{"id":9341,"date":"2017-05-02T21:44:36","date_gmt":"2017-05-02T21:44:36","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/prealgebra\/?post_type=chapter&#038;p=9341"},"modified":"2020-09-05T00:30:31","modified_gmt":"2020-09-05T00:30:31","slug":"notation-and-modeling-addition-of-integers","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/wm-prealgebra\/chapter\/notation-and-modeling-addition-of-integers\/","title":{"raw":"Notation and Modeling Addition of Integers","rendered":"Notation and Modeling Addition of Integers"},"content":{"raw":"<div class=\"textbox learning-objectives\">\r\n<h3>Learning Outcomes<\/h3>\r\n<ul>\r\n \t<li>Model addition of integers with color counters<\/li>\r\n<\/ul>\r\n<\/div>\r\nNow that we have located positive and negative numbers on the number line, it is time to discuss arithmetic operations with integers.\r\n\r\nMost students are comfortable with the addition and subtraction facts for positive numbers. But doing addition or subtraction with both positive and negative numbers may be more difficult. This difficulty relates to the way the brain learns.\r\n\r\nThe brain learns best by working with objects in the real world and then generalizing to abstract concepts. Toddlers learn quickly that if they have two cookies and their older brother steals one, they have only one left. This is a concrete example of [latex]2 - 1[\/latex]. Children learn their basic addition and subtraction facts from experiences in their everyday lives. Eventually, they know the number facts without relying on cookies.\r\n\r\nAddition and subtraction of negative numbers have fewer real world examples that are meaningful to us. Math teachers have several different approaches, such as number lines, banking, temperatures, and so on, to make these concepts real.\r\n\r\nWe will model addition and subtraction of negatives with two color counters. We let a purple counter represent a positive and a orange counter will represent a negative.<img class=\"aligncenter wp-image-15965 size-full\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2302\/2017\/05\/04235317\/IMG_01_03_019.png\" alt=\"One positive counter and one negative counter\" width=\"268\" height=\"183\" \/>\r\n\r\nIf we have one positive and one negative counter, the value of the pair is zero. They form a neutral pair. The value of this neutral pair is zero as summarized in the figure below.\r\n\r\nA purple counter represents [latex]+1[\/latex]. A orange counter represents [latex]-1[\/latex]. Together they add to zero.<img class=\"aligncenter wp-image-15966 size-full\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2302\/2017\/05\/04235341\/IMG_01_03_020.png\" alt=\"One positive counter and one negative counter equals zero. The positive counter and negative counter are circled, indicating a neutral pair.\" width=\"305\" height=\"273\" \/>\r\n\r\nIn the following examples, we will model four different addition facts using the numbers [latex]5,-5\\text{ and }3,-3[\/latex].\r\n<div class=\"textbox exercises\">\r\n<h3>example<\/h3>\r\nModel: [latex]5+3[\/latex]\r\n\r\nSolution:\r\n<table id=\"eip-id1168469436825\" class=\"unnumbered unstyled\" summary=\"This figure has five rows. The first row states interpret the expression and has 5 plus 3 means the sum of 5 and 3. The second row states to model the first number and start with 5 positives. It has 5 blue circles. The third row states model the second number and add 3 positives. It has 5 blue circles followed by 3 blue circles. The third row states count the total number of counters. It is followed by 8 blue circles. The fifth row states the sum of 5 and 3 is 8 and is followed by the expression 5 plus 3 equals 8.\">\r\n<tbody>\r\n<tr>\r\n<td>Interpret the expression.<\/td>\r\n<td>[latex]5+3[\/latex] means the sum of [latex]5[\/latex] and [latex]3[\/latex] .<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Model the first number. Start with [latex]5[\/latex] positives.<\/td>\r\n<td><img class=\"alignleft wp-image-15967 size-full\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2302\/2017\/05\/04235428\/IMG_01_03_021.png\" alt=\"Five positive counters\" width=\"282\" height=\"100\" \/><\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Model the second number. Add [latex]3[\/latex] positives.<\/td>\r\n<td><img class=\"alignleft wp-image-15968 size-full\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2302\/2017\/05\/04235452\/IMG_01_03_022.png\" alt=\"Five positive counters and three positive counters\" width=\"488\" height=\"101\" \/><\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Count the total number of counters.<\/td>\r\n<td><img class=\"alignleft wp-image-15969 size-full\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2302\/2017\/05\/04235514\/IMG_01_03_023.png\" alt=\"Eight positive counters\" width=\"448\" height=\"101\" \/><\/td>\r\n<\/tr>\r\n<tr>\r\n<td>The sum of [latex]5[\/latex] and [latex]3[\/latex] is [latex]8[\/latex].<\/td>\r\n<td>[latex]5+3=8[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<\/div>\r\n&nbsp;\r\n<div class=\"textbox key-takeaways\">\r\n<h3>try it<\/h3>\r\nModel the expression.\r\n\r\n[latex]2+4[\/latex]\r\n[reveal-answer q=\"510181\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"510181\"]\r\n\r\n<img class=\"alignleft wp-image-15970 size-full\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2302\/2017\/05\/04235538\/IMG_01_03_024.png\" alt=\"Two positive counters and four positive counters\" width=\"394\" height=\"75\" \/>\r\n\r\n&nbsp;\r\n\r\n&nbsp;\r\n\r\n&nbsp;\r\n\r\n[latex]6[\/latex]\r\n\r\n[\/hidden-answer]\r\n\r\n&nbsp;\r\n\r\nModel the expression.\r\n\r\n[latex]2+5[\/latex]\r\n[reveal-answer q=\"565122\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"565122\"]\r\n\r\n<img class=\"alignleft wp-image-15971 size-full\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2302\/2017\/05\/04235626\/IMG_01_03_025.png\" alt=\"Two positive counters and five positive counters\" width=\"431\" height=\"76\" \/>\r\n\r\n&nbsp;\r\n\r\n&nbsp;\r\n\r\n&nbsp;\r\n\r\n[latex]7[\/latex]\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n&nbsp;\r\n<div class=\"textbox exercises\">\r\n<h3>example<\/h3>\r\nModel: [latex]-5+\\left(-3\\right)[\/latex]\r\n[reveal-answer q=\"78976\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"78976\"]\r\n\r\nSolution:\r\n<table id=\"eip-id1168466019221\" class=\"unnumbered unstyled\" summary=\"This figure has five rows. The first row states to interpret the expression and is followed by negative 5 plus negative 3 means the sum of negative 5 and negative 3. The second row states to model the first number and start with 5 negatives followed by 5 red circles. The third row states model the second number and add 3 negatives followed by a row with 5 red circles and 3 red circles. The fourth row states count the total number of counters followed by 8 red circles. The fifth row states the sum of negative 5 and negative 3 is negative 8, and is followed by the expression negative 5 plus negative 3 equals negative 8.\">\r\n<tbody>\r\n<tr>\r\n<td>Interpret the expression.<\/td>\r\n<td>[latex]-5+\\left(-3\\right)[\/latex] means the sum of [latex]-5[\/latex] and [latex]-3[\/latex] .<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Model the first number. Start with [latex]5[\/latex] negatives.