{"id":9395,"date":"2017-05-02T22:12:02","date_gmt":"2017-05-02T22:12:02","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/prealgebra\/?post_type=chapter&#038;p=9395"},"modified":"2020-09-05T04:08:48","modified_gmt":"2020-09-05T04:08:48","slug":"notation-and-modeling-subtraction-of-integers","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/wm-prealgebra\/chapter\/notation-and-modeling-subtraction-of-integers\/","title":{"raw":"Notation and Modeling Subtraction of Integers","rendered":"Notation and Modeling Subtraction of Integers"},"content":{"raw":"<div class=\"textbox learning-objectives\">\r\n<h3>Learning Outcomes<\/h3>\r\n<ul>\r\n \t<li>Model integer subtraction with color counters<\/li>\r\n<\/ul>\r\n<\/div>\r\nYou learn as a child how to subtract numbers through everyday experiences. For example, if you have 10 animal cookies and eat 6 of them, you will have 4 animal cookies left.\r\n\r\nReal-life experiences serve as models for subtracting positive numbers, and in some cases, such as temperature, for adding negative as well as positive numbers. But it is difficult to relate subtracting negative numbers to common life experiences. Most people do not have an intuitive understanding of subtraction when negative numbers are involved. Math teachers use several different models to explain subtracting negative numbers.\r\n\r\nWe will continue to use counters to model subtraction. Remember, the purple counters represent positive numbers and the orange counters represent negative numbers.\r\n\r\nPerhaps when you were younger, you read [latex]5 - 3[\/latex] as <em>five take away three<\/em>. When we use counters, we can think of subtraction the same way.\r\n\r\nWe will model four subtraction scenarios\u00a0using the numbers [latex]5[\/latex] and [latex]3[\/latex].\r\n<ul>\r\n \t<li>[latex]5 - 3[\/latex]<\/li>\r\n \t<li>[latex]- 5-\\left(-3\\right)[\/latex]<\/li>\r\n \t<li>[latex]-5 - 3[\/latex]<\/li>\r\n \t<li>[latex]5-\\left(-3\\right)[\/latex]<\/li>\r\n<\/ul>\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-id1168469608926\" class=\"unnumbered unstyled\" style=\"width: 75%;\" summary=\"This figure has five rows. The first row states interpret the expression. It is followed by 5 minus 3 means 5 take away 3. The second row states model the first number, start with 5 positives. It is followed by 5 blue circles. The third row states take away the second number, so take away 3 positives. It is followed by 5 blue circles with the first three circled. The fourth row states find the number of counters that are left. It is followed by 2 blue circles that are labeled two positives. The fifth row states the difference of 5 and 3 is 2. It is followed by the expression 5 minus 3 equals 2.\">\r\n<tbody>\r\n<tr>\r\n<td>Interpret the expression.<\/td>\r\n<td>[latex]5 - 3[\/latex] means [latex]5[\/latex] take away [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=\"size-full wp-image-16011 alignleft\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2302\/2017\/05\/05023929\/IMG_01_03_063.png\" alt=\"Five positive counters\" width=\"299\" height=\"109\" \/><\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Take away the second number. So take away [latex]3[\/latex] positives.<\/td>\r\n<td><img class=\"wp-image-16013 size-full alignleft\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2302\/2017\/05\/05023953\/IMG_01_03_064.png\" alt=\"Three positive counters are circled out of five total positive counters\" width=\"360\" height=\"122\" \/><\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Find the counters that are left.<\/td>\r\n<td><img class=\"alignleft wp-image-16014 size-full\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2302\/2017\/05\/05024045\/IMG_01_03_065.png\" alt=\"Two positive counters\" width=\"126\" height=\"73\" \/><\/td>\r\n<\/tr>\r\n<tr>\r\n<td><\/td>\r\n<td>[latex]5 - 3=2[\/latex]\r\n\r\nThe difference between [latex]5[\/latex] and [latex]3[\/latex] is [latex]2[\/latex] .<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<\/div>\r\nNow you can try a similar problem.\r\n<div class=\"textbox key-takeaways\">\r\n<h3>try it<\/h3>\r\nModel the expression:\r\n\r\n[latex]6 - 4[\/latex]\r\n[reveal-answer q=\"49141\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"49141\"]\r\n\r\n<img class=\"alignleft wp-image-16015 size-full\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2302\/2017\/05\/05024109\/IMG_01_03_066.png\" alt=\"A total of six positive counters and four of them are circled\" width=\"370\" height=\"109\" \/>\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\r\n[latex]7 - 4[\/latex]\r\n[reveal-answer q=\"400845\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"400845\"]\r\n\r\n<img class=\"alignleft wp-image-16016 size-full\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2302\/2017\/05\/05024209\/IMG_01_03_067.png\" alt=\"A total of seven positive counters and four are circled\" width=\"405\" height=\"109\" \/>\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\nIn the previous example we subtracted [latex]3[\/latex] positives from positive [latex]5[\/latex]. Now we will subtract\u00a0[latex]3[\/latex] negatives from\u00a0negative [latex]5[\/latex]. Compare the results of this example to the previous example after you read through it.\r\n<div class=\"textbox exercises\">\r\n<h3>example<\/h3>\r\nModel: [latex]-5-\\left(-3\\right)[\/latex]\r\n[reveal-answer q=\"511456\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"511456\"]\r\n\r\nSolution:\r\n<table id=\"eip-id1168469693755\" class=\"unnumbered unstyled\" style=\"width: 75%;\" summary=\"This figure has five rows. The first row states interpret the expression. It is followed by 5 minus negative 3 means 5 take away negative 3. The second row states model the first number, start with 5 negatives. It is followed by 5 red circles. The third row states take away the second number, so take away 3 negatives. It is followed by 5 red circles with the first three circled. The fourth row states find the number of counters that are left. It is followed by 2 red circles which are labeled two negatives. The fifth row states the difference of negative 5 and negative 3 is negative 2. Next to this is the expression negative 5 minus negative 3 equals negative two.\">\r\n<tbody>\r\n<tr>\r\n<td>Interpret the expression.<\/td>\r\n<td>[latex]-5-\\left(-3\\right)[\/latex] means [latex]-5[\/latex] take away [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=\"size-full wp-image-16017 alignleft\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2302\/2017\/05\/05024302\/IMG_01_03_068.png\" alt=\"Five negative counters\" width=\"299\" height=\"109\" \/><\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Take away the second number. So take away [latex]3[\/latex] negatives.<\/td>\r\n<td><img class=\"alignleft wp-image-16018 size-full\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2302\/2017\/05\/05024322\/IMG_01_03_069.png\" alt=\"A total of five negative counters with three of the five circled\" width=\"359\" height=\"122\" \/><\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Find the number of counters that are left.<\/td>\r\n<td><img class=\"alignleft wp-image-16019 size-full\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2302\/2017\/05\/05024352\/IMG_01_03_070.png\" alt=\"Two negative counters\" width=\"126\" height=\"73\" \/><\/td>\r\n<\/tr>\r\n<tr>\r\n<td><\/td>\r\n<td>[latex]-5-\\left(-3\\right)=-2[\/latex]\r\n\r\nThe difference between [latex]-5[\/latex] and [latex]-3[\/latex] is [latex]-2[\/latex] .<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\nYou can try a similar problem.\r\n<div class=\"textbox key-takeaways\">\r\n<h3>try it<\/h3>\r\nModel the expression:\r\n\r\n[latex]-6-\\left(-4\\right)[\/latex]\r\n[reveal-answer q=\"720789\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"720789\"]\r\n\r\n<img class=\"alignleft wp-image-16020 size-full\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2302\/2017\/05\/05024414\/IMG_01_03_071.png\" alt=\"A total of six negative counters with four of the six circled.\" width=\"355\" height=\"110\" \/>\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:\r\n\r\n[latex]-7-\\left(-4\\right)[\/latex]\r\n[reveal-answer q=\"188555\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"188555\"]\r\n\r\n<img class=\"alignleft wp-image-16021 size-full\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2302\/2017\/05\/05024455\/IMG_01_03_072.png\" alt=\"A total of seven negative counters with four of the seven counters circled\" width=\"404\" height=\"109\" \/>\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\nNotice that the previous two examples\u00a0are very much alike.\r\n<ul id=\"fs-id801766\">\r\n \t<li>First, we subtracted [latex]3[\/latex] positives from [latex]5[\/latex] positives to get [latex]2[\/latex] positives.<\/li>\r\n \t<li>Then we subtracted [latex]3[\/latex] negatives from [latex]5[\/latex] negatives to get [latex]2[\/latex] negatives.<\/li>\r\n<\/ul>\r\nEach example used counters of only one color, and the \"take away\" model of subtraction was easy to apply.\r\n\r\n<img class=\"aligncenter wp-image-16022 size-full\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2302\/2017\/05\/05024540\/IMG_01_03_073.png\" alt=\"Five positive counters minus three positive counters equals two. Five negative counters minus three negative counters equals negative two.