The Number Line in the Common Core

Rationale

Many of my 8^th grade students come to my classroom with gaps in content knowledge, especially in understanding basic mathematical concepts and operations.  Since Shirakawa is a K-8 school, all three 8^th grade classes have homeroom teachers and students are not regrouped by math performing levels.  The academic gap seems even larger when there are 32 students with heterogeneous mathematical backgrounds.  A lot of 8^th grade students have difficulty developing a conceptual understanding of decimals and decimal operations and often treat them as an isolated mathematical concept.  Students should understand that mathematics is just a collection of rules, but it is logical and makes sense.  Most of the mathematical mistakes my students produce are due to memorizing steps and rules when solving problems.  One dominant misconception my students have when solving decimal problems is about dealing with the decimal point.  I conducted an informal survey online asking students what the most confusing part is when working on decimal operations.  Here are some of the most common responses I have received: ⁴

When I first started learning about decimals, I didn’t know how to line up the decimals for addition and subtraction.  And for multiplication, I didn’t know where to put the decimal points when I got the product.  – Student A

The most confusing thing for me when I started learning about decimal operations was when “carrying over” in addition and subtraction over the decimal point.  For example, 1.37 + 15.8 =?  – Student B

Multiplication and division were worse for me because of how the decimal point behaved.  All of a sudden, instead of remainders, you have to move a decimal point and add zeroes everywhere and the re-put the decimal point. – Student C

It feels like someone up made up the rules as we went along when it came to where the decimal was placed after the calculations.  Sometimes it goes in the first spot and sometimes it depends on how many decimal places there were.  – Student D

According to my students’ comments, the origin of trouble comes from not paying close attention to the decimal point.  Decimal points give concrete values to each digit of the number; as the decimal point gets moved one digit to th­­e left, the place value becomes ten times multiplied and as it gets moved one digit to the right, the place value becomes one-tenth of the previous digit’s value.  This relation stays exactly the same in every scale in all numbers.  For example:

Many complain when they do not reach the correct answer adding and subtracting the decimals and would say, “I line up the numbers on the right as I was taught before,” but the ‘lining up the numbers on the right’ rule only applies to whole number operations.  When performing decimal computations, numbers must be lined up by the decimal to add digits that are multiplying the same base-ten unit.  This rule, however, often is not made clear to most students.  Students’ lack of understanding of decimal place values results in (1) not counting the correct number of decimal places, (2) not aligning the decimal point when adding or subtracting decimals, (3) not understanding each decimal place is divided into tenths (the magnitude of base-ten scales), and (4) not being able to explain “how” and “why” when operating with decimal points.  The number line can be a useful tool in showing the interrelationships of decimal numbers by identifying each decimal point and demonstrating decimal expansions to illustrate how decimals fill in each number line segment. 

Our current curriculum, CPM Core Connections, encourages students to be responsible for their own learning through working together in small heterogeneous groups of four.  My role as a teacher in this unit is to provide support and guidance in student team discussions and present an opportunity to think and examine how decimals are recorded on the number line system.  Students will (1) demonstrate their ideas, (2) listen to what group members have to say, (3) see how the same problem can be solved in different ways by seeing their teammates’ work, and (4) ask questions to resolve mathematical conflicts together.  This curriculum unit will assist me to meet goals of the Common Core Standards and to design a more student-centered learning environment to encourage students to take an active role in cooperative learning.