<\/td>\r\n<td><img class=\"alignleft wp-image-15972 size-full\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2302\/2017\/05\/04235658\/IMG_01_03_026.png\" alt=\"Five negative counters\" width=\"282\" height=\"100\" \/><\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Model the second number. Add [latex]3[\/latex] negatives.<\/td>\r\n<td><img class=\"alignleft wp-image-15973 size-full\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2302\/2017\/05\/04235723\/IMG_01_03_027.png\" alt=\"Five negative counters and three negative counters\" width=\"503\" height=\"101\" \/><\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Count the total number of counters.<\/td>\r\n<td><img class=\"alignleft wp-image-15974 size-full\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2302\/2017\/05\/04235747\/IMG_01_03_028.png\" alt=\"Eight negative counters\" width=\"449\" height=\"101\" \/><\/td>\r\n<\/tr>\r\n<tr>\r\n<td>The sum of [latex]\u22125[\/latex] and [latex]\u22123[\/latex] is [latex]\u22128[\/latex].<\/td>\r\n<td>[latex]-5+-3=-8[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n&nbsp;\r\n<div class=\"textbox key-takeaways\">\r\n<h3>try it<\/h3>\r\nModel the expression.\r\n\r\n[latex]-2+\\left(-4\\right)[\/latex]\r\n[reveal-answer q=\"818374\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"818374\"]\r\n\r\n<img class=\"alignleft wp-image-15975 size-full\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2302\/2017\/05\/04235811\/IMG_01_03_029.png\" alt=\"Two negative counters and four negative counters\" width=\"395\" height=\"75\" \/>\r\n\r\n&nbsp;\r\n\r\n&nbsp;\r\n\r\n&nbsp;\r\n\r\n[latex]\u22126[\/latex]\r\n\r\n[\/hidden-answer]\r\n\r\n&nbsp;\r\n\r\nModel the expression.\r\n\r\n[latex]-2+\\left(-5\\right)[\/latex]\r\n[reveal-answer q=\"875730\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"875730\"]\r\n\r\n<img class=\"alignleft wp-image-15976 size-full\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2302\/2017\/05\/04235842\/IMG_01_03_030.png\" alt=\"Two negative counters and five negative counters\" width=\"431\" height=\"76\" \/>\r\n\r\n&nbsp;\r\n\r\n&nbsp;\r\n\r\n&nbsp;\r\n\r\n[latex]\u22127[\/latex]\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\nThe first and second examples\u00a0are very similar. The first example adds [latex]5[\/latex] positives and [latex]3[\/latex] positives\u2014both positives. The second example adds [latex]5[\/latex] negatives and [latex]3[\/latex] negatives\u2014both negatives. In each case, we got a result of [latex]8[\/latex], either [latex]8[\/latex] positives or [latex]8[\/latex] negatives. When the signs are the same, the counters are all the same color.\r\n\r\nNow let\u2019s see what happens when the signs are different.\r\n<div class=\"textbox exercises\">\r\n<h3>example<\/h3>\r\nModel: [latex]-5+3[\/latex]\r\n[reveal-answer q=\"229832\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"229832\"]\r\n\r\nSolution:\r\n<table id=\"eip-id1168468228028\" class=\"unnumbered unstyled\" summary=\"This figure has six rows. The first row states interpret the expression and is followed by negative 5 plus 3 means the sum of negative 5 and 3. The second row states model the first number and start with 5 negatives. It is followed by 5 red circles. The third row states model the second number and add 3 positives. It is followed by 5 red circles and 3 blue circles. The fourth row states remove any neutral pairs. It is followed by a row of 5 red circles and under that a row of 3 blue circles. The first three red and blue circles are grouped together. The fifth row states count the result and has 2 red circles. The sixth row states the sum of negative 5 and 3 is negative 2. It is followed by the expressions negative 5 plus 3 equals negative 2\">\r\n<tbody>\r\n<tr>\r\n<td>Interpret the expression.<\/td>\r\n<td>[latex]-5+3[\/latex] means the sum of [latex]-5[\/latex] and [latex]3[\/latex] .<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Model the first number. Start with [latex]5[\/latex] negatives.<\/td>\r\n<td><img class=\"alignleft wp-image-15977 size-full\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2302\/2017\/05\/04235914\/IMG_01_03_031.png\" alt=\"Five negative counters\" width=\"272\" height=\"80\" \/><\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Model the second number. Add [latex]3[\/latex] positives.<\/td>\r\n<td><img class=\"alignleft wp-image-15978 size-full\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2302\/2017\/05\/04235939\/IMG_01_03_032.png\" alt=\"Five negative counters and three positive counters\" width=\"276\" height=\"127\" \/><\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Remove any neutral pairs.<\/td>\r\n<td><img class=\"alignleft wp-image-15980 size-full\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2302\/2017\/05\/05000326\/IMG_01_03_033.png\" alt=\"Five negative counters and three positive counters with three neutral pairs circled\" width=\"321\" height=\"151\" \/><\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Count the result.<\/td>\r\n<td><img class=\"alignleft wp-image-15979 size-full\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2302\/2017\/05\/05000037\/IMG_01_03_034.png\" alt=\"Two negative counters\" width=\"152\" height=\"100\" \/><\/td>\r\n<\/tr>\r\n<tr>\r\n<td>The sum of [latex]\u22125[\/latex] and [latex]3[\/latex] is [latex]\u22122[\/latex].<\/td>\r\n<td>[latex]-5+3=-2[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\nNotice that there were more negatives than positives, so the result is negative.\r\n<div class=\"textbox key-takeaways\">\r\n<h3>try it<\/h3>\r\nModel the expression, and then simplify:\r\n[latex]2+\\left(-4\\right)[\/latex]\r\n[reveal-answer q=\"637985\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"637985\"]\r\n\r\n<img class=\"alignleft wp-image-15981 size-full\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2302\/2017\/05\/05000406\/IMG_01_03_035.png\" alt=\"Four negative counters and two positive counters with two neutral pairs circled.\" width=\"276\" height=\"151\" \/>\r\n\r\n&nbsp;\r\n\r\n&nbsp;\r\n\r\n&nbsp;\r\n\r\n&nbsp;\r\n\r\n&nbsp;\r\n\r\n&nbsp;\r\n\r\n[latex]\u22122[\/latex]\r\n\r\n[\/hidden-answer]\r\n\r\n&nbsp;\r\n\r\nModel the expression, and then simplify:\r\n[latex]2+\\left(-5\\right)[\/latex]\r\n[reveal-answer q=\"955006\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"955006\"]\r\n\r\n<img class=\"alignleft wp-image-15982 size-full\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2302\/2017\/05\/05000452\/IMG_01_03_036.png\" alt=\"Five negative counters and two positive counters with two neutral pairs circled.\" width=\"330\" height=\"151\" \/>\r\n\r\n&nbsp;\r\n\r\n&nbsp;\r\n\r\n&nbsp;\r\n\r\n&nbsp;\r\n\r\n&nbsp;\r\n\r\n&nbsp;\r\n\r\n[latex]\u22123[\/latex]\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n&nbsp;\r\n<div class=\"textbox exercises\">\r\n<h3>example<\/h3>\r\nModel: [latex]5+\\left(-3\\right)[\/latex]\r\n[reveal-answer q=\"862636\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"862636\"]\r\n\r\nSolution:\r\n<table id=\"eip-id1168468738226\" class=\"unnumbered unstyled\" summary=\"This figure has six rows. The first row states interpret the expression and is followed by 5 plus negative 3 means the sum of 5 and negative 3. The second row states model the first number and start with 5 positives. It is followed by 5 blue circles. The third row states model the second number and add 3 negatives. It is followed by 5 blue circles and 3 red circles. The fourth row states remove any neutral pairs. It is followed by a row of 5 blue circles and under that a row of 3 red circles. The first three red and blue circles are grouped together. The fifth row states count the result and has 2 blue circles. The sixth row states the sum of 5 and negative 3 is 2. It is followed by the expression five plus negative 3 equals 2.