Lesson 2Using Diagrams to Represent Addition and Subtraction

Learning Goal

Let’s represent addition and subtraction of decimals.

Learning Targets

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

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

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

Warm Up: Changing Values

Problem 1

Here is a rectangle.

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

A rectangle divided vertically into 10 equal squares.

  1. 1

  2. 0.1

  3. 0.01

  4. 0.001

Problem 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

Activity 1: Squares and Rectangles

Problem 1

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.

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

  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.

  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.

Print Version

You may be familiar with base-ten blocks that represent ones, tens, and hundreds. Here are some diagrams that we will use to represent base-ten units.

  • A large square represents 1 one.

  • A medium rectangle represents 1 tenth.

  • A medium square represents 1 hundredth.

  • A small rectangle represents 1 thousandth.

  • A small square represents 1 ten-thousandth.

A diagram of base-ten units: 1 large square labeled "one.” 1 medium rectangle labeled “zero point one, or tenth.” 1 medium square labeled “ 0 point 0 one, or hundredth.” 1 tiny rectangle labeled “0 point 0 0 1, or thousandth.” 1 tiny square labeled “0 point 0 0 0 1, or ten-thousandth.”

  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.

    A base-ten diagram of 13 medium squares.

  2. Here is the diagram that Han drew to represent 0.025. Draw a different diagram that represents 0.025. Explain why your diagram and Han’s diagram represent the same number.

    A base-ten diagram of 2 medium squares and 5 tiny rectangles.

Problem 2

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

  1. 0.1

  2. 0.02

  3. 0.43

Problem 3

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.

Activity 2: 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 3: Representing Subtraction

Problem 1

Here are the diagrams you used to represent ones, tenths, hundredths, thousandths, and ten-thousandths.

A diagram of base-ten units: 1 large square labeled "one.” 1 medium rectangle labeled “zero point one, or tenth.” 1 medium square labeled “ 0 point 0 one, or hundredth.” 1 tiny rectangle labeled “0 point 0 0 1, or thousandth.” 1 tiny square labeled “0 point 0 0 0 1, or ten-thousandth.”

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.

  1. 4 tenth pieces. 3 crossed out.
  2. 8 thousandth pieces, 3 crossed out.
  3. 1 tenth piece and 5 hundredth pieces, 4 hundredth pieces crossed out.

Problem 2

Express each subtraction in words.

Problem 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.

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.

Ten hundredths pieces bundled into one tenth piece

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 .

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 .

An addition problem of 0.013 + 0.008 = 0.021 shown vertically and lining up the decimals.