Lesson 14

Using Diagrams to Represent Addition and Subtraction

Let’s represent addition and subtraction of decimals.

14.1: 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.

    1. \(34.56000 = 34.56\)
    2. \(25 = 25.0\)
    3. \(2.405 = 2.45\)

14.2: Finding Sums in Different Ways

  1. Here are two ways to calculate the value of \(0.26 + 0.07\). In the diagram, each rectangle represents 0.1 and each square represents 0.01.

    Two strategies used to calculate addition expression.

    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.

    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.

    Image of a green square.

    One

    Image of a green rectangle.

    Tenth

    image of a green square.

    Hundredth

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

    The Move tool

  2. Find the value of \(0.38 + 0.69\) by drawing a diagram. Can you find the sum without bundling? Would it be useful to bundle some pieces? Explain your reasoning.

  3. Calculate \(0.38 + 0.69\). Check your calculation against your diagram in the previous question.

  4. Find each sum. The larger square represents 1, the rectangle represents 0.1, and the smaller square represents 0.01.

    1. Green base 10 pieces. On the left: 2 large squares, 5 rectangles, 9 small squares. On the right: 3 rectangles, 1 small square.
    2. 6 and 3 hundredths + 98 thousandths


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?

14.3: Subtracting Decimals of Different Lengths

To represent \(0.4 - 0.03\), Diego and Noah drew different diagrams. Each rectangle represented 0.1. Each square represented 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.” 

  • Noah started by drawing 4 rectangles to represent 0.4. He then crossed out 3 of rectangles to represent the subtraction, leaving 1 rectangle in his diagram.

    Noah's Method. tenths. 4 tenth pieces. 3 crossed out.

  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 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.” 
  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 (the arrow) when you are done choosing blocks.
  • Subtract by deleting with the delete tool (the trash can), not by crossing out.



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. 

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 \(0.08 + 0.13\). Here is a diagram where a square represents 0.01 and a rectangle (made up of ten squares) represents 0.1.

Base ten diagram. 

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

Base ten diagram. 

We now have 2 tenths and 1 hundredth, so \(0.08 + 0.13 = 0.21\).

Base ten diagram. 0 point 21. Two rectangles. 1 small square.

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

Vertical addition. First line. 0 point 13. Second line. Plus 0 point 0 8. Horizontal line. Third line. 0 point 21. Above the 1 in the first line is 1.


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 \(0.008 + 0.013\). Here is a diagram where a small rectangle represents 0.001.

Base 10 diagram. 

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

Base ten diagram. 

The sum is 2 hundredths and 1 thousandth.

Base ten diagram. 0 point 0 2 1. Two small squares. 1 small rectangle.

Here is a vertical calculation of \(0.008 + 0.013\).

Vertical addition. 

Base-ten diagrams can help us understand subtraction as well. Suppose we are finding \(0.23 - 0.07\). Here is a diagram showing 0.23, or 2 tenths and 3 hundredths.

Base ten diagram. 0 point 23. Two rectangles in the tenths column. 3 small squares in the hundredths column.

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).

Base ten diagram. 

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

Base ten diagram. 

We have 1 tenth and 6 hundredths remaining, so \(0.23 - 0.07 = 0.16\).

Base ten diagram. 0 point 16. One rectangle in the tenths column. 6 small squares in the hundredths column.

Here is a vertical calculation of \(0.23 - 0.07\).

Vertical subtraction. 

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 \(0.023 - 0.007\). Here is a diagram showing 0.023.

Base 10 diagram. 0 point 0 2 3. Two small squares in the hundredths column. Three small rectangles in the thousandths column.

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

Base 10 diagram. 

Now we can remove 7 thousandths.

Base 10 diagram. 

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

Base ten diagram. 0 point 0 1 6. One small square in the hundredths column. 6 small rectangles in the thousandths column.

Here is a vertical calculation of \(0.023 - 0.007\).

Vertical subtraction.