{"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":"2017-12-11T23:36:18","modified_gmt":"2017-12-11T23:36:18","slug":"notation-and-modeling-addition-of-integers","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/csn-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 blue counter represent a positive and a red counter will represent a negative.\r\n\r\n<img class=\"aligncenter wp-image-15438\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2302\/2017\/05\/11223428\/AddingIntegers_1.png\" alt=\"This figure has a blue circle labeled positive and a red circle labeled negative.\" width=\"300\" height=\"159\" \/>\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 blue counter represents [latex]+1[\/latex]. A red counter represents [latex]-1[\/latex]. Together they add to zero.\r\n\r\n<img class=\"aligncenter wp-image-15439\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2302\/2017\/05\/11223451\/AddingIntegers_2.png\" alt=\"This figure has a blue circle over a red circle. Beside them is the statement 1 plus negative 1 equals 0.\" width=\"300\" height=\"219\" \/>\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=\"alignnone wp-image-15440\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2302\/2017\/05\/11223536\/AddingIntegers_3.png\" alt=\"five blue circles\" width=\"228\" height=\"75\" \/><\/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=\"alignnone  wp-image-15441\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2302\/2017\/05\/11223630\/AddingIntegers_4.png\" alt=\"five blue circles with an additional three circles\" width=\"402\" height=\"74\" \/><\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Count the total number of counters.<\/td>\r\n<td><img class=\"alignnone  wp-image-15442\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2302\/2017\/05\/11223730\/AddingIntegers_5.png\" alt=\"eight blue circles labelled eight positives\" width=\"359\" height=\"74\" \/><\/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=\"alignnone wp-image-15443\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2302\/2017\/05\/11223837\/AddingIntegers_6.png\" alt=\"This figure has six blue circles in a row, representing positive counters. The first two circles are separated from the following four circles.\" width=\"378\" height=\"68\" \/>\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=\"alignnone wp-image-15444\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2302\/2017\/05\/11223927\/AddingIntegers_7.png\" alt=\"This figure has seven blue circles in a row, representing positive counters. Two circles are separated from the following 5 circles.\" width=\"380\" height=\"69\" \/>\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=\"alignnone wp-image-15446\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2302\/2017\/05\/11224115\/AddingIntegers_8.png\" alt=\"five pink circles representing negative counters\" width=\"209\" height=\"65\" \/><\/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=\"alignnone wp-image-15445\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2302\/2017\/05\/11224008\/AddingIntegers_9.png\" alt=\"five pink circles representing negative counters, separated by another three pink circles.\" width=\"359\" height=\"66\" \/><\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Count the total number of counters.<\/td>\r\n<td><img class=\"alignnone wp-image-15447\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2302\/2017\/05\/11224136\/AddingIntegers_10.png\" alt=\"eight pink circles representing negative counters; labelled eight negatives\" width=\"322\" height=\"65\" \/><\/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=\"alignnone wp-image-15448\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2302\/2017\/05\/11224225\/AddingIntegers_11.png\" alt=\"This figure has six pink circles in a row, representing positive counters. Two circles are separated from the following four circles.\" width=\"351\" height=\"79\" \/>\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=\"alignnone wp-image-15449\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2302\/2017\/05\/11224435\/AddingIntegers_12.png\" alt=\"This figure has seven pink circles in a row, representing positive counters. Two circles are separated from the following five circles.\" width=\"352\" height=\"64\" \/>\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=\"alignnone wp-image-15450\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2302\/2017\/05\/11224515\/AddingIntegers_13.png\" alt=\"five red negative markers\" width=\"356\" height=\"66\" \/><\/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=\"alignnone wp-image-15488\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2302\/2017\/05\/11231233\/Artboard_14.png\" alt=\"five red negative markers with three blue positive markers\" width=\"354\" height=\"109\" \/><\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Remove any neutral pairs.<\/td>\r\n<td><img class=\"alignnone wp-image-15452\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2302\/2017\/05\/11224546\/AddingIntegers_15.png\" alt=\"five red negative markers with three blue positive markers; three negative markers are cancelled out by three positive markers, leaving two negative markers on their own\" width=\"354\" height=\"122\" \/><\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Count the result.