Lesson 3: Adding and Subtracting Decimals with Few Non-Zero Digits

Let’s add and subtract decimals.

3.1: Do the Zeros Matter?

  1. Evaluate mentally: $1.009+0.391$

  2. Decide if each equation is true or false. Be prepared to explain your reasoning.

    a. $34.56000 = 34.56$

    b. $25 = 25.0$

    c. $2.405 = 2.45$

3.2: Calculating Sums

  1. Tyler and Jada drew base-ten diagrams to represent $0.007 + 0.004$. Tyler drew 11 small rectangles. Jada drew only two figures: a square and a small rectangle.

    1. If both students represented the sum correctly, what value does each small rectangle represent? What value does each square represent?
    1. Draw or describe a diagram that could represent the sum $0.008 + 0.07$.
  1. Here are two calculations of $0.2 + 0.05$. Which is correct? Explain why one is correct and the other is incorrect.

Two calculations of zero point 2 plus zero point zero five are indicated.  The calculation on the left adds zero point 2 and zero point zero 5 by aligning the ones units, tenths unit, and hundredths unit. The sum is zero point 2 5. The calculation on the right adds zero point 2 and zero point zero five by aligning the hundredths unit under the tenths unit. The sum is zero point zero 7.

  1. Compute each sum. If you get stuck, draw base-ten diagrams to help you.
    1. The vertical calculation of zero point 1 1 plus zero point zero zero 5 is indicated by aligning the ones units, tenths units, hundredths units, and thousandths units.
    1. $0.209 + 0.01$
    1. $3.02 + 1.14$
  • The applet has tools that create each of the base-ten blocks. This time you need to decide the value of each block before you begin.
  • Select a Block tool, and then click on the screen to place it.

 

 

 

Click on the Move tool when you are done choosing blocks.

GeoGebra Applet FXEZD466

3.3: Subtracting Decimals of Different Lengths

To represent $0.4 - 0.03$, Diego and Noah drew different diagrams. Each rectangle shown here represents 0.1. Each square represents 0.01.

  • Diego started by drawing 4 rectangles for 0.4. He then replaced 1 rectangle with 10 squares and crossed out 3 squares for the subtraction of 0.03, leaving 3 rectangles and 7 squares in his drawing.

    A base-ten diagram labeled “Diego’s Method.” There are 2 columns for the diagram. The first column header is labeled "tenths" and there are 4 rectangles. The second column header is labeled "hundredths" and there are 10 squares in that column. The last rectangle is circled with a dashed line and an arrow pointing from the rectangle to the column of squares is labeled “unbundle.” The last three squares are crossed out.
  • Noah started by drawing 4 rectangles for 0.4. He then crossed out 3 of them to represent the subtraction, leaving 1 rectangle in his drawing.

  1. Do you agree that either diagram correctly represents $0.4 - 0.03$? Discuss your reasoning with a partner.

  2. To represent $0.4 - 0.03$, Elena drew another diagram. She also started by drawing 4 rectangles. She then replaced all 4 rectangles with 40 squares and crossed out 3 squares for the subtraction of 0.03, leaving 37 squares in her drawing. Is her diagram correct? Discuss your reasoning with a partner.

    A base-ten diagram labeled “Elena's Method.” There are 2 columns for the diagram. The first column header is labeled "tenths" and there are 4 rectangles. The second column header is labeled "hundredths" and there are 40 squares in that column. All four rectangles are circled with a dashed line and an arrow pointing from the rectangles to the column of squares is labeled “unbundle.” The last three squares are crossed out.
  3. Find each difference. If you get stuck, you can use the applet to represent each expression and find its value.
    1. $0.3 - 0.05$
    2. $2.1 - 0.4$
    3. $1.03 - 0.06$
    4. $0.02 - 0.007$

Be prepared to explain your reasoning.

  • The applet has tools that create each of the base-ten blocks. This time you need to decide the value of each block before you begin.
  • Select a Block tool, and then click on the screen to place it.

 

 

 

Click on the Move tool when you are done choosing blocks.

An image of a trash can labeled delete tool.

Subtract by deleting with the delete tool, not crossing out.

GeoGebra Applet WfP5JSmE

Summary

Base-ten diagrams can help us understand subtraction as well as addition. Suppose we are finding $0.023 - 0.007$. Here is a diagram showing 0.023, or 2 hundredths and 3 thousandths. 

Subtracting 7 thousandths means removing 7 small squares, but we do not have enough to remove. Because 1 hundredth is equal to 10 thousandths, we can “unbundle” (or decompose) one of the hundredths (1 rectangle) into 10 thousandths (10 small squares).

We now have 1 hundredth and 13 thousandths, from which we can remove 7 thousandths.

We have 1 hundredth and 6 thousandths remaining, so $0.023 - 0.007 = 0.016$.

Here is a vertical calculation of $0.023 - 0.007$.

In both calculations, notice that a hundredth is unbundled (or decomposed) into 10 thousandths in order to subtract 7 thousandths. 

Practice Problems ▶