Math at the beginning of this week was a bit different because Diane was still out of town for her older daughter’s wedding, so we couldn’t have our usual groups meet on Monday. Instead, we did some activities with everyone that involved learning more about tests for divisibility (that is, determining if a given number is a multiple of a smaller number). Most students were already able to explain how they would decide if a number could be divided by 2, 5, or 10 and leave no remainder.

But what about 3? As it turns out, the divisibility tests for 3 and also for 9 involve finding the digital root of the number in question. You find the digital root by adding all of the digits of the number without regard for place value. For example, we can find the digital root of 12,591 by  calculating 1+2+5+9+1. It adds up to 18, which we add again so that we end up with a single digit: 1+8=9. Since the digital root of that original number is a multiple of 3, we know that 3 will go into it without a remainder. That number is also divisible by 9 because the digital root is 9.

Is there a test for 4? Yes. If the last 2 digits of the number are a multiple of 4, the number is a multiple of 4. For example, we can see that 5,616 is divisible by 4 because 16 (the last two digits) is a multiple of 4. Why does that work? It works because all of the higher places are always divisible by 4 because 4 goes into any number of hundreds, thousands, and so on. We only need to check the tens and ones. The test for 8 is similar, but the final 3 digits need to be a multiple of 8 because not all hundreds are divisible but all thousands and upward are. That test usually involves too much computation to be helpful, so we mention it but don’t practice it.

What about 6? Six is built from 2 times 3. So a number is divisible by six if it is even and is also a multiple of three. So first we check the ones digit to see if it’s even. If it is, we then apply the threes test by finding the digital root.

There is a test for 7, but it’s a lot more complicated, so we don’t teach it.

The chart below summarizes what students explored (image source: http://math.tutorvista.com). Everyone needs a lot more opportunity to use the newer rules before it all becomes automatic.

There are several reasons that all of this is useful. First, it’s one way to do a quick check on a division computation. If you are not expecting a remainder but end up with one, there’s a mistake somewhere. It also builds students’ understanding of number relationships. There are no “tricks” involved. Everything has a mathematical reason. For example, the digital root of a number is actually what you would get as a remainder if you divided the number by 9. (The exception to that is any number that is a multiple of 9, because the remainder would be zero, not 9.) Another use of divisibility rules is practical. Suppose we have 212 pencils to give out to 6 classrooms. Will every room get the same number? No, because the digital root of 212 is 5, which is not a multiple of 3. If it’s not a multiple of 3, it can’t be a multiple of 6. How many more pencils would we need in order to make the shares equal? We’d need 4 more, because the next even number that is a multiple of 3 is 216. Do you see any connection between that first digital root of 5 and the number of extra pencils needed (4) to end up with an even multiple of three?  (Hint: what is 5 + 4?) As we worked with divisibility, these were the kinds of things that came into the conversation. All of it is part of number theory, and all of it builds number sense.

On Tuesday, Diane and Jeri’s group went on an all-day trip, so we did some work just in our classroom with short division, using our divisibility rules to predict whether there would be a remainder. Short division is a very useful algorithm when you are dividing by a single digit or any number for which you can easily do mental calculations. The image below compares long and short division by showing the same problem worked both ways (image source: http://eisforexplore.blogspot.com/).

The second example involves the following thinking: Six doesn’t go into 2, so I’ll start with 25. It goes 4 times, with 1 left over. Regroup the left-over 1 with the 3 to make 13. Six goes into 13 twice, with 1 left over. Regroup the 1 with the 2 to make 12. Six goes into 12 twice, with nothing left over. All done.

One of the things we like about short division is that it is clearly the reverse of the standard multiplication algorithm. That is, if we multiply 422 by 6 in order to check our division work, we will get the same regrouping (“carrying”) going the opposite way. Six times 2 is 12, so we would write down the 2 and regroup the 1 (ten) to the tens place, and so on. Long division is also a good algorithm to know, but it isn’t the only way to divide and is often more time-consuming than the task actually requires.

We spent some time this week learning a bit more about the world beyond our own neighborhoods. First, we introduced the students to Kiva. This is an organization that arranges micro-loans in many parts of the world, and we contribute part of our lunch sale profits to them. Students looked at the information on several people who had applied for loans and, as a group, decided on whom to support. We read some biographical information that increased students’ understanding of how some people were living in distant places such as El Salvador, Senegal, and Colombia. (One woman had never been to school because she was the oldest child in a large family, and her mother kept her home to help with the housework and care for her siblings. She wanted a loan to buy supplies for her small food-selling business. She uses the profits from it to help pay for her own children’s education.) Because the money is a loan and not a donation, it gets repaid, so we can lend it out again. In fact, what we loaned this week was what had been repaid on loans made by last year’s group. Next week, we’ll chose a new recipient of a $25 loan from the lunch sale we just completed. To learn more about this organization, visit www.kiva.org and see how it all works.

