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?