<\/td>\r\n<td><img class=\"alignnone wp-image-15453\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2302\/2017\/05\/11224556\/AddingIntegers_16.png\" alt=\"two red negative markers labelled two negatives\" width=\"356\" height=\"86\" \/><\/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=\"alignnone wp-image-15454\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2302\/2017\/05\/11224613\/AddingIntegers_17.png\" alt=\"four red negative markers with two blue positive markers; two negative markers are cancelled out by two positive markers, leaving two negative markers on their own\" width=\"356\" height=\"137\" \/>\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=\"alignnone wp-image-15455\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2302\/2017\/05\/11224625\/AddingIntegers_18.png\" alt=\"five red negative markers with two blue positive markers; two negative markers are cancelled out by two positive markers, leaving three negative markers on their own\" width=\"356\" height=\"135\" \/>\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=\"alignnone wp-image-15456\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2302\/2017\/05\/11224640\/AddingIntegers_19.png\" alt=\"five blue positive markers\" width=\"357\" height=\"54\" \/><\/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=\"alignnone wp-image-15457\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2302\/2017\/05\/11224651\/AddingIntegers_20.png\" alt=\"five blue positive markers with two red negative markers\" width=\"356\" height=\"110\" \/><\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Remove any neutral pairs.<\/td>\r\n<td><img class=\"alignnone wp-image-15458\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2302\/2017\/05\/11224704\/AddingIntegers_21.png\" alt=\"five blue positive markers with two red negative markers; two positive markers are cancelled out by two negative markers, leaving three positive markers on their own\" width=\"355\" height=\"111\" \/><\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Count the result.<\/td>\r\n<td><img class=\"alignnone wp-image-15459\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2302\/2017\/05\/11224716\/AddingIntegers_22.png\" alt=\"two blue positive markers labelled two positives\" width=\"355\" height=\"81\" \/><\/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=\"alignnone wp-image-15460\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2302\/2017\/05\/11224730\/AddingIntegers_23.png\" alt=\"four blue positive markers with two red negative markers; two positive markers are cancelled out by two negative markers, leaving two positive markers on their own\" width=\"355\" height=\"137\" \/>\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=\"alignnone wp-image-15461\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2302\/2017\/05\/11224744\/AddingIntegers_24.png\" alt=\"five blue positive markers with two red negative markers; two positive markers are cancelled out by two negative markers, leaving three positive markers on their own\" width=\"356\" height=\"142\" \/>\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=\"alignnone wp-image-15469\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2302\/2017\/05\/11224958\/AddingIntegers_31.png\" alt=\"four blue positive markers\" width=\"375\" height=\"70\" \/><\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Add two positives.<\/td>\r\n<td><img class=\"alignnone wp-image-15487\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2302\/2017\/05\/11231204\/AddingIntegers_45.png\" alt=\"four blue positive markers separated from two additional blue positive markers\" width=\"389\" height=\"79\" \/><\/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=\"alignnone wp-image-15463\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2302\/2017\/05\/11224814\/AddingIntegers_26.png\" alt=\"three red negative markers\" width=\"196\" height=\"76\" \/><\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Add [latex]6[\/latex] positives.<\/td>\r\n<td><img class=\"alignnone wp-image-15464\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2302\/2017\/05\/11224832\/AddingIntegers_27.png\" alt=\"three red negative markers next to six additional blue positive markers\" width=\"355\" height=\"127\" \/><\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Remove neutral pairs.<\/td>\r\n<td>\u00a0<img class=\"alignnone wp-image-15465\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2302\/2017\/05\/11224900\/AddingIntegers_28.png\" alt=\"three red negative markers next to six additional blue positive markers; the three negative markers are cancelled out by three positive markers, leaving three positive markers on their own\" width=\"354\" height=\"192\" \/><\/td>\r\n<\/tr>\r\n<tr>\r\n<td>How many are left?