Absolute and Relative Thinking

Books:

Lamon, S.J. (2010). Teaching fractions and ratios for understanding: Essential content knowledge and instructional strategies for teachers, 2^nd ed. New York: Routledge.

•         Chapter 3 (pages 29-38) offers further explanation of what additive and multiplicative (or absolute and relative) thinking are and how teachers can encourage students to use multiplicative (relative) thinking in their classrooms.  Several activities for classroom use and discussion are provided. 
•     You can find this book on Amazon by clicking here if you're interested in purchasing it or would like more information.

Links to Lessons and Teaching Activities:

NCTM Illuminations 5 lesson unit taking students through the ideas of renaming and changing the unit whole using pattern blocks to get them ready for more challenging relative thinking problems.
There are four different activity sheets that you can download and use in your classroom as well as an interactive "Patch Tool" which allows you to create patterns with pattern blocks or use them to illustrate fractional concepts.

·      http://illuminations.nctm.org/LessonDetail.aspx?ID=U113

Teaching Activities:

Figuring out how to walk students through the process of absolute thinking into more relative thinking using resources already available to you can be tricky!  On slide 17 of the Absolute and Relative Thinking Learning block we give an example of a typical problem you might see in a text book and suggest that you can push students to think more relatively by asking them to interpret the picture in terms of thirds.  Easier said that done right?  Here are some tips on how you might try this in your classroom.

• Begin by connecting the idea from the renaming activity (shown in The Equal Sign and Equality learning block) to fractions.
• Tie in the idea of unitizing with fractions (discussed in more detail in the Units, Ratios, and Rates learning block).  

This activity uses Unifix Cube fraction bars to help guide students in their exploration of renaming fractions to allow them to begin thinking more relatively when dealing with fractions.  By the end of this activity students should be able to:

• demonstrate ways to rename fractions using the fraction bars 
• find relative values when the unit whole changes

1. Ask your students to use the fraction bars to find as many different names for 2/3 as they can.

2. If a student finds that 8/12 is another name for 2/3, ask them what fraction of each third does 1/12 represent?

In order to make this less confusing, you may want to turn the fraction bars so that the numbers are no longer showing.  

What you are doing is changing the unit whole to 1/3 and asking students to think relatively in order to determine the relationship between 1/3 and the twelfth pieces.  

Once they've figured out that 1 black piece is 1/4 of one pink, help them talk about this in relative terms by reintroducing the numbers from the fraction bars so that they understand that 1/12 is a fourth of 1/3.  

Another way of thinking about this is to say that one fourth of 1/3 is another name for 1/12.  This will also come in handy once you start working with multiplying fractions.

3.  Now if you really want to challenge their thinking change the unit whole to 3/4.  

(Again you may want to turn the fraction bars so that the numbers don't show if you find that your students are having a hard time thinking of this as 1 whole instead of 3/4.)  

Ask them to use what they have just discovered about the relative relationship between the twelfths and 1/3 to figure out how many thirds there are in this new unit whole (3/4).

The will see that 3 thirds is too much.

Then they will likely figure out that 2 thirds is not enough.  

Encourage them to use their other fraction bar pieces to find the missing piece that will make both sets of fraction bars equal in length.

They will see that the 1/8 piece is too big or too much.

They will see that the 1/10 piece is still just a bit too big, but closer than the 1/8 piece.

Eventually they will find that one of the twelfth piece fits just right.  Help them state how many thirds there are in this new whole.

There are 2 thirds and a black twelfth piece.  

Ask your students to think about what the relative value of that black piece was when 1 pink piece was the whole.  If the black twelfth piece is 1/4 of a third, then this unit whole of 3 yellow pieces (or 3/4) can also be named 2 and 1/4 thirds.