Back in mid-September, we described making and flying gliders in this blog. We averaged the data we collected from their flights, as that was one of the math concepts that we were trying to deepen and secure.

This week, we returned to the project to take it a bit further. Tony had done this sort of thing in the science room with students for many years, and he offered to come in and join us when we created “darts” and used PVC-pipe launchers to send them flying. After discussing it, we decided that having the students build the launchers would be a wonderful learning experience but would take more time than we could give it, so Tony built them for us and gave students a short explanation of the construction process. Because he used pieces of yardsticks for the track that directs the flight of the dart (or glider), students were able to gather data about how far back they were pulling the rubber band that powered the flight. The launch arm can be raised or lowered, so our next step when we return to this activity in a few weeks will be to measure and record the launch angle in order get the desired result — maximum distance or success at hitting a target, depending on our goal. We will launch gliders as well as darts, as the two will (should!) behave quite differently from each other.

In anticipation of working with launch angles, we spent some time in math class with protractors. Students became familiar with “benchmark” angles of 90 degrees and 45 degrees to help with self-checking and estimation. We reviewed (or for some, introduced) the skills required to use a protractor to measure and draw angles. We found that any quadrilateral that we drew had an angle sum of 360 degrees and talked about why. Some students remembered discovering last year that all triangles have an angle sum of 180 degrees. We’ll take this further when we do more geometry after Conference Week.

We took advantage of Tony’s visit to do some singing with him, too. We’ve been having a weekly sing in our building on Wednesday mornings, and it’s been a wonderful way to begin the middle of the week. If your child has come home singing fragments of “I’m My Own Grandpa” or “This Land is Your Land” — that’s why.

On the other hand, we hope your child is NOT bringing home any of the racist terms or profanity that we are encountering as we read To Kill A Mockingbird to the class. It’s a powerful story, and we are just getting into the court case that is the central event. We have started discussing what seems to be a theme in story — belief versus reality. Although it clearly applies to Jem and Scout’s characterization of their neighbor “Boo” Radley, we will soon see that it’s a theme which will emerge in the court case as well. And if you haven’t read this book recently, you might want to re-visit it yourselves. We’ll be seeing the film after we have completed the book.

One of our ongoing math topics is proportional reasoning, which we first discussed in this blog on October 7.  This concept has application in many places. We can use it to test the equivalency of fractions by “cross-multiplying.”  For example, if we want to see if 6/8 is equivalent to 9/12, we can write down the two fractions as a proportion and multiply each numerator by the opposite denominator. If the products are equivalent, the fractions are equivalent:

There are other ways to find this out, and at least one of then also involves proportional reasoning. I can see that 9 equals 6 plus half of 6. There is the same relationship between the denominators: 12 equals 8 plus half of 8. So the fractions (or ratios) are in proportion and are therefore equivalent.  (A third way to test for equivalency is to simplify both fractions, of course, which works well here but may not be so helpful with different numbers.)

Cross multiplication gives us more information than just simple equivalency. Let’s first look at why it works. There are no “tricks” in mathematics, by the way. Everything has a reason behind it, and nothing is based on magic or coincidence.

One way to create fractions with identical denominators so they can be compared, sequenced, added, or subtracted is to use the “quick common denominator” method. That is, we can change sixths and eighths into forty-eighths because both numbers are factors of 48. We know that because we got 48 by multiplying them. There is a lesser common denominator for 6 and 8: 24. But the “quick common denominator” is also valid, and it’s what comes into cross multiplying. The products we get are what the numerators of the fractions would be if we used their denominators’ product as the common denominator. In the case above, we would have multiplied the first fraction by 12/12 and the second fraction by 8/8. Since those are both equal to 1, we haven’t changed the value of the fractions — just the size and number of the parts.

If the cross-products are not the same, we get some more information. We find out which of the two fractions is greater. Often, that’s all we need to know about those fractions. We don’t always need to know how much greater, which would require more computational steps:

From the above, we can see that 6/8 would have a greater numerator than 8/12 when they were changed to having the same “quick” denominator, so 6/8 must be greater.

