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Number Sense: Addition

14 Addition Exploration 2: Multiple Ways of Adding

Mary Ann Esteban

Addition Exploration 2: Multiple Ways of Adding

For many years, the “traditional way” of adding numbers has been ingrained so that perhaps we might think this is the only way to add numbers.  Our previous exploration on number talks will hopefully show that there are different approaches to adding numbers with some flexibility.  Remember to keep a growth mindset as you approach these different ways of adding.  Hawaii is a great melting pot with many people of different cultures living here.  One important aspect about growth mindset is to be open to several other methods.  Think of these as additional tools you can add to your current knowledge of math materials.  You might find a method that appeals to you.

Alternative algorithms:

Let’s add 294 + 486.

Using Partial Sums (developed in India)

Note here that it is important to really understand your place value.

    \[\begin{aligned} &294\\ + &\underline{486} \end{aligned}\]

 

    \[\begin{aligned} 4+6=&10\\ 90+80=&170\\ 200+400=&600\\ &780 \end{aligned}\]

 

Starting from the right side, 6 + 4 gives you 10.

In the middle column, 9 + 8 really means 90 + 80, which gives you 170.

In the third column, 2 + 4 really means 200 + 400, which gives you 600.

Adding up these values 10 + 170 + 600 gives you 780.

This video shows the partial sums method but going from left to right instead.

 

Cross out Method

The cross out method allows you to add from left to right.

Let’s add 294 + 486.

    \[\begin{aligned} &294\\ + &\underline{486} \end{aligned}\]

Starting from the left, 2 + 4 = 6.  Write down the 6. Looking to the next column, 9 + 8 = 17.  Since this number is greater than 9, we cross out the 6 and change it to a 7.  Looking at the last column, 4 + 6 = 10. This number is greater than 9, so we cross out the 7 and change it to an 8.  Write down the 0 in the last column.

Lattice Algorithm

The lattice algorithm was developed by the Chinese and Indians.  There has been some popularity in re-introducing this technique in school.  Many students realize that they like this algorithm the most.

First create several boxes below the two numbers.  Draw a diagonal line extending from the upper right to the lower left of each box.  To obtain the sum, add along the diagonal.

Let’s add 294 + 486.

 

 

Discuss how this technique addresses the idea of place value.

Below is a video to reference regarding the lattice method for addition.

Can you find other methods of adding two numbers?

We’re able to find another one called the Scratch Algorithm.  This is an interesting method that allows you to add two numbers without going over 19.  Why is that?  Check out the video and see why.

 

Visual Exploding Dots Method

The videos below emphasize place value and how you can use this concept of dots and boxes to add numbers.  But before we dive into that, please take the time to look at the framework for exploding dots.  In the first video, the instructor shows you how to use dots and boxes using different bases.  First in base 2, base 3, then in base 10.  The instuctor illustrates the importance of putting down the correct number of dots in the correct place value.

In this second video, we will learn how to add using dots and boxes.  Notice that in this visual manner of adding, certain accommodations are made according to our place value rules.  See below.

 

Addition Exploration Homework

For the following problems, show how to add using the traditional algorithm, cross out, partial sums, and lattice algorithms.

  1. 943 + 285
  2. 1608 + 8327
  3. 9624 + 356
  4. 7285 + 2148 + 3491
  5. Discuss your understanding of place value.
  6. Share your experiences with learning addition.  Explain whether and how these new algorithms have changed your experience with adding numbers.
  7. What are some advantages and disadvantages you see with each of the techniques?
    • Traditional
    • Cross out
    • Partial Sums
    • Lattice

 

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Addition Exploration 2: Multiple Ways of Adding Copyright © by Mary Ann Esteban is licensed under a Creative Commons Attribution-NonCommercial 4.0 International License, except where otherwise noted.

