# Discovering Decimals Part 2: Addition & Subtraction

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.

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

Last time I focused on some basics about learning the number bonds (combinations) of 10 as well as adding 10 to any number. Today I want to show the benefits of making a 10 when adding numbers with sums greater than 10 (such as 8 + 5).  Then I’ll show how to help students add up to apply that to addition and subtraction of larger numbers. I’ll model this using concrete and pictorial representations (which are both important before starting abstract forms).

Using a 10 Frame:

A ten frame is an excellent manipulative for students to experience ways to “Make a 10.” I am attaching a couple of videos I like to illustrate the point.

• Model this process with your students using 2 ten frames.
• Put 8 counters on one ten frame. (I love using 2-color counters.)
• Put 5 counters (in another color) on the second ten frame.
• Determine how many counters to move from one ten frame to the other to “make a 10.” In this example, I moved 2 to join the 8 to make a 10. That left 3 on the second ten frame. 10 + 3 = 13 (and 8 + 5 = 13).

The example below shows the same problem, but this time move 5 from the first ten frame to the second ten frame to “make a 10.” That left 3 on the first ten frame. 3 + 10 = 13 (and 8 + 5 = 13). Continue reading

# Fractions Part 5: Equivalent Fractions

This is part 5 of a series of fractions posts. Thanks for sticking around! Through explorations with fraction manipulatives, pictures, and drawings, we hope students begin to notice there may be different ways to express the same area using fractional terms. To cut a sandwich into halves and eat one of the halves is the same as cutting the same sandwich into fourths and eating two of the fourths. Read on for several freebies about equivalent fractions.

Students can gain experience finding equivalent fractions using models in several ways: Fraction strips, area models, set models, bar / length models, and number lines. Then with a strong understanding using concrete and pictorial models, the student is ready to apply paper-pencil methods to name equivalent fractions. But remember to use the same size whole: As in this picture using pattern blocks, 2/3 of a trapezoid is NOT the same as 2/3 of a hexagon.

Fraction Strips: If you don’t have sets of fraction strips, here is a free resource Fraction, decimal and percent strips charts via Kim Tran (TPT).  Commercially available strips are also nice. But, with some 1″ strips of construction paper, students can create their own and probably learn a lot about the relationship between halves, fourths, and eighths as well as thirds, sixths, and twelfths in the process of partitioning and cutting them.

TIP:  Make sure students know how to read a fraction strip chart. A couple of years ago I was working with a third grade class and assumed they could readily see that 5/10 was equivalent to 1/2. But after confusing looks, I realized that I needed to physically show them how to follow a line vertically down the chart to find other fractions that were in line (by placing a ruler or long pencil along the vertical line). Another example.  “To find another fraction equivalent to 1/3, find the line at the end of the 1/3 section and trace it vertically down the page to see if there are other fractions that stop along that same line. You should see in the sixth’s line that 2/6 lines up, and in the ninth’s line that 3/9 lines up and in the twelfths line that 4/12 lines up.”

After cutting and labeling strips, then explore equivalent fractions (those with the same size length). Students should be able to generalize that different fractions can used to represent the same area.

Area models:  By covering or partitioning shapes, students should notice that even though the same area is covered or shaded,  the number and size of the parts can change. Continue reading

# Discovering Decimals Part 2: Addition & Subtraction

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.

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

# Discovering Decimals Part 1: Basic Concepts

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.

Anchor charts are excellent ways to highlight strategies in pictorial form. Here are some examples of anchor charts to help students relate decimals to fractions, location on a number line, word form, and equivalencies. Get the free pdf version here: Discovering decimals anchor charts. It includes a blank form to create your own.

In this model, I chose the 1000 cube to model 356 thousandths. It’s a little tricky – be sure to see that the 300 part is shaded all the way (front and top – picture 3 slices of 100), the 50 part is shaded (front and half the top – picture half of a 100 slice), and the 6 part is just shaded in the front (picture 6 individual parts). The entire cube would represent 1 whole.

Here’s a matching activity / game in which students match decimal to fraction, word form, expanded form, money, and pictorial form. Included is a blank page so you can make your own or have students take notes. Click here for the FREE activity:  Decimal, Fraction, & Money Match

Another resource (\$2.50 at TPT from Joanne Miller) to help students relate the decimal to the pictorial form:Decimal 100 grid Scoot

Finally, below is an activity to practice or reinforce decimal concepts. The page showing can be used as a “Number of the Day” practice. I also created a game using this model, and the whole packet is included in this free pdf. Click here: Discovering Decimals number of the day and game

For more teaching help (videos and interactive models) for decimals, check out the following 3 free resources. These are also listed in my resources section of the blog (top black bar):

As always, you are welcome to share your decimal discovery ideas. Just click the comment box speech bubble at the top of the article or the comment box at the end of the article.