Another application of this method (and of proportional reasoning in general) occurs when we are looking for an equivalent fraction or ratio. For example, suppose a recipe serves 8 people, but we want to serve 12 people. The first ingredient is flour. We need 3 cups of flour for 8 servings. How many cups of flour do we need for the larger number of servings?

Cross-multiplication gives us a target numerator of 36 (3 times 12). So 8 times something (?) must also equal 36 if our recipe is going to turn out as expected. If we divide 36 by 8, we get the missing number in the proportion: 4  and 1/2. Again, we could consider the proportion another way: 12 servings equal 8 plus half of 8. So we need 3 plus half of 3 for the flour. Whichever way we do it, we are keeping the numbers in proportion to each other.

Finally, this finds one of its most useful applications in percentage calculations. Again, there are other ways, but setting a percentage problem up as a proportion always works. We always have 100 as one of the denominators because percent means parts out of 100. We are missing either the percentage, which is the number over 100, or the part (which is the numerator of other ratio) or the whole (which is the denominator of the other ratio). We can use cross-multiplication or our recognition of the relationships going across to find the missing part. Every time. One method for three types of problems. For example, let’s find the missing parts in this relationship: 17% of 544 is 32.

So we can use cross-multiplication to replace any one of the missing pieces:

On Grandparents’ Day this week, many of our guests were first bewildered and then delighted to see what our students were doing with this process. Some saw correctly that not every student is secure with this yet, but they also saw the value in it and participated in our children’s efforts to analyze our profits for each item in our lunch sale. The students were given an incomplete chart that told them what each item (sandwich, pudding, individual cookie, etc.) had cost for our previous Friday’s lunch sale. They were also told what we charge for each one. They were asked to calculate both the cash amount of our profit on each and the percentage of the selling price that was profit.

As they worked to complete this later in the day, they noticed that we are making a lot more money on puddings than on fruit cup. They also began to realize that the box lunch price raised some questions. We moved it along by asking them to calculate what our box lunch would cost if it were bought as separate items. It turned out —  much to my surprise a few days before as I planned this lesson — that we were charging too much for the box lunch! We had a good discussion about how this happened — it was mainly due to raising the box lunch price a couple of times when we raised other prices without looking closely enough at the real costs. Then we talked about what we should do. We agreed that the box lunch was intended to be a bit of a bargain, as bundled items often are, but that it also involved more labor and materials because I create and wrap the smaller sandwiches from the larger ones that our vendor makes. So we agreed on a price drop that still gives us a decent (but not indecent) profit.

The more we can make our math lessons real and practical, the more our children will value and comprehend the skills that they are getting.

Although a lot of our time in this 4-day week was taken up by the Terra Nova assessment, we did fit in some of our “usual” stuff. The wagon train adventure was one of them. One of the continuing elements of the multi-month simulation is the keeping of a diary. We are observing that many students have greatly increased the length, elaboration, and overall quality of their writing as they return again and again to reporting on the adventure through their chosen character’s eyes.

Some of our students began the year as enthusiastic, confident authors. Creating a simple “word problem” for math resulted in a page of rich narrative with the mathematics appropriately embedded. For them, the diary-keeping is just another opportunity to apply their skills and creativity and challenge themselves to stretch a bit higher.  But others were less at home with this kind of writing at the beginning, and many of them are showing impressive growth.

A couple of things are contributing to this, I believe. First, there is a bare-bones script in place. That is, instead of being faced with a blank page, students have notes about the day’s events (or have trusted to their memory, which is not as reliable an approach). Things have happened — difficult river crossings, missing children, snakebites, clogged trails, celebrations and losses — so they have something to talk about. At the same time, their narrator’s identity is clearly defined: age and gender, family structure, personality (created by them in their first entry and embellished since then). reasons for going west, and more. They have things — a wagon full of things — some of which are important to their journey, such as barrels for water and tools to clear the trail. So the richness of detail and presence of a narrator’s voice that helps to define a strong writer are partly in place, still subject to their own creative ideas and way of storytelling, but not needing to be invented in their entirety every time. And the repetition is a third important component. It’s not one story one time. It’s a continuing and repeated opportunity to spin a tale, add depth to another character, reveal more of one’s own history and interests, and paint a word picture that the reader can see.  Finally, the first-person books students chose to read have added one more piece to their resource pile. That is, they have a model — a book in which a character is reporting on a wagon train journey that is in at least some ways similar to their own. Decisions must be made, unexpected events must be coped with, and the individual personalities in the train must find a way to function as a community or bear the consequences of discord. What they have read (as several students have noted in their book write-ups) gives them ideas for what to write.