<\/td>\r\n<td><img class=\"alignnone wp-image-15467\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2302\/2017\/05\/11224923\/AddingIntegers_29.png\" alt=\"three blue positive markers\" width=\"194\" height=\"77\" \/><\/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=\"alignnone wp-image-15462\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2302\/2017\/05\/11224803\/AddingIntegers_25.png\" alt=\"four blue positive markers\" width=\"221\" height=\"75\" \/><\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Add [latex]5[\/latex] negatives.<\/td>\r\n<td><img class=\"alignnone wp-image-15468\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2302\/2017\/05\/11224945\/AddingIntegers_30.png\" alt=\"four blue positive markers separated from five red negative markers\" width=\"355\" height=\"123\" \/><\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Remove neutral pairs.<\/td>\r\n<td><img class=\"alignnone wp-image-15470\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2302\/2017\/05\/11225039\/AddingIntegers_32.png\" alt=\"four blue positive markers separated from five red negative markers; the four positive markers are cancelled out by four negative markers, leaving one negative marker on its own\" width=\"356\" height=\"183\" \/><\/td>\r\n<\/tr>\r\n<tr>\r\n<td>How many are left?<\/td>\r\n<td><img class=\"alignnone wp-image-15471\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2302\/2017\/05\/11225051\/AddingIntegers_33.png\" alt=\"a single red negative marker\" width=\"354\" height=\"56\" \/><\/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=\"alignnone wp-image-15472\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2302\/2017\/05\/11225105\/AddingIntegers_34.png\" alt=\"two red negative markers\" width=\"354\" height=\"50\" \/><\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Add [latex]3[\/latex] negatives.<\/td>\r\n<td><img class=\"alignnone wp-image-15475\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2302\/2017\/05\/11225132\/AddingIntegers_36.png\" alt=\"two red negative markers separated from three additional red negative markers\" width=\"356\" height=\"51\" \/><\/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=\"alignnone wp-image-15477\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2302\/2017\/05\/11225146\/AddingIntegers_37.png\" alt=\"three blue positive markers separated from four additional blue positive markers\" width=\"356\" height=\"130\" \/>\r\n\r\n2. [latex]3[\/latex]\r\n\r\n<img class=\"alignnone wp-image-15478\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2302\/2017\/05\/11225202\/AddingIntegers_38.png\" alt=\"one red negative marker separated from four blue positive markers\" width=\"357\" height=\"62\" \/>\r\n\r\n3. [latex]-2[\/latex]\r\n\r\n<img class=\"alignnone  wp-image-15490\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2302\/2017\/05\/11233032\/Artboard_31.png\" alt=\"four blue positive markers separated from six red negative markers\" width=\"355\" height=\"123\" \/>\r\n\r\n4. [latex]-4[\/latex]\r\n\r\n<img class=\"alignnone wp-image-15480\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2302\/2017\/05\/11225226\/AddingIntegers_40.png\" alt=\"two red negative markers separated from two additional red negative markers\" width=\"369\" height=\"64\" \/>\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=\"alignnone wp-image-15481\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2302\/2017\/05\/11225237\/AddingIntegers_41.png\" alt=\"five blue positive markers separated from one additional blue positive marker\" width=\"424\" height=\"130\" \/>\r\n\r\n2. [latex]4[\/latex]\r\n\r\n<img class=\"alignnone wp-image-15482\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2302\/2017\/05\/11225249\/AddingIntegers_42.png\" alt=\"three red negative markers separated from seven blue positive markers\" width=\"424\" height=\"130\" \/>\r\n\r\n3. [latex]-6[\/latex]\r\n\r\n<img class=\"alignnone wp-image-15484\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2302\/2017\/05\/11225259\/AddingIntegers_43.png\" alt=\"two blue positive markers separated from eight red negative markers\" width=\"424\" height=\"130\" \/>\r\n\r\n4. [latex]-7[\/latex]\r\n\r\n<img class=\"alignnone wp-image-15485\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2302\/2017\/05\/11225312\/AddingIntegers_44.png\" alt=\"three red negative markers separated from four additional red negative markers\" width=\"422\" height=\"130\" \/>\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 blue counter represent a positive and a red counter will represent a negative.<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter wp-image-15438\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2302\/2017\/05\/11223428\/AddingIntegers_1.png\" alt=\"This figure has a blue circle labeled positive and a red circle labeled negative.\" width=\"300\" height=\"159\" \/><\/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 blue counter represents [latex]+1[\/latex]. A red counter represents [latex]-1[\/latex]. Together they add to zero.<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter wp-image-15439\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2302\/2017\/05\/11223451\/AddingIntegers_2.png\" alt=\"This figure has a blue circle over a red circle. Beside them is the statement 1 plus negative 1 equals 0.