Friday, October 27, 2017

Melons to Market, Reviewing with Khan Academy

Sam's 4th-6th Grade Math

This week in addition to the usual work in the texts we spent quite a bit of time investigating this situation:

A boy has 45 watermelons in the desert. He needs to get them across to the Oasis fair, 15 miles away. He can only carry 15 watermelons at a time, and he eats one watermelon every mile he walks, including walking back to where he started from. He can also leave watermelons at any mile he has walked, but no fractions of a mile. How many watermelons can he possibly take to the fair? How did you arrive at your conclusion?

The class spent a good amount of time, in groups of 2 or 3, working out how to even begin to approach this problem. Much of the class initially believed it to be impossible. By the end of the first day, everyone agreed that it was possible to get at least 2 melons to the market; the task for day two was how to determine the maximum number. Investigating this problem required much more than straightforward arithmetic; it's more of a logic puzzle and while working through it our class ended up discussing ideas like resupply/refuel depots for Antarctic exploration, note-taking/tracking techniques (how do you keep track of where all the melons are, and how many are left) and the important strategies of using manipulatives, working backwards, and looking at simpler cases.

Evie, Jacob and Ben used a number
line and manipulatives (beads) to help
them think about the problem.
Ishan and Juliana also used beads
to represent the melons. 

Algebra and Pre-Algebra

The seventh and eighth graders had a lot going on this week--two days with field trips and an extended OWL class that cut into our math time. Almost all the students completed an assessment this week and are either working on reviewing concepts using Khan Academy or are moving on to the next major unit of study: solving single-variable linear equations for the Pre-algebra group; linear equations in two-variables for the Beginning Algebra group; a detour into linear programming for the Completing Algebra Group.

Thursday, October 19, 2017

10/16 - 10/20 -- A Math Classic

Sam's 4-6th Math Class

This week in addition to our usual work in the texts, we looked at a classic math puzzle from antiquity. The common phrasing of the puzzle comes to us (translated from Latin) through the author Metrodorus:

'Here lies Diophantus,' the wonder behold.
Through art algebraic, the stone tells how old:
'God gave him his boyhood one-sixth of his life,
One twelfth more as youth while whiskers grew rife;
And then yet one-seventh ere marriage begun;
In five years there came a bouncing new son.
Alas, the dear child of master and sage,
After attaining half the measure of his father's life chill fate took him.
After consoling his fate by the science of numbers for four years, he ended his life.

We spent a good portion of the class decoding--discussing the phrasing of the problem and how to parse the words into mathematical notation--a skill critical for real world problem solving. The students rapidly found fractions they could extract from the text and with a little direction were generally able to make some kind of model to represent the situation.

Most of the students choose to treat the problem as they would a word problem from the Singapore texts and drew a model something like this:

But what do the orange regions represent?

The difficulty with this approach was how to think about the extra 5 years between his marriage and the birth of his son and the four years at the end of his life. Eventually, we got to the idea that the orange regions in the above picture must be those 9 (5 +4) years. By figuring out what fraction of his life these 9 years represented, the students could then figure out his age.

One group of students who have some experience with algebra and variables was asked to express and solve the problem algebraically. They, correctly, arrived at:
x/6 + x/12 + x/7 + 5 + x/2 + 4 = x 
which can be solved for the correct answer.

At the end of the lesson we spent some time talking about the similarities between the algebraic and pictorial representations. One of the foundational principles of the Singapore Math program is the idea that student learning should progress from concrete models (physical objects) through pictorial representations (bar models, other drawings) before students are asked to work on fully abstract, algebraic notation. Explicitly showing the students the connection between the bar models and algebra facilitates this transition.

Pre-Algebra and Algebra

The 7/8s had a number of special activities this week that preempted or cut into math, you can read about those in the 7/8 class section. Nevertheless,  all the students were at the end of a unit of study and are independently reviewing or working on a assessment (in this case a chapter test).  

Thursday, October 12, 2017

Counterfeit Coin Puzzles -- What's the Real Value?

There is a great tradition of mathematicians sending each other puzzles with simple guidelines and potentially deep mathematical content. One category of these are the counterfeit coin problems.