\" width=\"728\" height=\"132\" \/>Now let\u2019s see what happens when we subtract one positive and one negative number. We will need to use both positive and negative counters and sometimes some neutral pairs, too. Adding a neutral pair does not change the value.\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-id1168468477542\" class=\"unnumbered unstyled\" style=\"width: 75%;\" summary=\"This figure has seven rows. The first row states interpret the expression. It is followed by negative 5 minus 3 means negative 5 take away 3. The second row states to model the first number, start with 5 negatives. It is followed by 5 red circles. The third row states take away the second number, so we need to take away 3 positives. It is followed by a row of 5 red circles, beside them 3 red circles. Below the three red circles are 3 blue circles. The fourth row states there are no positives to take away so add neutral pairs until you have 3 positives. The fifth row states now take away three positives. It is followed by a row of 5 red circles, beside them 3 red circles. Below the three red circles are 3 blue circles. The blue shaded circles are circled. The sixth row states count the number of counters that are left. It is followed by 8 red circles. The seventh row states the difference of negative five and three is negative 8. It is followed by the expression negative 5 minus 3 equals negative 8.\">\r\n<tbody>\r\n<tr>\r\n<td>Interpret the expression.<\/td>\r\n<td>[latex]-5 - 3[\/latex] means [latex]-5[\/latex] take away [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=\"size-full wp-image-16017 alignleft\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2302\/2017\/05\/05024302\/IMG_01_03_068.png\" alt=\"Five negative counters\" width=\"299\" height=\"109\" \/><\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Take away the second number.\r\n\r\nSo we need to take away [latex]3[\/latex] positives.<\/td>\r\n<td><\/td>\r\n<\/tr>\r\n<tr>\r\n<td>But there are no positives to take away.\r\n\r\nAdd neutral pairs until you have [latex]3[\/latex] positives.<\/td>\r\n<td><img class=\"alignleft wp-image-16024 size-full\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2302\/2017\/05\/05024730\/IMG_01_03_075.png\" alt=\"Five negative counters and three neutral pairs (a neutral pair is made of one negative counter and one positive counter)\" width=\"508\" height=\"132\" \/><\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Now take away [latex]3[\/latex] positives.<\/td>\r\n<td><img class=\"alignleft wp-image-16025 size-full\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2302\/2017\/05\/05024813\/IMG_01_03_076.png\" alt=\"Five negative counters and three neutral pairs where the positive counter in all three neutral pairs is circled.\" width=\"508\" height=\"142\" \/><\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Count the number of counters that are left.<\/td>\r\n<td><img class=\"alignleft wp-image-16026 size-full\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2302\/2017\/05\/05024852\/IMG_01_03_077.png\" alt=\"Eight negative counters\" width=\"450\" height=\"88\" \/><\/td>\r\n<\/tr>\r\n<tr>\r\n<td><\/td>\r\n<td>[latex]-5 - 3=-8[\/latex]\r\n\r\nThe difference of [latex]-5[\/latex] and [latex]3[\/latex] is [latex]-8[\/latex] .<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n&nbsp;\r\n\r\n<\/div>\r\nNow you can try a similar problem.\r\n<div class=\"textbox key-takeaways\">\r\n<h3>try it<\/h3>\r\nModel the expression:\r\n\r\n[latex]-6 - 4[\/latex]\r\n[reveal-answer q=\"647548\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"647548\"]\r\n\r\n<img class=\"alignleft wp-image-16027 size-full\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2302\/2017\/05\/05024927\/IMG_01_03_078.png\" alt=\"Six negative counters and four neutral pairs. The positive counters in the four neutral pairs are circled.\" width=\"592\" height=\"150\" \/>\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]\u221210[\/latex]\r\n\r\n[\/hidden-answer]\r\n\r\n&nbsp;\r\n\r\nModel the expression:\r\n\r\n[latex]-7 - 4[\/latex]\r\n[reveal-answer q=\"391259\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"391259\"]\r\n\r\n<img class=\"alignleft wp-image-16028 size-full\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2302\/2017\/05\/05025022\/IMG_01_03_079.png\" alt=\"Seven negative counters and four neutral pairs. The four positive counters in the neutral pairs are circled.\" width=\"638\" height=\"150\" \/>\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]\u221211[\/latex]\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\nNow we will subtract a negative number from a positive number. Think of this as taking away the negative.\r\n<div class=\"textbox exercises\">\r\n<h3>example<\/h3>\r\nModel: [latex]5-\\left(-3\\right)[\/latex]\r\n[reveal-answer q=\"943073\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"943073\"]\r\n\r\nSolution:\r\n<table id=\"eip-id1168467234371\" class=\"unnumbered unstyled\" style=\"width: 75%;\" summary=\"This figure has seven rows. The first row states interpret the expression. It is followed by 5 minus negative 3 means 5 take away negative 3. The second row states to model the first number, start with 5 positives. It is followed by 5 blue circles. The third row states take away the second number, so we need to take away 3 negatives. It is followed by a row of 5 blue circles, beside them 3 blue circles. Below the three blue circles are 3 red circles. The fourth row states there are no negatives to take away so add neutral pairs until you have 3 negatives. The fifth row states now take away three negatives. It is followed by a row of 5 blue circles, beside them 3 blue circles. Below the three blue circles are 3 red circles. The red circles are circled. The sixth row states count the number of counters that are left. It is followed by 8 blue circles. The seventh row states the difference of 5 and negative 3 is eight, beside it the expression 5 minus negative 3 equals eight.\">\r\n<tbody>\r\n<tr>\r\n<td>Interpret the expression.<\/td>\r\n<td>[latex]5-\\left(-3\\right)[\/latex] means [latex]5[\/latex] take away [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=\"size-full wp-image-16011 alignleft\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2302\/2017\/05\/05023929\/IMG_01_03_063.png\" alt=\"Five positive counters\" width=\"299\" height=\"109\" \/><\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Take away the second number, so take away [latex]3[\/latex] negatives.<\/td>\r\n<td><\/td>\r\n<\/tr>\r\n<tr>\r\n<td>But there are no negatives to take away.\r\n\r\nAdd neutral pairs until you have [latex]3[\/latex] negatives.<\/td>\r\n<td><img class=\"alignleft wp-image-16029 size-full\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2302\/2017\/05\/05025128\/IMG_01_03_081.png\" alt=\"Five positive counters and three neutral pairs \" width=\"508\" height=\"132\" \/><\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Then take away [latex]3[\/latex] negatives.<\/td>\r\n<td><img class=\"alignleft wp-image-16030 size-full\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2302\/2017\/05\/05025159\/IMG_01_03_082.png\" alt=\"Five positive counters and three neutral pairs, the three negative counters of the neutral pairs are circled.\" width=\"508\" height=\"142\" \/><\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Count the number of counters that are left.<\/td>\r\n<td><img class=\"size-full wp-image-16031 alignleft\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2302\/2017\/05\/05025401\/IMG_01_03_083.png\" alt=\"Eight positive counters\" width=\"450\" height=\"89\" \/><\/td>\r\n<\/tr>\r\n<tr>\r\n<td><\/td>\r\n<td>The difference of [latex]5[\/latex] and [latex]-3[\/latex] is [latex]8[\/latex] .\r\n\r\n[latex]5-\\left(-3\\right)=8[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\nNow you can try a similar problem.\r\n<div class=\"textbox key-takeaways\">\r\n<h3>try it<\/h3>\r\nModel the expression:\r\n\r\n[latex]6-\\left(-4\\right)[\/latex]\r\n[reveal-answer q=\"459252\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"459252\"]\r\n\r\n<img class=\"alignleft wp-image-16033 size-full\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2302\/2017\/05\/05032449\/IMG_01_03_084.png\" alt=\"Six positive counters and four neutral pairs, the four negative counters associated with the neutral pairs are circled.\" width=\"593\" height=\"150\" \/>\r\n\r\n[latex]10[\/latex]\r\n\r\n[\/hidden-answer]\r\n\r\n&nbsp;\r\n\r\nModel the expression:\r\n[latex]7-\\left(-4\\right)[\/latex]\r\n[reveal-answer q=\"632721\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"632721\"]\r\n\r\n<img class=\"alignleft wp-image-16034 size-full\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2302\/2017\/05\/05032604\/IMG_01_03_085.png\" alt=\"Seven positive counters and four neutral pairs, the four negative counters associated with the neutral pairs are circled.\" width=\"639\" height=\"151\" \/>\r\n\r\n[latex]11[\/latex]\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\nNow we will do an example that summarizes the situations above, with different numbers. Recall the different scenarios:\r\n<ul>\r\n \t<li>subtracting a positive number from a positive number<\/li>\r\n \t<li>subtracting a positive number from a negative number<\/li>\r\n \t<li>subtracting a negative number from a positive number<\/li>\r\n \t<li>subtracting a negative number from a negative number<\/li>\r\n<\/ul>\r\n<div class=\"textbox exercises\">\r\n<h3>example<\/h3>\r\nModel each subtraction.