How to Teach Combinations Using Drink Markers
Steps

Prepare your cups and markers.
 Collect some plastic soda straws. Try to collect a diverse variety of colors, but you will need a minimum of three colors.
 Cut the straws into small pieces of approximately 1 centimeter (0.4 in) length each.
 Open the small pieces along the sides as shown in the demonstration using a pair of scissors.

Gather the group of people you're interested in demonstrating the mathematical concepts to in one place.

Introduce the main idea behind the activity to the group of people.
 The main idea is to use the straw pieces to create markers which in turn will be used to distinguish each paper cup from the others using a distinct combination of colors.
 Each player will be assigned a specific combination of straw colors so as to distinguish his/her own cup in order to reuse it (environmental aspect).

Start teaching/explaining the mathematical concepts using the same example but different hypothetical situations.Use the subsequent sections to demonstrate the desired mathematical concept.

Introduce the mathematics after you finish.Do not use the scientific names of methods and concepts while doing the activities. Rather, at the end of the session, state them as a surprise to the group. Something in the lines of"can you believe what we just did is called the rule of product in mathematics?"

Hide straw pieces from all colors but two, say Red (R) and Green (G).

Start by asking (pretending to be kidding preferably) "how many cups can we distinguish by marking each cup with only one straw marker?". The normal answer will be two cups.

Start increasing the number of markers, asking questions like:"what if we are to useexactlytwo markers?" and "exactlythree markers?" for which the answers should be 4 and 8 respectively.

Generalize by starting to use symbols like "what if we useexactlynmarkers?". The answers should be guided towards things like:
 2 time 2 times 2 .....ntimes
 2 raised to the power ofn.
 2 multiplied by itselfntimes.

Repeat the last set of steps, but changing the number of colors you allow for the straws.For example:
 Allow the colors Red, Blue, and Green. Start all over asking about usingexactlytwo markers, then three. Make the group notice that the results changed into things like 3 multiplied by itselfntimes, 3n

Generalize the idea toncolors using exactlymmarkers which should lead to multiplying thenalternativesof each marker times the number of markers.

Ask questions of the likes of:"What if we intend to choose the first marker to be one of red or green and the second marker to be one of green, red, or blue?" This should be guided to the answer of 2 * 3 = 6

Generalize the last example into the form:"the number of alternatives available for each marker multiplied" or "the result of the multiplication of the number of alternatives each marker has"

Hide straw pieces from all colors but two, say Red (R) and Green (G)

Start by asking the following question:"How many cups can we distinguish if we are to usea maximumof 1 marker?". If you do get the answer of 2, remind the group that one option is to mark an additional cup simply by not putting any marker on it (zero markers)

Repeat the last question for an increased number of markers.Emphasize thea maximum ofpart of your question. The answers for two markers should be 1 (no markers) + 2 (one marker, either R or G) + 4 (two markers, either RR, RG, GR, GG) = 7 different cups.

Start asking the group about why we had to add?What was different in this case? What difference did the word "a maximum of n markers" make to the standard question of "exactly n markers"?

Guide the group towards the answer which is:in this case, we had the option to use less than the exact number of markers, however, we can not use both 2 markers and 3 markers at the same time, which is the real reason why we added instead of multiplying.

Point out the fact that up to this point, the group has been able to use as many straws from the same color as they wanted.

Ask the opening question of "What if we limited the number of straws to use from each color?" Start by limiting all straw colors to using only one straw.

Use simple examples at first.For instance: "Using one red, one green and one blue straws, how many cups can we distinguish if we are to use one marker on each cup?". Then ask the question again using two and three markers for each cup. The answers should be three, three and one respectively.

Direct the group to the right direction by asking "What changed?". The answer is to be that each straw used is not replaceable like the previous situations.

Hide all the straws except the ones with Red (R), Green (G) and Blue (B) colors.

Ask the group to find the number of cups they can distinguish if they are not allowed to use the same color twice, and are to use exactly two markers?Guide the discussion to the right answer of 6 (2 markers, either RG, RB, GR, GB, BR or BG  RR, BB and GG are not allowed)

Increase the number of markers to three.Guide the discussion into the right answer of 6 (three markers, either RGB, RBG, GRB, GBR, BRG, BGR  RGG, RBB, GRR, GBB, BRR, BGG, RRR, GGG, BBB, RRG, RRB, GGR, GGB, BBR, BBG, RGR, RBR, GRG, GBG, BRB and BGB are not allowed because of repetition)

Direct the group to an alternative method of thinking:When putting the first marker, they have three free options to choose from, but once the first marker is specified, they only have two options for the second and once this is specified, they're left with only one option for the last marker. Multiplying yields 3*2*1 = 6

Increase the number of colors available suddenly and noticeably (for example, here use 6 colors)

Repeat the previous reasoning.This should yield 6*5*4 = 120

Make the group notice that there were three multiplication (the same number as the markers) starting with the number of colors (alternatives) and decreasing by 1 each time.

Finally, demonstrate the 6*5*4 = (6*5*4*3*2*1)/(3*2*1) = 6!/3! = (number of alternatives)!/(number of places or slots)!

Repeat same steps as in permutations.

Restrict the group further by assuming you can't distinguish when the cup has (RGB) and when it has (BGR) or even (GBR)
 You might have to use playing cards for combinations as they clarify the concept further (a hand of 2,3,4,5 and 6 is no different than a hand of 4, 3, 2, 6 and 5.

Clarify to the group that under the new restriction, the order of a certain combination of straws is irrelevant so the number of combinations having the same set of colors should be removed from the original permutation.
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 Do not explicitly state that you intend to teach the group about the concepts. In many situations, that might cause the group's enthusiasm and receptiveness to go down.
 Pretend to be asking the questions to actually find the answers. Do not directly answer questions, let the group think about the answer while playing the role of the invisible guide.
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Date: 03.12.2018, 07:22 / Views: 91282