Secret Numbers (Addition Boomerang variants)

We have been enjoying Mathpickle activities in our math games classes recently. This week, the 3rd and 4th graders will be playing with some of the more advanced Addition Boomerang variations.

During our planning, we came up with one extra pointer to tie the activity more closely with multiplication. Also, we had ideas for variations and wanted to record our notes so we can use them again in the future.

Tie with multiplication

In Gord's video explanation, he sometimes records in the center of each loop how many times that loop has been used. We emphasize this and write some related equations to help draw out the connection between the repeated additions in this activity and multiplication.

The first way we do this is by making tally marks inside the loop every time that branch is chosen. At any time, you can pause and write down an equation for the current total in a form

AxN + BxM = Total

where A and B are the values of the loops, N and M are the number of times each loop has been used.

Alternatively, we can show a "completed" round of throws by simply writing the number of passes for each loop in the middle and, again, write out an equation showing the total as the sum of two products.

Secret Number Variations

We start with a basic addition boomerang lay-out, either with 2 or 4 branches, both players (or teams) share a common set of addends and take turns adding on to a common running total. In our variants, the players choose and write down a secret number that helps inform their target for the game:

• Version A: players pick a number between 70 and 100. This is their target for the game and they win if the common total hits that value, whether the target is reached on their turn or their opponents turn.
• Version B: players pick a number larger than 15. They win if the total hits a multiple of their secret number. For example, if they choose 17 and the running total hits 51 (aka 17 x 3) then they would win. If the total is a multiple of both secret numbers, then the player who chose the larger secret number wins.
• Version C: players pick any number. They win if the total hits a multiple of their secret number that is larger than 60 (not equal to 60). If the total is a multiple of both secret numbers, then the player who chose the larger secret number wins.

Version B is, I think, the most directly playable.

Possible issues
I'm not sure how to deal with the case where both players choose the same secret number.

For Version A, it will be interesting to see what modification kids can find that will deal with the fact that it is very easy to miss any particular target. In the basic game, once the total is larger than your target, there is no hope of recovery. There are several ways to address this. I would be eager to hear any rule sets that kids create and hear about the experiences.

In Version C, I wonder if choosing 2 as the secret number is too strong a move?

Broken Ruler and Multiplication refresh

Ruler Explorations

We noticed that one of our tools, a ruler, had gotten broken.

Is it still useful? As a challenge, J2 looked at measuring a noodle from his soup.

There were two ideas:

1. the noodles were too long, so had to be broken in pieces to measure with the remaining ruler
2. Our ruler doesn't have to start at 0, we can use subtraction!

While we were talking about this, I recalled the idea of Golomb Rulers. We came up with a ruler that was marked only with 0, 1, 3, 7, 11, 12 cm. This lets us measure 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, and 12cm distances.

What if this ruler gets broken? For example, we imagined cutting our ruler between the 1cm and 3cm markings. What measurements are still possible? Is there anything interesting in the relationships between how many ways there were to measure a distance before the break and how many ways to measure after?

Multiplication Refresh

We recently re-watched Graham Fletcher's Progression of Multiplication. Both J1 and J2 did some practice around this. The most interesting point was J2's reaction to Graham's comment at 4:54: "This sucks!"

"Why did kid's say that?"  "Hmm, let's try out a couple of examples..."

We rolled dice to randomly generate digits for an example and were lucky to get 35 x 34. J2 quickly saw this as 35 x 35 - 35 and knows a pattern that let him quickly calculate 35 x 35 = 1225. As a result, 35 x 34 was pretty easy for him to calculate.

Then, he worked through a graphical representation and a powers-of-ten version. At the end, we got to compare and contrast the different approaches.

Continuing to play with some old activities

Fold-and-punch
We did some more fold-and-punch activities. This time, we folded the paper, then drew a location for the punch, and tried to figure out how many holes would result and where they would be. We broke out our serious hole-puncher:

Unfortunately, must be operated by an adult

In this example, we got a small surprise that the result wasn't a power of 2:

Chairs (and tables)
Another round of building chairs, following the NRICH activity. This time, with J3:

We got to compare and contrast our designs:

• how many cubes were used for the legs? Which one had more and how many more?
• How many cubes were used for the whole chair? How did they compare?