\r\n<ol>\r\n \t<li>[latex]8 \u2212 2[\/latex]<\/li>\r\n \t<li>[latex]\u22125 \u2212 4[\/latex]<\/li>\r\n \t<li>[latex]6 \u2212 (\u22126)[\/latex]<\/li>\r\n \t<li>[latex]\u22128 \u2212 (\u22123)[\/latex]<\/li>\r\n<\/ol>\r\n[reveal-answer q=\"214895\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"214895\"]\r\n<table id=\"eip-id1168468308805\" class=\"unnumbered unstyled\" style=\"width: 75%;\" 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]8 - 2[\/latex]\r\n\r\nThis means [latex]8[\/latex] take away [latex]2[\/latex] .<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Start with [latex]8[\/latex] positives.<\/td>\r\n<td><img class=\"alignleft wp-image-16035 size-full\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2302\/2017\/05\/05032641\/IMG_01_03_086.png\" alt=\"Eight positive counters\" width=\"450\" height=\"75\" \/><\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Take away [latex]2[\/latex] positives.<\/td>\r\n<td><img class=\"alignleft wp-image-16036 size-full\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2302\/2017\/05\/05032702\/IMG_01_03_087.png\" alt=\"Eight positive counters, two of the eight counters are circled.\" width=\"450\" height=\"75\" \/><\/td>\r\n<\/tr>\r\n<tr>\r\n<td>How many are left?<\/td>\r\n<td>[latex]6[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td><\/td>\r\n<td>[latex]8 - 2=6[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<table id=\"eip-id1168466328467\" class=\"unnumbered unstyled\" style=\"width: 75%;\" 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]-5 - 4[\/latex]\r\n\r\nThis means [latex]-5[\/latex] take away [latex]4[\/latex] .<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Start with [latex]5[\/latex] negatives.<\/td>\r\n<td><img class=\"alignleft wp-image-16037 size-full\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2302\/2017\/05\/05032736\/IMG_01_03_088.png\" alt=\"Five negative counters\" width=\"296\" height=\"75\" \/><\/td>\r\n<\/tr>\r\n<tr>\r\n<td>You need to take away [latex]4[\/latex] positives.\r\n\r\nAdd [latex]4[\/latex] neutral pairs to get [latex]4[\/latex] positives.<\/td>\r\n<td><img class=\"alignleft wp-image-16039 size-full\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2302\/2017\/05\/05033026\/IMG_01_03_089.png\" alt=\"Five negative counters and four neutral pairs.\" width=\"542\" height=\"150\" \/><\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Take away [latex]4[\/latex] positives.<\/td>\r\n<td><img class=\"alignleft wp-image-16040 size-full\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2302\/2017\/05\/05033052\/IMG_01_03_090.png\" alt=\"Five negative counters and four neutral pairs, the positive counters associated with the neutral pairs are circled.\" width=\"539\" height=\"150\" \/><\/td>\r\n<\/tr>\r\n<tr>\r\n<td>How many are left?<\/td>\r\n<td><img class=\"alignleft wp-image-16041 size-full\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2302\/2017\/05\/05033143\/IMG_01_03_091.png\" alt=\"Nine negative counters\" width=\"505\" height=\"75\" \/><\/td>\r\n<\/tr>\r\n<tr>\r\n<td><\/td>\r\n<td>[latex]-9[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td><\/td>\r\n<td>[latex]-5 - 4=-9[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<table id=\"eip-id1168469609690\" class=\"unnumbered unstyled\" style=\"width: 75%;\" 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]6-\\left(-6\\right)[\/latex]\r\n\r\nThis means [latex]6[\/latex] take away [latex]-6[\/latex] .<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Start with [latex]6[\/latex] positives.<\/td>\r\n<td><img class=\"alignleft wp-image-16042 size-full\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2302\/2017\/05\/05033215\/IMG_01_03_092.png\" alt=\"Six positive counters\" width=\"346\" height=\"75\" \/><\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Add [latex]6[\/latex] neutrals to get [latex]6[\/latex] negatives to take away.<\/td>\r\n<td><img class=\"alignleft wp-image-16043 size-full\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2302\/2017\/05\/05033243\/IMG_01_03_093.png\" alt=\"Six positive counters and six neutral pairs\" width=\"685\" height=\"151\" \/><\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Remove [latex]6[\/latex] negatives.<\/td>\r\n<td><img class=\"alignleft wp-image-16044 size-full\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2302\/2017\/05\/05033308\/IMG_01_03_094.png\" alt=\"Six positive counters and six neutral pairs, the negative counters associated with the six neutral pairs are circled.\" width=\"697\" height=\"151\" \/><\/td>\r\n<\/tr>\r\n<tr>\r\n<td>How many are left?<\/td>\r\n<td><img class=\"alignleft wp-image-16045 size-full\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2302\/2017\/05\/05033342\/IMG_01_03_095.png\" alt=\"Twelve positive counters\" width=\"638\" height=\"75\" \/><\/td>\r\n<\/tr>\r\n<tr>\r\n<td><\/td>\r\n<td>[latex]12[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td><\/td>\r\n<td>[latex]6-\\left(-6\\right)=12[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<table id=\"eip-id1168468534936\" class=\"unnumbered unstyled\" style=\"width: 75%;\" 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]-8-\\left(-3\\right)[\/latex]\r\n\r\nThis means [latex]-8[\/latex] take away [latex]-3[\/latex] .<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Start with [latex]8[\/latex] negatives.<\/td>\r\n<td><img class=\"alignleft wp-image-16046 size-full\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2302\/2017\/05\/05033438\/IMG_01_03_096.png\" alt=\"Eight negative counters\" width=\"450\" height=\"75\" \/><\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Take away [latex]3[\/latex] negatives.<\/td>\r\n<td><img class=\"alignleft wp-image-16047 size-full\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2302\/2017\/05\/05033506\/IMG_01_03_097.png\" alt=\"Eight negative counters, three are circled\" width=\"498\" height=\"88\" \/><\/td>\r\n<\/tr>\r\n<tr>\r\n<td>How many are left?<\/td>\r\n<td><img class=\"alignleft wp-image-16048 size-full\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2302\/2017\/05\/05033543\/IMG_01_03_098.png\" alt=\"Five negative counters\" width=\"298\" height=\"87\" \/><\/td>\r\n<\/tr>\r\n<tr>\r\n<td><\/td>\r\n<td>[latex]-5[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td><\/td>\r\n<td>[latex]-8-\\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\nNow you can try a similar problem.\r\n<div class=\"textbox key-takeaways\">\r\n<h3>try it<\/h3>\r\nModel each subtraction.\r\n\r\n1. [latex]7 - (-8)[\/latex]\r\n2. [latex]-7 - (-2)[\/latex]\r\n3. [latex]4 - 1[\/latex]\r\n4. [latex]-6 - 8[\/latex]\r\n[reveal-answer q=\"324876\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"324876\"]\r\n\r\n1. [latex]15[\/latex]\r\n\r\n<img class=\"alignleft wp-image-16050 size-full\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2302\/2017\/05\/05033632\/IMG_01_03_099.png\" alt=\"Seven positive counters and eight neutral pairs with the negative counters circled.\" width=\"865\" height=\"150\" \/>\r\n\r\n2. [latex]-5[\/latex]\r\n\r\n<img class=\"alignleft wp-image-16051 size-full\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2302\/2017\/05\/05033718\/IMG_01_03_100.png\" alt=\"Seven total negative counters with two of those negative counters circled.\" width=\"465\" height=\"88\" \/>\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]3[\/latex]\r\n\r\n<img class=\"alignleft wp-image-16052 size-full\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2302\/2017\/05\/05033801\/IMG_01_03_101.png\" alt=\"Four positive counters with one of those counters circled\" width=\"359\" height=\"88\" \/>\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]-14[\/latex]\r\n\r\n<img class=\"alignleft wp-image-16054 size-full\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2302\/2017\/05\/05034110\/IMG_01_03_102.png\" alt=\"Six negative counters and eight neutral pairs with the positive counters in the neutral pairs circled.\" width=\"829\" height=\"151\" \/>\r\n\r\n[\/hidden-answer]\r\n\r\n&nbsp;\r\n\r\nModel each subtraction.\r\n\r\n1. [latex]4 - (-6)[\/latex]\r\n2. [latex]-8 - (-1)[\/latex]\r\n3. [latex]7 - 3[\/latex]\r\n4. [latex]-4 - 2[\/latex]\r\n[reveal-answer q=\"677150\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"677150\"]\r\n\r\n1. [latex]10[\/latex]\r\n\r\n<img class=\"alignleft wp-image-16055 size-full\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2302\/2017\/05\/05034207\/IMG_01_03_103.png\" alt=\"Four positive counters and six neutral pairs with the negative counters in the neutral pairs circled.\" width=\"724\" height=\"151\" \/>\r\n\r\n2. [latex]-7[\/latex]\r\n\r\n<img class=\"alignleft wp-image-16056 size-full\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2302\/2017\/05\/05034256\/IMG_01_03_104.png\" alt=\"Eight total negative counters with one negative counter circled\" width=\"510\" height=\"88\" \/>\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]4[\/latex]\r\n\r\n<img class=\"alignleft wp-image-16058 size-full\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2302\/2017\/05\/05034609\/IMG_01_03_105.png\" alt=\"Seven total positive counters with three of those counters circled.