The second activity was focused closer to home. Students read and summarized the article in their current issue of Junior Scholastic magazine about the fatal shooting of Michael Brown in Ferguson, Missouri. We then discussed it in half-groups the next day. The discussions involved not only the events in Ferguson but also brought out students’ questions and observations about such things as racism, fairness, stereotypes, courts and juries, and differences in the neighborhoods in which they each live. One of our goals was to help students recognize that not everyone has the same experiences or the same concerns. It’s not easy to step outside of your own assumptions to see things from another perspective. One student commented, “I think that everyday people probably don’t understand how much racism there is.” We asked her whom she had in mind when she said everyday people. She stopped and thought for a minute or two and then said, with some surprise in her voice, “Me!” That gave us a good opportunity to talk about our biases and unconscious behaviors. For example, if a character in a story is not described physically and there is no other guiding information, what do you assume that person looks like? White, black, Asian, or something else? We didn’t all have the same answer. We hope that you’ll continue such conversations at home.

As always, we did many more things than this, but it seems more informative to give you a closer view of part of our week than a quick sketch of all of it.

The week was dominated by the Terra Nova standardized tests. We began by having every student fill in the cover page of the provided answer sheet, although only the sixth graders would be using that for their answers. Fifths are given a separate answer sheet for each subtest so that it is less visually-overwhelming and easier to deal with. But the official answer sheet brought out a lot of questions and contained some challenges. Students had to write their names in boxes across the top and then fill in circles that matched the letters in columns below each box. (These are the answer sheets used for machine-scoring of tests, although our teachers score them by hand.) Some had difficulty staying in the correct column.Then they needed to indicate their birth date in 2 numbered columns for the month, 2 columns for the day, and 2 columns for the year. Several students again were confused by the layout. And then came one more problem to solve — there was an “ethnicity” section. Which category are you? We explained that a lot of this information is needed only by larger schools and school districts that collect this kind of data, that we don’t collect it, but that it was good practice to learn how to do it. Several children were again baffled — What’s ethnicity? What does “Hispanic” mean? What is “multi-ethnic”? One child had no idea what he should choose, and we suggested that he have a talk about it at home. All of this challenge and learning went on before we had even opened the test booklet!

But there were lots of other things this week, too.

Our ongoing efforts to do things with our hands as well as our heads took an old-fashioned turn when some of the girls started making clothespin dolls during choice on a rainy day. Glue guns, fabric, yarn, craft sticks, and — of course — classic clothespins quickly became little people.

There continues to be a lot of blending of our two groups at choice time. The board games our group has made brought in some more players on that same rainy day.

We spent some time with Junior Scholastic magazine to learn more about Ebola. As is always true with current events, some children had more knowledge, more questions, and/or more interest than others. The main messages we wanted to convey related to how devastating this epidemic is in many parts of West Africa and why it was difficult for those countries to respond effectively to it. The article also made the point that the virus is difficult to catch.

Our week ended with a morning of wonderful visitors — mostly grandparents along with some other adult friends and relatives. We began by bringing the whole building together for a sing. “I’m my Own Grandpa” had everyone laughing. “Passing Through” had an easy chorus, and the room rang with voices old and young. Students asked for two more from their songbooks. “The Horses Run Around” brought in a bit of absurdity, and we ended with “Love Potion Number Nine” — a hit song from the ancient days of rock’n’roll.

The two classes then divided, and our group spent about 20 minutes engaging our guests with dulcimers, the games they have made, and explanations of our wagon train adventure. Then the entire group went to science, where they had a chance to make crash helmets for eggs. Sad to say, some eggs did not survive the drop, but others did. At 11:00, we said goodbye to our visitors, who seemed to have had a very good time.

We wrapped up the day with the last of the Terra Nova testing, art class, and a crossword puzzle. It will be good to get back to our usual schedule next week!