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Jan 14, 2020
Unchanged: <h1>Addition Exploration 2: Multiple Ways of Adding</h1>Unchanged: <h1>Addition Exploration 2: Multiple Ways of Adding</h1>
Unchanged: For many years, the "traditional way" of adding numbers has been ingrained so that perhaps we might think this is the only way to add numbers.  Our previous exploration on number talks will hopefully show that there are different approaches to adding numbers with some flexibility.  Remember to keep a growth mindset as you approach these different ways of adding.  Hawaii is a great melting pot with many people of different cultures living here.  One important aspect about growth mindset is to be open to several other methods.  Think of these as additional tools you can add to your current knowledge of math materials.  You might find a method that appeals to you.Unchanged: For many years, the "traditional way" of adding numbers has been ingrained so that perhaps we might think this is the only way to add numbers.  Our previous exploration on number talks will hopefully show that there are different approaches to adding numbers with some flexibility.  Remember to keep a growth mindset as you approach these different ways of adding.  Hawaii is a great melting pot with many people of different cultures living here.  One important aspect about growth mindset is to be open to several other methods.  Think of these as additional tools you can add to your current knowledge of math materials.  You might find a method that appeals to you.
Unchanged: Alternative algorithms:Unchanged: Alternative algorithms:
Unchanged: Let’s add 294 + 486.Unchanged: Let’s add 294 + 486.
Unchanged: <h3><strong>Using Partial Sums (developed in India)</strong></h3>Unchanged: <h3><strong>Using Partial Sums (developed in India)</strong></h3>
Unchanged: Note here that it is important to really understand your place value.Unchanged: Note here that it is important to really understand your place value.
Unchanged: $$\begin{aligned} &amp;294\\Unchanged: $$\begin{aligned} &amp;294\\
Unchanged: + &amp;\underline{486}Unchanged: + &amp;\underline{486}
Unchanged: \end{aligned}$$Unchanged: \end{aligned}$$
Unchanged: <pre class="default prettyprint prettyprinted"><code><span class="pln"></span></code></pre>Unchanged: <pre class="default prettyprint prettyprinted"><code><span class="pln"></span></code></pre>
Unchanged: &nbsp;Unchanged: &nbsp;
Unchanged: $$\begin{aligned}Unchanged: $$\begin{aligned}
Unchanged: 4+6=&amp;10\\Unchanged: 4+6=&amp;10\\
Unchanged: 90+80=&amp;170\\Unchanged: 90+80=&amp;170\\
Unchanged: 200+400=&amp;600\\Unchanged: 200+400=&amp;600\\
Unchanged: &amp;780Unchanged: &amp;780
Unchanged: \end{aligned}$$Unchanged: \end{aligned}$$
Unchanged: &nbsp;Unchanged: &nbsp;
Unchanged: <div>Unchanged: <div>
Unchanged: Starting from the right side, 6 + 4 gives you 10.Unchanged: Starting from the right side, 6 + 4 gives you 10.
Unchanged: In the middle column, 9 + 8 really means 90 + 80, which gives you 170.Unchanged: In the middle column, 9 + 8 really means 90 + 80, which gives you 170.
Unchanged: In the third column, 2 + 4 really means 200 + 400, which gives you 600.Unchanged: In the third column, 2 + 4 really means 200 + 400, which gives you 600.
Unchanged: Adding up these values 10 + 170 + 600 gives you 780.Unchanged: Adding up these values 10 + 170 + 600 gives you 780.
Unchanged: This video shows the <a href="https://www.youtube.com/watch?v=yzdUIOJAEpI">partial sums method</a> but going from left to right instead.Unchanged: This video shows the <a href="https://www.youtube.com/watch?v=yzdUIOJAEpI">partial sums method</a> but going from left to right instead.
Unchanged: &nbsp;Unchanged: &nbsp;
Unchanged: <h3><strong>Cross out Method</strong></h3>Unchanged: <h3><strong>Cross out Method</strong></h3>
Unchanged: The cross out method allows you to add from left to right.Unchanged: The cross out method allows you to add from left to right.
Unchanged: Let’s add 294 + 486.Unchanged: Let’s add 294 + 486.
Unchanged: $$\begin{aligned} &amp;294\\Unchanged: $$\begin{aligned} &amp;294\\
Unchanged: + &amp;\underline{486}Unchanged: + &amp;\underline{486}
Unchanged: \end{aligned}$$Unchanged: \end{aligned}$$
Unchanged: </div>Unchanged: </div>
Unchanged: <div>Unchanged: <div>
Unchanged: <span style="font-size: 14pt">Starting from the left, 2 + 4 = 6.  Write down the 6. Looking to the next column, 9 + 8 = 17.  Since this number is greater than 9, we cross out the 6 and change it to a 7.  Looking at the last column, 4 + 6 = 10. This number is greater than 9, so we cross out the 7 and change it to an 8.  Write down the 0 in the last column.</span>Unchanged: <span style="font-size: 14pt">Starting from the left, 2 + 4 = 6.  Write down the 6. Looking to the next column, 9 + 8 = 17.  Since this number is greater than 9, we cross out the 6 and change it to a 7.  Looking at the last column, 4 + 6 = 10. This number is greater than 9, so we cross out the 7 and change it to an 8.  Write down the 0 in the last column.</span>
Unchanged: <h3><strong>Lattice Algorithm</strong></h3>Unchanged: <h3><strong>Lattice Algorithm</strong></h3>