\" width=\"459\" height=\"88\" \/>\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]-6[\/latex]\r\n\r\n<img class=\"alignleft wp-image-16057 size-full\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2302\/2017\/05\/05034358\/IMG_01_03_106.png\" alt=\"Five negative counters and two neutral pairs with the two positive counters circled.\" width=\"508\" height=\"142\" \/>\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\nEach of the examples so far have been carefully constructed so that the sign of the answer matched the sign of the first number in the expression. \u00a0For example, in\u00a0\u00a0[latex]\u22125 \u2212 4[\/latex], the result is [latex]-9[\/latex], which is the same sign as [latex]-5[\/latex]. Now we will see subtraction where the sign of the result is different from the starting number.\r\n<div class=\"textbox exercises\">\r\n<h3>example<\/h3>\r\nModel each subtraction expression:\r\n<ol id=\"eip-id1168468264062\" class=\"circled\">\r\n \t<li>[latex]2 - 8[\/latex]<\/li>\r\n \t<li>[latex]-3-\\left(-8\\right)[\/latex]<\/li>\r\n<\/ol>\r\n[reveal-answer q=\"485889\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"485889\"]\r\n\r\nSolution\r\n<table id=\"eip-id1168466285116\" class=\"unnumbered unstyled\" style=\"width: 75%;\" summary=\"This figure has six rows. The first row states we start with 2 positives. It is followed by 2 blue circles. The second row states we need to take away 8 positives but we have only 2. The third row states add neutral pairs until there are 8 positives to take away. It is followed by a row of 8 blue circles. The first two are separated from the last 6. Under the last six there is a row of 6 red circles. The fourth row states then take away the 8 positives. It is followed by a row of 8 blue circles. They are circled. The fifth row states find the number of counters that are left. There are six negatives. It is followed by 6 red circles labeled six negatives. The sixth row has 2 minus 8 equals negative 6.\">\r\n<tbody>\r\n<tr>\r\n<td>1.\r\n\r\nWe start with [latex]2[\/latex] positives.<\/td>\r\n<td><\/td>\r\n<\/tr>\r\n<tr>\r\n<td>We need to take away [latex]8[\/latex] positives, but we have only [latex]2[\/latex].<\/td>\r\n<td><img class=\"size-full wp-image-16060 alignleft\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2302\/2017\/05\/05035350\/IMG_01_03_107.png\" alt=\"Two positive counters\" width=\"125\" height=\"90\" \/><\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Add neutral pairs until there are [latex]8[\/latex] positives to take away.<\/td>\r\n<td><img class=\"alignleft wp-image-16061 size-full\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2302\/2017\/05\/05035411\/IMG_01_03_108.png\" alt=\"Two positive counters and six neutral pairs\" width=\"500\" height=\"151\" \/><\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Then take away [latex]8[\/latex]\u00a0positives.<\/td>\r\n<td><img class=\"alignleft wp-image-16062 size-full\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2302\/2017\/05\/05035452\/IMG_01_03_109.png\" alt=\"Two positive counters and six neutral pairs, all of the positive counters are circled.\" width=\"544\" height=\"172\" \/><\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Find the number of counters that are left.\r\n\r\nThere are [latex]6[\/latex] negatives.<\/td>\r\n<td><img class=\"alignleft wp-image-16063 size-full\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2302\/2017\/05\/05035523\/IMG_01_03_110.png\" alt=\"Six negative counters\" width=\"349\" height=\"89\" \/><\/td>\r\n<\/tr>\r\n<tr>\r\n<td><\/td>\r\n<td>[latex]2 - 8=-6[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<table id=\"eip-id1168468388588\" class=\"unnumbered unstyled\" style=\"width: 75%;\" summary=\"This figure has six rows. The first row states we start with 3 negatives. It is followed by 3 red circles. The second row states we need to take away 8 negatives but we have only 3. The third row states add neutral pairs until there are 8 negatives to take away. It is followed by a row of 8 red circles. The first three are separated from the last 5. Under the last 5 there is a row of 5 blue circles. The fourth row states then take away the 8 negatives. It is followed by a row of 8 red circles. They are circled. The fifth row states find the number of counters that are left. There are five positives. It is followed by 5 blue circles. The sixth row has negative 3 minus negative 8 equals 5.\">\r\n<tbody>\r\n<tr>\r\n<td>2.\r\n\r\nWe start with [latex]3[\/latex] negatives.<\/td>\r\n<td><img class=\"alignleft wp-image-16064 size-full\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2302\/2017\/05\/05035545\/IMG_01_03_111.png\" alt=\"Three negative counters\" width=\"192\" height=\"95\" \/><\/td>\r\n<\/tr>\r\n<tr>\r\n<td>We need to take away [latex]8[\/latex] negatives, but we have only [latex]3[\/latex].<\/td>\r\n<td><\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Add neutral pairs until there are [latex]8[\/latex] negatives to take away.<\/td>\r\n<td><img class=\"alignleft wp-image-16065 size-full\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2302\/2017\/05\/05035615\/IMG_01_03_112.png\" alt=\"Three negative counters and five neutral pairs\" width=\"473\" height=\"150\" \/><\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Then take away the [latex]8[\/latex] negatives.<\/td>\r\n<td><img class=\"alignleft wp-image-16066 size-full\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2302\/2017\/05\/05035650\/IMG_01_03_113.png\" alt=\"Three negative counters and five neutral pairs, all the negative counters are circled.\" width=\"547\" height=\"160\" \/><\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Find the number of counters that are left.\r\n\r\nThere are [latex]5[\/latex] positives.<\/td>\r\n<td><img class=\"alignleft wp-image-16067 size-full\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2302\/2017\/05\/05035734\/IMG_01_03_114.png\" alt=\"Five positive counters\" width=\"331\" height=\"87\" \/><\/td>\r\n<\/tr>\r\n<tr>\r\n<td><\/td>\r\n<td>[latex]-3-\\left(-8\\right)=5[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\nNow you can try a similar problem.\r\n<div class=\"textbox key-takeaways\">\r\n<h3>try it<\/h3>\r\nModel each subtraction expression.\r\n<ol id=\"eip-id1168466205125\" class=\"circled\">\r\n \t<li>[latex]7 - 9[\/latex]<\/li>\r\n \t<li>[latex]-5--9[\/latex]<\/li>\r\n<\/ol>\r\n[reveal-answer q=\"678756\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"678756\"]\r\n\r\n1.\r\n\r\n<img class=\"alignleft wp-image-16070 size-full\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2302\/2017\/05\/05040624\/IMG_01_03_115.png\" alt=\"Seven positive counters and two neutral pairs. All of the positive counters are circled.\" width=\"587\" height=\"147\" \/>\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\n2.\r\n\r\n<img class=\"alignleft wp-image-16071 size-full\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2302\/2017\/05\/05040700\/IMG_01_03_116.png\" alt=\"Five positive counters and four neutral pairs. All of the positive counters are circled.\" width=\"691\" height=\"155\" \/>\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]4[\/latex]\r\n\r\n[\/hidden-answer]\r\n\r\n&nbsp;\r\n\r\nModel each subtraction expression.\r\n<ol id=\"eip-id1168468470083\" class=\"circled\">\r\n \t<li>[latex]5 - 8[\/latex]<\/li>\r\n \t<li>[latex]-7-\\left(-10\\right)[\/latex]<\/li>\r\n<\/ol>\r\n[reveal-answer q=\"164098\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"164098\"]\r\n\r\n1.\r\n\r\n<img class=\"alignleft wp-image-16072 size-full\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2302\/2017\/05\/05040741\/IMG_01_03_117.png\" alt=\"Five positive counters and three neutral pairs. All positive counters are circled.\" width=\"579\" height=\"154\" \/>\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\n2.\r\n\r\n<img class=\"alignleft wp-image-16073 size-full\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2302\/2017\/05\/05040818\/IMG_01_03_118.png\" alt=\"Seven positive counters and three neutral pairs. All positive counters are circled.\" width=\"639\" height=\"150\" \/>\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\nWhen you subtract two integers, there are two possibilities, either the result will have a different sign from the starting number, or it will have the same sign.\r\n\r\nWatch the video below to see more examples of modeling integer subtraction with color counters.\r\n\r\nhttps:\/\/youtu.be\/ZjiL7GfxgiI","rendered":"<div class=\"textbox learning-objectives\">\n<h3>Learning Outcomes<\/h3>\n<ul>\n<li>Model integer subtraction with color counters<\/li>\n<\/ul>\n<\/div>\n<p>You learn as a child how to subtract numbers through everyday experiences. For example, if you have 10 animal cookies and eat 6 of them, you will have 4 animal cookies left.<\/p>\n<p>Real-life experiences serve as models for subtracting positive numbers, and in some cases, such as temperature, for adding negative as well as positive numbers. But it is difficult to relate subtracting negative numbers to common life experiences. Most people do not have an intuitive understanding of subtraction when negative numbers are involved. Math teachers use several different models to explain subtracting negative numbers.