Lesson 14 Do the Zeros Matter?

Learning Goal

Let’s represent addition and subtraction of decimals.

Learning Targets

  • I can use diagrams and vertical calculations to represent and reason about addition and subtraction of decimals.

  • I can use place value to explain addition and subtraction of decimals.

  • 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: Finding Sums in Different Ways

Problem 1

Here are two ways to calculate the value of . In the diagram, each rectangle represents 0.1 and each square represents 0.01.

A diagram of two strategies used to calculate an expression. The strategy on the left is a vertical equation of 0 point 2 6 plus 0 point 0 7 results in 0 point 3 3. A 1 is written above the tenths column. The strategy on the right is of a base-ten diagram. There are 2 large rectangles and 6 small squares indicated. Directly below, the squares are an additional 7 small squares indicated. A dashed circle contains 10 of the small squares with an arrow labeled bundle pointing to a third large rectangle. The third rectangle is drawn under the other two existing large rectangles.

The applet has tools that create each of the base-ten blocks. Select a Block tool, and then click on the screen to place it.

One

Tenth

Hundredth

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

open applet in presentation mode

Use what you know about base-ten units and addition of base-ten numbers to explain:

  1. Why ten squares can be “bundled” into a rectangle.

  2. How this “bundling” is reflected in the computation.

Print Version

Here are two ways to calculate the value of . In the diagram, each rectangle represents 0.1 and each square represents 0.01.

A diagram of two strategies used to calculate an expression. The strategy on the left is a vertical equation of 0 point 2 6 plus 0 point 0 7 results in 0 point 3 3. A 1 is written above the tenths column. The strategy on the right is of a base-ten diagram. There are 2 large rectangles and 6 small squares indicated. Directly below, the squares are an additional 7 small squares indicated. A dashed circle contains 10 of the small squares with an arrow labeled bundle pointing to a third large rectangle. The third rectangle is drawn under the other two existing large rectangles.

Use what you know about base-ten units and addition to explain:

  1. Why ten squares can be “bundled” into a rectangle.

  2. How this “bundling” is reflected in the computation.

Problem 2

Find the value of by drawing a diagram. Can you find the sum without bundling? Would it be useful to bundle some pieces? Explain your reasoning.

Problem 3

Calculate . Check your calculation against your diagram in the previous question.

Problem 4

Find each sum.

  1. The larger square represents 1, the larger rectangle represents 0.1, the smaller square represents 0.01, and the smaller rectangle represents 0.001.

    Two diagrams of base-ten blocks are indicated. The top diagram has 2 large squares, 5 large rectangles, and 9 small squares. The bottom diagram has 3 large rectangles, 1 small square, and 2 small rectangles.
  2. An addition problem of 6.03 and 0.098. The decimals are lined up.

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.

  1. If you had 500 violet jewels and wanted to trade so that you carried as few jewels as possible, which jewels would you have?

  2. Suppose you have 1 orange jewel, 2 yellow jewels, and 1 indigo jewel. If you’re given 2 green jewels and 1 yellow jewels, what is the fewest number of jewels that could represent the value of the jewels you have?

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.

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 represent collections of base-ten units—tens, ones, tenths, hundredths, etc. We can use them to help us understand sums of decimals.

Suppose we are finding . Here is a diagram where a square represents 0.01 and a rectangle (made up of ten squares) represents 0.1.

A base ten diagram of 0.08 + 0.13. 8 squares representing hundredths are on top. Below that is a long rectangle representing tenths and 3 squares representing hundredths.

To find the sum, we can “bundle” (or compose) 10 hundredths as 1 tenth.

A base ten diagram showing ten hundredths squares bundled into one-tenth rectangle.

We now have 2 tenths and 1 hundredth, so

Base ten diagram of 0.21 with 2 tenths bars and 1 hundredth square

We can also use vertical calculation to find 0.08+0.13.

An vertical addition problem showing 10 hundredths bundled as 1 tenth


Notice how this representation also shows 10 hundredths are bundled (or composed) as 1 tenth.

This works for any decimal place. Suppose we are finding. Here is a diagram where a small rectangle represents 0.001.

A base ten diagram showing 0.008 with 8 small rectangles in the thousandths place and 0.013 with a square in the hundredths place and 3 small rectangles in the thousandths place.

We can “bundle” (or compose) 10 thousandths as 1 hundredth.

A base ten diagram showing 10 thousandths bundled into one 1 hundredth.

The sum is 2 hundredths and 1 thousandth.

two hundredths squares and one thousandth line.

Here is a vertical calculation of .

A vertical calculation of 0.013 + 0.008 with decimals lined up and and answer of 0.021.

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 showing a hundredths square unbundled into 10 rectangles of thousandths.

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.