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:
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1
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0.1
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0.01
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0.001
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- Here is a square.
What number does the square represent if each small rectangle represents:
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10
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0.1
- 0.00001
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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%2F20240727%2Fus-east-1%2Fs3%2Faws4_request&X-Amz-Date=20240727T023801Z&X-Amz-Expires=604800&X-Amz-SignedHeaders=host&X-Amz-Signature=cc4e9a8c179649cef652555631f9a6131cc7f5d12c57c796659b5fe74c5ab62e)
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%2F20240727%2Fus-east-1%2Fs3%2Faws4_request&X-Amz-Date=20240727T023801Z&X-Amz-Expires=604800&X-Amz-SignedHeaders=host&X-Amz-Signature=af9fafd331849fc673a923603ab59b9958d72bb57d179380e7b210a12e91f1dd)
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%2F20240727%2Fus-east-1%2Fs3%2Faws4_request&X-Amz-Date=20240727T023801Z&X-Amz-Expires=604800&X-Amz-SignedHeaders=host&X-Amz-Signature=c3cf865e1c1115fa7d0c50a97edaf424266e86ec3c4843211754dbe56aa1f1bf)
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%2F20240727%2Fus-east-1%2Fs3%2Faws4_request&X-Amz-Date=20240727T023801Z&X-Amz-Expires=604800&X-Amz-SignedHeaders=host&X-Amz-Signature=c43cb0c588a8721eca46fd542f9180315c80fbd36a9ca5a4f8e811c15a66e8e5)
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%2F20240727%2Fus-east-1%2Fs3%2Faws4_request&X-Amz-Date=20240727T023801Z&X-Amz-Expires=604800&X-Amz-SignedHeaders=host&X-Amz-Signature=df479a698c493db345fe0f464b3684fd5af9329ac45a09f17748b5ea7e2f5395)
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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.
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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.
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For each of these numbers, draw or describe two different diagrams that represent it.
- 0.1
- 0.02
- 0.43
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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.
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\(0.03 + 0.05\)
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\(0.06 + 0.07\)
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\(0.4 + 0.7\)
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2.3: Finding Sums in Different Ways
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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:
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Why ten squares can be “bundled” into a rectangle.
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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.
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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.
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Calculate \(0.38 + 0.69\). Check your calculation against your diagram in the previous question.
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Find each sum. The larger square represents 1, the rectangle represents 0.1, and the smaller square represents 0.01.
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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.
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If you had 500 violet jewels and wanted to trade so that you carried as few jewels as possible, which jewels would you have?
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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
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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.
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Express each subtraction in words.
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\(0.05 - 0.02\)
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\(0.024 - 0.003\)
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\(1.26 - 0.14\)
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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.
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\(0.05 - 0.02\)
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\(0.024 - 0.003\)
- \(1.26 - 0.14\)
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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%2F20240727%2Fus-east-1%2Fs3%2Faws4_request&X-Amz-Date=20240727T023801Z&X-Amz-Expires=604800&X-Amz-SignedHeaders=host&X-Amz-Signature=67fa578338602436829f313001ac0f0088de8aeaf7be8dacded0009b25271c43)
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%2F20240727%2Fus-east-1%2Fs3%2Faws4_request&X-Amz-Date=20240727T023801Z&X-Amz-Expires=604800&X-Amz-SignedHeaders=host&X-Amz-Signature=b0d658d551a2d1dbcb98b2d1f9b313c61cdc5e231836b24d31ceb8e5cb0b8492)
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%2F20240727%2Fus-east-1%2Fs3%2Faws4_request&X-Amz-Date=20240727T023801Z&X-Amz-Expires=604800&X-Amz-SignedHeaders=host&X-Amz-Signature=137d27ad3dcc4e9281c8c437e45ba1e7b8c0240161dc438014e9c159b152f0f1)
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%2F20240727%2Fus-east-1%2Fs3%2Faws4_request&X-Amz-Date=20240727T023801Z&X-Amz-Expires=604800&X-Amz-SignedHeaders=host&X-Amz-Signature=5ab073e6172e56c81b588ae3f03b3072493244fa0402ddf674a0eeb14740f68a)
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%2F20240727%2Fus-east-1%2Fs3%2Faws4_request&X-Amz-Date=20240727T023801Z&X-Amz-Expires=604800&X-Amz-SignedHeaders=host&X-Amz-Signature=e93e1de2173080371da5c57055fff4e80232eccd26186fa2888b1d480463ebdc)
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%2F20240727%2Fus-east-1%2Fs3%2Faws4_request&X-Amz-Date=20240727T023801Z&X-Amz-Expires=604800&X-Amz-SignedHeaders=host&X-Amz-Signature=dc9a009cce6590caadaf03a75c22f10e7722d1f93bde583d1feaebab73722c3e)
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%2F20240727%2Fus-east-1%2Fs3%2Faws4_request&X-Amz-Date=20240727T023801Z&X-Amz-Expires=604800&X-Amz-SignedHeaders=host&X-Amz-Signature=c1439a0cea5b9411b486c5b83a14d0fa8950c55dabcf3842f2a4eca4e856a381)
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%2F20240727%2Fus-east-1%2Fs3%2Faws4_request&X-Amz-Date=20240727T023801Z&X-Amz-Expires=604800&X-Amz-SignedHeaders=host&X-Amz-Signature=308d2e1298ba5c92d2a2089f3cfc4faa417177360cad0fcf37a64433d959e426)