<\/p>\n<p>We will continue to use counters to model subtraction. Remember, the purple counters represent positive numbers and the orange counters represent negative numbers.<\/p>\n<p>Perhaps when you were younger, you read [latex]5 - 3[\/latex] as <em>five take away three<\/em>. When we use counters, we can think of subtraction the same way.<\/p>\n<p>We will model four subtraction scenarios\u00a0using the numbers [latex]5[\/latex] and [latex]3[\/latex].<\/p>\n<ul>\n<li>[latex]5 - 3[\/latex]<\/li>\n<li>[latex]- 5-\\left(-3\\right)[\/latex]<\/li>\n<li>[latex]-5 - 3[\/latex]<\/li>\n<li>[latex]5-\\left(-3\\right)[\/latex]<\/li>\n<\/ul>\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-id1168469608926\" class=\"unnumbered unstyled\" style=\"width: 75%;\" summary=\"This figure has five rows. The first row states interpret the expression. It is followed by 5 minus 3 means 5 take away 3. The second row states model the first number, start with 5 positives. It is followed by 5 blue circles. The third row states take away the second number, so take away 3 positives. It is followed by 5 blue circles with the first three circled. The fourth row states find the number of counters that are left. It is followed by 2 blue circles that are labeled two positives. The fifth row states the difference of 5 and 3 is 2. It is followed by the expression 5 minus 3 equals 2.\">\n<tbody>\n<tr>\n<td>Interpret the expression.<\/td>\n<td>[latex]5 - 3[\/latex] means [latex]5[\/latex] take away [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=\"size-full wp-image-16011 alignleft\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2302\/2017\/05\/05023929\/IMG_01_03_063.png\" alt=\"Five positive counters\" width=\"299\" height=\"109\" \/><\/td>\n<\/tr>\n<tr>\n<td>Take away the second number. So take away [latex]3[\/latex] positives.<\/td>\n<td><img loading=\"lazy\" decoding=\"async\" class=\"wp-image-16013 size-full alignleft\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2302\/2017\/05\/05023953\/IMG_01_03_064.png\" alt=\"Three positive counters are circled out of five total positive counters\" width=\"360\" height=\"122\" \/><\/td>\n<\/tr>\n<tr>\n<td>Find the counters that are left.<\/td>\n<td><img loading=\"lazy\" decoding=\"async\" class=\"alignleft wp-image-16014 size-full\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2302\/2017\/05\/05024045\/IMG_01_03_065.png\" alt=\"Two positive counters\" width=\"126\" height=\"73\" \/><\/td>\n<\/tr>\n<tr>\n<td><\/td>\n<td>[latex]5 - 3=2[\/latex]<\/p>\n<p>The difference between [latex]5[\/latex] and [latex]3[\/latex] is [latex]2[\/latex] .<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<p>Now you can try a similar problem.<\/p>\n<div class=\"textbox key-takeaways\">\n<h3>try it<\/h3>\n<p>Model the expression:<\/p>\n<p>[latex]6 - 4[\/latex]<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q49141\">Show Solution<\/span><\/p>\n<div id=\"q49141\" class=\"hidden-answer\" style=\"display: none\">\n<p><img loading=\"lazy\" decoding=\"async\" class=\"alignleft wp-image-16015 size-full\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2302\/2017\/05\/05024109\/IMG_01_03_066.png\" alt=\"A total of six positive counters and four of them are circled\" width=\"370\" height=\"109\" \/><\/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:<\/p>\n<p>[latex]7 - 4[\/latex]<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q400845\">Show Solution<\/span><\/p>\n<div id=\"q400845\" class=\"hidden-answer\" style=\"display: none\">\n<p><img loading=\"lazy\" decoding=\"async\" class=\"alignleft wp-image-16016 size-full\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2302\/2017\/05\/05024209\/IMG_01_03_067.png\" alt=\"A total of seven positive counters and four are circled\" width=\"405\" height=\"109\" \/><\/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>In the previous example we subtracted [latex]3[\/latex] positives from positive [latex]5[\/latex]. Now we will subtract\u00a0[latex]3[\/latex] negatives from\u00a0negative [latex]5[\/latex]. Compare the results of this example to the previous example after you read through it.<\/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=\"q511456\">Show Solution<\/span><\/p>\n<div id=\"q511456\" class=\"hidden-answer\" style=\"display: none\">\n<p>Solution:<\/p>\n<table id=\"eip-id1168469693755\" class=\"unnumbered unstyled\" style=\"width: 75%;\" summary=\"This figure has five rows. The first row states interpret the expression. It is followed by 5 minus negative 3 means 5 take away negative 3. The second row states model the first number, start with 5 negatives. It is followed by 5 red circles. The third row states take away the second number, so take away 3 negatives. It is followed by 5 red circles with the first three circled. The fourth row states find the number of counters that are left. It is followed by 2 red circles which are labeled two negatives. The fifth row states the difference of negative 5 and negative 3 is negative 2. Next to this is the expression negative 5 minus negative 3 equals negative two.\">\n<tbody>\n<tr>\n<td>Interpret the expression.<\/td>\n<td>[latex]-5-\\left(-3\\right)[\/latex] means [latex]-5[\/latex] take away [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=\"size-full wp-image-16017 alignleft\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2302\/2017\/05\/05024302\/IMG_01_03_068.png\" alt=\"Five negative counters\" width=\"299\" height=\"109\" \/><\/td>\n<\/tr>\n<tr>\n<td>Take away the second number. So take away [latex]3[\/latex] negatives.<\/td>\n<td><img loading=\"lazy\" decoding=\"async\" class=\"alignleft wp-image-16018 size-full\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2302\/2017\/05\/05024322\/IMG_01_03_069.png\" alt=\"A total of five negative counters with three of the five circled\" width=\"359\" height=\"122\" \/><\/td>\n<\/tr>\n<tr>\n<td>Find the number of counters that are left.<\/td>\n<td><img loading=\"lazy\" decoding=\"async\" class=\"alignleft wp-image-16019 size-full\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2302\/2017\/05\/05024352\/IMG_01_03_070.png\" alt=\"Two negative counters\" width=\"126\" height=\"73\" \/><\/td>\n<\/tr>\n<tr>\n<td><\/td>\n<td>[latex]-5-\\left(-3\\right)=-2[\/latex]<\/p>\n<p>The difference between [latex]-5[\/latex] and [latex]-3[\/latex] is [latex]-2[\/latex] .<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<\/div>\n<\/div>\n<p>You can try a similar problem.<\/p>\n<div class=\"textbox key-takeaways\">\n<h3>try it<\/h3>\n<p>Model the expression:<\/p>\n<p>[latex]-6-\\left(-4\\right)[\/latex]<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q720789\">Show Solution<\/span><\/p>\n<div id=\"q720789\" class=\"hidden-answer\" style=\"display: none\">\n<p><img loading=\"lazy\" decoding=\"async\" class=\"alignleft wp-image-16020 size-full\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2302\/2017\/05\/05024414\/IMG_01_03_071.png\" alt=\"A total of six negative counters with four of the six circled.\" width=\"355\" height=\"110\" \/><\/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:<\/p>\n<p>[latex]-7-\\left(-4\\right)[\/latex]<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q188555\">Show Solution<\/span><\/p>\n<div id=\"q188555\" class=\"hidden-answer\" style=\"display: none\">\n<p><img loading=\"lazy\" decoding=\"async\" class=\"alignleft wp-image-16021 size-full\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2302\/2017\/05\/05024455\/IMG_01_03_072.png\" alt=\"A total of seven negative counters with four of the seven counters circled\" width=\"404\" height=\"109\" \/><\/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>Notice that the previous two examples\u00a0are very much alike.<\/p>\n<ul id=\"fs-id801766\">\n<li>First, we subtracted [latex]3[\/latex] positives from [latex]5[\/latex] positives to get [latex]2[\/latex] positives.<\/li>\n<li>Then we subtracted [latex]3[\/latex] negatives from [latex]5[\/latex] negatives to get [latex]2[\/latex] negatives.<\/li>\n<\/ul>\n<p>Each example used counters of only one color, and the &#8220;take away&#8221; model of subtraction was easy to apply.<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter wp-image-16022 size-full\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2302\/2017\/05\/05024540\/IMG_01_03_073.png\" alt=\"Five positive counters minus three positive counters equals two. Five negative counters minus three negative counters equals negative two.\" width=\"728\" height=\"132\" \/>Now let\u2019s see what happens when we subtract one positive and one negative number. We will need to use both positive and negative counters and sometimes some neutral pairs, too. Adding a neutral pair does not change the value.<\/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-id1168468477542\" class=\"unnumbered unstyled\" style=\"width: 75%;\" summary=\"This figure has seven rows. The first row states interpret the expression. It is followed by negative 5 minus 3 means negative 5 take away 3. The second row states to model the first number, start with 5 negatives. It is followed by 5 red circles. The third row states take away the second number, so we need to take away 3 positives. It is followed by a row of 5 red circles, beside them 3 red circles. Below the three red circles are 3 blue circles. The fourth row states there are no positives to take away so add neutral pairs until you have 3 positives. The fifth row states now take away three positives. It is followed by a row of 5 red circles, beside them 3 red circles. Below the three red circles are 3 blue circles. The blue shaded circles are circled. The sixth row states count the number of counters that are left. It is followed by 8 red circles. The seventh row states the difference of negative five and three is negative 8. It is followed by the expression negative 5 minus 3 equals negative 8.