 

 

Using our heads and also our hands ~

We’ve been making more use of our new woodworking tools. We have a wonderful (and, sadly, out-of-print) book of traditional board games from around the world. It contains not only images of the game boards but also some information about the cultural origins and strategies required to play well. We started by playing on board patterns photocopied  from the book. Students had to read the directions and figure it all out for themselves. There were solitaire puzzles as well as games for two. Some games seemed too easy or biased in favor of one player at first . . . then they began to realize there was more to the game than first appeared as they started to develop strategies.

The next stage was choosing a game to make in wood. We went outside to our new student-built workbench and selected white pine boards that seemed to be the right width, ranging from 1 by 6 to 1 by 10. They needed to decide how much of the wood was needed and learned to use a steel square to draw a cutting line. Students could cut the board with a hand saw or saber saw:

Although cutting the board with a hand saw was a lot more work and — for most students — a new experience, it wasn’t easy to make a straight cut with the powered saw because of the narrow blade. This is an extremely safe saw for students to use and is a good introduction to power tools.

The next step was smoothing the cut sides with sandpaper or our belt sander. Once again, a lot of learning and problem-solving went on. Why do we wrap the sandpaper around a small block of wood? How do we keep a smooth edge when the board is wider than the sanding belt or sanding disk?

Then students transferred the game pattern to the wood. Some were fairly complex and would take a long time to measure and get the angles and intersections right if we drew them ourselves. One student said, “Couldn’t we poke something through the paper?”  Great idea. So we taped the papers to the boards, got a hammer and a nail, and put a hole in the middle of every circle on the pattern.

Lines connecting some of the piece locations were important in most of the games — they indicated how moves must be made. Some students drew the lines before drilling the sockets or holes for the game pieces, while others did it afterward.

Although most students reached for rulers on their own, one student looked at his wobbly lines and asked plaintively, “Were we supposed to use a ruler?” He solved his problem by getting a ruler and a wider marker so he could re-draw the lines more neatly. Teachers didn’t need to say a word.

We made the sockets for the round glass game pieces with a spade bit in our cordless drills, going just deep enough into the wood to create a hole in which the game piece could sit. Students who decided to use golf tees for playing pieces used a standard drill bit that was about the same diameter as the shaft of the tee. “How do we know what size to use?” Try comparing your game piece to the available bits. “How do I put the bit in the drill?”  Ask someone who is already drilling. They’ll show you.

Talking about the spade bits gave us a chance for a practical math lesson. We had bits that were 1/4, 3/4, 5/8, and 1/2 inch wide. Without picking them up and looking at them, could we figure out how they ranked in size? A number line on the chalkboard from 0 to 1 gave us a start. Students quickly placed a mark at 1/2, 1/4, and 3/4. But what about 5/8? There was a moment of silence. Then we started adding equivalent fraction names to each of the marks we had: 1/2 was also 2/4, 3/6, 4/8 . . . aha! When all three fractions were labeled as far as eighths, we could see where the 5/8 bit should go. None of this was new to any of our class, but things need to be retrieved over and over and applied in a range of ways before the learning is secure and able to be used independently in new situations.

By the end of our short week, we had a number of finished games. We’ll get the rest done next week, and some students will probably make more than one. We plan to share them with our guests on Grandparents’ Day.

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In search of our roots . . . square and cube, that is ~

Two of our math groups worked with square numbers and then extended that into cubes and general work with exponents over the course of several days in the past 2 weeks.  Some of this was described briefly in last week’s blog post when we did it with students only from our class. This week, we took it further with our new instructional groupings that combine our students with Diane’s.

Things we could hold in our hands were essential to building conceptual understanding, and we had a new use for the work we had done previously with factors, prime numbers, and prime factorization. First we worked with flat square tiles to make the first 5 square numbers: 1, 4, 9, 16, and 25. The “square root” is the length of one side. All of those numbers are perfect squares, because their square root is a whole number. Write the difference between each square number on a piece of paper and place it between each pair. Is there a pattern? It goes up by 3, 5, 7, 9 . . . Do you think that continues?

Why isn’t 12 a “square” number? Take 12 tiles and try to arrange them into a square. Oh. The closest we can get is a 3 by 4 rectangle. So what can we conclude about the square root of 12? After a little thought, the group agreed that it had to be more than 3 but less than 4. We’ll come back to that later.

Now let’s take a look at the prime factorization of a number that’s a perfect square, such as 36. Using either of the two methods that we’ve learned, students came up with the right factor string:  36 = 2 x 2 x 3 x 3.  Now we rearrange the factors into two identical sets:  (2 x 3) x (2 x 3) = 6 x 6. And so 6 is the square root of 36, built from half of the prime factors. What if we can’t make two matching sets of prime factors? Then we know that the number is not a perfect square.