\">\r\n<tbody>\r\n<tr>\r\n<td>Interpret the expression.<\/td>\r\n<td>[latex]5+\\left(-3\\right)[\/latex] means the sum of [latex]5[\/latex] and [latex]-3[\/latex] .<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Model the first number. Start with [latex]5[\/latex] positives.<\/td>\r\n<td><img class=\"alignleft wp-image-15983 size-full\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2302\/2017\/05\/05000525\/IMG_01_03_037.png\" alt=\"Five positive counters\" width=\"272\" height=\"79\" \/><\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Model the second number. Add [latex]3[\/latex] negatives.<\/td>\r\n<td><img class=\"alignleft wp-image-15984 size-full\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2302\/2017\/05\/05000548\/IMG_01_03_038.png\" alt=\"Five positive counters and three negative counters\" width=\"275\" height=\"127\" \/><\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Remove any neutral pairs.<\/td>\r\n<td><img class=\"alignleft wp-image-15986 size-full\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2302\/2017\/05\/05000830\/IMG_01_03_039.png\" alt=\"Five positive counters and three negative counters with three neutral pairs circled.\" width=\"320\" height=\"152\" \/><\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Count the result.<\/td>\r\n<td><img class=\"alignleft wp-image-15985 size-full\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2302\/2017\/05\/05000642\/IMG_01_03_040.png\" alt=\"Two positive counters\" width=\"152\" height=\"100\" \/><\/td>\r\n<\/tr>\r\n<tr>\r\n<td>The sum of [latex]5[\/latex] and [latex]\u22123[\/latex] is [latex]2[\/latex].<\/td>\r\n<td>[latex]5+\\left(-3\\right)=2[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n&nbsp;\r\n<div class=\"textbox key-takeaways\">\r\n<h3>try it<\/h3>\r\nModel the expression, and then simplify:\r\n\r\n[latex]\\left(-2\\right)+4[\/latex]\r\n[reveal-answer q=\"512582\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"512582\"]\r\n\r\n<img class=\"alignleft wp-image-15987 size-full\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2302\/2017\/05\/05000907\/IMG_01_03_041.png\" alt=\"Two negative counters and four positive counters with two neutral pairs circled\" width=\"281\" height=\"151\" \/>\r\n\r\n&nbsp;\r\n\r\n&nbsp;\r\n\r\n&nbsp;\r\n\r\n&nbsp;\r\n\r\n&nbsp;\r\n\r\n&nbsp;\r\n\r\n[latex]2[\/latex]\r\n\r\n[\/hidden-answer]\r\n\r\n&nbsp;\r\n\r\nModel the expression:\r\n[latex]\\left(-2\\right)+5[\/latex]\r\n[reveal-answer q=\"556256\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"556256\"]\r\n\r\n<img class=\"alignleft wp-image-15988 size-full\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2302\/2017\/05\/05000947\/IMG_01_03_042.png\" alt=\"Two negative counters and five positive counters with two neutral pairs circled.\" width=\"332\" height=\"151\" \/>\r\n\r\n&nbsp;\r\n\r\n&nbsp;\r\n\r\n&nbsp;\r\n\r\n&nbsp;\r\n\r\n&nbsp;\r\n\r\n&nbsp;\r\n\r\n[latex]3[\/latex]\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n&nbsp;\r\n<div class=\"textbox exercises\">\r\n<h3>example<\/h3>\r\nModel each addition.\r\n<ol>\r\n \t<li>[latex]4 + 2[\/latex]<\/li>\r\n \t<li>[latex]\u22123 + 6[\/latex]<\/li>\r\n \t<li>[latex]4 + (\u22125)[\/latex]<\/li>\r\n \t<li>[latex]-2 + (\u22123)[\/latex]<\/li>\r\n<\/ol>\r\n[reveal-answer q=\"480477\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"480477\"]\r\n\r\nSolution\r\n<table id=\"eip-id1168468294641\" class=\"unnumbered unstyled\" summary=\".\">\r\n<tbody>\r\n<tr>\r\n<td>1.<\/td>\r\n<td><\/td>\r\n<\/tr>\r\n<tr>\r\n<td><\/td>\r\n<td>[latex]4+2[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Start with [latex]4[\/latex] positives.<\/td>\r\n<td><img class=\"alignleft wp-image-15989 size-full\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2302\/2017\/05\/05001037\/IMG_01_03_043.png\" alt=\"Four positive counters\" width=\"242\" height=\"76\" \/><\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Add two positives.<\/td>\r\n<td><img class=\"alignleft wp-image-15990 size-full\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2302\/2017\/05\/05001109\/IMG_01_03_044.png\" alt=\"Four positive counters and two positive counters\" width=\"395\" height=\"76\" \/><\/td>\r\n<\/tr>\r\n<tr>\r\n<td>How many do you have?<\/td>\r\n<td>[latex]4+2=6[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<table id=\"eip-id1168468679001\" class=\"unnumbered unstyled\" summary=\".\">\r\n<tbody>\r\n<tr>\r\n<td>2.<\/td>\r\n<td><\/td>\r\n<\/tr>\r\n<tr>\r\n<td><\/td>\r\n<td>[latex]-3+6[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Start with [latex]3[\/latex] negatives.<\/td>\r\n<td><img class=\"alignleft wp-image-15992 size-full\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2302\/2017\/05\/05001612\/IMG_01_03_045.png\" alt=\"Three negative counters\" width=\"175\" height=\"70\" \/><\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Add [latex]6[\/latex] positives.<\/td>\r\n<td><img class=\"alignleft wp-image-15993 size-full\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2302\/2017\/05\/05001636\/IMG_01_03_046.png\" alt=\"Three negative counters and six positive counters\" width=\"330\" height=\"127\" \/><\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Remove neutral pairs.<\/td>\r\n<td>\u00a0<img class=\"alignleft wp-image-15994 size-full\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2302\/2017\/05\/05001728\/IMG_01_03_047.png\" alt=\"Three negative counters and six positive counters with three neutral pairs circled.\" width=\"339\" height=\"187\" \/><\/td>\r\n<\/tr>\r\n<tr>\r\n<td>How many are left?<\/td>\r\n<td><img class=\"alignleft wp-image-15995 size-full\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2302\/2017\/05\/05001803\/IMG_01_03_048.png\" alt=\"Three positive counters\" width=\"175\" height=\"70\" \/><\/td>\r\n<\/tr>\r\n<tr>\r\n<td>\u00a0[latex]-3+6=3[\/latex]<\/td>\r\n<td>[latex]3[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<table id=\"eip-id1168467662609\" class=\"unnumbered unstyled\" summary=\".\">\r\n<tbody>\r\n<tr>\r\n<td>3.<\/td>\r\n<td><\/td>\r\n<\/tr>\r\n<tr>\r\n<td><\/td>\r\n<td>[latex]4+\\left(-5\\right)[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Start with [latex]4[\/latex] positives.<\/td>\r\n<td><img class=\"alignleft wp-image-15996 size-full\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2302\/2017\/05\/05002104\/IMG_01_03_049.png\" alt=\"Four positive counters\" width=\"242\" height=\"76\" \/><\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Add [latex]5[\/latex] negatives.<\/td>\r\n<td><img class=\"alignleft wp-image-15997 size-full\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2302\/2017\/05\/05002129\/IMG_01_03_050.png\" alt=\"Four positive counters and five negative counters\" width=\"277\" height=\"128\" \/><\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Remove neutral pairs.<\/td>\r\n<td><img class=\"alignleft wp-image-15998 size-full\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2302\/2017\/05\/05002202\/IMG_01_03_051.png\" alt=\"Four positive counters and five negative counters with four neutral pairs circled.