Using Number Pieces for Work with Decimals

I've written before about Number Pieces - one of my favorite apps!  (It is free - I hope you give it a try!)  While there are obvious connections to the primary grades for this app, how might you use it in the intermediate grades?

One possibility is for addition and subtraction of decimals.  Using base-10 blocks for decimals is a shift for students.  Since early school experiences, many students have thought of the unit (or small cube) as "one".  Before using base-10 blocks with decimals, it is important to give students opportunities to explore changing the meaning of "one".  If "one" is the long/rod, what does the unit cube represent?  If "one" is the flat, what does the long/rod represent?

After students have a chance to explore these relationships, the base-10 blocks can be a useful tool.  For example, in the problem "Mary bought 9 pens.  Each pen cost $0.35.  How much did Mary spend on the pens?" students might use repeated addition to solve the problem.  They might set up the problem to show that the flat will represent $1.00 and 3 tenths and 5 hundredths represent the cost of one pen.

Then students might draw 9 representations of 3 tenths and 5 hundredths to show the 9 pens.  They can show the repeated addition as 0.35 x 9.

If you use the "lasso" tool, you can group all of the blocks together and then click the "join" button.  This will group the tenths into ones, as well as the hundredths into tenths.

Using the "lasso" again, the app will join the tenths into ones.  The student can then see that 0.35 x 9 is equal to 3.15.

This app provides students with a tool to visualize joining situations (and separating situations if you use the different colors) with decimals.  This can be an engaging tool for students and help students create mental representations of the problems they are solving!

Learning Progressions and CCSS

In the past few weeks, several of my friends have shared examples that have been posted about math problems.  A concern is that Common Core is changing the mathematics that we teach.  Here is one example:

My first thought about this exercise is that the "old fashion" way is obviously simpler - especially if one already knows the traditional algorithm (or method).  The "new" way is a form of counting on and takes more steps than the previous method.

I also notice that the problem is very simplistic - there is no need for regrouping in this example.  Both methods are valid ways to solve the problem.  Students that are fluent in using counting on would be able to solve this exercise quickly and efficiently by counting on.

Now, consider the problem 62 - 45.  When regrouping is involved, this is a more complex problem.  To use the "old fashion" way shown above, one would need to regroup from the 6 tens.  Now there are 5 tens and 12 ones (or we've changed 60 + 2 to 50 + 12).  Then one needs to subtract 12 - 5 in the ones place to get 7.  Next, one needs to subtract 50 - 40 to get 10.  So the answer is 17.

In using the "new" way, one could count up from 45 to 50 (adding 5), then from 50 to 60 (adding 10), and finally from 60 to 62 (adding 2).  5 + 10 + 2 is 17.

For students with strong number sense, the second method is quick and efficient.  This method is also less prone to regrouping errors.  Students should use a method that is efficient, mathematically valid, and generalizable according to Campbell, Rowan & Suarez (1998).  This description can be applied to both of the solution methods shown in the above photo.

Another concern I have heard from parents is that Common Core does not teach the standard (or traditional) algorithms.  A quick review of Common Core State Standards will demonstrate that this is NOT an accurate statement.  In grade 4, students are expected to add and subtract multidigit whole numbers using the standard algorithm.  In grade 5, students are expected to multiply multidigit whole numbers using the standard algorithm.  In grade 6, students are expected to divide multidigit numbers using the standard algorithm.

So, what are students doing prior to grade 4?  In the primary grades, students should be developing strong number sense and inventing strategies (student-invented, not teacher-demonstrated) to solve contextual problems.  As students' understanding of place value deepens and strategies become more sophisticated, they will transition to expanded algorithms.  These expanded algorithms reveal place value and what happens to numbers when we operate upon them.  When students have a strong understanding of these expanded algorithms, it is a natural transition to move to the standard algorithms.  Michael Battista writes extensively about these learning progressions in his work with Cognition-Based Assessment.

These progressions are based upon research about mathematics and how children learn.  As educators, we need to do a better job helping parents understand why our teaching methods have changed and demonstrate that their children will be proficient mathematicians!