Children's Math Book Review: How Big is a Million? (Penguins!)

Although our penguin unit is long over, I couldn't resist picking up one more related book. Pipkin, a young penguin, wants to know How Big is a Million? He finds ten fish, 100 penguins, and 1,000 snowflakes, but he cannot find a million. Finally, his mother takes him outside to look at the starry night. In a pocket in the back of the book, the reader finds a poster depicting exactly one million stars. In contrast to some other "million" books where the quantity is distributed throughout the book, this picture allows the reader a glimpse of a million of something in one shot. Sweet book for exploring the concept of one million. 

The only thing I wasn't sure about...  Does the book actually show 1,000 snowflakes? I know it shows 10, 100, and a million, but I wasn't sure about 1,000. A friend of mine found some information from the author, explaining how the book developed. It leaves me to assume that all the illustrations are accurately counted.

Keywords: million lesson plan, penguin lesson plan

Fabulous Math Website

I accidentally found Links Learning when I was searching for a math site. It's phenomenal. The "Illustrated Lessons" are short movies that explain math concepts and then allow students to follow-up with interactive quizzes. From the site:

In this section, you will find some fun ways to learn about math. You can start out with Estimation of Length, Place Value, Weight and Capacity. If you want something more challenging, take a look at Line Symmetry, Patterns and Tangrams. Try some of these at home with your family!

On the Tangram movie, it gives you step by step instructions on making your own tangram. You push the button when you're ready for the next step. The lessons are great for introducing or reinforcing a concept.

Braggin' 'Bout Place Value

The excitement over our last Place Value Structures class continues. Today I heard a little voice bragging to a big brother, "Didya know ten times 10,000 is 100,000????!!!!!" :)

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!) ;)

Incan Quipu Math (place value, history)

During the Inca empire, a system of knots was used for adding numbers. We've been reading about it in a fascinating math/activity/history book,  The Secret Life of Math; discover how (and why) numbers have survived from the cave dwellers to us!

We did a little quipu activity and quickly figured out why this math method--though effective--didn't last to the present day. At least not if everyone was as bad at knot tying as we are! ;) It is, however, a fun activity to explore history and reinforce knowledge of place value.



Inca Quipu Math activity

Materials needed:

a thin dowel or chopstick
thin cord
colored markers
scissors
paper/pencil (optional)

Procedure:

1. Tie four long cords onto the dowel. If the cord is too short, you won't be able to tie many knots, so leave yourself plenty of room. The three cords to your right are the addends, the cord on the left is the summation.

2. Though the Incas didn't do this, we added colored marks on the cord so we could easily see the divisions between each place value grouping. As you tie knots the cords won't continue to line up evenly, but it will help you be able to see that the hundreds are between the orange and purple marks, etc.

3. Decide on three numbers you'd like to add. Tie knots on your cord to represent the numbers. For example, on the cord on the far right (below) we chose the number 2,133. This is represented by two knots in the very top section, one knot in the next section, three knots in the third section, and three knots in the bottom section. We made number cards just to help us remember what we were doing. (Later it was fun to mix and match the cards with the cords!)

4. Add the knots in all the ones sections on all three cords. Record the total number of ones on the summation cord. For us, the ones totaled 11; therefore, we put 1 knot in the ones section and 1 knot on the tens section on the summation cord. Continue through each subsequent place value section.

We did this activity during a study of Inca history. My student, age 7, found it difficult to tie the knots, so I did most of the tying. He did the adding and writing. When we finished he said, "That was FUN!"

Another place value game that my children enjoy from time to time...Learning Resources Dino Math Tracks Place Value Game. Here's the Amazon product description:

"Prehistoric pals make this award-winning math game a blast. Roll the dice and move the dinosaurs around a delightfully illustrated game board. Learn all about numbers - from counting, addition and subtraction to place value skills. Includes game board, 16 dinosaurs, cards, number die and instructions for various levels of play. For 2 to 4 players."

I also found a little online activity to practice identifying place value in a number.