\" width=\"298\" height=\"219\" \/><\/td>\r\n<\/tr>\r\n<tr>\r\n<td>How many are left?<\/td>\r\n<td><img class=\"size-full wp-image-15999 alignleft\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2302\/2017\/05\/05002237\/IMG_01_03_052.png\" alt=\"One negative counter\" width=\"84\" height=\"75\" \/><\/td>\r\n<\/tr>\r\n<tr>\r\n<td>\u00a0[latex]4+\\left(-5\\right)=-1[\/latex]<\/td>\r\n<td>[latex]-1[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<table id=\"eip-id1168468388113\" class=\"unnumbered unstyled\" summary=\".\">\r\n<tbody>\r\n<tr>\r\n<td>4.<\/td>\r\n<td><\/td>\r\n<\/tr>\r\n<tr>\r\n<td><\/td>\r\n<td>[latex]-2+\\left(-3\\right)[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Start with [latex]2[\/latex] negatives.<\/td>\r\n<td><img class=\"size-full wp-image-16000 alignleft\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2302\/2017\/05\/05002444\/IMG_01_03_053.png\" alt=\"Two negative counters\" width=\"143\" height=\"75\" \/><\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Add [latex]3[\/latex] negatives.<\/td>\r\n<td><img class=\"alignleft wp-image-16001 size-full\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2302\/2017\/05\/05002505\/IMG_01_03_054.png\" alt=\"Two negative counters and three negative counters\" width=\"336\" height=\"76\" \/><\/td>\r\n<\/tr>\r\n<tr>\r\n<td>How many do you have?<\/td>\r\n<td>[latex]-2+\\left(-3\\right)=-5[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n&nbsp;\r\n<div class=\"textbox key-takeaways\">\r\n<h3>try it<\/h3>\r\nModel each addition.\r\n\r\n1. [latex]3 + 4[\/latex]\r\n2. [latex]\u22121 + 4[\/latex]\r\n3. [latex]4 + (\u22126)[\/latex]\r\n4. [latex]\u22122 + (\u22122)[\/latex]\r\n[reveal-answer q=\"767954\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"767954\"]\r\n\r\n1. [latex]7[\/latex]\r\n\r\n<img class=\"size-full wp-image-16002 alignleft\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2302\/2017\/05\/05002536\/IMG_01_03_055.png\" alt=\"Three positive counters and four positive counters\" width=\"221\" height=\"128\" \/>\r\n\r\n&nbsp;\r\n\r\n&nbsp;\r\n\r\n&nbsp;\r\n\r\n&nbsp;\r\n\r\n&nbsp;\r\n\r\n2. [latex]3[\/latex]\r\n\r\n<img class=\"size-full wp-image-16003 alignleft\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2302\/2017\/05\/05002603\/IMG_01_03_056.png\" alt=\"One negative counter and four positive counters\" width=\"341\" height=\"75\" \/>\r\n\r\n&nbsp;\r\n\r\n&nbsp;\r\n\r\n&nbsp;\r\n\r\n3. [latex]-2[\/latex]\r\n\r\n<img class=\"size-full wp-image-16004 alignleft\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2302\/2017\/05\/05002637\/IMG_01_03_057.png\" alt=\"Four positive counters and six negative counters\" width=\"320\" height=\"128\" \/>\r\n\r\n&nbsp;\r\n\r\n&nbsp;\r\n\r\n&nbsp;\r\n\r\n&nbsp;\r\n\r\n&nbsp;\r\n\r\n4. [latex]-4[\/latex]\r\n\r\n<img class=\"size-full wp-image-16005 alignleft\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2302\/2017\/05\/05002704\/IMG_01_03_058.png\" alt=\"Two negative counters and two negative counters\" width=\"291\" height=\"76\" \/>\r\n\r\n&nbsp;\r\n\r\n&nbsp;\r\n\r\n&nbsp;\r\n\r\n[\/hidden-answer]\r\n\r\n1. [latex]5 + 1[\/latex]\r\n2. [latex]\u22123 + 7[\/latex]\r\n3. [latex]2 + (\u22128)[\/latex]\r\n4. [latex]\u22123 + (\u22124)[\/latex]\r\n[reveal-answer q=\"388859\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"388859\"]\r\n\r\n1. [latex]6[\/latex]\r\n\r\n<img class=\"alignleft wp-image-16006 size-full\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2302\/2017\/05\/05002735\/IMG_01_03_059.png\" alt=\"Five positive counters and one positive counter\" width=\"270\" height=\"128\" \/>\r\n\r\n&nbsp;\r\n\r\n&nbsp;\r\n\r\n&nbsp;\r\n\r\n&nbsp;\r\n\r\n&nbsp;\r\n\r\n2. [latex]4[\/latex]\r\n\r\n<img class=\"alignleft wp-image-16007 size-full\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2302\/2017\/05\/05002808\/IMG_01_03_060.png\" alt=\"Three negative counters and seven positive counters\" width=\"372\" height=\"128\" \/>\r\n\r\n&nbsp;\r\n\r\n&nbsp;\r\n\r\n&nbsp;\r\n\r\n&nbsp;\r\n\r\n&nbsp;\r\n\r\n3. [latex]-6[\/latex]\r\n\r\n<img class=\"alignleft wp-image-16008 size-full\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2302\/2017\/05\/05002847\/IMG_01_03_061.png\" alt=\"Two positive counters and eight negative counters\" width=\"425\" height=\"128\" \/>\r\n\r\n&nbsp;\r\n\r\n&nbsp;\r\n\r\n&nbsp;\r\n\r\n&nbsp;\r\n\r\n&nbsp;\r\n\r\n4. [latex]-7[\/latex]\r\n\r\n<img class=\"size-full wp-image-16009 alignleft\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2302\/2017\/05\/05002919\/IMG_01_03_062.png\" alt=\"Three negative counters and four negative counters\" width=\"219\" height=\"127\" \/>\r\n\r\n&nbsp;\r\n\r\n&nbsp;\r\n\r\n&nbsp;\r\n\r\n&nbsp;\r\n\r\n&nbsp;\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\nIn the following video we present more examples of using color counters to model addition of integers.\r\n\r\nhttps:\/\/youtu.be\/beI5bQB9a7Y","rendered":"<div class=\"textbox learning-objectives\">\n<h3>Learning Outcomes<\/h3>\n<ul>\n<li>Model addition of integers with color counters<\/li>\n<\/ul>\n<\/div>\n<p>Now that we have located positive and negative numbers on the number line, it is time to discuss arithmetic operations with integers.<\/p>\n<p>Most students are comfortable with the addition and subtraction facts for positive numbers. But doing addition or subtraction with both positive and negative numbers may be more difficult. This difficulty relates to the way the brain learns.<\/p>\n<p>The brain learns best by working with objects in the real world and then generalizing to abstract concepts. Toddlers learn quickly that if they have two cookies and their older brother steals one, they have only one left. This is a concrete example of [latex]2 - 1[\/latex]. Children learn their basic addition and subtraction facts from experiences in their everyday lives. Eventually, they know the number facts without relying on cookies.<\/p>\n<p>Addition and subtraction of negative numbers have fewer real world examples that are meaningful to us. Math teachers have several different approaches, such as number lines, banking, temperatures, and so on, to make these concepts real.<\/p>\n<p>We will model addition and subtraction of negatives with two color counters. We let a purple counter represent a positive and a orange counter will represent a negative.<img loading=\"lazy\" decoding=\"async\" class=\"aligncenter wp-image-15965 size-full\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2302\/2017\/05\/04235317\/IMG_01_03_019.png\" alt=\"One positive counter and one negative counter\" width=\"268\" height=\"183\" \/><\/p>\n<p>If we have one positive and one negative counter, the value of the pair is zero. They form a neutral pair. The value of this neutral pair is zero as summarized in the figure below.<\/p>\n<p>A purple counter represents [latex]+1[\/latex]. A orange counter represents [latex]-1[\/latex]. Together they add to zero.<img loading=\"lazy\" decoding=\"async\" class=\"aligncenter wp-image-15966 size-full\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2302\/2017\/05\/04235341\/IMG_01_03_020.png\" alt=\"One positive counter and one negative counter equals zero. The positive counter and negative counter are circled, indicating a neutral pair.\" width=\"305\" height=\"273\" \/><\/p>\n<p>In the following examples, we will model four different addition facts using the numbers [latex]5,-5\\text{ and }3,-3[\/latex].