The Last Number--an exploration

Just found this problem recently, and thought about what a great exploration it would be--for nearly any age!

Consider the string 1, 2, 3, 4, 5, 6, 7, 8, 9, 10. Cross out any two numbers in this list and add the difference to the end of the list. This new number is now part of the list. Continue the process of crossing out two number on the list and adding the difference until there remains only one number. What can you say about the last number? Explore. [from Richard Hoshino]

This problem offers a rich exploration of number while practicing basic subtraction facts. (Making it great for 2nd grade!).  

Once your students have a conjecture about the final number, can they try that conjecture out on a different string of numbers?  What happens when your string goes from 1-15?  How about a string from 3-12?  Can they figure out what the pattern is and why the difference ends up like it does?  Even if they can't, they should have some rich time practicing subtraction, working together, and looking for patterns!

Can you figure out what is happening?  If you are like me, you might spend a lot of your free time working on this problem to see if you can make sense of it!

Building Conceptual Understanding of Fractions

In 3rd, 4th, and 5th grades, fractions dominate much of our instructional time, and it is important that we work to help our students gain a firm grasp on the concepts of fractions. Going deep now will help them down the road as they use fractions in more complicated mathematical situations.

A reminder of the progression of fractions in elementary school by Graham Fletcher allows us to remember how important our models are:



Sometimes I'm afraid that teachers steer clear of fraction exploration because they are uncomfortable with them themselves.  Imagine all of the learning that could occur if we went out of our own comfort zone with our students?  

I have shared Fraction resources before in this blog.  I have also shared an exploration or two.  Here are some more resources that might help your students (and you) to make more sense of fractions.

When and when not to give the answer:  This Marilyn Burns' post offers an opportunity for your students to build their own understanding of fractions.

Fractions, Decimals, & Percentages:  A number talk which begins with fraction addition.

Exploring Fractions:  An article from nrich.maths.org which offers links to rich tasks that develop a deep understanding of fractions.

Illustrative Mathematics also has some resources that will help you better understand the horizontal progression of fraction skills.

Maybe you find one or two ideas above to help you stretch during your fractions unit--that's great!  Don't try to change everything all at once. As always, let me know if there is something I can do to help!

Does it look right? Does it sound right? Does it make sense?

These words were always such a big part of my instruction of reading workshop.  Students used hand signals to help them remember the three questions as they reviewed what they had just read.  Many times, they realized the word they had just read did not sound right or make sense in the context they had just read.  So---they worked to figure out what the new word was that they were struggling to read.

Do we teach our students to do this same kind of thinking when solving a math problem?  In my experience, it is a less common practice.  Sometimes, a teacher will ask the students if the answer makes sense, but really, shouldn't we be teaching our students to ask themselves that question?

Tying our math instruction to real world situations allows for the students to better make sense of their answer.  We need to be intentional in our plans to teach students to make sense of situations and solutions. One of our biggest goals during math instruction is for our students to EXPECT that their answer will make sense.