\">\n<tbody>\n<tr>\n<td>Interpret the expression.<\/td>\n<td>[latex]-5 - 3[\/latex] means [latex]-5[\/latex] take away [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=\"size-full wp-image-16017 alignleft\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2302\/2017\/05\/05024302\/IMG_01_03_068.png\" alt=\"Five negative counters\" width=\"299\" height=\"109\" \/><\/td>\n<\/tr>\n<tr>\n<td>Take away the second number.<\/p>\n<p>So we need to take away [latex]3[\/latex] positives.<\/td>\n<td><\/td>\n<\/tr>\n<tr>\n<td>But there are no positives to take away.<\/p>\n<p>Add neutral pairs until you have [latex]3[\/latex] positives.<\/td>\n<td><img loading=\"lazy\" decoding=\"async\" class=\"alignleft wp-image-16024 size-full\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2302\/2017\/05\/05024730\/IMG_01_03_075.png\" alt=\"Five negative counters and three neutral pairs (a neutral pair is made of one negative counter and one positive counter)\" width=\"508\" height=\"132\" \/><\/td>\n<\/tr>\n<tr>\n<td>Now take away [latex]3[\/latex] positives.<\/td>\n<td><img loading=\"lazy\" decoding=\"async\" class=\"alignleft wp-image-16025 size-full\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2302\/2017\/05\/05024813\/IMG_01_03_076.png\" alt=\"Five negative counters and three neutral pairs where the positive counter in all three neutral pairs is circled.\" width=\"508\" height=\"142\" \/><\/td>\n<\/tr>\n<tr>\n<td>Count the number of counters that are left.<\/td>\n<td><img loading=\"lazy\" decoding=\"async\" class=\"alignleft wp-image-16026 size-full\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2302\/2017\/05\/05024852\/IMG_01_03_077.png\" alt=\"Eight negative counters\" width=\"450\" height=\"88\" \/><\/td>\n<\/tr>\n<tr>\n<td><\/td>\n<td>[latex]-5 - 3=-8[\/latex]<\/p>\n<p>The difference of [latex]-5[\/latex] and [latex]3[\/latex] is [latex]-8[\/latex] .<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>&nbsp;<\/p>\n<\/div>\n<p>Now you can try a similar problem.<\/p>\n<div class=\"textbox key-takeaways\">\n<h3>try it<\/h3>\n<p>Model the expression:<\/p>\n<p>[latex]-6 - 4[\/latex]<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q647548\">Show Solution<\/span><\/p>\n<div id=\"q647548\" class=\"hidden-answer\" style=\"display: none\">\n<p><img loading=\"lazy\" decoding=\"async\" class=\"alignleft wp-image-16027 size-full\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2302\/2017\/05\/05024927\/IMG_01_03_078.png\" alt=\"Six negative counters and four neutral pairs. The positive counters in the four neutral pairs are circled.\" width=\"592\" height=\"150\" \/><\/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]\u221210[\/latex]<\/p>\n<\/div>\n<\/div>\n<p>&nbsp;<\/p>\n<p>Model the expression:<\/p>\n<p>[latex]-7 - 4[\/latex]<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q391259\">Show Solution<\/span><\/p>\n<div id=\"q391259\" class=\"hidden-answer\" style=\"display: none\">\n<p><img loading=\"lazy\" decoding=\"async\" class=\"alignleft wp-image-16028 size-full\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2302\/2017\/05\/05025022\/IMG_01_03_079.png\" alt=\"Seven negative counters and four neutral pairs. The four positive counters in the neutral pairs are circled.\" width=\"638\" height=\"150\" \/><\/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]\u221211[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<p>Now we will subtract a negative number from a positive number. Think of this as taking away the negative.<\/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=\"q943073\">Show Solution<\/span><\/p>\n<div id=\"q943073\" class=\"hidden-answer\" style=\"display: none\">\n<p>Solution:<\/p>\n<table id=\"eip-id1168467234371\" class=\"unnumbered unstyled\" style=\"width: 75%;\" summary=\"This figure has seven rows. The first row states interpret the expression. It is followed by 5 minus negative 3 means 5 take away negative 3. The second row states to model the first number, start with 5 positives. It is followed by 5 blue circles. The third row states take away the second number, so we need to take away 3 negatives. It is followed by a row of 5 blue circles, beside them 3 blue circles. Below the three blue circles are 3 red circles. The fourth row states there are no negatives to take away so add neutral pairs until you have 3 negatives. The fifth row states now take away three negatives. It is followed by a row of 5 blue circles, beside them 3 blue circles. Below the three blue circles are 3 red circles. The red circles are circled. The sixth row states count the number of counters that are left. It is followed by 8 blue circles. The seventh row states the difference of 5 and negative 3 is eight, beside it the expression 5 minus negative 3 equals eight.\">\n<tbody>\n<tr>\n<td>Interpret the expression.<\/td>\n<td>[latex]5-\\left(-3\\right)[\/latex] means [latex]5[\/latex] take away [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=\"size-full wp-image-16011 alignleft\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2302\/2017\/05\/05023929\/IMG_01_03_063.png\" alt=\"Five positive counters\" width=\"299\" height=\"109\" \/><\/td>\n<\/tr>\n<tr>\n<td>Take away the second number, so take away [latex]3[\/latex] negatives.<\/td>\n<td><\/td>\n<\/tr>\n<tr>\n<td>But there are no negatives to take away.<\/p>\n<p>Add neutral pairs until you have [latex]3[\/latex] negatives.<\/td>\n<td><img loading=\"lazy\" decoding=\"async\" class=\"alignleft wp-image-16029 size-full\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2302\/2017\/05\/05025128\/IMG_01_03_081.png\" alt=\"Five positive counters and three neutral pairs\" width=\"508\" height=\"132\" \/><\/td>\n<\/tr>\n<tr>\n<td>Then take away [latex]3[\/latex] negatives.<\/td>\n<td><img loading=\"lazy\" decoding=\"async\" class=\"alignleft wp-image-16030 size-full\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2302\/2017\/05\/05025159\/IMG_01_03_082.png\" alt=\"Five positive counters and three neutral pairs, the three negative counters of the neutral pairs are circled.\" width=\"508\" height=\"142\" \/><\/td>\n<\/tr>\n<tr>\n<td>Count the number of counters that are left.<\/td>\n<td><img loading=\"lazy\" decoding=\"async\" class=\"size-full wp-image-16031 alignleft\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2302\/2017\/05\/05025401\/IMG_01_03_083.png\" alt=\"Eight positive counters\" width=\"450\" height=\"89\" \/><\/td>\n<\/tr>\n<tr>\n<td><\/td>\n<td>The difference of [latex]5[\/latex] and [latex]-3[\/latex] is [latex]8[\/latex] .<\/p>\n<p>[latex]5-\\left(-3\\right)=8[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<\/div>\n<\/div>\n<p>Now you can try a similar problem.<\/p>\n<div class=\"textbox key-takeaways\">\n<h3>try it<\/h3>\n<p>Model the expression:<\/p>\n<p>[latex]6-\\left(-4\\right)[\/latex]<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q459252\">Show Solution<\/span><\/p>\n<div id=\"q459252\" class=\"hidden-answer\" style=\"display: none\">\n<p><img loading=\"lazy\" decoding=\"async\" class=\"alignleft wp-image-16033 size-full\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2302\/2017\/05\/05032449\/IMG_01_03_084.png\" alt=\"Six positive counters and four neutral pairs, the four negative counters associated with the neutral pairs are circled.\" width=\"593\" height=\"150\" \/><\/p>\n<p>[latex]10[\/latex]<\/p>\n<\/div>\n<\/div>\n<p>&nbsp;<\/p>\n<p>Model the expression:<br \/>\n[latex]7-\\left(-4\\right)[\/latex]<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q632721\">Show Solution<\/span><\/p>\n<div id=\"q632721\" class=\"hidden-answer\" style=\"display: none\">\n<p><img loading=\"lazy\" decoding=\"async\" class=\"alignleft wp-image-16034 size-full\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2302\/2017\/05\/05032604\/IMG_01_03_085.png\" alt=\"Seven positive counters and four neutral pairs, the four negative counters associated with the neutral pairs are circled.\" width=\"639\" height=\"151\" \/><\/p>\n<p>[latex]11[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<p>Now we will do an example that summarizes the situations above, with different numbers. Recall the different scenarios:<\/p>\n<ul>\n<li>subtracting a positive number from a positive number<\/li>\n<li>subtracting a positive number from a negative number<\/li>\n<li>subtracting a negative number from a positive number<\/li>\n<li>subtracting a negative number from a negative number<\/li>\n<\/ul>\n<div class=\"textbox exercises\">\n<h3>example<\/h3>\n<p>Model each subtraction.<\/p>\n<ol>\n<li>[latex]8 \u2212 2[\/latex]<\/li>\n<li>[latex]\u22125 \u2212 4[\/latex]<\/li>\n<li>[latex]6 \u2212 (\u22126)[\/latex]<\/li>\n<li>[latex]\u22128 \u2212 (\u22123)[\/latex]<\/li>\n<\/ol>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q214895\">Show Solution<\/span><\/p>\n<div id=\"q214895\" class=\"hidden-answer\" style=\"display: none\">\n<table id=\"eip-id1168468308805\" class=\"unnumbered unstyled\" style=\"width: 75%;\" summary=\".