### 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:

We spent a good portion of the class'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.

**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).