Note-taking is an important part of this activity, too. Even if students choose to take no or barely minimal notes for their diary as the events of the day unfold, they must take notes for trail decisions. Writing only what is essential but writing it accurately and completely enough is a skill that begins at this level and continues to develop into middle school.  It requires a metacognitive process that must go along with attentive listening — what do I extract from all that I am hearing, what are the relationships to note, what is irrelevant? Not every sixth grader is ready to do that — it’s a maturational issue and may need more time. So then you go to plan B — who in your group takes good notes? Get them to share. In time, you’ll be able to do it for yourself and help others. As you might expect, our students span a wide range of readiness for this, and we are seeing a lot of good teamwork evolving as a result.

So we’re feeling pleased with the way this entire learning opportunity is being embraced by many of our students. We hope you are following the westward adventure at home. We have a lot of the journey ahead of us, in every sense.

 

As always, we did a lot of different things this week. We chose recipients for our first Kiva loans from our hoagie profits, made a BIG decision in our wagon train simulation, and took a close look at the difference between the mean and median for a set of data (using statistics of the number of school days in various states in 1900 — quite a range!).

One of the major things we did was in math class.

We began a whole-group foray into proportional reasoning this week. (Separation into smaller, differentiated math groups is coming very soon, but we are doing things together right now. That means some students are reviewing, and some are venturing into new waters.) Proportional reasoning is incredibly important and practical. Students engage with it when they re-size an image on a computer. If they don’t keep the dimensions in proportion, it looks funny. But proportional reasoning presents a cognitive challenge as we bring it into math activities. First, it requires students to identify a relationship between two things (it could be more, but two are all we tackle), quantify it for one set of numbers, and create an equivalent set of numbers that have the same relationship. It connects with fractions, percentages, time/rate/distance, and much, much more. It may be one of the most useful concepts and strategies our students can master.

We started with a story, because narrative is always a good hook. Suppose we need firewood along the trail as we journey west. We have stopped for the evening, unhitched our animals, started the cooking fires. Then someone notices a dead tree not far from our campsite. It would be really good to cut down for firewood. But it’s tall. Will it fall on one of the wagons? It seems to be leaning in their direction, and no one is eager to re-hitch their animals and move to a secure place. But maybe it will miss. Maybe it won’t. Trees can fall in unexpected directions. We can measure the distance from the tree to the nearest wagon, but how tall is the tree? Opinions differ. Is there a way to be at least pretty sure?

We then went out and looked at a couple of trees on the wood chip field and asked how we might determine their height. One child suggested tying a string to a rock and throwing the rock as high as the tree, using the length of string required as the height. We didn’t try it, but most of us agreed it was probably not going to give us the information we needed. One suggested having someone climb the tree and drop down a rope or string, but we said the tree was dead and the branches were likely to be brittle. In one of our half-groups, one student remembered doing something like this last year with angles or triangles but wasn’t quite sure what it had been. After instructing students in a different method that is too complicated to explain here, we then turned to proportional reasoning. There was, unfortunately, no sufficiently clear shadow for any of our nearby trees, so we set up an analogous situation.

First, a student held a meter stick straight up, resting on the ground. Other students measured the length of its shadow. We could see that, at that time of day, the shadow was roughly half again as long as the meter stick. Then we measured a student’s shadow (while pretending he was our troublesome tree). We predicted his height, based on the previous activity, and we came up with a number that was accurate within a couple of centimeters when we then measured him.

