# Monthly Archives: November 2012

## Seventh-graders don’t understand decimals

I mean, they really don’t fundamentally get what decimals are at their core, or how a numbering system works when it gets down to quantities less than 1.  This quarter, my intervention/enrichment time is spent teaching the Decimals and Powers of 10 module from Pearson’s Mathematics Navigator program.  It used to amaze me how many seventh-graders who I perceived as reasonably competent at math could not answer the screener questions for the module.

I love having the opportunity to work with students on filling these gaps, and I wanted to share some activities and ideas for building a strong sense of decimal understanding in students.  You can still build the connections to your grade-level topics –  algebra, and rates and ratios, and personal finance, and fraction operations – while continuing to strengthen students’ numeracy in our fundamental number system.

1)  Teach them another number system.  Have you considered allowing every student to learn binary, or hex, or even base-5 arithmetic?  Binary was a great launching pad for my Navigator class, because it brought out what really underpins a numbering system.  Each place value is the base raised to another power.  I started with this fun binary game that I had the kids pair up and play.  Then we moved on to the activities I found in this computer science worksheet.  The kids found it engaging and fun.  I led them through some follow-up problems with binary numbers that have a “decimal point”.  It’s possible!  The place values to the right of the “ones” place would be the 1/2 place, the 1/4 place, the 1/8 place, and so on.  I gave them a sheet of shaded areas and challenged them to write the shaded fractions as binary “decimals”.  Again, the whole purpose of this is to understand how place-value works in general and get a sense of how each place to the right of a “decimal” is a smaller chunk of a whole.  I use the word “decimal” in quotes because the point wouldn’t be called a “decimal point” in binary.  What is it called?  Let me know if you know.

2)  Break out the base-ten blocks.  The second-grade teachers in your building may have hoarded them to teach kids their place values that are bigger than ones.  You need to take a turn.  Hold up the big cube and tell the kids “this is one whole”.  Put them to work figuring out what fraction the other pieces represent.  They’ll likely struggle with this as kids who have difficulty with decimals are also not perceiving fractions well, but with support and creative questioning, they’ll be able to see that 10 of the flats make up the unit whole, so they’re one-tenth.  100 of the rods go into the whole, so they are one-hundredth.  1000 of the units go into the whole, so they’re one-thousandth.

I could not find a blank place value chart that lets you assign what place value the blocks are, so I made one.

The students put a big decimal to the right of the “ones” place.

Start by having the kids identify which amounts of blocks would match certain decimals.

Write on the board 0.1.  Say “Show me one-tenth of a whole.”
Write 0.01.  say “show me one one-hundredth of a whole.”
Write 0.05.  “Show me five one-hundredths.”
Write 0.008.  “Show me eight one-thousandths.”
Write 0.102  “Show me one-tenth and two one-thousandths.”
Next, you can have them try adding fractional amounts together to show how groups of smaller decimals combine to make bigger decimals (and, for example, why 0.06 is a larger amount than 0.012, which many don’t understand).

Have the students start with a smaller amount and then add on smaller amounts until they need to “swap” for the next place value up.

How many groups of 0.2 can you put together to make one whole?

How many groups of 0.012 can you add together until you have a group of thousandths you have to swap out?  What decimal do you have then?

How many groups of 0.125 add up to make one whole?  What fraction must 0.125 represent?

My fourth-grade daughter has brought home assignments using base-ten blocks, so I know her teacher is using them to introduce decimals.  I have enjoyed tying the middle-school concepts of fraction/decimal conversions and operations on decimals in with the blocks.

3)  Redefine what the “whole” is.  I gave the students transparency blocks and told them “one” had now been redefined as a flat piece.  They now had to search for the tenths and hundredths… and describe what a thousandth would look like.  They had surprisingly little trouble identifying tenths and hundredths, but struggled to define what kind of shape would be one one-thousandth of the flat.  Eventually, and with lots of creative questioning, they agreed a thousandth would be what you’d get if you cut a hundredth into ten equal pieces.

4)  Use number lines.  More than just a tool for elementary students to practice their counting, they’re an essential part of number sense.  Good mathematicians carry a number line in their head and use it to make sense of whole numbers, fractional numbers, and also negative numbers. I like this blank number line:

This number line, showing divisions down to the hundredths, can assist understanding of fractions and decimals.

I start by directing students to label the first thick line as “0” and the middle thick line as “1”.  Then, just as we did with the blocks, students work first to just be able to represent decimals on the number line.  Where is 0.1 (one tenth)?  How do you know it’s one-tenth of a whole?  where is 0.01?  Where would 0.001 be?  Find 0.4 on the number line.  Identify where 0.84 is.  Where would you put 0.25?  0.025?  0.785?

Ask students to put decimals in order using the number line.

The number line is powerful when you use it for skip-counting.  Ask the students to skip-count by 0.03 or 0.4 and read out the decimal representations of the numbers as they go.

One of my favorite uses of the number line is to convert fractions to decimals.  Ask a student to show where one-half is on the number line and identify what that is in terms of tenths, hundredths, and thousandths.  Look for harder fractions.  One-fourth must be 0.25 because if one-half is at 0.5, and you cut five-tenths in half, on the number line that would be halfway between 0.2 and 0.3, and so it’s two-tenths and five one-hundredths.  Could they find one-eighth on the number line?
5)  Use the blocks and number lines to reinforce fraction/decimal conversions.  We become so used to long division as an algorithm that we tend to forget what each step means.  Try long division with blocks!
Give a group of three a large cube and instruct them to share it evenly.  Hard to do, isn’t it?  Would you like me to swap it out for something smaller?  How many of the flats would you like so you can share it?
The group now has ten tenths.  these can be divided up so that each person has three tenths, or 0.3 for each person.  However there is one flat left.  Can I swap that flat out and give you hundredths?
There are now ten hundredths to go around, and each person can take three of them.  So each student now has three tenths and three hundredths, or 0.33.  However, there is one hundredth left.  Swap it out for ten thousandths, and then each person can have three thousandths and has a total of 0.333 with one thousandth left over.  By now we have run out of blocks to swap (and I guarantee at least one energetic student will volunteer to try and cut the thousandth into smaller pieces), but see if students have noticed the pattern and can describe the decimal representation of 1 divided by 3, or 1/3.

You might consider demonstrating the base-ten block division with long division in parallel, as it emphasizes what each step in the long division actually does.

Try base-ten block division with 1/6, 1/4, 1/8, or even 1/12.

Give your students the gift of being able to make sense of decimal numbers and what they mean.  Allow them to touch and hold and experience what our number system is all about.  Let me know if you have other ideas or resources for helping math make sense!