<\/p>\n<div class=\"textbox exercises\">\n<h3>example<\/h3>\n<p>Model: [latex]5+3[\/latex]<\/p>\n<p>Solution:<\/p>\n<table id=\"eip-id1168469436825\" class=\"unnumbered unstyled\" summary=\"This figure has five rows. The first row states interpret the expression and has 5 plus 3 means the sum of 5 and 3. The second row states to model the first number and start with 5 positives. It has 5 blue circles. The third row states model the second number and add 3 positives. It has 5 blue circles followed by 3 blue circles. The third row states count the total number of counters. It is followed by 8 blue circles. The fifth row states the sum of 5 and 3 is 8 and is followed by the expression 5 plus 3 equals 8.\">\n<tbody>\n<tr>\n<td>Interpret the expression.<\/td>\n<td>[latex]5+3[\/latex] means the sum of [latex]5[\/latex] and [latex]3[\/latex] .<\/td>\n<\/tr>\n<tr>\n<td>Model the first number. Start with [latex]5[\/latex] positives.<\/td>\n<td><img loading=\"lazy\" decoding=\"async\" class=\"alignleft wp-image-15967 size-full\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2302\/2017\/05\/04235428\/IMG_01_03_021.png\" alt=\"Five positive counters\" width=\"282\" height=\"100\" \/><\/td>\n<\/tr>\n<tr>\n<td>Model the second number. Add [latex]3[\/latex] positives.<\/td>\n<td><img loading=\"lazy\" decoding=\"async\" class=\"alignleft wp-image-15968 size-full\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2302\/2017\/05\/04235452\/IMG_01_03_022.png\" alt=\"Five positive counters and three positive counters\" width=\"488\" height=\"101\" \/><\/td>\n<\/tr>\n<tr>\n<td>Count the total number of counters.<\/td>\n<td><img loading=\"lazy\" decoding=\"async\" class=\"alignleft wp-image-15969 size-full\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2302\/2017\/05\/04235514\/IMG_01_03_023.png\" alt=\"Eight positive counters\" width=\"448\" height=\"101\" \/><\/td>\n<\/tr>\n<tr>\n<td>The sum of [latex]5[\/latex] and [latex]3[\/latex] is [latex]8[\/latex].<\/td>\n<td>[latex]5+3=8[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<p>&nbsp;<\/p>\n<div class=\"textbox key-takeaways\">\n<h3>try it<\/h3>\n<p>Model the expression.<\/p>\n<p>[latex]2+4[\/latex]<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q510181\">Show Solution<\/span><\/p>\n<div id=\"q510181\" class=\"hidden-answer\" style=\"display: none\">\n<p><img loading=\"lazy\" decoding=\"async\" class=\"alignleft wp-image-15970 size-full\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2302\/2017\/05\/04235538\/IMG_01_03_024.png\" alt=\"Two positive counters and four positive counters\" width=\"394\" height=\"75\" \/><\/p>\n<p>&nbsp;<\/p>\n<p>&nbsp;<\/p>\n<p>&nbsp;<\/p>\n<p>[latex]6[\/latex]<\/p>\n<\/div>\n<\/div>\n<p>&nbsp;<\/p>\n<p>Model the expression.<\/p>\n<p>[latex]2+5[\/latex]<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q565122\">Show Solution<\/span><\/p>\n<div id=\"q565122\" class=\"hidden-answer\" style=\"display: none\">\n<p><img loading=\"lazy\" decoding=\"async\" class=\"alignleft wp-image-15971 size-full\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2302\/2017\/05\/04235626\/IMG_01_03_025.png\" alt=\"Two positive counters and five positive counters\" width=\"431\" height=\"76\" \/><\/p>\n<p>&nbsp;<\/p>\n<p>&nbsp;<\/p>\n<p>&nbsp;<\/p>\n<p>[latex]7[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<p>&nbsp;<\/p>\n<div class=\"textbox exercises\">\n<h3>example<\/h3>\n<p>Model: [latex]-5+\\left(-3\\right)[\/latex]<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q78976\">Show Solution<\/span><\/p>\n<div id=\"q78976\" class=\"hidden-answer\" style=\"display: none\">\n<p>Solution:<\/p>\n<table id=\"eip-id1168466019221\" class=\"unnumbered unstyled\" summary=\"This figure has five rows. The first row states to interpret the expression and is followed by negative 5 plus negative 3 means the sum of negative 5 and negative 3. The second row states to model the first number and start with 5 negatives followed by 5 red circles. The third row states model the second number and add 3 negatives followed by a row with 5 red circles and 3 red circles. The fourth row states count the total number of counters followed by 8 red circles. The fifth row states the sum of negative 5 and negative 3 is negative 8, and is followed by the expression negative 5 plus negative 3 equals negative 8.\">\n<tbody>\n<tr>\n<td>Interpret the expression.<\/td>\n<td>[latex]-5+\\left(-3\\right)[\/latex] means the sum of [latex]-5[\/latex] and [latex]-3[\/latex] .<\/td>\n<\/tr>\n<tr>\n<td>Model the first number. Start with [latex]5[\/latex] negatives.<\/td>\n<td><img loading=\"lazy\" decoding=\"async\" class=\"alignleft wp-image-15972 size-full\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2302\/2017\/05\/04235658\/IMG_01_03_026.png\" alt=\"Five negative counters\" width=\"282\" height=\"100\" \/><\/td>\n<\/tr>\n<tr>\n<td>Model the second number. Add [latex]3[\/latex] negatives.<\/td>\n<td><img loading=\"lazy\" decoding=\"async\" class=\"alignleft wp-image-15973 size-full\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2302\/2017\/05\/04235723\/IMG_01_03_027.png\" alt=\"Five negative counters and three negative counters\" width=\"503\" height=\"101\" \/><\/td>\n<\/tr>\n<tr>\n<td>Count the total number of counters.<\/td>\n<td><img loading=\"lazy\" decoding=\"async\" class=\"alignleft wp-image-15974 size-full\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2302\/2017\/05\/04235747\/IMG_01_03_028.png\" alt=\"Eight negative counters\" width=\"449\" height=\"101\" \/><\/td>\n<\/tr>\n<tr>\n<td>The sum of [latex]\u22125[\/latex] and [latex]\u22123[\/latex] is [latex]\u22128[\/latex].<\/td>\n<td>[latex]-5+-3=-8[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<\/div>\n<\/div>\n<p>&nbsp;<\/p>\n<div class=\"textbox key-takeaways\">\n<h3>try it<\/h3>\n<p>Model the expression.<\/p>\n<p>[latex]-2+\\left(-4\\right)[\/latex]<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q818374\">Show Solution<\/span><\/p>\n<div id=\"q818374\" class=\"hidden-answer\" style=\"display: none\">\n<p><img loading=\"lazy\" decoding=\"async\" class=\"alignleft wp-image-15975 size-full\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2302\/2017\/05\/04235811\/IMG_01_03_029.png\" alt=\"Two negative counters and four negative counters\" width=\"395\" height=\"75\" \/><\/p>\n<p>&nbsp;<\/p>\n<p>&nbsp;<\/p>\n<p>&nbsp;<\/p>\n<p>[latex]\u22126[\/latex]<\/p>\n<\/div>\n<\/div>\n<p>&nbsp;<\/p>\n<p>Model the expression.<\/p>\n<p>[latex]-2+\\left(-5\\right)[\/latex]<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q875730\">Show Solution<\/span><\/p>\n<div id=\"q875730\" class=\"hidden-answer\" style=\"display: none\">\n<p><img loading=\"lazy\" decoding=\"async\" class=\"alignleft wp-image-15976 size-full\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2302\/2017\/05\/04235842\/IMG_01_03_030.png\" alt=\"Two negative counters and five negative counters\" width=\"431\" height=\"76\" \/><\/p>\n<p>&nbsp;<\/p>\n<p>&nbsp;<\/p>\n<p>&nbsp;<\/p>\n<p>[latex]\u22127[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<p>The first and second examples\u00a0are very similar. The first example adds [latex]5[\/latex] positives and [latex]3[\/latex] positives\u2014both positives. The second example adds [latex]5[\/latex] negatives and [latex]3[\/latex] negatives\u2014both negatives. In each case, we got a result of [latex]8[\/latex], either [latex]8[\/latex] positives or [latex]8[\/latex] negatives. When the signs are the same, the counters are all the same color.