm? Can probably be found in sampling. Don Cohen from "Calculus for seven year olds" has an activity expanding a sharing model into (infinite) strings of fractions, say, for sharing 6 cookies among 7 people: http://www.mathman.biz/html/prob2.html
Can use for dividing negative numbers by positive, but not for dividing by negatives, then? Some researchers, or kids, also call this and the next model "splitting".
Find how many people can get the given share. Natural for dividing by whole numbers and fractions, when the answer is whole. E.g. if you have three pizzas and want to give 1/6 per person, how many people can you feed. You have 21 cookies, and want to give each child 2, how many kids can you share them between? - Sue. How about negative numbers?
Breaking up into equal sized groups - Vicki. Natural for dividing by whole numbers of fractions when the answer is whole. How many thirds are in five apples? How many pairs can twenty kids form? 5 foot board cut into 2/3 foot sections will give you how many sections? (5 ÷ 2/3 = ?) - LauraLyn 4 ÷ ½ means “How many halves in 4? A concrete model can be used if you make an area model worth 4 and want to know how many halves are in 4 (partition 4 into half sized pieces) A string can represent a linear measurement (4 inch piece has how many half inch pieces in it?) and a favorite model of money (how many half dollars are in $4?). The length of a race will be ¾ of a mile. Each runner will run ¼ mile. How
many runners are needed to complete the race? The easiest model for kids to immediately connect with (IMO) is money. This gives you these fractions to work with ½, ¼, 1/10, 1/20, 1/100 and you can solve many problems and investigate patterns.- Vicki. Steve Demme's explanation in Math U See, but he called this "gozinta" (goes into) - how many halves "gozinta" 4. - Julie.
Velocity, unit price and other intrinsic rates or "intensive quantities" (per)
Find the velocity given distance and time. Works with all real numbers. Humans don't have grounding time metaphors, and kids may not measure distance much, so it's less intuitive for some, depending on driving/measuring experience. Liping Ma uses in her book—we’re building a road of a certain length (say 5 miles) and we know we can only build so much a day (say 2/3 of a mile). How many days will it take to build the road? (again 5 ÷ 2/3 = ?) - LauraLyn. Can be extended into negative numbers with past time and reverse direction.
Sports applications such as figuring out a player's batting average, or a horse's odds of winning a race. Or odds for cards (like poker hands) and rolls of a pair of dice. - Kari.
5:(-1/3). Place a marker on 5 units positive (to the right). Place a marker on -1/3 (to the left). If we continue to add markers of amount -1/3 to the left, we are multiplying -1/3 a positive number of times, and it is obvious that the result is going in
the wrong direction and will never reach the 5 marker on the right. So we must reverse direction by multiplying the -1/3 a negative number of times, and it will reach the 5 marker after -15 increments. - John. I've done something like this with kids physically taking steps across blocks in a small group, with negative and positive indicating opposite directions. A very good explanation of this is in a book called Playing with Infinity. - Julie.
Multiplicative invariance and other number rules, combined
Use several rules of operations together. Let's say the students first learn how to divide 5 / 3. They learn how to do 6 / (-2) and do all their integer operations. (Models could be used with those). They learn decimal operations. Here we'd study 5 / 0.2 and 5 / 0.3 and find that they are the same as dividing 50/2 and 50/3. One would need to justify this idea, of course. Then it's just a natural extension of the previously studied ideas that 5 / (-0.3) is equal to 50 / (-3) and then use what you know about integers. - Maria M. For example, in problems such as "divide 163 by 0.3," children are taught to change the decimal divisor 0.3 to the whole-number divisor 3 by multiplying by 10, divide 163 by 3, and, multiply the result by 10. What constitutes the understanding of this procedure is the awareness that the equality relation between the dividend 163, the divisor 0.3, and their quotient is not invariant under the change of the divisor 0.3 to 3, and that the "multiply by 10" transformation — applied to the quotient of 163÷3 — is an appropriate compensation for this change. - The Rational Number Project (a research group; this is from their review of the Multiplicative Conceptual Field here http://cehd.umn.edu/rationalnumberproject/90_1.html)
Iterations: Repeated subtraction ("the measurement model")
Many elementary level textbooks introduce division as repeated subtraction - how many 2s are in 6? - Research ideas for the classroom: Middle grades mathematics (a book). This model uses either the "number as a collection of objects" metaphor, or the "number as an object of a particular size" metaphor. C:B=A can mean either of the two possibilities. One: the repeated subtraction of collections of size B from an initial collection of size C until the initial collection is exhausted. Or two: The repeated subtraction of parts of the size B from an initial object of size C until the initial object is exhausted. The result, A, is the number of times the subtraction occurs. - Where Mathematics Comes From (a book).
Fraction bars or measuring sticks model
For ½ ÷ 1/3, it is easy to use the [fraction] bars to illustrate as follows: Lay out ½ bar. Underneath, lay out the 1/3 bar. Take another 1/3 bar and fold it in half to fill the remaining space of the ½ bar. Easy to see that you have 1 ½ of the 1/3 bars under (or covering) the half bar. - Vicki. C:B=A Dividing up: the splitting up of a physical segment C into A parts of length B. Iteration: the repeated subtraction of physical segments of length B from an initial physical segment of length C until nothing is left of the initial segment. The result, A, is the number of times the subtraction occurs. - Where Mathematics Comes From (a book).
Use any multiplication model with a known multiplier and a known product. For example, given the area of a rectangle is 6 square feet and one side 1.5 feet, find the other side. Kids would use multiplication and approximation, trying different numbers and seeing if they get the result close to the desired product.
Because the world immediately surrounding you is the place to find and apply math.
As you go
- Compare division models for their qualities. Which ones will work with fractions or negative numbers? Which ones you like and dislike, and why?
Higher and deeper