We moved back into the classroom to summarize it and apply it to a series of problems, including some that had been collected from children’s copybooks from the 1840s. The principle is called “The Rule of Three.”  That means, if you have three numbers of a two-part proportion, you can find the fourth by some simple computation. It shows up as “cross-multiplying” when we create equivalent fractions or test for equivalency. Sometimes the relationship can be analyzed easily in other ways, but cross-multiplying always works. We encourage students to share other solution strategies when we discuss the answers.

Typical questions that students are solving include:

1)  If you can get 12 toys for $35.76, what will 10 toys cost?

2)  If a big truck can travel 85 miles on 7 gallons of gas, how much gas will it need to travel 100 miles?

3)  Chloe has saved $39 toward a zoom lens for her camera. If that is 30% of what she needs, how much does the lens cost?

During our classroom time with one of the half-groups, as we talked about the rate oxen walked (2 miles an hour) and said that going 15 miles a day was a typical goal. a student asked how far our wagon train was going to travel. We said about 2,000 miles. “How long would it take?” someone asked. Students picked up on the question and worked out a proportion that the constrained formatting of Google Docs won’t let us show here as we did it in class. Essentially, we figured that 15 miles a day meant that we would need about 133 or 134 days to travel 2,000 miles. That converted to about 19 weeks or about 5 months. Since we have already seen that there are some days with little or no travel, 6 months is a realistic estimate for the journey to Hacker’s Valley. This was a student- inspired, useful illustration of the application of proportional reasoning and another connection with our social studies topic.

We seem to be off to a good start, noting that some of our students also worked with this last year, while others are just making sense of it.  This is a skill and a broad concept with which we will be working in many ways through the year.

Thanks to Mark’s ever-present camera, we have some photos of one group’s tree-measuring activity.

 

 

Our week began, as it usually will, with our “Monday Chat.” We gather in the breakout room and hear things people want to share about their weekend. We are working on getting to know all of us more deeply as we learn about what people are doing and dealing with outside of school, and we are also working on developing more elaborated language. Tell the story. Give us some details. Describe instead of list. And we want questions and comments from the group — what did that make you wonder about? What else would you like to know? Do you have something to share that seems related? This is something that parents can help to support at home, too. Some of our children are very good at speaking fully and extemporaneously, but others need some help with making their narrative interesting, coherent, and complete.

 

The trip to the Franklin Institute was a high point of the week. Jonathan wrote about it this way:

So the other day I went to the Franklin Institute with my classmates and Mark and the new science teacher, Sue .  My class and I took two vans to downtown Philadelphia. When we got to the museum, we got our tickets and went to the exhibit.  When we got to the exhibit, there was a lady at the gate. She took us into a room. In the room there was a intro being played off a projector. The intro was so amazing!  It had John F Kennedy, Ronald Reagan , and more. It talked about spies in World War II. It also talked about the Central Intelligence Agency, also known as the CIA. They  talked about the FBI and Secret Service. I would go back any time!

 

Spelling has become an integrated part of our week, and we will start customizing the practice lists next week. Students who miss a lot of words won’t be asked to work on all of them. Students who get most of the pre-test right will practice only what they missed and perhaps a few challenges that we add in just for them. Whatever the assignment, students should SPREAD OUT their practice over the four days so that frequent recall with breaks in between will make the learning more durable. This is really important and may need parental reminding — you want to know these words a month and a year and ten years from now — not just for the week. Frequent short practice is the way to achieve that. We do give some class time for it, but it’s also an ongoing part of homework. Right now, most of the words in the list are ones children need to write for social studies. We spent time this week looking at the list and noting some of the rules and patterns that one or more of the words represents. The biggest stumbling block for some of our group is really a reading rule. The difference between HOPPING and HOPING or DINNER and DINER should be a solid concept by now, but we have some for whom it remains elusive. They tend to assume that what they meant is what it says instead of really reading it to check. We’re going to be working on these kinds of things all year. and everyone will move along from where they are.

The biographical posters of our wagon train individuals and families are up in our center hallway. They’re delightful. Please stop up and see them when you have a chance.