Lesson 2
Using Diagrams to Represent Addition and Subtraction
Let’s represent addition and subtraction of decimals.
2.1: Changing Values
- Here is a rectangle.
What number does the rectangle represent if each small square represents:
-
1
-
0.1
-
0.01
-
0.001
-
- Here is a square.
What number does the square represent if each small rectangle represents:
-
10
-
0.1
- 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.](https://cms-im.s3.amazonaws.com/ayETisDeKGcYNMcNKSzXGsxm?response-content-disposition=inline%3B%20filename%3D%226-6.5.B1.2_Squares.png%22%3B%20filename%2A%3DUTF-8%27%276-6.5.B1.2_Squares.png&response-content-type=image%2Fpng&X-Amz-Algorithm=AWS4-HMAC-SHA256&X-Amz-Credential=AKIAXQCCIHWF3XOEFOW4%2F20240907%2Fus-east-1%2Fs3%2Faws4_request&X-Amz-Date=20240907T233040Z&X-Amz-Expires=604800&X-Amz-SignedHeaders=host&X-Amz-Signature=136d8368129ec48c21058da71f5303267ca94a7a4140aae6d14737b229262444)
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.](https://cms-im.s3.amazonaws.com/6XwbJZ3iF6mu5m7GDBLDKRMq?response-content-disposition=inline%3B%20filename%3D%226-6.5_Base_Ten_1_green.png%22%3B%20filename%2A%3DUTF-8%27%276-6.5_Base_Ten_1_green.png&response-content-type=image%2Fpng&X-Amz-Algorithm=AWS4-HMAC-SHA256&X-Amz-Credential=AKIAXQCCIHWF3XOEFOW4%2F20240907%2Fus-east-1%2Fs3%2Faws4_request&X-Amz-Date=20240907T233040Z&X-Amz-Expires=604800&X-Amz-SignedHeaders=host&X-Amz-Signature=6cac3ae3d475e10b545b365ce857087141dc1e61c6bce2222bd6e0c20e51d604)
One
![Image of a green rectangle.](https://cms-im.s3.amazonaws.com/9YsEabVXg5YgmynE9QRjece6?response-content-disposition=inline%3B%20filename%3D%226-6.5_Base_Ten_0.1_green.png%22%3B%20filename%2A%3DUTF-8%27%276-6.5_Base_Ten_0.1_green.png&response-content-type=image%2Fpng&X-Amz-Algorithm=AWS4-HMAC-SHA256&X-Amz-Credential=AKIAXQCCIHWF3XOEFOW4%2F20240907%2Fus-east-1%2Fs3%2Faws4_request&X-Amz-Date=20240907T233040Z&X-Amz-Expires=604800&X-Amz-SignedHeaders=host&X-Amz-Signature=95bd54d90e062b99c443110a5c221c531c5add922d6ebb41f0606180ca74d9b1)
Tenth
![image of a green square.](https://cms-im.s3.amazonaws.com/xP5o8Gj1ZFaNFqpNQAZ6J9k5?response-content-disposition=inline%3B%20filename%3D%226-6.5_Base_Ten_0.01_green.png%22%3B%20filename%2A%3DUTF-8%27%276-6.5_Base_Ten_0.01_green.png&response-content-type=image%2Fpng&X-Amz-Algorithm=AWS4-HMAC-SHA256&X-Amz-Credential=AKIAXQCCIHWF3XOEFOW4%2F20240907%2Fus-east-1%2Fs3%2Faws4_request&X-Amz-Date=20240907T233040Z&X-Amz-Expires=604800&X-Amz-SignedHeaders=host&X-Amz-Signature=a5fcb9fe7955d95b5ac3e716a1abed675d6deb3f331c732e750f03b471f2fea9)
Hundredth
Click on the Move tool when you are done choosing blocks.
