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

Learning Goal

Let’s add and subtract decimals.

Learning Targets

  • I can tell whether writing or removing a zero in a decimal will change its value.

  • I know how to solve subtraction problems with decimals that require “unbundling” or “decomposing.”

Warm Up: Do the Zeros Matter?

Problem 1

Evaluate mentally:

Problem 2

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

Activity 1: Calculating Sums

Problem 1

Andre and Jada drew base-ten diagrams to represent . Andre drew 11 small rectangles. Jada drew only two figures: a square and a small rectangle.

Andre 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?

  2. Draw or describe a diagram that could represent the sum .

Print Version

Andre and Jada drew base-ten diagrams to represent . Andre drew 11 small rectangles. Jada drew only two figures: a square and a small rectangle.

Andre 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?

  2. Draw or describe a diagram that could represent the sum .

Problem 2

Here are two calculations of . 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.

Problem 3

Compute each sum. If you get stuck, consider drawing base-ten diagrams to help you.

  • 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.

open applet in presentation mode
  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.

Print Version

Compute each sum. If you get stuck, consider drawing 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.

Activity 2: Subtracting Decimals of Different Lengths

Problem 1

Diego and Noah drew different diagrams to represent . Each rectangle represents 0.1. Each square represents 0.01.

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

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 to represent 0.4. He then crossed out 3 rectangles to represent the subtraction, leaving 1 rectangle in his diagram.

Noah's method is four rectangles representing tenths with three crossed out.

Do you agree that either diagram correctly represents ? Discuss your reasoning with a partner.

Print Version

Diego and Noah drew different diagrams to represent . Each rectangle represents 0.1. Each square represents 0.01.

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

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 to represent 0.4. He then crossed out 3 rectangles to represent the subtraction, leaving 1 rectangle in his diagram.

Noah's method is four rectangles representing tenths with three crossed out.

Do you agree that either diagram correctly represents ? Discuss your reasoning with a partner.

Problem 2

Elena also drew a diagram to represent . She started by drawing 4 rectangles. She then replaced all 4 rectangles with 40 squares and crossed out 3 squares to represent subtraction of 0.03, leaving 37 squares in her diagram. 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.

Problem 3

Find each difference. If you get stuck, you can use the applet to represent each expression and find its value.

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.

open applet in presentation mode

Print Version

Find each difference. Explain or show your reasoning.

Are you ready for more?

Problem 1

A distant, magical land uses jewels for their bartering system. The jewels are valued and ranked in order of their rarity. Each jewel is worth 3 times the jewel immediately below it in the ranking. The ranking is red, orange, yellow, green, blue, indigo, and violet. So a red jewel is worth 3 orange jewels, a green jewel is worth 3 blue jewels, and so on.

At the Auld Shoppe, a shopper buys items that are worth 2 yellow jewels, 2 green jewels, 2 blue jewels, and 1 indigo jewel. If they came into the store with 1 red jewel, 1 yellow jewel, 2 green jewels, 1 blue jewel, and 2 violet jewels, what jewels do they leave with? Assume the shopkeeper gives them their change using as few jewels as possible.

Lesson Summary

Base-ten diagrams can help us understand subtraction as well. Suppose we are finding . Here is a diagram showing 0.23, or 2 tenths and 3 hundredths.

A base ten diagram representing 0.23 using two rectangles to represent tenths and three squares to represent hundredths.

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

A base ten diagram representing 0.23 using two rectangles to represent tenths and three squares to represent hundredths. One of the tenths is unbundled to be 10 green squares.

We now have 1 tenth and 13 hundredths, from which we can remove 7 hundredths.

A diagram of 0.23 with one tenth rectangle and thirteen hundredths squares. Seven hundredths squares are crossed out.

We have 1 tenth and 6 hundredths remaining, so .

A tape diagram of 0.16 with one tenth rectangle and six hundredths squares.

Here is a vertical calculation of .

A vertical calculation of 0.23 - 0.07 showing unbundling a tenth into 10 hundredths in order to subtract 7 hundredths.

Notice how this representation also shows a tenth is unbundled (or decomposed) into 10 hundredths in order to subtract 7 hundredths.

This works for any decimal place. Suppose we are finding . Here is a diagram showing 0.023.

A base ten diagram of 0.023 with 2 hundredths squares and 3 small thousandths rectangles

We want to remove 7 thousandths (7 small rectangles). We can “unbundle” (or decompose) one of the hundredths into 10 thousandths.

A base ten diagram of 0.023 with 2 hundredths squares and 3 small thousandths rectangles with the second hundredth square unbundled into 10 small thousandths rectangles.

Now we can remove 7 thousandths.

A base ten diagram for 0.023 with 1 square in the hundredths and 13 small rectangles in the thousandths with 7 of them crossed out.

We have 1 hundredth and 6 thousandths remaining, so .

A base ten diagram for 0.016 showing 0 hundredths and 6 thousandths bars.

Here is a vertical calculation of .

A vertical subtraction problem of 0.023 - 0.007 showing unbundling.