Back to that troublesome 12 . . . Although not every number is a perfect square, every number does have a square root. It just won’t be a whole number.  So, if the square root of 12 is somewhere between 3 and 4, let’s get out our calculators and go hunting for it. The first suggestion is 3.5, so we do a quick lesson on the keys to use for putting an exponent into our calculators and give it a try:  3.5 squared = 12.25. Students got out some graph paper and drew a square that was 3 and 1/2 squares on each side. And yes, we could see that it comes out to be 12 and 1/4 squares in area. It’s important to keep going back to visual, tactile representations and not just numerical processes. If 3.5 is too big, let’s try 3.4, someone suggests. (We keep reminding students that that number is not “3 point 4” but “3 and 4 tenths.” An understanding of decimals requires a firm mastery of place value. If we always read 214 as two-one-four, we might not realize that it was a bit more than two hundred. It would just be a string of digits.) But 3.4 squared = 11.56. Too small. So the square root must be between 3.4 and 3.5, someone asks? Someone else has an idea — how about 3.45? When a third child expressed confusion, we rewrote the bracketing numbers as 3.40 and 3.50. A short discussion of decimal places and equivalencies brought up some forgotten learning, and we moved on. We gave out sheets on which students could record their guess and its squared value.

Gradually students got closer and closer to the elusive square root. “I have 12 with five zeroes!”  “I have 11 with four nines!”   As students worked, I observed their strategies. Most were systematic, but a few wobbled around the target without much method. I made a few suggestions that pointed to what other students were doing, and they began to see that a more analytical approach gave them better results. No one got a full display of zeroes or nines by the time class ended, but the excitement and exasperation made for a dramatic end. “Just one more!” several pleaded when I said it was time to pack up.

On to cubes . . . first we picked up some manipulatives again and built the first 4 cubic numbers: 1, 8, 27, and 64. Some students did what we had done with the flat tiles, making only one layer. As they realized they didn’t yet have a cube, another conceptual element clicked into place.

We discussed perfect cubes, relating them to perfect squares. When we asked how we might use prime factorization to find the root of a perfect cube, students quickly suggested that we would need three matching sets of prime factors. We tested it with 1000 and came up with 2 x 2 x 2 x 5 x 5 x 5. That re-grouped into (2 x 5) x (2 x 5) x (2 x 5). It works!Would it work for finding the roots of numbers raised to the fourth, fifth, or sixth power? We agreed that it would.

Our calculators have a square root key, of course. But this exploration made the concept of powers and roots clear in a way that just pushing one button to see what came up on the display could not possibly do. And it was a lot more fun.

Our two wagon trains have started on their simulated journey to Oregon and encountered the first of many challenges. As they arrived at a small settlement, they expected to fill up their water barrels at the town’s wells. Water had been given freely to travelers in the past. However, because of a severe drought, the town was charging for water, and they were enforcing this new policy by posting armed guards. Our groups had to make a decision. Should they pay for water even though some of their families said they couldn’t afford it? Go on without water and hope to find a spring or stream along the trail? Attack the guards and take the water without paying? If they decided to pay, what should be done about the people who didn’t have enough money?

Decision-making is a key element in this multi-week activity. The learning goals include being willing to look deeply at the options and consider all of the possible outcomes, not just the ones you would like. Students are given some likely choices (such as those listed above) and also invited to come up with alternatives of their own. Then they write brief pros and cons for each. After deciding on a personal preference, they meet as a group and come up with a team decision. The intention is that students will learn to apply this approach to real-life situations as well as the ones that come up during this imaginary adventure. Both of our teams did well with this first event. They each had a good discussion, listened to each others’ ideas thoughtfully, and came up with plans that ensured that every family would be able to continue on the trail with the water they would need. It bodes well for challenges that are bound to come up in the future.

Everyone has finished a biographical poster for their individual identity and, in many cases, their family. Although they didn’t end up putting as much personal information on the posters as we have typically had in past years, they all worked hard to collect or create authentic images — especially in terms of clothing and other aspects of general appearance.

