Discovering Decimals Part 2: Addition & Subtraction

by C. Elkins, OK Math and Reading Lady

Last week, we looked at some ways to gain number sense about decimals. This post will address using decimals in the operations of addition and subtraction . . . and how to model concretely and pictorially. You can also download the color grid pages along with a free decimal math game in this post. Part 3 (future post) will address multiplication and division of decimals.

If you missed last week’s post, please review it first before continuing with this one. Before performing various operations with decimals, students must have a basic understanding of how to represent them concretely, pictorially and numerically.  Example:  .8 = .80 can be proven with base ten blocks and with 100 grid drawings. This understanding should also be linked to fractions: 8/10 is equivalent to 80/100. Click here for pdf of Representing Decimals page.

Addition  

For concrete practice, use a 100 base ten block to represent the whole (ones), the tens rod to represent tenths, and unit blocks to represent hundredths. Construct each addend and then combine them. Ten tenths’ rods become one whole. Ten hundredths cubes become one tenth.

In a pictorial model, shade in the ones, tenths and hundreds on hundred grids. Use different colors to represent each addend. Click here for pdf of Adding Decimals page.

The concrete and pictorial models will also prove .8 = .80, reinforcing the concept that adding a zero to the right of a decimal does not change its value. This will be an important factor when moving to the standard algorithm vertical addition model.

Using an open number line is also a good pictorial model to use when adding decimals, especially if students are already familiar with its use regarding addition of whole numbers. This method reinforces number sense of the base ten system because you continually think, “What goes with .07 to make a tenth?” (answer = .03); or “What goes with .9 to make a whole?” (answer: .1).

A method called partial sums should also help students gain place value number sense along with addition of decimals. Each step is broken down (decomposed). 

Estimating will also be a critical addition problem solving step focusing on number sense. If adding 34.78 plus 24.12, does the student realize their answer should be somewhere close to 35 + 24? Are they thinking, “Is my first number closer to 34 or 35?”  What is halfway between 34 and 35 (34.5 or 34.50)? Where would 34.78 fall if it was on a numberline between 34 and 35? (See my post on rounding for more details.) Continue reading

Decimals: Part 1 – The Basics (revised)

by C. Elkins, OK Math and Reading Lady

Number sense regarding decimals usually starts with fourth grade and continues with more complex operations involving decimals in fifth grade and beyond. It is this extension of the place value system and then relating them to fractions and percentages that often perplex our students (and the teachers, too)!  Read ahead to get your freebies (Decimal practice notes, anchor charts, and Discovering Decimals Number of the Day / Game activity).  I have revised this previous post and included some more freebies below.

Students must understand  this base-ten value system extends in both directions — between any two values the 10-to-1 ratio remains the same. When using place value blocks in primary grades, students recognize the 100 square as 100, the tens strip as 10, and the units cube as 1.  Then with decimals, we ask them to reverse their thinking as the 100 square represents 1 whole, the tens strip represents a tenth, and the unit cube represents a hundredth.  This may take repeated practice to make the shift in thinking — but don’t leave it out. Remember the progression from concrete (hands-on) to pictorial to abstract is heavily grounded in research. Students will likely gain better understanding of decimals by beginning with concrete and pictorial representations.

I am sharing my decimal practice notes, which highlight some of the basic concepts to consider when teaching. Pronouncing the names for the decimals is not in these notes, but be sure to emphasize correct pronunciation — .34 is not “point three four.” It is “thirty-four hundredths.” Use the word and for the decimal point when combining with a whole number.  Example: 25.34 is pronounced “Twenty-five and thirty-four hundredths.” I know as adults we often use the term “point,” but we need to model correct academic language when teaching. You can get also the pdf version of these notes by clicking here: Decimal practice teaching notes. Continue reading

Discovering Decimals Part 3: Multiplication and Division

by C. Elkins, OK Math and Reading Lady

This is the last part of number operations dealing with decimals: multiplication and division in a concrete and pictorial method. This is actually not part of the KG-5th grade OAS standards, but it is addressed in 6th grade and for those of you utilizing the CCSS, you will find multiplication and division with decimals starts in 5th grade.  Or you may have advanced students who are ready to explore this concept. There are a couple of freebies included in this post. Read on to find them!

Multiplying Decimals:

Typically we teach our students that when you multiply 2 numbers together, the product is larger than the 2 factors. And when we divide two numbers, the quotient is smaller than the dividend.

Be careful about stating this generalization: This remains true when multiplying whole numbers (or even a combination of whole and decimal such as 5.2 x 6.4 = 33.28 in which the product is larger than either factor), but NOT with decimals or fractions less than 1 (example: .7 x .2 = .14 in which .14 is less than either factor .7 or .2). With division of decimals less than 1, the answer is often a whole number larger than either the divisor or the dividend (such as 5 divided by 1/2 = 10). This is a difficult concept, but modeling and practicing with concrete and pictorial models helps to see the reasoning. So, be careful not to say, “When you multiply two numbers together you always get a bigger number.” because it’s not always true.

To further understand this principal, let’s look at the use of the times (x) sign with whole numbers. The problem 5 x 2 could be shown in an array as 5 rows with 2 in each row (phrased as 5 “rows of” 2).  It can be shown in a set model such as 5 groups and each group has 2 (phrased as 5 “groups of” 2) . It can also be shown in an area model (box) as a shape divided equally into 5 rows and 2 columns (phrased as 5 “by” 2). Continue reading

Discovering Decimals Part 2: Addition & Subtraction

by C. Elkins, OK Math and Reading Lady

Last week, we looked at some ways to gain number sense about decimals. This post will address using decimals in the operations of addition and subtraction . . . and how to model concretely and pictorially. You can also download the color grid pages along with a free decimal math game in this post. Part 3 (future post) will address multiplication and division of decimals.

If you missed last week’s post, please review it first before continuing with this one. Before performing various operations with decimals, students must have a basic understanding of how to represent them concretely, pictorially and numerically.  Example:  .8 = .80 can be proven with base ten blocks and with 100 grid drawings. This understanding should also be linked to fractions: 8/10 is equivalent to 80/100. Click here for pdf of Representing Decimals page.

Addition  

For concrete practice, use a 100 base ten block to represent the whole (ones), the tens rod to represent tenths, and unit blocks to represent hundredths. Construct each addend and then combine them. Ten tenths’ rods become one whole. Ten hundredths cubes become one tenth. Continue reading

Number Lines and Rounding

by C. Elkins, OK Math and Reading Lady

I get requests from many teachers to help with instructional strategies regarding rounding, so I am happy to share my thoughts (and freebies) with you. Difficulty with rounding usually means students lack number sense. The essential goal of rounding is: Can you name a benchmark number (whole, tens, hundreds, thousands, tenths, hundredths, etc.) that a given number is closer to? I have found the more experience a student has with number lines, the better they will be with number sense, and the better they are with rounding to the nearest ___.  Then this rounding practice must be applied to real world problems to estimate sums, differences, products, or quotients.

When doing a google search for tips on rounding (ie Pinterest), you very often find an assortment of rhymes (such as “5 or more let it soar, 4 or less let it rest”) and graphics showing underlining of digits and arrows pointing to other digits. These steps are supposed to help children think about how to change (or round) a number to one with a zero. Many students can recite the rhyme, but then misunderstand the intent, often applying the steps to the wrong digit, showing they really don’t have number sense but are just trying to follow steps.

My answer (and that of other math specialists) is teaching students how to place any number on a number line, and then determining which benchmark number it is closest to. Continue reading to see examples and get some free activities.

Continue reading