Minimalist Math Musings...

During our lessons on base 5, we talked quite a bit about "minimal collection." Apparently, the concept has stuck...

Yesterday, my 8yo ds walks up to me and says, "In Sleeping Queens, the minimal collection to get 50 is two 15s and one 20."

In the game you have to get 50 points and can get it by collecting various groups of cards. And, yes indeedy, the least number of cards you need to get 50 points is by the set of cards he mentioned. :)

I'm amused. Not minimally. ;)

Place Value Structures, Class #4

Few things are more thrilling that seeing a child with an "ah ha" moment. Times that by several children and you've made my day!

Today was our last class on place value structures using material from Bridges. For the last three weeks we've been considering place value in base five, looking at units (1), strips (5), mats (25), and stripmats (125). Today I asked students to imagine what the pieces might look like if we were working in base four. They began by building the pieces with tile. They debated over whether the mat should have 16 pieces or 24 pieces. (I think that a few jumped from the logic that if base 5 had 25 pieces, then base 4 should have 24 pieces.) We looked at the base five chart we'd made. Students recalled that with base five, each place value was x5, so decided that with base four, each would be x4 with a total of 16 for a mat. No easy task.

They cut a unit, strip and mat from graph paper. I then asked them to cut out a strip mat. When they couldn't agree on the area, I asked them to build mats with tile. Ah-ha! Then they remembered how to follow the place value pattern and had no trouble building a stripmat of 64.

Since the base five building didn't come easy, I wondered how the next question (and one of my big goals for the whole class!) would go...  "Close your eyes. Picture base ten. What would a unit look like? A strip? A mat?" The quietest child in the class couldn't hold back..."100!" They all immediately told me that base ten would have a unit of 1, a strip of 10, and a mat of 100!!!  I gave them base ten pieces and asked them to create a strip mat. With shining eyes and wide grins they asked me for TEN MATS and told me that a strip mat would have 1000 units. They built a strip mat and then proceeded to tell me what the next piece would look like and that it would have 10,000 units. "And then 100,000 and then 1,000,000...!!!!!!!!!!!!!" They helped me to complete a venn diagram comparing base five and base ten.

We read Sir Cumference and All the King's Tens and talked about place value in base ten. I also read The 329th Friend and asked them to build 329 (and a few additional numbers) with the base ten pieces.

It isn't unusual for teachers to ask, "Why do we teach other bases to children?" It's elementary, my dear! When we teach other bases, we give children an opportunity to develop conceptual understanding of place value. That deep understanding transfers to work in "our base," base ten. The lightbulbs in my classroom were going off so fast today that the electric meter must have been smokin'! Base ten place value means something new and exciting.

It's elementary. :)

You've been a wonderful class! Can't wait to see you again!

Place Value Structures, Class #2

We're continuing our exploration of Base 5, along with several other mathematical concepts. We covered a lot today...
While waiting for students to arrive, we enjoyed the book Easy Math Puzzles by David Adler. This is a fun book that I've used with several classes. Each puzzle is short but takes a significant amount of thought.

We opened the regular session with Moja Means One; Swahili Counting Book by Muriel Feelings. I chose this book after reading this idea connecting it to base 5:

"People have been counting on their hands for centuries. In some languages, the word "hand" means "five." If you ask a farmer how many goats he had and the farmer answered, "hand," how many goats would that be? (Five.) Suppose you have not yet thought to use two hands but just use one hand for counting. You can use toothpicks for this, and put down one I, then another 11, and another III, and one more IIII, until you have five IIIII. Now, how do you represent six if you can't count on your other hand? (Maybe you could just use the same hand over again, or cross the picks.). Lead children to discover the following:
6 is 5 and one more
7 is 5 and 2 more
8 is 5 and 3 more
9 is 5 and 4 more
10 is two 5s"

We discussed ways that larger numbers could be represented if hand = five. How could we represent 25? 50? I asked them to consider how this idea related to the exploration we did last week with our place value mats and cubes.

We continued with a very brief review of the Base 5/place value mat/cube counting. I asked them to give me a "thumbs-up" as long as I counted correctly and a "thumbs-down" if I made a mistake. They were all quick to catch my errors!

We used lessons from Bridges (Math Learning Center) looking at collections of Base 5 pieces. How many strips and units does it take to make a given number? Then we traded our strips for more and more pieces until we used only units.

We defined "minimal collection" and built more examples of base 5 collections. We recorded our work and looked for patterns.

 After learning to build minimal collections, we played "Up to the Mat and Back," racing to build and dismantle (add/subtract) Base 5 collections.

Along with all the Base 5 work we looked at odd/even numbers and played a subtraction/probability game, concluding with an enjoyable reading of Math Curse by Jon Scieszka (which does have some base work in it!)

Busy day! Thanks for all your hard work!

Place Value Structures, Class #1

Yesterday I began a series of 4 homeschool classes (2 hours each), focusing on "Place Value Structures." Sadly, I was so excited thinking about the class that I forgot to take photos. So a few posed photos will have to suffice...

As children arrived, they settled in to explore polydrons as they took turns completing a chart of information about themselves. After everyone had a chance to fill in their information, we talked through the chart, both to introduce ourselves and to consider what characteristics in our class came up the most/least often, comparing #s of family members, pets, etc.

We then set off to explore a "Foreign Counting System" using materials from Bridges (produced by The Math Learning Center.) I counted centimeter cubes onto a place value mat and asked students to try to figure out what rule or pattern I was using to count.

As students caught on, I gave them their own manipulatives and mats and they continued to count with me until we had 4 in each column. (Base 5)

Children discussed what they noticed about this counting system as I took notes on a class chart. One child immediately noticed that it was like counting money using pennies, nickels, and quarters.

I handed out base 5 pieces and challenged students to make observations with a partner. We then recorded the information on a class chart. Children worked together to figure out what the next piece in the sequence would look like. We finished up by drawing base 5 arrangements on graph paper.

Extra activity: In the middle of class we looked at a hundreds board and counted by 5s from 46 to 81. This was quite a change from the usual counting by 2s, 5s, and 10s. (Also from Bridges/Number Corner materials.)

Children's Math Books: No class could be complete without a few good books. We read Earth Day-Hooray! (place value) and Once Upon a Dime (money).

I have a delightful class of students. I'm looking forward to seeing you all again next week. (With a camera!) ;)