Week of Oct. 10: wagon train and “You need algebra to solve that!”

Our simulated journey west: We’ve been on the trail for a couple of (virtual) months now. Students have faced a number of decisions, some of which were individual and some made by the entire team. One of their ongoing decisions relates to the differences among the families and lone travelers in each 9-wagon train. Some people have more money than others and can buy essential supplies without a lot of worry. Some people packed more wisely and have tools and equipment to clear the trail, deal with medical emergencies, repair wagons, and more. So the question becomes one of community support. You have what seems like a lot of food. Should you share it with a family that didn’t bring enough or lost things in the first river crossing? Or should you hold onto it in case you need all of it? Does your family come first? Will the people with insufficient supplies slow down the whole train? Sticking together seems to be safer for everyone, but that means you can go only as fast as the slowest wagon. As dilemmas such as these are discussed by the students, we look for opportunities to point out that this has connection to our world today. As they know from our Kiva donation research, many people live with much less than our families have. Are we all part of the same metaphorical wagon train?

A second type of decision involves choices. On the far side of the Sand Flats River, the trail splits. One was shorter on the map but is said to be very dry and is dangerous because it goes through a Native American burial ground. The other is much longer but parallels a river for much of its distance and is said to have no issues with the local people. Which one to take? Do we have to all agree? How will we decide? After lengthy discussions and a night to think about it, two of the trains chose the shorter trail while the other went for the longer one. As the simulation has progressed, both of those decisions turned out to contain some unexpected events, both good and bad.

A third type of decision our students have made requires some collective problem-solving. Some people have dogs with them, and they have been barking at night as they run around the prairie chasing coyotes and other animals. But people (and oxen) need their sleep. How can we manage the dogs? The two teams that chose the shorter trail have now been stopped by a large group of armed warriors who demand that the pioneers turn back. They explained that the last wagon trains that came through brought diseases that devastated their people. (We discussed this at some length — why might we feel perfectly well but make the people around us seriously or fatally ill?) So those two groups will share what they have decided to do in response and find the outcome next week.

There are several important learning experiences embedded in opportunities to make decisions such as these. They include: listening to other people’s ideas as well as promoting your own; remaining supportive of each other when a decision turns out to be a poor one (whether you were in favor of it or not); and weighing your options carefully, not only considering what may be good about them but also what might happen that is unwanted.

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“You need algebra to solve that!” Our blended-building math groups got started on Monday. Our 32 fifth and sixth graders have been split into 8 instructional groups that are led by the four teachers. One of our learning specialists (Jen) joins us when she can fit our morning classes into her schedule. The group sizes range from as few as two students to as many as five. This allows a lot of individual attention and lesson differentiation as well as giving every child many opportunities to explain their ideas and ask questions.

One of the things that all of our groups will be doing is working with a visual problem solving strategy often called bar modeling. It’s a distinctive part of the highly-respected mathematics curriculum developed in Singapore and is an instructional component in the Primary Mathematics series that is a major classroom resource. Frequently, when students ask for parental help with some of the problems they get for homework, they end up with a page that has a collection of equations (in the parent’s handwriting) and return to school with what may be a correct solution but no idea how it was derived. In such cases, after we have found other ways to solve the troublesome problem, we strongly encourage those students to go home and show their parents that it can, in fact, be done without formal algebra. Here is an example similar to a problem that one of our groups wrestled with last week:

Lisa and Sandy started out with the same amount of money. After Lisa spent $18 and Sandy spent $25, Lisa had twice as much money as Sandy. How much did each one have in the beginning?

The first step we ask students to do is to write a “solution sentence” with a blank for the answer:

Lisa and Sandy each had $___________ at first.

This is especially helpful to students when there are likely to be several steps needed to solve the problem because it ensures that they will complete the work and identify which of several numbers that they may have generated is the one they actually need to answer the question.

For this problem, a good first step is to draw 2 bars of approximately the same length and label them with all of the information we have. We can place question marks for the unknown value (what they started with).

bar1Then we can show on the bars what they each spent because that information was already provided. No need for any calculations yet. The drawings don’t need to be precise in their proportions — they’re just creating a visual representation of the information and the process.

bar2

We know that Lisa spent less than Sandy. How much less? Time to calculate. If she spent $18 and Sandy spent $25, Lisa spent $7 less. So she now has $7 more than Sandy. We also know she now has twice as much as Sandy. Time for a new set of bars, perhaps, where we show that Sandy’s bar is half as long as Lisa’s and that Lisa’s bar is $7 longer than Sandy’s:

bar3

If half of Lisa’s bar is $7, the whole bar must be worth $14, and Sandy’s must be $7. We can then check with our original numbers. They each had $32 to start. All done — and no formal algebra was required!

barmodel1

The primary value of this kind of work is in the step-by-step reasoning required. We aren’t expecting students to quickly have a solution strategy and an instantaneous answer. We’re expecting them to slow down and think. Students have to engage first with understanding the problem situation so they can draw it. Often, they have no idea of what steps may be involved to solve it. But writing the solution sentence gives everyone a place to start, and that’s important for confidence as well as for understanding. I don’t know what the answer actually is, but this is what I’m looking for. Then they make a diagram to illustrate the information that has been provided and note where there are any unknown values. Taking the time to do this makes it more likely that they will pay attention to all of the information and its relationship to the question. They next adjust or add to the diagram as they do calculations or record more information from the problem text. Sometimes, as in this example, more than one diagram may be helpful. There’s rarely just one successful way to model a problem visually. Students (and their teachers) learn a lot by analyzing and discussing each other’s way of thinking.

 

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