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.](../../../../../../embeds/1eb95ea7--6.5.B1.Image.01.png)
1
0.1
0.01
0.001
Problem 2
Here is a square.
![A square divided horizontally into 10 equal rectangles.](../../../../../../embeds/5155c229--6.5.B1.Image.02.png)
What number does the square represent if each small rectangle represents:
10
0.1
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.
![](../../../../../../embeds/5c73e10f--6.5_Base_Ten_1_green.png)
One
![](../../../../../../embeds/a2eab14b--6.5_Base_Ten_0.1_green.png)
Tenth
![](../../../../../../embeds/a394e3c6--6.5_Base_Ten_0.01_green.png)
Hundredth
![](../../../../../../embeds/09e91ec8--6.5.mode_move.png)
Click on the Move tool when you are done choosing blocks.
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.
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.
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.
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.
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.
Problem 2
For each of these numbers, draw or describe two different diagrams that represent it.
0.1
0.02
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
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.
![](../../../../../../embeds/5c73e10f--6.5_Base_Ten_1_green.png)
One
![](../../../../../../embeds/a2eab14b--6.5_Base_Ten_0.1_green.png)
Tenth
![](../../../../../../embeds/a394e3c6--6.5_Base_Ten_0.01_green.png)
Hundredth
![](../../../../../../embeds/09e91ec8--6.5.mode_move.png)
Click on the Move tool when you are done choosing blocks.
Use what you know about base-ten units and addition of base-ten numbers to explain:
Why ten squares can be “bundled” into a rectangle.
How this “bundling” is reflected in the computation.
Print Version
Here are two ways to calculate the value of
Use what you know about base-ten units and addition to explain:
Why ten squares can be “bundled” into a rectangle.
How this “bundling” is reflected in the computation.
Problem 2
Find the value of
Problem 3
Calculate
Problem 4
Find each sum.
The larger square represents 1, the larger rectangle represents 0.1, the smaller square represents 0.01, and the smaller rectangle represents 0.001.
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.
If you had 500 violet jewels and wanted to trade so that you carried as few jewels as possible, which jewels would you have?
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.
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.
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
![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.](../../../../../../embeds/89c9033d--6.5.B1.Image.22a-01-(1).png)
To find the sum, we can “bundle” (or compose) 10 hundredths as 1 tenth.
![Ten hundredths pieces bundled into one tenth piece](../../../../../../embeds/5a659501--6.5.B1.Image.23a_01.png)
We now have 2 tenths and 1 hundredth, so
![Base ten diagram of 0.21 with 2 tenths bars and 1 hundredth square](../../../../../../embeds/5127ea51--6.5.B1.Image.24a_01.png)
We can also use vertical calculation to find
![An vertical addition problem showing 10 hundredths bundled as 1 tenth](../../../../../../embeds/48a3c091--6.5.B1.Image.24b-(1).png)
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
![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.](../../../../../../embeds/32aedf54--6.5.B1.Image.22a-02-(1).png)
We can “bundle” (or compose) 10 thousandths as 1 hundredth.
![A base ten diagram showing 10 thousandths bundled into one 1 hundredth.](../../../../../../embeds/501c248d--6.5.B1.Image.23a_02.png)
The sum is 2 hundredths and 1 thousandth.
![two hundredths squares and one thousandth line.](../../../../../../embeds/7196da1c--6.5.B1.Image.24a_02.png)
Here is a vertical calculation of
![An addition problem of 0.013 + 0.008 = 0.021 shown vertically and lining up the decimals.](../../../../../../embeds/8aceb2f6--6.5.B1.Image.24b-(2).png)