Penny Jars and Cube Towers in Math

Our Math Workshop emphasis in recent weeks has been centered around analyzing relationships between two quantities in situations of constant change. These situations have started out very concretely, through using actual pennies and linking cubes, and then representing these situations more abstractly with arithmetic expressions.

In the above "penny jar" situation, there is a starting number of 5 pennies in the "jar". Then, 6 pennies are added each "round". The total number of pennies shown is (5 x 6) + 5, or 35 pennies. Students have been challenged to extend patterns such as this one, in order to determine the total number of pennies for any round. If asked to identify the number of pennies in the 20th round, for example, students would determine that there are 20 groups of 6 pennies (20 x 6), and then 5 more (+5) when including the "starting" pennies in the jar: 20 x 6 + 5 = 125. There would be 125 pennies in the jar after 20 rounds of adding 6 pennies each round. The most general arithmetic expression to represent "any" round might be n x 6 + 5, or 6n + 5.

These Investigations (from our Scott Foresman curriculum tool) have provided tables, such as the example shown above, in helping students make sense of (and represent) these situations algebraically.

In the above "double tower" situation, there are two skylights. As new "floors" are added to the double tower, the number of skylights never changes. Students have learned to represent this as "+2". Each time a floor is added, however, the double tower gains 6 new windows. The example above has 2 floors. This would be represented as 6 + 6 + 2, or 2 x 6 + 2. There are a total of 14 windows on this tower. Again, students have learned through active exploration (by actually constructing various towers with linking cube manipulatives as they fill in provided charts) how to represent this situation with a general arithmetic expression: n x 6 + 2 (6n + 2). How many windows would be on the 100th floor of this tower? To solve, students would simply calculate 100 x 6 + 2. A 100 floor double tower would have 602 windows.

Consider the two towers shown below: The Square Tower and Corner Tower.

How many windows (including skylights) would each of these towers have if they were 100 floors tall? Leave a comment with your answers. Be sure to share your thinking on how you arrive at your totals!!

Happy mathematizing!

The Narrative Side of Nonfiction

Readers have done lots of digging into nonfiction this year! Recently, we started to uncover another side of nonfiction, a genre called Narrative Nonfiction. These are true stories where we learn factual information, but they are told in the format of a narrative (story). We usually don't see many text features in them like we do in our "all about" books.

We can summarize a narrative nonfiction text in one sentence,
just like a narrative, by using the
"Somebody, Wanted, But, So" sentence frame.

We have also learned that narrative nonfiction, or biography, books can be divided into two piles: achievement texts and disaster texts. An achievement text will tell us how a character overcomes conundrums and obstacles to achieve something. In the beginning, we are usually introduced to traits of the character, sometimes shown through ministories about their childhood. The character usually goes on to use these trails to overcome difficulties. Disaster texts tell us about a situation that gets worse and worse for the character(s) and does not end well. We looked at the book, Pompeii: Buried Alive, and noticed some structures of a disaster text. It usually begins with an "all is well" mood, where the character is in a perfect environment. It gives us a sense that something is going to happen to ruin the perfection that is being described to us. The beginning of this book is all about the city of Pompeii and how daily life was moving on as normal. We could feel that something big was about to happen to make it a disaster text, and not an achievement! We used our reading sense!

Narrative nonfiction texts have underlying ideas. We
can uncover those ideas by investigating a character and
their choices, challenges, and lessons!

We have read MANY mentor texts to help us expand our reading minds into this genre! There are lots of biographies to explore, also! Some of us have realized that this is our favorite genre. It has definitely deepened our love of nonfiction!

The choices are endless when it comes to finding narrative
nonfiction in our classroom!

Readers, which books have you come to love in this unit? Has it opened your eyes to anything new? What are some cool things you have learned from reading these books?