Another part of our study of changes in 19th century America is learning about the material world of the typical citizen and the economy that supported it. We did the first part of a short research activity on inventions. Students were given a list of ordinary objects (zippers, matches. gunpowder, traffic lights, cameras, and the like) and asked to make an educated guess about the period of time in which they were invented — ranging from before 1000 CE to the current century. Then the list was divided among the students and they were challenged to find the actual invention date. As we expected, students who had been assigned the same invention did not always come up with the same date. We briefly discussed why — different inventors given credit for the same item, unreliable sources, different definitions of just what a “camera” or other object was, and more. The final stage of this will be to do some group research on the conflicting dates and see if we can find the reason(s) for the inconsistency. This is an important part of learning to do research of any kind as well as understanding more about the nature of invention and attribution.

The economy of the time is also an important topic. Students rolled dice to see how much money they were taking on their journey.  The sum of two dice times ten gave them their nest egg. But how much was that money really worth? We looked at an editorial from the NY Daily Tribune that was published in 1851, describing a carpenters’ strike in Philadelphia. The carpenters wanted another 25 cents per day. The editorial described the typical living expenses for a family of five — food, clothing, rent, fuel, etc. As children read through the list, they could compare the average carpenters’ weekly salary ($9) with what that money would buy, such as “butcher’s meat” at $0.14 per pound. We noted that the entire family of 5 was described as consuming 2 pounds of that meat in a week. We were able to make some comparisons with modern costs for the same goods and also talk about differences in how much we owned or consumed.  It became clear that the carpenters were striking because their current rate of pay did not meet basic needs. (We also talked a bit about what a “strike” was and why workers joined a union.) Gradually, students came to understand that the value of money at any given time depends on how long you had to work to earn it and what it would buy. The belief that “things were really cheap back then” was true only if you imagined shopping with today’s typical wages. A primary source such as the one we used is much more informative than simply telling students that people earned less. They could see it for themselves.

We have 4 small math groups at the moment (with a change coming next week). Jeri has been working with her students on place value, rounding, and understanding large numbers. At the of the week, they spent two days using data from the U. S. Census bureau to work with large numbers while learning about regions and some of the states within them. We are one of the “early adopter” classrooms who are helping the Census Bureau develop a rich series of lessons for students of all ages.

The two groups that I teach have been working with factors and multiples, prime numbers, and exponents. They are all realizing how important it is to be quick with number facts for these activities. At the end of the week, we spent 2 days using our calculators to find non-integer square roots without the use of the square root key, and we also did the same for cube roots. This activity helps to develop number sense about orders of magnitude, and it illustrates the infinite nature of the base-ten system in terms of decimal places. Asked to find the square root of 12, students knew that it had to be more than 3 and less than 4 because 3 is the square root of 9 and 4 is the square root of 16. They put a guess into their calculator, squared it, and refined their guess when they saw the result. So a series of guesses might be 3.5, 3.4, 3.45, 3.452, and so on — going deeper into the decimal side of the place value chart and getting closer and closer with each new attempt. Everyone became very involved in the pursuit and begged for time at the end of class to try once more. “I’m almost there! I have 11 and 5 nines!”

 

We began working on spelling and vocabulary development this week. The foundation of our work comes from a series called Vocabulary from Classical Roots. Each of the lessons is based on a theme; this week it was seeing. Students were taught about the roots vis and spect, given a sample list of common and unfamiliar words that contain those parts, and completed a set of brief activities in the workbook that reinforced the meanings. At the same time, we put most of the word list into the week’s assignment at the website called Spelling City. (We also included a few unrelated words that students were going to need to write often, such as Oregon and prairie.) Parents are encouraged to look at the website and see not only what their child is doing with the assignment but also to learn about the many other language-based activities that are available there and which they may want to encourage their child to explore.

We have started reading a book to the class: The Amazing Maurice and his Educated Rodents by Terry Pratchett. The author is a prolific writer who combines satire, word play, memorable characters, and convoluted plots in all of his novels, most of which are set on a flat planet called Discworld. The novel that we are reading is a variation of the pied piper story, which most of the students vaguely recognized but needed explained. Pratchett’s humor is somewhat subtle, so it’s important for our students to be listening carefully and staying engaged to get the full breadth and depth of the book. There are likely to be other books by Terry Pratchett available at the upcoming book fair during conference week, and some of our class might be interested in getting one for their own pleasure reading.

Next week, we’ll be doing some preparation for the Terra Nova assessment that will take place during the week of October 13.  We’ll be sending parents information about that testing sometime next week. We’ll also be starting our newly-formed math groups that will blend students from both 5th/6th classrooms, as we explained in the message that went home on Friday.