\" width=\"300\" height=\"219\" \/><\/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=\"alignnone wp-image-15440\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2302\/2017\/05\/11223536\/AddingIntegers_3.png\" alt=\"five blue circles\" width=\"228\" height=\"75\" \/><\/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=\"alignnone  wp-image-15441\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2302\/2017\/05\/11223630\/AddingIntegers_4.png\" alt=\"five blue circles with an additional three circles\" width=\"402\" height=\"74\" \/><\/td>\n<\/tr>\n<tr>\n<td>Count the total number of counters.<\/td>\n<td><img loading=\"lazy\" decoding=\"async\" class=\"alignnone  wp-image-15442\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2302\/2017\/05\/11223730\/AddingIntegers_5.png\" alt=\"eight blue circles labelled eight positives\" width=\"359\" height=\"74\" \/><\/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=\"alignnone wp-image-15443\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2302\/2017\/05\/11223837\/AddingIntegers_6.png\" alt=\"This figure has six blue circles in a row, representing positive counters. The first two circles are separated from the following four circles.\" width=\"378\" height=\"68\" \/><\/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=\"alignnone wp-image-15444\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2302\/2017\/05\/11223927\/AddingIntegers_7.png\" alt=\"This figure has seven blue circles in a row, representing positive counters. Two circles are separated from the following 5 circles.\" width=\"380\" height=\"69\" \/><\/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=\"alignnone wp-image-15446\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2302\/2017\/05\/11224115\/AddingIntegers_8.png\" alt=\"five pink circles representing negative counters\" width=\"209\" height=\"65\" \/><\/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=\"alignnone wp-image-15445\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2302\/2017\/05\/11224008\/AddingIntegers_9.png\" alt=\"five pink circles representing negative counters, separated by another three pink circles.\" width=\"359\" height=\"66\" \/><\/td>\n<\/tr>\n<tr>\n<td>Count the total number of counters.<\/td>\n<td><img loading=\"lazy\" decoding=\"async\" class=\"alignnone wp-image-15447\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2302\/2017\/05\/11224136\/AddingIntegers_10.png\" alt=\"eight pink circles representing negative counters; labelled eight negatives\" width=\"322\" height=\"65\" \/><\/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=\"alignnone wp-image-15448\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2302\/2017\/05\/11224225\/AddingIntegers_11.png\" alt=\"This figure has six pink circles in a row, representing positive counters. Two circles are separated from the following four circles.\" width=\"351\" height=\"79\" \/><\/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=\"alignnone wp-image-15449\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2302\/2017\/05\/11224435\/AddingIntegers_12.png\" alt=\"This figure has seven pink circles in a row, representing positive counters. Two circles are separated from the following five circles.\" width=\"352\" height=\"64\" \/><\/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=\"alignnone wp-image-15450\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2302\/2017\/05\/11224515\/AddingIntegers_13.png\" alt=\"five red negative markers\" width=\"356\" height=\"66\" \/><\/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=\"alignnone wp-image-15488\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2302\/2017\/05\/11231233\/Artboard_14.png\" alt=\"five red negative markers with three blue positive markers\" width=\"354\" height=\"109\" \/><\/td>\n<\/tr>\n<tr>\n<td>Remove any neutral pairs.<\/td>\n<td><img loading=\"lazy\" decoding=\"async\" class=\"alignnone wp-image-15452\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2302\/2017\/05\/11224546\/AddingIntegers_15.png\" alt=\"five red negative markers with three blue positive markers; three negative markers are cancelled out by three positive markers, leaving two negative markers on their own\" width=\"354\" height=\"122\" \/><\/td>\n<\/tr>\n<tr>\n<td>Count the result.<\/td>\n<td><img loading=\"lazy\" decoding=\"async\" class=\"alignnone wp-image-15453\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2302\/2017\/05\/11224556\/AddingIntegers_16.png\" alt=\"two red negative markers labelled two negatives\" width=\"356\" height=\"86\" \/><\/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=\"alignnone wp-image-15454\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2302\/2017\/05\/11224613\/AddingIntegers_17.png\" alt=\"four red negative markers with two blue positive markers; two negative markers are cancelled out by two positive markers, leaving two negative markers on their own\" width=\"356\" height=\"137\" \/><\/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=\"alignnone