\">\n<tbody>\n<tr>\n<td>1.<\/td>\n<td><\/td>\n<\/tr>\n<tr>\n<td><\/td>\n<td>[latex]8 - 2[\/latex]<\/p>\n<p>This means [latex]8[\/latex] take away [latex]2[\/latex] .<\/td>\n<\/tr>\n<tr>\n<td>Start with [latex]8[\/latex] positives.<\/td>\n<td><img loading=\"lazy\" decoding=\"async\" class=\"alignleft wp-image-16035 size-full\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2302\/2017\/05\/05032641\/IMG_01_03_086.png\" alt=\"Eight positive counters\" width=\"450\" height=\"75\" \/><\/td>\n<\/tr>\n<tr>\n<td>Take away [latex]2[\/latex] positives.<\/td>\n<td><img loading=\"lazy\" decoding=\"async\" class=\"alignleft wp-image-16036 size-full\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2302\/2017\/05\/05032702\/IMG_01_03_087.png\" alt=\"Eight positive counters, two of the eight counters are circled.\" width=\"450\" height=\"75\" \/><\/td>\n<\/tr>\n<tr>\n<td>How many are left?<\/td>\n<td>[latex]6[\/latex]<\/td>\n<\/tr>\n<tr>\n<td><\/td>\n<td>[latex]8 - 2=6[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<table id=\"eip-id1168466328467\" class=\"unnumbered unstyled\" style=\"width: 75%;\" summary=\".\">\n<tbody>\n<tr>\n<td>2.<\/td>\n<td><\/td>\n<\/tr>\n<tr>\n<td><\/td>\n<td>[latex]-5 - 4[\/latex]<\/p>\n<p>This means [latex]-5[\/latex] take away [latex]4[\/latex] .<\/td>\n<\/tr>\n<tr>\n<td>Start with [latex]5[\/latex] negatives.<\/td>\n<td><img loading=\"lazy\" decoding=\"async\" class=\"alignleft wp-image-16037 size-full\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2302\/2017\/05\/05032736\/IMG_01_03_088.png\" alt=\"Five negative counters\" width=\"296\" height=\"75\" \/><\/td>\n<\/tr>\n<tr>\n<td>You need to take away [latex]4[\/latex] positives.<\/p>\n<p>Add [latex]4[\/latex] neutral pairs to get [latex]4[\/latex] positives.<\/td>\n<td><img loading=\"lazy\" decoding=\"async\" class=\"alignleft wp-image-16039 size-full\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2302\/2017\/05\/05033026\/IMG_01_03_089.png\" alt=\"Five negative counters and four neutral pairs.\" width=\"542\" height=\"150\" \/><\/td>\n<\/tr>\n<tr>\n<td>Take away [latex]4[\/latex] positives.<\/td>\n<td><img loading=\"lazy\" decoding=\"async\" class=\"alignleft wp-image-16040 size-full\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2302\/2017\/05\/05033052\/IMG_01_03_090.png\" alt=\"Five negative counters and four neutral pairs, the positive counters associated with the neutral pairs are circled.\" width=\"539\" height=\"150\" \/><\/td>\n<\/tr>\n<tr>\n<td>How many are left?<\/td>\n<td><img loading=\"lazy\" decoding=\"async\" class=\"alignleft wp-image-16041 size-full\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2302\/2017\/05\/05033143\/IMG_01_03_091.png\" alt=\"Nine negative counters\" width=\"505\" height=\"75\" \/><\/td>\n<\/tr>\n<tr>\n<td><\/td>\n<td>[latex]-9[\/latex]<\/td>\n<\/tr>\n<tr>\n<td><\/td>\n<td>[latex]-5 - 4=-9[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<table id=\"eip-id1168469609690\" class=\"unnumbered unstyled\" style=\"width: 75%;\" summary=\".\">\n<tbody>\n<tr>\n<td>3.<\/td>\n<td><\/td>\n<\/tr>\n<tr>\n<td><\/td>\n<td>[latex]6-\\left(-6\\right)[\/latex]<\/p>\n<p>This means [latex]6[\/latex] take away [latex]-6[\/latex] .<\/td>\n<\/tr>\n<tr>\n<td>Start with [latex]6[\/latex] positives.<\/td>\n<td><img loading=\"lazy\" decoding=\"async\" class=\"alignleft wp-image-16042 size-full\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2302\/2017\/05\/05033215\/IMG_01_03_092.png\" alt=\"Six positive counters\" width=\"346\" height=\"75\" \/><\/td>\n<\/tr>\n<tr>\n<td>Add [latex]6[\/latex] neutrals to get [latex]6[\/latex] negatives to take away.<\/td>\n<td><img loading=\"lazy\" decoding=\"async\" class=\"alignleft wp-image-16043 size-full\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2302\/2017\/05\/05033243\/IMG_01_03_093.png\" alt=\"Six positive counters and six neutral pairs\" width=\"685\" height=\"151\" \/><\/td>\n<\/tr>\n<tr>\n<td>Remove [latex]6[\/latex] negatives.<\/td>\n<td><img loading=\"lazy\" decoding=\"async\" class=\"alignleft wp-image-16044 size-full\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2302\/2017\/05\/05033308\/IMG_01_03_094.png\" alt=\"Six positive counters and six neutral pairs, the negative counters associated with the six neutral pairs are circled.\" width=\"697\" height=\"151\" \/><\/td>\n<\/tr>\n<tr>\n<td>How many are left?<\/td>\n<td><img loading=\"lazy\" decoding=\"async\" class=\"alignleft wp-image-16045 size-full\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2302\/2017\/05\/05033342\/IMG_01_03_095.png\" alt=\"Twelve positive counters\" width=\"638\" height=\"75\" \/><\/td>\n<\/tr>\n<tr>\n<td><\/td>\n<td>[latex]12[\/latex]<\/td>\n<\/tr>\n<tr>\n<td><\/td>\n<td>[latex]6-\\left(-6\\right)=12[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<table id=\"eip-id1168468534936\" class=\"unnumbered unstyled\" style=\"width: 75%;\" summary=\".\">\n<tbody>\n<tr>\n<td>4.<\/td>\n<td><\/td>\n<\/tr>\n<tr>\n<td><\/td>\n<td>[latex]-8-\\left(-3\\right)[\/latex]<\/p>\n<p>This means [latex]-8[\/latex] take away [latex]-3[\/latex] .<\/td>\n<\/tr>\n<tr>\n<td>Start with [latex]8[\/latex] negatives.<\/td>\n<td><img loading=\"lazy\" decoding=\"async\" class=\"alignleft wp-image-16046 size-full\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2302\/2017\/05\/05033438\/IMG_01_03_096.png\" alt=\"Eight negative counters\" width=\"450\" height=\"75\" \/><\/td>\n<\/tr>\n<tr>\n<td>Take away [latex]3[\/latex] negatives.<\/td>\n<td><img loading=\"lazy\" decoding=\"async\" class=\"alignleft wp-image-16047 size-full\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2302\/2017\/05\/05033506\/IMG_01_03_097.png\" alt=\"Eight negative counters, three are circled\" width=\"498\" height=\"88\" \/><\/td>\n<\/tr>\n<tr>\n<td>How many are left?<\/td>\n<td><img loading=\"lazy\" decoding=\"async\" class=\"alignleft wp-image-16048 size-full\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2302\/2017\/05\/05033543\/IMG_01_03_098.png\" alt=\"Five negative counters\" width=\"298\" height=\"87\" \/><\/td>\n<\/tr>\n<tr>\n<td><\/td>\n<td>[latex]-5[\/latex]<\/td>\n<\/tr>\n<tr>\n<td><\/td>\n<td>[latex]-8-\\left(-3\\right)=-5[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<\/div>\n<\/div>\n<p>Now you can try a similar problem.<\/p>\n<div class=\"textbox key-takeaways\">\n<h3>try it<\/h3>\n<p>Model each subtraction.<\/p>\n<p>1. [latex]7 - (-8)[\/latex]<br \/>\n2. [latex]-7 - (-2)[\/latex]<br \/>\n3. [latex]4 - 1[\/latex]<br \/>\n4. [latex]-6 - 8[\/latex]<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q324876\">Show Solution<\/span><\/p>\n<div id=\"q324876\" class=\"hidden-answer\" style=\"display: none\">\n<p>1. [latex]15[\/latex]<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"alignleft wp-image-16050 size-full\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2302\/2017\/05\/05033632\/IMG_01_03_099.png\" alt=\"Seven positive counters and eight neutral pairs with the negative counters circled.\" width=\"865\" height=\"150\" \/><\/p>\n<p>2. [latex]-5[\/latex]<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"alignleft wp-image-16051 size-full\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2302\/2017\/05\/05033718\/IMG_01_03_100.png\" alt=\"Seven total negative counters with two of those negative counters circled.\" width=\"465\" height=\"88\" \/><\/p>\n<p>&nbsp;<\/p>\n<p>&nbsp;<\/p>\n<p>&nbsp;<\/p>\n<p>&nbsp;<\/p>\n<p>3. [latex]3[\/latex]<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"alignleft wp-image-16052 size-full\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2302\/2017\/05\/05033801\/IMG_01_03_101.png\" alt=\"Four positive counters with one of those counters circled\" width=\"359\" height=\"88\" \/><\/p>\n<p>&nbsp;<\/p>\n<p>&nbsp;<\/p>\n<p>&nbsp;<\/p>\n<p>&nbsp;<\/p>\n<p>4. [latex]-14[\/latex]<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"alignleft wp-image-16054 size-full\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2302\/2017\/05\/05034110\/IMG_01_03_102.png\" alt=\"Six negative counters and eight neutral pairs with the positive counters in the neutral pairs circled.\" width=\"829\" height=\"151\" \/><\/p>\n<\/div>\n<\/div>\n<p>&nbsp;<\/p>\n<p>Model each subtraction.<\/p>\n<p>1. [latex]4 - (-6)[\/latex]<br \/>\n2. [latex]-8 - (-1)[\/latex]<br \/>\n3. [latex]7 - 3[\/latex]<br \/>\n4. [latex]-4 - 2[\/latex]<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q677150\">Show Solution<\/span><\/p>\n<div id=\"q677150\" class=\"hidden-answer\" style=\"display: none\">\n<p>1. [latex]10[\/latex]<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"alignleft wp-image-16055 size-full\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2302\/2017\/05\/05034207\/IMG_01_03_103.png\" alt=\"Four positive counters and six neutral pairs with the negative counters in the neutral pairs circled.\" width=\"724\" height=\"151\" \/><\/p>\n<p>2. [latex]-7[\/latex]<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"alignleft wp-image-16056 size-full\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2302\/2017\/05\/05034256\/IMG_01_03_104.png\" alt=\"Eight total negative counters with one negative counter circled\" width=\"510\" height=\"88\" \/><\/p>\n<p>&nbsp;<\/p>\n<p>&nbsp;<\/p>\n<p>&nbsp;<\/p>\n<p>&nbsp;<\/p>\n<p>3. [latex]4[\/latex]<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"alignleft wp-image-16058 size-full\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2302\/2017\/05\/05034609\/IMG_01_03_105.png\" alt=\"Seven total positive counters with three of those counters circled.