Unchanged: The lattice algorithm was developed by the Chinese and Indians.  There has been some popularity in re-introducing this technique in school.  Many students realize that they like this algorithm the most.Unchanged: The lattice algorithm was developed by the Chinese and Indians.  There has been some popularity in re-introducing this technique in school.  Many students realize that they like this algorithm the most.
Unchanged: First create several boxes below the two numbers.  Draw a diagonal line extending from the upper right to the lower left of each box.  To obtain the sum, add along the diagonal.Unchanged: First create several boxes below the two numbers.  Draw a diagonal line extending from the upper right to the lower left of each box.  To obtain the sum, add along the diagonal.
Unchanged: Let’s add 294 + 486.Unchanged: Let’s add 294 + 486.
Unchanged: </div>Unchanged: </div>
Unchanged: &nbsp;Unchanged: &nbsp;
Unchanged: &nbsp;Unchanged: &nbsp;
Unchanged: <div>Unchanged: <div>
Unchanged: Discuss how this technique addresses the idea of place value.Unchanged: Discuss how this technique addresses the idea of place value.
Unchanged: Below is a video to reference regarding the lattice method for addition.Unchanged: Below is a video to reference regarding the lattice method for addition.
Deleted: https://www.youtube.com/watch?v=8-QHQWd4uAs 
Unchanged: Can you find other methods of adding two numbers?Unchanged: Can you find other methods of adding two numbers?
Unchanged: We're able to find another one called the Scratch Algorithm.  This is an interesting method that allows you to add two numbers without going over 19.  Why is that?  Check out the video and see why.Unchanged: We're able to find another one called the Scratch Algorithm.  This is an interesting method that allows you to add two numbers without going over 19.  Why is that?  Check out the video and see why.
Deleted: [embed]https://www.youtube.com/embed/AJBFxOBxMz8[/embed] 
Unchanged: &nbsp;Unchanged: &nbsp;
 Added: <h3><strong>Visual Exploding Dots Method</strong></h3>
 Added: The videos below emphasize place value and how you can use this concept of dots and boxes to add numbers.  But before we dive into that, please take the time to look at the framework for exploding dots.  In the first video, the instructor shows you how to use dots and boxes using different bases.  First in base 2, base 3, then in base 10.  The instuctor illustrates the importance of putting down the correct number of dots in the correct place value.
 Added: https://youtu.be/Fqq10lt9X-M
 Added: </div>
 Added: In this second video, we will learn how to add using dots and boxes.  Notice that in this visual manner of adding, certain accommodations are made according to our place value rules.  See below.
 Added: https://youtu.be/2M9w79Y5Wn0
 Added: <div>
Unchanged: &nbsp;Unchanged: &nbsp;
Unchanged: <div class="bcc-box bcc-info">Unchanged: <div class="bcc-box bcc-info">
Unchanged: <h3 itemprop="educationalUse">Addition Exploration Homework</h3>Unchanged: <h3 itemprop="educationalUse">Addition Exploration Homework</h3>
Unchanged: For the following problems, show how to add using the traditional algorithm, cross out, partial sums, and lattice algorithms.Unchanged: For the following problems, show how to add using the traditional algorithm, cross out, partial sums, and lattice algorithms.
Unchanged: <ol>Unchanged: <ol>
Unchanged: <li style="font-weight: 400">943 + 285</li>Unchanged: <li style="font-weight: 400">943 + 285</li>
Unchanged: <li style="font-weight: 400">1608 + 8327</li>Unchanged: <li style="font-weight: 400">1608 + 8327</li>
Unchanged: <li style="font-weight: 400">9624 + 356</li>Unchanged: <li style="font-weight: 400">9624 + 356</li>
Unchanged: <li style="font-weight: 400">7285 + 2148 + 3491</li>Unchanged: <li style="font-weight: 400">7285 + 2148 + 3491</li>
Unchanged: <li style="font-weight: 400">Discuss your understanding of place value.</li>Unchanged: <li style="font-weight: 400">Discuss your understanding of place value.</li>
Unchanged: <li style="font-weight: 400">Share your experiences with learning addition.  Explain whether and how these new algorithms have changed your experience with adding numbers.</li>Unchanged: <li style="font-weight: 400">Share your experiences with learning addition.  Explain whether and how these new algorithms have changed your experience with adding numbers.</li>
Unchanged: <li style="font-weight: 400">What are some advantages and disadvantages you see with each of the techniques?Unchanged: <li style="font-weight: 400">What are some advantages and disadvantages you see with each of the techniques?
Unchanged: <ul>Unchanged: <ul>
Unchanged: <li>Traditional</li>Unchanged: <li>Traditional</li>
Unchanged: <li>Cross out</li>Unchanged: <li>Cross out</li>
Unchanged: <li>Partial Sums</li>Unchanged: <li>Partial Sums</li>
Unchanged: <li>Lattice</li>Unchanged: <li>Lattice</li>
Unchanged: </ul>Unchanged: </ul>
Unchanged: </li>Unchanged: </li>
Unchanged: </ol>Unchanged: </ol>
Unchanged: </div>Unchanged: </div>
Unchanged: &nbsp;Unchanged: &nbsp;
Unchanged: </div>Unchanged: </div>