wp-image-15455\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2302\/2017\/05\/11224625\/AddingIntegers_18.png\" alt=\"five red negative markers with two blue positive markers; two negative markers are cancelled out by two positive markers, leaving three negative markers on their own\" width=\"356\" height=\"135\" \/><\/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=\"alignnone wp-image-15456\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2302\/2017\/05\/11224640\/AddingIntegers_19.png\" alt=\"five blue positive markers\" width=\"357\" height=\"54\" \/><\/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=\"alignnone wp-image-15457\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2302\/2017\/05\/11224651\/AddingIntegers_20.png\" alt=\"five blue positive markers with two red negative markers\" width=\"356\" height=\"110\" \/><\/td>\n<\/tr>\n<tr>\n<td>Remove any neutral pairs.<\/td>\n<td><img loading=\"lazy\" decoding=\"async\" class=\"alignnone wp-image-15458\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2302\/2017\/05\/11224704\/AddingIntegers_21.png\" alt=\"five blue positive markers with two red negative markers; two positive markers are cancelled out by two negative markers, leaving three positive markers on their own\" width=\"355\" height=\"111\" \/><\/td>\n<\/tr>\n<tr>\n<td>Count the result.<\/td>\n<td><img loading=\"lazy\" decoding=\"async\" class=\"alignnone wp-image-15459\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2302\/2017\/05\/11224716\/AddingIntegers_22.png\" alt=\"two blue positive markers labelled two positives\" width=\"355\" height=\"81\" \/><\/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=\"alignnone wp-image-15460\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2302\/2017\/05\/11224730\/AddingIntegers_23.png\" alt=\"four blue positive markers with two red negative markers; two positive markers are cancelled out by two negative markers, leaving two positive markers on their own\" width=\"355\" height=\"137\" \/><\/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=\"alignnone wp-image-15461\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2302\/2017\/05\/11224744\/AddingIntegers_24.png\" alt=\"five blue positive markers with two red negative markers; two positive markers are cancelled out by two negative markers, leaving three positive markers on their own\" width=\"356\" height=\"142\" \/><\/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=\"alignnone wp-image-15469\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2302\/2017\/05\/11224958\/AddingIntegers_31.png\" alt=\"four blue positive markers\" width=\"375\" height=\"70\" \/><\/td>\n<\/tr>\n<tr>\n<td>Add two positives.<\/td>\n<td><img loading=\"lazy\" decoding=\"async\" class=\"alignnone wp-image-15487\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2302\/2017\/05\/11231204\/AddingIntegers_45.png\" alt=\"four blue positive markers separated from two additional blue positive markers\" width=\"389\" height=\"79\" \/><\/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=\"alignnone wp-image-15463\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2302\/2017\/05\/11224814\/AddingIntegers_26.png\" alt=\"three red negative markers\" width=\"196\" height=\"76\" \/><\/td>\n<\/tr>\n<tr>\n<td>Add [latex]6[\/latex] positives.<\/td>\n<td><img loading=\"lazy\" decoding=\"async\" class=\"alignnone wp-image-15464\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2302\/2017\/05\/11224832\/AddingIntegers_27.png\" alt=\"three red negative markers next to six additional blue positive markers\" width=\"355\" height=\"127\" \/><\/td>\n<\/tr>\n<tr>\n<td>Remove neutral pairs.<\/td>\n<td>\u00a0<img loading=\"lazy\" decoding=\"async\" class=\"alignnone wp-image-15465\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2302\/2017\/05\/11224900\/AddingIntegers_28.png\" alt=\"three red negative markers next to six additional blue positive markers; the three negative markers are cancelled out by three positive markers, leaving three positive markers on their own\" width=\"354\" height=\"192\" \/><\/td>\n<\/tr>\n<tr>\n<td>How many are left?<\/td>\n<td><img loading=\"lazy\" decoding=\"async\" class=\"alignnone wp-image-15467\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2302\/2017\/05\/11224923\/AddingIntegers_29.png\" alt=\"three blue positive markers\" width=\"194\" height=\"77\" \/><\/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=\"alignnone wp-image-15462\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2302\/2017\/05\/11224803\/AddingIntegers_25.png\" alt=\"four blue positive markers\" width=\"221\" height=\"75\" \/><\/td>\n<\/tr>\n<tr>\n<td>Add [latex]5[\/latex] negatives.<\/td>\n<td><img loading=\"lazy\" decoding=\"async\" class=\"alignnone wp-image-15468\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2302\/2017\/05\/11224945\/AddingIntegers_30.png\" alt=\"four blue positive markers separated from five red negative markers\" width=\"355\" height=\"123\" \/><\/td>\n<\/tr>\n<tr>\n<td>Remove neutral pairs.