<\/p>\n<p>Now let\u2019s see what happens when the signs are different.<\/p>\n<div class=\"textbox exercises\">\n<h3>example<\/h3>\n<p>Model: [latex]-5+3[\/latex]<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q229832\">Show Solution<\/span><\/p>\n<div id=\"q229832\" class=\"hidden-answer\" style=\"display: none\">\n<p>Solution:<\/p>\n<table id=\"eip-id1168468228028\" class=\"unnumbered unstyled\" summary=\"This figure has six rows. The first row states interpret the expression and is followed by negative 5 plus 3 means the sum of negative 5 and 3. The second row states model the first number and start with 5 negatives. It is followed by 5 red circles. The third row states model the second number and add 3 positives. It is followed by 5 red circles and 3 blue circles. The fourth row states remove any neutral pairs. It is followed by a row of 5 red circles and under that a row of 3 blue circles. The first three red and blue circles are grouped together. The fifth row states count the result and has 2 red circles. The sixth row states the sum of negative 5 and 3 is negative 2. It is followed by the expressions negative 5 plus 3 equals negative 2\">\n<tbody>\n<tr>\n<td>Interpret the expression.<\/td>\n<td>[latex]-5+3[\/latex] means the sum of [latex]-5[\/latex] and [latex]3[\/latex] .<\/td>\n<\/tr>\n<tr>\n<td>Model the first number. Start with [latex]5[\/latex] negatives.<\/td>\n<td><img loading=\"lazy\" decoding=\"async\" class=\"alignleft wp-image-15977 size-full\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2302\/2017\/05\/04235914\/IMG_01_03_031.png\" alt=\"Five negative counters\" width=\"272\" height=\"80\" \/><\/td>\n<\/tr>\n<tr>\n<td>Model the second number. Add [latex]3[\/latex] positives.<\/td>\n<td><img loading=\"lazy\" decoding=\"async\" class=\"alignleft wp-image-15978 size-full\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2302\/2017\/05\/04235939\/IMG_01_03_032.png\" alt=\"Five negative counters and three positive counters\" width=\"276\" height=\"127\" \/><\/td>\n<\/tr>\n<tr>\n<td>Remove any neutral pairs.<\/td>\n<td><img loading=\"lazy\" decoding=\"async\" class=\"alignleft wp-image-15980 size-full\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2302\/2017\/05\/05000326\/IMG_01_03_033.png\" alt=\"Five negative counters and three positive counters with three neutral pairs circled\" width=\"321\" height=\"151\" \/><\/td>\n<\/tr>\n<tr>\n<td>Count the result.<\/td>\n<td><img loading=\"lazy\" decoding=\"async\" class=\"alignleft wp-image-15979 size-full\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2302\/2017\/05\/05000037\/IMG_01_03_034.png\" alt=\"Two negative counters\" width=\"152\" height=\"100\" \/><\/td>\n<\/tr>\n<tr>\n<td>The sum of [latex]\u22125[\/latex] and [latex]3[\/latex] is [latex]\u22122[\/latex].<\/td>\n<td>[latex]-5+3=-2[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<\/div>\n<\/div>\n<p>Notice that there were more negatives than positives, so the result is negative.<\/p>\n<div class=\"textbox key-takeaways\">\n<h3>try it<\/h3>\n<p>Model the expression, and then simplify:<br \/>\n[latex]2+\\left(-4\\right)[\/latex]<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q637985\">Show Solution<\/span><\/p>\n<div id=\"q637985\" class=\"hidden-answer\" style=\"display: none\">\n<p><img loading=\"lazy\" decoding=\"async\" class=\"alignleft wp-image-15981 size-full\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2302\/2017\/05\/05000406\/IMG_01_03_035.png\" alt=\"Four negative counters and two positive counters with two neutral pairs circled.\" width=\"276\" height=\"151\" \/><\/p>\n<p>&nbsp;<\/p>\n<p>&nbsp;<\/p>\n<p>&nbsp;<\/p>\n<p>&nbsp;<\/p>\n<p>&nbsp;<\/p>\n<p>&nbsp;<\/p>\n<p>[latex]\u22122[\/latex]<\/p>\n<\/div>\n<\/div>\n<p>&nbsp;<\/p>\n<p>Model the expression, and then simplify:<br \/>\n[latex]2+\\left(-5\\right)[\/latex]<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q955006\">Show Solution<\/span><\/p>\n<div id=\"q955006\" class=\"hidden-answer\" style=\"display: none\">\n<p><img loading=\"lazy\" decoding=\"async\" class=\"alignleft wp-image-15982 size-full\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2302\/2017\/05\/05000452\/IMG_01_03_036.png\" alt=\"Five negative counters and two positive counters with two neutral pairs circled.\" width=\"330\" height=\"151\" \/><\/p>\n<p>&nbsp;<\/p>\n<p>&nbsp;<\/p>\n<p>&nbsp;<\/p>\n<p>&nbsp;<\/p>\n<p>&nbsp;<\/p>\n<p>&nbsp;<\/p>\n<p>[latex]\u22123[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<p>&nbsp;<\/p>\n<div class=\"textbox exercises\">\n<h3>example<\/h3>\n<p>Model: [latex]5+\\left(-3\\right)[\/latex]<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q862636\">Show Solution<\/span><\/p>\n<div id=\"q862636\" class=\"hidden-answer\" style=\"display: none\">\n<p>Solution:<\/p>\n<table id=\"eip-id1168468738226\" class=\"unnumbered unstyled\" summary=\"This figure has six rows. The first row states interpret the expression and is followed by 5 plus negative 3 means the sum of 5 and negative 3. The second row states model the first number and start with 5 positives. It is followed by 5 blue circles. The third row states model the second number and add 3 negatives. It is followed by 5 blue circles and 3 red circles. The fourth row states remove any neutral pairs. It is followed by a row of 5 blue circles and under that a row of 3 red circles. The first three red and blue circles are grouped together. The fifth row states count the result and has 2 blue circles. The sixth row states the sum of 5 and negative 3 is 2. It is followed by the expression five plus negative 3 equals 2.\">\n<tbody>\n<tr>\n<td>Interpret the expression.<\/td>\n<td>[latex]5+\\left(-3\\right)[\/latex] means the sum of [latex]5[\/latex] and [latex]-3[\/latex] .<\/td>\n<\/tr>\n<tr>\n<td>Model the first number. Start with [latex]5[\/latex] positives.<\/td>\n<td><img loading=\"lazy\" decoding=\"async\" class=\"alignleft wp-image-15983 size-full\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2302\/2017\/05\/05000525\/IMG_01_03_037.png\" alt=\"Five positive counters\" width=\"272\" height=\"79\" \/><\/td>\n<\/tr>\n<tr>\n<td>Model the second number. Add [latex]3[\/latex] negatives.<\/td>\n<td><img loading=\"lazy\" decoding=\"async\" class=\"alignleft wp-image-15984 size-full\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2302\/2017\/05\/05000548\/IMG_01_03_038.png\" alt=\"Five positive counters and three negative counters\" width=\"275\" height=\"127\" \/><\/td>\n<\/tr>\n<tr>\n<td>Remove any neutral pairs.<\/td>\n<td><img loading=\"lazy\" decoding=\"async\" class=\"alignleft wp-image-15986 size-full\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2302\/2017\/05\/05000830\/IMG_01_03_039.png\" alt=\"Five positive counters and three negative counters with three neutral pairs circled.\" width=\"320\" height=\"152\" \/><\/td>\n<\/tr>\n<tr>\n<td>Count the result.<\/td>\n<td><img loading=\"lazy\" decoding=\"async\" class=\"alignleft wp-image-15985 size-full\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2302\/2017\/05\/05000642\/IMG_01_03_040.png\" alt=\"Two positive counters\" width=\"152\" height=\"100\" \/><\/td>\n<\/tr>\n<tr>\n<td>The sum of [latex]5[\/latex] and [latex]\u22123[\/latex] is [latex]2[\/latex].