5 choose 2.

Figure out how many ways we can put two balls in 5 spaces. This activity is taken from Math From Three to Seven.

Task 1

Supplies: 30 pennies and 20 dimes. Or you can also have 30 objects of one color and 20 of another color (beads, pegs, etc.). 

What to do: 
- Invite your child to form as many different parades as she can with 3 pennies and 2 dimes. 
- If she hasn't found 10 yet, encourage her to look at patterns or work systematically. 
- When she gets to 10 parades, ask her if there could be any more. 

Task 2

Supplies: Graph paper. Draw a bunch (at least 10) 3 by 4 grids. Or print out the download below. 
What to do: The goal is to go from the bottom left to the top right by moving only right or up. 
- Invite your child to figure out how many different ways she can do it. 
- What if we recorded all the paths with R when it goes right and for U when it goes up. (A path along the bottom edge then up along the right edge would be RRRUU). Can she write all her paths?  

Let your child discover if there is a connection between the two tasks. If she can't see it right now, it's ok too. Trust that at some point (maybe in a few days, weeks, or months), she will think back on it. We will also do another 5 choose 2 activity in a few days. 

How it went

Nia has a tendency to let her older sister do the work and just copy so I separated Bel and Nia so they wouldn’t look at each other’s progress. They both wanted the pink and teal dominoes and there was a little dispute at the beginning. Bel set out to work on her own. Nia put down two “parades” where the colors where all together (3blue-2pink; 2pink-3blue) and said she was done. She wanted to keep dominoes of the same color together and was unhappy when I suggested she found more combinations. She put two or three more down (1blue-2pink-2blue and I don’t quite remember which other ones) but when I suggested the 2blue-2pink-1blue, she said it was the same as the 1blue-2pink-2blue and didn’t want it. I was puzzled but when asked what made it the same, Nia used the word “symmetric”. I hadn’t planned this activity to be about symmetries and I was quite surprised she brought it up.

Nia’s work. She and I paired them to show her symmetries.

I then asked her to build the symmetric ones to the ones in front of her and was met with much reluctance. I showed her how she could make the exact copy and turn it upside down (rotate by 180 degrees), she got a little more interested. All and all, there was much pushing from me than I would have liked. Maybe a good story behind the activity would have helped?

Bel’s work. This is the order in which she created them.

Bel, working on her own, declared she found 13 of them. She commented that it was difficult to come up with different patterns. I asked her to check that she didn’t have any doubles – and if she found a way to systematically build the parades. After a checking, she did get 10. She did not find a pattern for creating all of them. We did not discuss whether there could be more than 10 and did not have time for the second part with the rectangles.

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