# Conference Seating -3Act Math

ACT 1:

Recently I had to go to a conference and when I walked in, this is what I saw

What questions do you have when you look at this picture?

Ask students to write down their questions, I normally ask students to find at least 3.  When I observe that most students have questions written, I ask them to share those questions with their neighbor.  I then throw up a Microsoft Word document and start typing down questions students supply.  Students from my classroom came up with all sorts of different questions, some we can easily answer and others that we can’t.  I am looking for a key question or questions to start this lesson.  If students do not ask one of these questions, I tell them that I hope I can answer most of the questions provided, but that I need them to consider one of these questions first.

• How many tables are there?
• How many people are there?
• How many people are in a row?
• How many people can sit at a table?

Any of these type of questions will lead students down the inquiry I hope to explore with them.

There are 512 attendees at this conference, how many tables are needed to seat them all?

How would this change if I needed to seat 900 attendees?

ACT 2:

Linear Equations is the topic I wish to address with my students.  Students will need to think of counting strategies needed to determine the number of people at the conference.  There are many different combinations of ways to figure this number out, but here are the stats:

There are 8 people seated at the head table, but they needed to use 6 tables (extra space was used for the podiums). Otherwise there are 4 sections of tables, 9 rows of tables to a section, 14 people to a row, and the table size is 6′.

These are hard to identify by the picture so the students really work on the following pieces of information: 6′ tables, 9 rows and 512 attendees.

I also ask students to create an equation for the number of tables that are needed.

ACT 3:

For comfort reasons, 2 people are seated per table.

You will need 7 tables per row.

4 x 9 x 7+ 6 = 258 tables.

Students will typically write this type of equation based on attendees:

512 = 2x + 8

504 = 2x

252 = x

Then I typically ask them how many tables per row, and what type of configurations makes sense for the conference (does one LONG row really work?).

Extensions:

After we decide the general equation for the conference, I then ask the students how many tables are needed for varying amounts of attendees: 200, 900, 1200, 2500.  What type of configurations would work for these new conventions?

# Using Pictures to Solve Compound Fractions

Today I find that Bridging the Gap stole my thunder (sort of) by posting on a method I have been using in class recently.  We have been looking at compound fractions when writing equivalent forms of equations.  Changing a standard form equation such as 2x + 1/2y = 4 to slope-intercept has really thrown a curve to my students, and it’s all about one concept- fractions.  My students admit they struggle with fractions and then to create a compound fraction?  Most just put the pencil down, worry about other things and try to nap in class.  I have tackled the problem by using pictures of fractions to help students visualize what is going on.

I first start with an improper fraction example:

We work on dialogue on what 4/2 means, and students came up with the wording you now see on the board.  They then draw 4 circles and split them into groups of 2.

We then look at an example when there is a fraction in the denominator:

Students once again write their wording to the fraction and attempt to complete it as instructed.  I have had students struggle on how to make groups of 1/2, but when you ask them to draw 1/2 they easily give you a semi-circle.  They they have that aha! moment and quickly draw lines bisecting the four wholes.  They created eight 1/2 pictures.

Students really grasp the first two examples quickly, and we then throw them the curve, a fraction in the numerator:

Many students automatically say they can’t split a half of a circle into a group of four, so I will go back and change the fraction to a whole number, 4.  they quickly draw four lines between the wholes and say that this problem is nothing like the one I asked them.  I have them explain their process to find 4/4, and then write it as 1/4.  Some students will see that you can cut the circle into four pieces, but others will still struggle- and this is where as an instructor I have to realize that inadvertently I gave them a specific context in which to look at creating groups.  I gave them a circle, so I ask the students if we need to start with a circle, and many say no.  They draw a rectangle on the board and are able to make it into a group of four:

Once they see this, they quickly come to the idea that they can cut the half into four pieces with either a rectangle or circular representation.  They identify the new pieces as eighths.

We do a few other problems where students practice with both cases, and then I ask students to find a pattern.  Although they do not tell me to multiply by the reciprocal, students understand what numbers to combine and why they are doing so.

The next step, of course is to extend this concept beyond the unit fraction…