<\/td>\n<td>[latex]5+\\left(-3\\right)=2[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<\/div>\n<\/div>\n<p>&nbsp;<\/p>\n<div class=\"textbox key-takeaways\">\n<h3>try it<\/h3>\n<p>Model the expression, and then simplify:<\/p>\n<p>[latex]\\left(-2\\right)+4[\/latex]<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q512582\">Show Solution<\/span><\/p>\n<div id=\"q512582\" class=\"hidden-answer\" style=\"display: none\">\n<p><img loading=\"lazy\" decoding=\"async\" class=\"alignleft wp-image-15987 size-full\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2302\/2017\/05\/05000907\/IMG_01_03_041.png\" alt=\"Two negative counters and four positive counters with two neutral pairs circled\" width=\"281\" height=\"151\" \/><\/p>\n<p>&nbsp;<\/p>\n<p>&nbsp;<\/p>\n<p>&nbsp;<\/p>\n<p>&nbsp;<\/p>\n<p>&nbsp;<\/p>\n<p>&nbsp;<\/p>\n<p>[latex]2[\/latex]<\/p>\n<\/div>\n<\/div>\n<p>&nbsp;<\/p>\n<p>Model the expression:<br \/>\n[latex]\\left(-2\\right)+5[\/latex]<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q556256\">Show Solution<\/span><\/p>\n<div id=\"q556256\" class=\"hidden-answer\" style=\"display: none\">\n<p><img loading=\"lazy\" decoding=\"async\" class=\"alignleft wp-image-15988 size-full\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2302\/2017\/05\/05000947\/IMG_01_03_042.png\" alt=\"Two negative counters and five positive counters with two neutral pairs circled.\" width=\"332\" height=\"151\" \/><\/p>\n<p>&nbsp;<\/p>\n<p>&nbsp;<\/p>\n<p>&nbsp;<\/p>\n<p>&nbsp;<\/p>\n<p>&nbsp;<\/p>\n<p>&nbsp;<\/p>\n<p>[latex]3[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<p>&nbsp;<\/p>\n<div class=\"textbox exercises\">\n<h3>example<\/h3>\n<p>Model each addition.<\/p>\n<ol>\n<li>[latex]4 + 2[\/latex]<\/li>\n<li>[latex]\u22123 + 6[\/latex]<\/li>\n<li>[latex]4 + (\u22125)[\/latex]<\/li>\n<li>[latex]-2 + (\u22123)[\/latex]<\/li>\n<\/ol>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q480477\">Show Solution<\/span><\/p>\n<div id=\"q480477\" class=\"hidden-answer\" style=\"display: none\">\n<p>Solution<\/p>\n<table id=\"eip-id1168468294641\" class=\"unnumbered unstyled\" summary=\".\">\n<tbody>\n<tr>\n<td>1.<\/td>\n<td><\/td>\n<\/tr>\n<tr>\n<td><\/td>\n<td>[latex]4+2[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Start with [latex]4[\/latex] positives.<\/td>\n<td><img loading=\"lazy\" decoding=\"async\" class=\"alignleft wp-image-15989 size-full\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2302\/2017\/05\/05001037\/IMG_01_03_043.png\" alt=\"Four positive counters\" width=\"242\" height=\"76\" \/><\/td>\n<\/tr>\n<tr>\n<td>Add two positives.<\/td>\n<td><img loading=\"lazy\" decoding=\"async\" class=\"alignleft wp-image-15990 size-full\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2302\/2017\/05\/05001109\/IMG_01_03_044.png\" alt=\"Four positive counters and two positive counters\" width=\"395\" height=\"76\" \/><\/td>\n<\/tr>\n<tr>\n<td>How many do you have?<\/td>\n<td>[latex]4+2=6[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<table id=\"eip-id1168468679001\" class=\"unnumbered unstyled\" summary=\".\">\n<tbody>\n<tr>\n<td>2.<\/td>\n<td><\/td>\n<\/tr>\n<tr>\n<td><\/td>\n<td>[latex]-3+6[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Start with [latex]3[\/latex] negatives.<\/td>\n<td><img loading=\"lazy\" decoding=\"async\" class=\"alignleft wp-image-15992 size-full\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2302\/2017\/05\/05001612\/IMG_01_03_045.png\" alt=\"Three negative counters\" width=\"175\" height=\"70\" \/><\/td>\n<\/tr>\n<tr>\n<td>Add [latex]6[\/latex] positives.<\/td>\n<td><img loading=\"lazy\" decoding=\"async\" class=\"alignleft wp-image-15993 size-full\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2302\/2017\/05\/05001636\/IMG_01_03_046.png\" alt=\"Three negative counters and six positive counters\" width=\"330\" height=\"127\" \/><\/td>\n<\/tr>\n<tr>\n<td>Remove neutral pairs.<\/td>\n<td>\u00a0<img loading=\"lazy\" decoding=\"async\" class=\"alignleft wp-image-15994 size-full\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2302\/2017\/05\/05001728\/IMG_01_03_047.png\" alt=\"Three negative counters and six positive counters with three neutral pairs circled.\" width=\"339\" height=\"187\" \/><\/td>\n<\/tr>\n<tr>\n<td>How many are left?<\/td>\n<td><img loading=\"lazy\" decoding=\"async\" class=\"alignleft wp-image-15995 size-full\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2302\/2017\/05\/05001803\/IMG_01_03_048.png\" alt=\"Three positive counters\" width=\"175\" height=\"70\" \/><\/td>\n<\/tr>\n<tr>\n<td>\u00a0[latex]-3+6=3[\/latex]<\/td>\n<td>[latex]3[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<table id=\"eip-id1168467662609\" class=\"unnumbered unstyled\" summary=\".\">\n<tbody>\n<tr>\n<td>3.<\/td>\n<td><\/td>\n<\/tr>\n<tr>\n<td><\/td>\n<td>[latex]4+\\left(-5\\right)[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Start with [latex]4[\/latex] positives.<\/td>\n<td><img loading=\"lazy\" decoding=\"async\" class=\"alignleft wp-image-15996 size-full\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2302\/2017\/05\/05002104\/IMG_01_03_049.png\" alt=\"Four positive counters\" width=\"242\" height=\"76\" \/><\/td>\n<\/tr>\n<tr>\n<td>Add [latex]5[\/latex] negatives.<\/td>\n<td><img loading=\"lazy\" decoding=\"async\" class=\"alignleft wp-image-15997 size-full\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2302\/2017\/05\/05002129\/IMG_01_03_050.png\" alt=\"Four positive counters and five negative counters\" width=\"277\" height=\"128\" \/><\/td>\n<\/tr>\n<tr>\n<td>Remove neutral pairs.<\/td>\n<td><img loading=\"lazy\" decoding=\"async\" class=\"alignleft wp-image-15998 size-full\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2302\/2017\/05\/05002202\/IMG_01_03_051.png\" alt=\"Four positive counters and five negative counters with four neutral pairs circled.\" width=\"298\" height=\"219\" \/><\/td>\n<\/tr>\n<tr>\n<td>How many are left?<\/td>\n<td><img loading=\"lazy\" decoding=\"async\" class=\"size-full wp-image-15999 alignleft\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2302\/2017\/05\/05002237\/IMG_01_03_052.png\" alt=\"One negative counter\" width=\"84\" height=\"75\" \/><\/td>\n<\/tr>\n<tr>\n<td>\u00a0[latex]4+\\left(-5\\right)=-1[\/latex]<\/td>\n<td>[latex]-1[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<table id=\"eip-id1168468388113\" class=\"unnumbered unstyled\" summary=\".\">\n<tbody>\n<tr>\n<td>4.<\/td>\n<td><\/td>\n<\/tr>\n<tr>\n<td><\/td>\n<td>[latex]-2+\\left(-3\\right)[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Start with [latex]2[\/latex] negatives.<\/td>\n<td><img loading=\"lazy\" decoding=\"async\" class=\"size-full wp-image-16000 alignleft\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2302\/2017\/05\/05002444\/IMG_01_03_053.png\" alt=\"Two negative counters\" width=\"143\" height=\"75\" \/><\/td>\n<\/tr>\n<tr>\n<td>Add [latex]3[\/latex] negatives.<\/td>\n<td><img loading=\"lazy\" decoding=\"async\" class=\"alignleft wp-image-16001 size-full\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2302\/2017\/05\/05002505\/IMG_01_03_054.png\" alt=\"Two negative counters and three negative counters\" width=\"336\" height=\"76\" \/><\/td>\n<\/tr>\n<tr>\n<td>How many do you have?<\/td>\n<td>[latex]-2+\\left(-3\\right)=-5[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<\/div>\n<\/div>\n<p>&nbsp;<\/p>\n<div class=\"textbox key-takeaways\">\n<h3>try it<\/h3>\n<p>Model each addition.