<\/td>\n<td><img loading=\"lazy\" decoding=\"async\" class=\"alignnone wp-image-15470\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2302\/2017\/05\/11225039\/AddingIntegers_32.png\" alt=\"four blue positive markers separated from five red negative markers; the four positive markers are cancelled out by four negative markers, leaving one negative marker on its own\" width=\"356\" height=\"183\" \/><\/td>\n<\/tr>\n<tr>\n<td>How many are left?<\/td>\n<td><img loading=\"lazy\" decoding=\"async\" class=\"alignnone wp-image-15471\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2302\/2017\/05\/11225051\/AddingIntegers_33.png\" alt=\"a single red negative marker\" width=\"354\" height=\"56\" \/><\/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=\"alignnone wp-image-15472\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2302\/2017\/05\/11225105\/AddingIntegers_34.png\" alt=\"two red negative markers\" width=\"354\" height=\"50\" \/><\/td>\n<\/tr>\n<tr>\n<td>Add [latex]3[\/latex] negatives.<\/td>\n<td><img loading=\"lazy\" decoding=\"async\" class=\"alignnone wp-image-15475\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2302\/2017\/05\/11225132\/AddingIntegers_36.png\" alt=\"two red negative markers separated from three additional red negative markers\" width=\"356\" height=\"51\" \/><\/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=\"alignnone wp-image-15477\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2302\/2017\/05\/11225146\/AddingIntegers_37.png\" alt=\"three blue positive markers separated from four additional blue positive markers\" width=\"356\" height=\"130\" \/><\/p>\n<p>2. [latex]3[\/latex]<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"alignnone wp-image-15478\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2302\/2017\/05\/11225202\/AddingIntegers_38.png\" alt=\"one red negative marker separated from four blue positive markers\" width=\"357\" height=\"62\" \/><\/p>\n<p>3. [latex]-2[\/latex]<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"alignnone  wp-image-15490\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2302\/2017\/05\/11233032\/Artboard_31.png\" alt=\"four blue positive markers separated from six red negative markers\" width=\"355\" height=\"123\" \/><\/p>\n<p>4. [latex]-4[\/latex]<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"alignnone wp-image-15480\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2302\/2017\/05\/11225226\/AddingIntegers_40.png\" alt=\"two red negative markers separated from two additional red negative markers\" width=\"369\" height=\"64\" \/><\/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=\"alignnone wp-image-15481\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2302\/2017\/05\/11225237\/AddingIntegers_41.png\" alt=\"five blue positive markers separated from one additional blue positive marker\" width=\"424\" height=\"130\" \/><\/p>\n<p>2. [latex]4[\/latex]<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"alignnone wp-image-15482\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2302\/2017\/05\/11225249\/AddingIntegers_42.png\" alt=\"three red negative markers separated from seven blue positive markers\" width=\"424\" height=\"130\" \/><\/p>\n<p>3. [latex]-6[\/latex]<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"alignnone wp-image-15484\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2302\/2017\/05\/11225259\/AddingIntegers_43.png\" alt=\"two blue positive markers separated from eight red negative markers\" width=\"424\" height=\"130\" \/><\/p>\n<p>4. [latex]-7[\/latex]<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"alignnone wp-image-15485\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2302\/2017\/05\/11225312\/AddingIntegers_44.png\" alt=\"three red negative markers separated from four additional red negative markers\" width=\"422\" height=\"130\" \/><\/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\/csn-prealgebra\/wp-json\/pressbooks\/v2\/chapters\/9341","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/courses.lumenlearning.com\/csn-prealgebra\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/courses.lumenlearning.com\/csn-prealgebra\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/csn-prealgebra\/wp-json\/wp\/v2\/users\/17535"}],"version-history":[{"count":23,"href":"https:\/\/courses.lumenlearning.com\/csn-prealgebra\/wp-json\/pressbooks\/v2\/chapters\/9341\/revisions"}],"predecessor-version":[{"id":15491,"href":"https:\/\/courses.lumenlearning.com\/csn-prealgebra\/wp-json\/pressbooks\/v2\/chapters\/9341\/revisions\/15491"}],"part":[{"href":"https:\/\/courses.lumenlearning.com\/csn-prealgebra\/wp-json\/pressbooks\/v2\/parts\/6350"}],"metadata":[{"href":"https:\/\/courses.lumenlearning.com\/csn-prealgebra\/wp-json\/pressbooks\/v2\/chapters\/9341\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/courses.lumenlearning.com\/csn-prealgebra\/wp-json\/wp\/v2\/media?parent=9341"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/csn-prealgebra\/wp-json\/pressbooks\/v2\/chapter-type?post=9341"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/csn-prealgebra\/wp-json\/wp\/v2\/contributor?post=9341"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/csn-prealgebra\/wp-json\/wp\/v2\/license?post=9341"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}