Getting Fancy with a LFHC Telescoping Task

Blog readers know that I have a definite "thing" for low floor high ceiling tasks as a way to unite a class and engage all students in high depth problem solving in a heterogeneous classroom (click here if you're not convinced).  Low floor, high ceiling problems are also an excellent means of formative assessment and a way to emphasize math norms that encourage growth mindset in a math classroom.

As we start off the year, we take stock of our students.  What do they do well and what do they need to work on?  We see students at different places, and we wonder how to create a structure that includes, engages and challenges all learners.

In this blog post, I offer up a telescoping task as a low floor, high ceiling instructional tool.  It is linked below, and for lack of a more creative name, I'm calling it: 

"Creating Fancy Duck Tape Problems"

This task could be used for any grade level from second grade (although maybe the number would need to change) up through algebra.  It can be used as a formative assessment to gauge your students' ability to apply math concepts along a progression ranging from place value to additive and multiplicative reasoning to fractions to ratios and percents all the way up to algebraic systems.  Teachers and students can assess based on conceptual mastery as well as use of these math practices:

• CCSS.MATH.PRACTICE.MP1 Make sense of problems and persevere in solving them.
• CCSS.MATH.PRACTICE.MP2 Reason abstractly and quantitatively.
• CCSS.MATH.PRACTICE.MP4 Model with mathematics.

I would launch this task with the whole class after (or maybe while) introducing students to Jo Boaler's "Positive Norms in Math Class" (presented in the Prezi below or you can use the 8.5 by 11 posters linked here). 

I would use math mindsets to frame the task and explicitly focus on these positive norms:

• Norm 3. Questions are really important.
• Norm 4. Math is about creativity and making sense.
• Norm 5. Math is about Connections and Communicating
• Norm 6. Value depth over speed.

To kick off the task, students are given some information about the amounts of (yes, you guessed it) fancy duct tape left on rolls in an art classroom.  The low floor, high ceiling prompt is to ask students what they notice about the data:

Students who are unaccustomed to open questions like this may balk.  So you may need to get them going.  I would work with the class to clarify what a "statement" is (soliciting some examples from the class) and then what a "math statement" is.  We'd come up with something along the lines of:

A statement is a sentence that is either true or false (but not both). A math statement is a statement that relates to mathematical relationships.  It can be written using words and/or symbols.

I would read the task information as whole class (regardless of the age level) and ask them to spend time independently noticing and jotting down math statements reminding them with the math practices and norms when appropriate.  I would use this first question as a formative assessment, checking to see the richness of the math language in the room.

After working independently, I would have students share their ideas, and then I would use popsicle sticks to randomly select folks to share out what they discussed.  We would collect these statements on an anchor chart (a big piece of poster paper that I would post electronically on a class site).  We would also create working definitions of the terms "expression", "equation", and "visual representation".  I would look for ways to bring in this language, post examples for students to take note.  For example:

Here are is a math statement in words: The length of cheetah tape roll is more than the combined lengths of the dotted tape and the paisley tape rolls.  

Here is a math statement written in symbols:  359 > 281 + 57.  This math statement is an inequality.  "359" and "281 + 57" are both expressions.  The ">" sign shows us that the expression on the left is greater than the expression on the right.  Is this math statement true?  How do you know? 

Hopefully, a student explains that 338 = 281 + 57.  Then you can post the equation and define the term equation as: a statement that two expressions are equal. 

When unpacking the term "visual representation," it may be worth having students watch Jo Boaler's "brain crossing" video.  This emphasizes the importance of multiple representations and modalities.

As you move to the task itself, rather than simply providing students with all of the levels  at once, you may want to feed them to students one by one to emphasize the "depth over speed" and "creativity and connecting" and "questions are important" norms.  It's important that students have a math as slow food rather than fast food mentality as they work through the task progression.  They will be asked to create questions starting from additive reasoning...

 on up to algebra...

 Teachers would likely decide to focus on just a section of the progression within that range.  Hopefully, the breadth and depth of this telescoping task can be a useful tool in a heterogeneous classroom.  It could be an extended exploration or an anchor activity.

If anyone wants help figuring out how to use it in your classroom or if you have any brilliant ideas, please send me an email.

Have fun and do great math!