<\/p>\n<p>1. [latex]3 + 4[\/latex]<br \/>\n2. [latex]\u22121 + 4[\/latex]<br \/>\n3. [latex]4 + (\u22126)[\/latex]<br \/>\n4. [latex]\u22122 + (\u22122)[\/latex]<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q767954\">Show Solution<\/span><\/p>\n<div id=\"q767954\" class=\"hidden-answer\" style=\"display: none\">\n<p>1. [latex]7[\/latex]<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"size-full wp-image-16002 alignleft\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2302\/2017\/05\/05002536\/IMG_01_03_055.png\" alt=\"Three positive counters and four positive counters\" width=\"221\" height=\"128\" \/><\/p>\n<p>&nbsp;<\/p>\n<p>&nbsp;<\/p>\n<p>&nbsp;<\/p>\n<p>&nbsp;<\/p>\n<p>&nbsp;<\/p>\n<p>2. [latex]3[\/latex]<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"size-full wp-image-16003 alignleft\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2302\/2017\/05\/05002603\/IMG_01_03_056.png\" alt=\"One negative counter and four positive counters\" width=\"341\" height=\"75\" \/><\/p>\n<p>&nbsp;<\/p>\n<p>&nbsp;<\/p>\n<p>&nbsp;<\/p>\n<p>3. [latex]-2[\/latex]<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"size-full wp-image-16004 alignleft\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2302\/2017\/05\/05002637\/IMG_01_03_057.png\" alt=\"Four positive counters and six negative counters\" width=\"320\" height=\"128\" \/><\/p>\n<p>&nbsp;<\/p>\n<p>&nbsp;<\/p>\n<p>&nbsp;<\/p>\n<p>&nbsp;<\/p>\n<p>&nbsp;<\/p>\n<p>4. [latex]-4[\/latex]<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"size-full wp-image-16005 alignleft\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2302\/2017\/05\/05002704\/IMG_01_03_058.png\" alt=\"Two negative counters and two negative counters\" width=\"291\" height=\"76\" \/><\/p>\n<p>&nbsp;<\/p>\n<p>&nbsp;<\/p>\n<p>&nbsp;<\/p>\n<\/div>\n<\/div>\n<p>1. [latex]5 + 1[\/latex]<br \/>\n2. [latex]\u22123 + 7[\/latex]<br \/>\n3. [latex]2 + (\u22128)[\/latex]<br \/>\n4. [latex]\u22123 + (\u22124)[\/latex]<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q388859\">Show Solution<\/span><\/p>\n<div id=\"q388859\" class=\"hidden-answer\" style=\"display: none\">\n<p>1. [latex]6[\/latex]<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"alignleft wp-image-16006 size-full\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2302\/2017\/05\/05002735\/IMG_01_03_059.png\" alt=\"Five positive counters and one positive counter\" width=\"270\" height=\"128\" \/><\/p>\n<p>&nbsp;<\/p>\n<p>&nbsp;<\/p>\n<p>&nbsp;<\/p>\n<p>&nbsp;<\/p>\n<p>&nbsp;<\/p>\n<p>2. [latex]4[\/latex]<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"alignleft wp-image-16007 size-full\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2302\/2017\/05\/05002808\/IMG_01_03_060.png\" alt=\"Three negative counters and seven positive counters\" width=\"372\" height=\"128\" \/><\/p>\n<p>&nbsp;<\/p>\n<p>&nbsp;<\/p>\n<p>&nbsp;<\/p>\n<p>&nbsp;<\/p>\n<p>&nbsp;<\/p>\n<p>3. [latex]-6[\/latex]<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"alignleft wp-image-16008 size-full\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2302\/2017\/05\/05002847\/IMG_01_03_061.png\" alt=\"Two positive counters and eight negative counters\" width=\"425\" height=\"128\" \/><\/p>\n<p>&nbsp;<\/p>\n<p>&nbsp;<\/p>\n<p>&nbsp;<\/p>\n<p>&nbsp;<\/p>\n<p>&nbsp;<\/p>\n<p>4. [latex]-7[\/latex]<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"size-full wp-image-16009 alignleft\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2302\/2017\/05\/05002919\/IMG_01_03_062.png\" alt=\"Three negative counters and four negative counters\" width=\"219\" height=\"127\" \/><\/p>\n<p>&nbsp;<\/p>\n<p>&nbsp;<\/p>\n<p>&nbsp;<\/p>\n<p>&nbsp;<\/p>\n<p>&nbsp;<\/p>\n<\/div>\n<\/div>\n<\/div>\n<p>In the following video we present more examples of using color counters to model addition of integers.<\/p>\n<p><iframe loading=\"lazy\" id=\"oembed-1\" title=\"Adding Integers with the Same Sign Using Color Counters\" width=\"500\" height=\"281\" src=\"https:\/\/www.youtube.com\/embed\/beI5bQB9a7Y?feature=oembed&#38;rel=0\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/p>\n\n\t\t\t <section class=\"citations-section\" role=\"contentinfo\">\n\t\t\t <h3>Candela Citations<\/h3>\n\t\t\t\t\t <div>\n\t\t\t\t\t\t <div id=\"citation-list-9341\">\n\t\t\t\t\t\t\t <div class=\"licensing\"><div class=\"license-attribution-dropdown-subheading\">CC licensed content, Shared previously<\/div><ul class=\"citation-list\"><li>Adding Integers with the Same Sign Using Color Counters. <strong>Authored by<\/strong>: James Sousa (Mathispower4u.com). <strong>Located at<\/strong>: <a target=\"_blank\" href=\"https:\/\/youtu.be\/beI5bQB9a7Y\">https:\/\/youtu.be\/beI5bQB9a7Y<\/a>. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em><\/li><\/ul><div class=\"license-attribution-dropdown-subheading\">CC licensed content, Specific attribution<\/div><ul class=\"citation-list\"><li>Prealgebra. <strong>Provided by<\/strong>: OpenStax. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em>. <strong>License Terms<\/strong>: Download for free at http:\/\/cnx.org\/contents\/caa57dab-41c7-455e-bd6f-f443cda5519c@9.757<\/li><\/ul><\/div>\n\t\t\t\t\t\t <\/div>\n\t\t\t\t\t <\/div>\n\t\t\t <\/section>","protected":false},"author":17535,"menu_order":9,"template":"","meta":{"_candela_citation":"[{\"type\":\"cc\",\"description\":\"Adding Integers with the Same Sign Using Color Counters\",\"author\":\"James Sousa (Mathispower4u.com)\",\"organization\":\"\",\"url\":\"https:\/\/youtu.be\/beI5bQB9a7Y\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"\"},{\"type\":\"cc-attribution\",\"description\":\"Prealgebra\",\"author\":\"\",\"organization\":\"OpenStax\",\"url\":\"\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"Download for free at http:\/\/cnx.org\/contents\/caa57dab-41c7-455e-bd6f-f443cda5519c@9.757\"}]","CANDELA_OUTCOMES_GUID":"b70b5f52-4571-4992-b2ab-e9051b1e46bf","pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"class_list":["post-9341","chapter","type-chapter","status-publish","hentry"],"part":6350,"_links":{"self":[{"href":"https:\/\/courses.lumenlearning.com\/wm-prealgebra\/wp-json\/pressbooks\/v2\/chapters\/9341","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/courses.lumenlearning.com\/wm-prealgebra\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/courses.lumenlearning.com\/wm-prealgebra\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/wm-prealgebra\/wp-json\/wp\/v2\/users\/17535"}],"version-history":[{"count":25,"href":"https:\/\/courses.lumenlearning.com\/wm-prealgebra\/wp-json\/pressbooks\/v2\/chapters\/9341\/revisions"}],"predecessor-version":[{"id":16010,"href":"https:\/\/courses.lumenlearning.com\/wm-prealgebra\/wp-json\/pressbooks\/v2\/chapters\/9341\/revisions\/16010"}],"part":[{"href":"https:\/\/courses.lumenlearning.com\/wm-prealgebra\/wp-json\/pressbooks\/v2\/parts\/6350"}],"metadata":[{"href":"https:\/\/courses.lumenlearning.com\/wm-prealgebra\/wp-json\/pressbooks\/v2\/chapters\/9341\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/courses.lumenlearning.com\/wm-prealgebra\/wp-json\/wp\/v2\/media?parent=9341"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/wm-prealgebra\/wp-json\/pressbooks\/v2\/chapter-type?post=9341"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/wm-prealgebra\/wp-json\/wp\/v2\/contributor?post=9341"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/wm-prealgebra\/wp-json\/wp\/v2\/license?post=9341"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}