A Favorite Task: Painted Cube for "Exploring the MTBoS"

I imagine most of you have heard of the "Painted Cube" task...if you create a cube out of "x" 1 inch blocks, paint it, and take it apart, how many of the block will be painted on 3 faces, 2 faces, 1 face, or no face.  I've used this task in a variety of different places in the curriculum...patterns, surface area, writing expressions, writing functions.

In the past I've given students a copy of a prompt along with a table to collect all of their data, and lots of guiding questions of what to look at, count, extend, etc. When students struggled (or just needed the extra visual or kinesthetic support) I brought out sets of cubes and had students build models. 

Last year I wanted to really get students asking the questions so I created the following models and let students decide what we were trying to find out....

What a difference this made....rather than filling out a chart because they had to, students were genuinely engaged in trying to figure out what was going on with these cubes.  Some students wanted to know how many of each color there were, others wanted to be able to build the next model in the series.

After students came up with their question, I did provide the chart to any students who asked for something to organize their data.  By then, most were interested in completing the chart and seemed to see it as a tool for answering their question rather than just a worksheet to fill out.

This year I'm just going to share this image and off we'll go!

Subtle Differentiation

This year we switched from 90 minute periods to 47 minute periods.  I am finding that the hardest part of the transition for me is differentiating the learning for students.  In a 90 minute period I felt like I had a lot of flexibility in how I structured lessons.  We had 90 one day and 45 the next, and I would plan the two together, doing most of the exploration, practice, discussion on day one and more application on day two.

Right now my classes are working on the 7th grade common core ratios and proportions domain.  Specifically we are exploring 7.RP.3 multi step ratio and percent problems.  I've introduced several different models- ratio tables, proportions, equations, and a visual method using a 100 square grid of graph paper.  Students are welcome to use any strategy when solving problems...most seem to be gravitating towards the ratio table as it gives them a little more structure when trying to interpret the problem.

Yesterday I wanted students to practice solving ratio/percent problems in a variety of contexts.  We used individual dry erase boards (actually Communicators- clear plastic sleeves- I like these better as I can slide graph paper or a blank ratio table in them).  I had my problems on a PowerPoint, and the process was to put a new problem up, walk around and question assumptions/strategies, and share the answer.

  Here is where the "subtle differentiation" came in.  I wanted to challenge students who were able to answer the question, but still be able to work with students who made a mistake or had a misconception. This year I've been making a conscious effort to focus on SMP 3- critiquing reasoning of others, so for each problem I created a "common misconception" response.  I asked students who had the correct solution to try and determine what had gone wrong. I've used a similar strategy using actual student work but wasn't sure I would have enough time to identify one during every problem.  I also used a variety of strategies in my misconceptions, so even though a student solved it with a ratio table, they had to analyze an error I made writing an equation to represent the situation.

It led to some interesting discussion and I was able to work with a few more students on the original problem.  Concept assessment Monday, so we'll see!

180- Day 5 Connecting methods for dividing fractions

First of all, I am still over-planning for my lessons.  I'm so used to planning for a 90 minute block instead of a 47 minute period.  I've been starting each class with Estimate180, which is fantastic!  Totally zero entry for the kids and fun!  It is turning out to be perfect for the beginning of class- takes just a few minutes so I can take attendance and maybe glance around for homework.  One class is just obsessed with Mr. Stadel though...they want to know where the pictures were taken, whether the picture is of his house/yard/etc, and one even asked if he was my brother :)

Today we did a little bit more exploring with dividing fractions and unit rate...worked on connecting the graphing method we did yesterday to the standard algorithm for dividing fractions.  I used 4 divided by 1/3 so we were doing a lot of circling of groups and they were happy for a more efficient method.

I've been using kind of an exit ticket in the INB each day...depending on where we get to it either ends the period or starts the next day.  Here's the example from today...I wanted students to summarize what they learned about unit rate and dividing fractions in order to compare two dripping buckets.  I had them use both the graphing method and the standard algorithm- from here they will be able to choose whatever method they find most efficient for any given problem.

180- Day Four- Visual Graph Paper Method for Dividing Fractions
In 7.RP.a.1, students need to be able to calculate unit rate, including rates that have complex fractions.  I tried a method for dividing fractions using graph paper that I found at Fawn Nyugen's site.  Definitely great for my visual learners.  Through the process I found students really just had trouble understanding whether the quotient should be a number greater than 1 or a number less than one, so that became our focus.  By drawing out the fractions using the same wholes, they could see which was larger.  I kept coming back to...is this similar to a "10/2" problem, which is more than 1, or a "2/10" problem, which is less than one.  I did really like the method though!

Some of my not so very visual students did really struggle with it.  Common misconceptions:  When students created the rectangles to represent the fractions some of them really struggled to draw an accurate representation of the fraction.  For example, if they chose a 4x5 rectangle because they were trying to model 4ths and 5ths, when they tried to shade 3/4 they would shade 3 sets of 4 squares rather than 3 rows out of the  4 rows in their rectangle.


This links to the activity sheet I used with my students.  The end of the document has pictures from Fawn's website!