![The Move tool](https://cms-im.s3.amazonaws.com/yWAc7KBdGw2HZgRkoFfDv7ky?response-content-disposition=inline%3B%20filename%3D%226.5.mode_move%20copy.png%22%3B%20filename%2A%3DUTF-8%27%276.5.mode_move%2520copy.png&response-content-type=image%2Fpng&X-Amz-Algorithm=AWS4-HMAC-SHA256&X-Amz-Credential=AKIAXQCCIHWF3XOEFOW4%2F20240907%2Fus-east-1%2Fs3%2Faws4_request&X-Amz-Date=20240907T233040Z&X-Amz-Expires=604800&X-Amz-SignedHeaders=host&X-Amz-Signature=003ccd60b9e290295ff07499a7e253a45a0e03a969aa78d85d9a6146190cd46b)
-
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 0.25. Draw a different diagram that represents 0.25. Explain why your diagram and Han’s diagram represent the same number.
-
For each of these numbers, draw or describe two different diagrams that represent it.
- 0.1
- 0.02
- 0.43
-
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.
-
\(0.03 + 0.05\)
-
\(0.06 + 0.07\)
-
\(0.4 + 0.7\)
-
2.3: Finding Sums in Different Ways
-
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.
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.
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.
-
-
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.
-
Calculate \(0.38 + 0.69\). Check your calculation against your diagram in the previous question.
-
Find each sum. The larger square represents 1, the rectangle represents 0.1, and the smaller square represents 0.01.
-
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?
2.4: Representing Subtraction
-
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.
-
Express each subtraction in words.
-
\(0.05 - 0.02\)
-
\(0.024 - 0.003\)
-
\(1.26 - 0.14\)
-
-
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.
-
\(0.05 - 0.02\)
-
\(0.024 - 0.003\)
- \(1.26 - 0.14\)
-
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.](https://cms-im.s3.amazonaws.com/pubrE26H5izVogThEq6PEmYs?response-content-disposition=inline%3B%20filename%3D%226.5.B1.Image.22a-01%20%25281%2529.png%22%3B%20filename%2A%3DUTF-8%27%276.5.B1.Image.22a-01%2520%25281%2529.png&response-content-type=image%2Fpng&X-Amz-Algorithm=AWS4-HMAC-SHA256&X-Amz-Credential=AKIAXQCCIHWF3XOEFOW4%2F20240907%2Fus-east-1%2Fs3%2Faws4_request&X-Amz-Date=20240907T233041Z&X-Amz-Expires=604800&X-Amz-SignedHeaders=host&X-Amz-Signature=ce70fa94bb397b951a86f0ce053608e9fee0b4f64ef0f27b37212ddfeeec9b1b)
To find the sum, we can “bundle” (or compose) 10 hundredths as 1 tenth.
![Base ten diagram.](https://cms-im.s3.amazonaws.com/pMSRcX2cWNuCuc6QnBPRthne?response-content-disposition=inline%3B%20filename%3D%226.5.B1.Image.23a_01.png%22%3B%20filename%2A%3DUTF-8%27%276.5.B1.Image.23a_01.png&response-content-type=image%2Fpng&X-Amz-Algorithm=AWS4-HMAC-SHA256&X-Amz-Credential=AKIAXQCCIHWF3XOEFOW4%2F20240907%2Fus-east-1%2Fs3%2Faws4_request&X-Amz-Date=20240907T233041Z&X-Amz-Expires=604800&X-Amz-SignedHeaders=host&X-Amz-Signature=bdb58d7c84dc39f0cd326c5cd5d45e258180bd0e13f002a4b11b30ea678116ea)
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.](https://cms-im.s3.amazonaws.com/BGfHABxzfXtSWZp3RWwd7Ree?response-content-disposition=inline%3B%20filename%3D%226.5.B1.Image.24a_01.png%22%3B%20filename%2A%3DUTF-8%27%276.5.B1.Image.24a_01.png&response-content-type=image%2Fpng&X-Amz-Algorithm=AWS4-HMAC-SHA256&X-Amz-Credential=AKIAXQCCIHWF3XOEFOW4%2F20240907%2Fus-east-1%2Fs3%2Faws4_request&X-Amz-Date=20240907T233041Z&X-Amz-Expires=604800&X-Amz-SignedHeaders=host&X-Amz-Signature=6b749fcf83de071c2346486bec307274f45a1c4c614fd7ee6b837ee5b5b82231)