The basic phrasing is given N coins, a simple double-pan balance, and the knowledge that one coin is counterfeit, how many weighings is the least that will be required to be guaranteed we can determine the identity of the bogus coin.

We began looking at a few simple cases:
--often in math it is useful to consider a simple cases and then build up to more complex ones, even if the ultimate goal is to solve the more complex problem. (In this problem it is very useful to know how many weighings it will take with 2, 3 and 4 coins if we want to analyze larger numbers because a situation often arises in the middle of solving for a larger number that results in needing to identify the coin from a remaining group of 3, 4 or 5 coins. 

Case of 3 coins -- how many weighings will it take?

This one is pretty straightforward and the kids all figured it out. I'll leave it as an exercise for the reader.

Case of 9 coins, one is heavier. How many?
(If you work through this note how useful it is to already know how to do 3 coins.)

The added restriction makes this problem a little simpler. Again, I will leave it to the reader.

What is the most than can be done in 2 weighings?

This is not a trivial question, though it can be solved exhaustively. 

Throughout these exercises I emphasized the importance of accurate data recording and the value of finding useful ways of displaying the data. For this type of problem we learned about branching (tree) diagrams and how to work back through or data to check the validity of each conclusion.

It is doubly important for the children to see math as not always an open and shut, algorithmic process. While this problem does have an algorithmic solution, developing that algorithm is as much art as an exhaustive search for every possibility. Through these kinds of exercises the students can come to see the value of trying out different tactics because sometimes what doesn't work for one situation actually informs the next step and leads to a process that does work. Interestingly, in class we also tried to figure out the maximum number of coins that could be done in three weighings, given that we didn't know if the bogus coin was lighter or heavier; I hadn't looked at the solution in a while and we tried to re-derive it in class, but failed to arrive at the correct answer. Sometimes, 55 minutes just isn't enough time to try out all the possibilities.

Thursday, October 5, 2017

10/2 - 10/5 -- Fractions, Negative Numbers, Story Problems and Isolating a Variable

Sam's 4-6th Math Class

Another shortened week meant that many of the kids didn't get to finish their sticky note. The kids should still try to have their Singapore assignments completed for Monday 10/9.
Following on last week's discussions about the Bar Model strategy, on Monday and Tuesday we talked about using the bar model strategy to solve problems with fractions. The bar model is particularly useful for fractions and ratios as they are easy to represent pictorially; if Hermione has 3/4 as many Bertie Bott's as Ron, the kids can generally figure out to draw 3 boxes for Hermione and 4 for Ron. It is relatively straightforward, though it does require paying significant attention to the exact wording, for the kids to take this drawing and figure out the answers to a variety of different questions like:
Q: If Ron has 20 more than Hermione, how many do they have all together.
A: Ron has one more box than Hermione, so each box is 20; there are seven boxes, so the kids have 140 candies.

Q: If Ron gives Hermione 5 candies, they will have the same amount. How many candies did Hermione start with.
A: Ron has one extra box, to make him and Hermione even he'd need to give her half a box,; so half a box is 5, which means a box is 10, therefore Hermione has 10 x 4 = 40 candies to start.
The algebra to solve these kinds of problems is further removed from tangible objects. For the above problems you start with something like H = (3/4)R, which isn't too bad, but it gets rapidly more complex -- can you figure out how to represent and solve the second example problem?
It's in these years (8-12) that kids start being able to handle increasing levels of abstraction, but the more it connects back to a concrete understanding of the world, the easier time they tend to have.

Pre-Algebra and Algebra

The 7/8s had four full math classes this week and as a group are making good progress on their missions in the textbooks.

This week I introduced a bit more Khan Academy, particularly for the Pre-Algebra group who all benefitted from some reviewing of concepts about negative numbers which hadn't been introduced on the Singapore series.

The Beginning Algebra group continues to work with linear equations; the hot new topic for this week was solving generic equations for a specified variable.

​From my past experience I have found that this is a difficult topic for many children and requires a strong ability to think abstractly.

An example of isolating a variable by "undistributing".

The Completing Algebra Group has been looking at complex story problems in one and two variables and technique using charts for working with specific types of problems, particularly distance-rate-time problems. Feel free to ask them about the train leaving Chicago and the other leaving New York.