Lesson 2

Using Diagrams to Represent Addition and Subtraction

Let’s represent addition and subtraction of decimals.

2.1: Changing Values

  1. Here is a rectangle.
    A rectangle divided vertically into 10 equal squares.

    What number does the rectangle represent if each small square represents:

    1. 1

    2. 0.1

    3. 0.01

    4. 0.001

  2. Here is a square.
    A square divided horizontally into 10 equal rectangles.

    What number does the square represent if each small rectangle represents:

    1. 10

    2. 0.1

    3. 0.00001

2.2: Squares and Rectangles

You may be familiar with base-ten blocks that represent ones, tens, and hundreds. Here are some diagrams that we will use to represent digital base-ten units. A large square represents 1 one. A rectangle represents 1 tenth. A small square represents 1 hundredth.

large square, labeled 1. rectangle, labeled 1 tenth. small square, labeled 1 hundredth.

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

  1. Here is the diagram that Priya drew to represent 0.13. Draw a different diagram that represents 0.13. Explain why your diagram and Priya’s diagram represent the same number.

    3 rows of green squares. Top row, 5 squares. Middle row, 5 squares. Bottom row, 3 squares.

  2. Here is the diagram that Han drew to represent 0.25. Draw a different diagram that represents 0.25. Explain why your diagram and Han’s diagram represent the same number.

    2 green rectangles. 5 green squares.

  3. For each of these numbers, draw or describe two different diagrams that represent it.

    1. 0.1
    2. 0.02
    3. 0.43

  4. Use diagrams of base-ten units to represent the following sums and find their values. Think about how you could use as few units as possible to represent each number.

    1. \(0.03 + 0.05\)

    2. \(0.06 + 0.07\)

    3. \(0.4 + 0.7\)

2.3: 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?

2.4: Representing Subtraction

  1. Here are diagrams that represent differences. Removed pieces are marked with Xs. The larger rectangle represents 1 tenth. For each diagram, write a numerical subtraction expression and determine the value of the expression.

    option a, 4 tenth pieces. 3 crossed out. option b, 8 thousandths pieces, 3 crossed out. option c, 1 tenth piece and 5 hundredth pieces, 4 hundredth pieces crossed out.

  2. Express each subtraction in words.

    1. \(0.05 - 0.02\)

    2. \(0.024 - 0.003\)

    3. \(1.26 - 0.14\)

  3. Find each difference by drawing a diagram and by calculating with numbers. Make sure the answers from both methods match. If not, check your diagram and your numerical calculation.

    1. \(0.05 - 0.02\)

    2. \(0.024 - 0.003\)

    3. \(1.26 - 0.14\)

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.