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.](https://cms-im.s3.amazonaws.com/EHxAD8TAEfBVKMfUebt2kaL5?response-content-disposition=inline%3B%20filename%3D%226.5.B1.Image.24b%20%25281%2529.png%22%3B%20filename%2A%3DUTF-8%27%276.5.B1.Image.24b%2520%25281%2529.png&response-content-type=image%2Fpng&X-Amz-Algorithm=AWS4-HMAC-SHA256&X-Amz-Credential=AKIAXQCCIHWF3XOEFOW4%2F20240907%2Fus-east-1%2Fs3%2Faws4_request&X-Amz-Date=20240907T233041Z&X-Amz-Expires=604800&X-Amz-SignedHeaders=host&X-Amz-Signature=481ccf90c07f6293601c9d3283db9b319de85f5f08348741f2541bb2cccb4bb3)
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.](https://cms-im.s3.amazonaws.com/PW6d51pwRZXaZ1Dixp2BFftn?response-content-disposition=inline%3B%20filename%3D%226.5.B1.Image.22a-02%20%25281%2529.png%22%3B%20filename%2A%3DUTF-8%27%276.5.B1.Image.22a-02%2520%25281%2529.png&response-content-type=image%2Fpng&X-Amz-Algorithm=AWS4-HMAC-SHA256&X-Amz-Credential=AKIAXQCCIHWF3XOEFOW4%2F20240907%2Fus-east-1%2Fs3%2Faws4_request&X-Amz-Date=20240907T233041Z&X-Amz-Expires=604800&X-Amz-SignedHeaders=host&X-Amz-Signature=ad6f146fb5cb53a8cad8983a09031c0640bc10ede7204e79c87df5dace20ae53)
We can “bundle” (or compose) 10 thousandths as 1 hundredth.
![Base ten diagram.](https://cms-im.s3.amazonaws.com/V8b16XxJDRMY4NueoVFoEie1?response-content-disposition=inline%3B%20filename%3D%226.5.B1.Image.23a_02.png%22%3B%20filename%2A%3DUTF-8%27%276.5.B1.Image.23a_02.png&response-content-type=image%2Fpng&X-Amz-Algorithm=AWS4-HMAC-SHA256&X-Amz-Credential=AKIAXQCCIHWF3XOEFOW4%2F20240907%2Fus-east-1%2Fs3%2Faws4_request&X-Amz-Date=20240907T233041Z&X-Amz-Expires=604800&X-Amz-SignedHeaders=host&X-Amz-Signature=bc4b000c0ec9116de3d0e0a4aceee26192497dc8781a5e9a382332ce41544b7e)
The sum is 2 hundredths and 1 thousandth.
![Base ten diagram. 0 point 0 2 1. Two small squares. 1 small rectangle.](https://cms-im.s3.amazonaws.com/t8gDJGBMHbCqsoLUihzBnqC7?response-content-disposition=inline%3B%20filename%3D%226.5.B1.Image.24a_02.png%22%3B%20filename%2A%3DUTF-8%27%276.5.B1.Image.24a_02.png&response-content-type=image%2Fpng&X-Amz-Algorithm=AWS4-HMAC-SHA256&X-Amz-Credential=AKIAXQCCIHWF3XOEFOW4%2F20240907%2Fus-east-1%2Fs3%2Faws4_request&X-Amz-Date=20240907T233041Z&X-Amz-Expires=604800&X-Amz-SignedHeaders=host&X-Amz-Signature=21db106e57718c729b1e7276ca697a3b45684f822e6d84038b6f77ec5e22ef7c)
Here is a vertical calculation of \(0.008 + 0.013\).
![Vertical addition.](https://cms-im.s3.amazonaws.com/EMbjxiiKKCWiT8zCHkWZ2BXq?response-content-disposition=inline%3B%20filename%3D%226.5.B1.Image.24b%20%25282%2529.png%22%3B%20filename%2A%3DUTF-8%27%276.5.B1.Image.24b%2520%25282%2529.png&response-content-type=image%2Fpng&X-Amz-Algorithm=AWS4-HMAC-SHA256&X-Amz-Credential=AKIAXQCCIHWF3XOEFOW4%2F20240907%2Fus-east-1%2Fs3%2Faws4_request&X-Amz-Date=20240907T233041Z&X-Amz-Expires=604800&X-Amz-SignedHeaders=host&X-Amz-Signature=a8a544b0eaa8edc34e0576272be8ef3965df2b07844ef19bfccf977756e60602)