\" width=\"459\" height=\"88\" \/><\/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]-6[\/latex]<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"alignleft wp-image-16057 size-full\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2302\/2017\/05\/05034358\/IMG_01_03_106.png\" alt=\"Five negative counters and two neutral pairs with the two positive counters circled.\" width=\"508\" height=\"142\" \/><\/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>Each of the examples so far have been carefully constructed so that the sign of the answer matched the sign of the first number in the expression. \u00a0For example, in\u00a0\u00a0[latex]\u22125 \u2212 4[\/latex], the result is [latex]-9[\/latex], which is the same sign as [latex]-5[\/latex]. Now we will see subtraction where the sign of the result is different from the starting number.<\/p>\n<div class=\"textbox exercises\">\n<h3>example<\/h3>\n<p>Model each subtraction expression:<\/p>\n<ol id=\"eip-id1168468264062\" class=\"circled\">\n<li>[latex]2 - 8[\/latex]<\/li>\n<li>[latex]-3-\\left(-8\\right)[\/latex]<\/li>\n<\/ol>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q485889\">Show Solution<\/span><\/p>\n<div id=\"q485889\" class=\"hidden-answer\" style=\"display: none\">\n<p>Solution<\/p>\n<table id=\"eip-id1168466285116\" class=\"unnumbered unstyled\" style=\"width: 75%;\" summary=\"This figure has six rows. The first row states we start with 2 positives. It is followed by 2 blue circles. The second row states we need to take away 8 positives but we have only 2. The third row states add neutral pairs until there are 8 positives to take away. It is followed by a row of 8 blue circles. The first two are separated from the last 6. Under the last six there is a row of 6 red circles. The fourth row states then take away the 8 positives. It is followed by a row of 8 blue circles. They are circled. The fifth row states find the number of counters that are left. There are six negatives. It is followed by 6 red circles labeled six negatives. The sixth row has 2 minus 8 equals negative 6.\">\n<tbody>\n<tr>\n<td>1.<\/p>\n<p>We start with [latex]2[\/latex] positives.<\/td>\n<td><\/td>\n<\/tr>\n<tr>\n<td>We need to take away [latex]8[\/latex] positives, but we have only [latex]2[\/latex].<\/td>\n<td><img loading=\"lazy\" decoding=\"async\" class=\"size-full wp-image-16060 alignleft\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2302\/2017\/05\/05035350\/IMG_01_03_107.png\" alt=\"Two positive counters\" width=\"125\" height=\"90\" \/><\/td>\n<\/tr>\n<tr>\n<td>Add neutral pairs until there are [latex]8[\/latex] positives to take away.<\/td>\n<td><img loading=\"lazy\" decoding=\"async\" class=\"alignleft wp-image-16061 size-full\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2302\/2017\/05\/05035411\/IMG_01_03_108.png\" alt=\"Two positive counters and six neutral pairs\" width=\"500\" height=\"151\" \/><\/td>\n<\/tr>\n<tr>\n<td>Then take away [latex]8[\/latex]\u00a0positives.<\/td>\n<td><img loading=\"lazy\" decoding=\"async\" class=\"alignleft wp-image-16062 size-full\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2302\/2017\/05\/05035452\/IMG_01_03_109.png\" alt=\"Two positive counters and six neutral pairs, all of the positive counters are circled.\" width=\"544\" height=\"172\" \/><\/td>\n<\/tr>\n<tr>\n<td>Find the number of counters that are left.<\/p>\n<p>There are [latex]6[\/latex] negatives.<\/td>\n<td><img loading=\"lazy\" decoding=\"async\" class=\"alignleft wp-image-16063 size-full\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2302\/2017\/05\/05035523\/IMG_01_03_110.png\" alt=\"Six negative counters\" width=\"349\" height=\"89\" \/><\/td>\n<\/tr>\n<tr>\n<td><\/td>\n<td>[latex]2 - 8=-6[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<table id=\"eip-id1168468388588\" class=\"unnumbered unstyled\" style=\"width: 75%;\" summary=\"This figure has six rows. The first row states we start with 3 negatives. It is followed by 3 red circles. The second row states we need to take away 8 negatives but we have only 3. The third row states add neutral pairs until there are 8 negatives to take away. It is followed by a row of 8 red circles. The first three are separated from the last 5. Under the last 5 there is a row of 5 blue circles. The fourth row states then take away the 8 negatives. It is followed by a row of 8 red circles. They are circled. The fifth row states find the number of counters that are left. There are five positives. It is followed by 5 blue circles. The sixth row has negative 3 minus negative 8 equals 5.\">\n<tbody>\n<tr>\n<td>2.<\/p>\n<p>We start with [latex]3[\/latex] negatives.<\/td>\n<td><img loading=\"lazy\" decoding=\"async\" class=\"alignleft wp-image-16064 size-full\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2302\/2017\/05\/05035545\/IMG_01_03_111.png\" alt=\"Three negative counters\" width=\"192\" height=\"95\" \/><\/td>\n<\/tr>\n<tr>\n<td>We need to take away [latex]8[\/latex] negatives, but we have only [latex]3[\/latex].<\/td>\n<td><\/td>\n<\/tr>\n<tr>\n<td>Add neutral pairs until there are [latex]8[\/latex] negatives to take away.<\/td>\n<td><img loading=\"lazy\" decoding=\"async\" class=\"alignleft wp-image-16065 size-full\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2302\/2017\/05\/05035615\/IMG_01_03_112.png\" alt=\"Three negative counters and five neutral pairs\" width=\"473\" height=\"150\" \/><\/td>\n<\/tr>\n<tr>\n<td>Then take away the [latex]8[\/latex] negatives.<\/td>\n<td><img loading=\"lazy\" decoding=\"async\" class=\"alignleft wp-image-16066 size-full\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2302\/2017\/05\/05035650\/IMG_01_03_113.png\" alt=\"Three negative counters and five neutral pairs, all the negative counters are circled.\" width=\"547\" height=\"160\" \/><\/td>\n<\/tr>\n<tr>\n<td>Find the number of counters that are left.<\/p>\n<p>There are [latex]5[\/latex] positives.<\/td>\n<td><img loading=\"lazy\" decoding=\"async\" class=\"alignleft wp-image-16067 size-full\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2302\/2017\/05\/05035734\/IMG_01_03_114.png\" alt=\"Five positive counters\" width=\"331\" height=\"87\" \/><\/td>\n<\/tr>\n<tr>\n<td><\/td>\n<td>[latex]-3-\\left(-8\\right)=5[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<\/div>\n<\/div>\n<p>Now you can try a similar problem.<\/p>\n<div class=\"textbox key-takeaways\">\n<h3>try it<\/h3>\n<p>Model each subtraction expression.<\/p>\n<ol id=\"eip-id1168466205125\" class=\"circled\">\n<li>[latex]7 - 9[\/latex]<\/li>\n<li>[latex]-5--9[\/latex]<\/li>\n<\/ol>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q678756\">Show Solution<\/span><\/p>\n<div id=\"q678756\" class=\"hidden-answer\" style=\"display: none\">\n<p>1.<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"alignleft wp-image-16070 size-full\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2302\/2017\/05\/05040624\/IMG_01_03_115.png\" alt=\"Seven positive counters and two neutral pairs. All of the positive counters are circled.\" width=\"587\" height=\"147\" \/><\/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<p>2.<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"alignleft wp-image-16071 size-full\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2302\/2017\/05\/05040700\/IMG_01_03_116.png\" alt=\"Five positive counters and four neutral pairs. All of the positive counters are circled.\" width=\"691\" height=\"155\" \/><\/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]4[\/latex]<\/p>\n<\/div>\n<\/div>\n<p>&nbsp;<\/p>\n<p>Model each subtraction expression.<\/p>\n<ol id=\"eip-id1168468470083\" class=\"circled\">\n<li>[latex]5 - 8[\/latex]<\/li>\n<li>[latex]-7-\\left(-10\\right)[\/latex]<\/li>\n<\/ol>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q164098\">Show Solution<\/span><\/p>\n<div id=\"q164098\" class=\"hidden-answer\" style=\"display: none\">\n<p>1.<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"alignleft wp-image-16072 size-full\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2302\/2017\/05\/05040741\/IMG_01_03_117.png\" alt=\"Five positive counters and three neutral pairs. All positive counters are circled.\" width=\"579\" height=\"154\" \/><\/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<p>2.<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"alignleft wp-image-16073 size-full\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2302\/2017\/05\/05040818\/IMG_01_03_118.png\" alt=\"Seven positive counters and three neutral pairs. All positive counters are circled.\" width=\"639\" height=\"150\" \/><\/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>When you subtract two integers, there are two possibilities, either the result will have a different sign from the starting number, or it will have the same sign.<\/p>\n<p>Watch the video below to see more examples of modeling integer subtraction with color counters.<\/p>\n<p><iframe loading=\"lazy\" id=\"oembed-1\" title=\"Subtracting Integers with Color Counters (No Extra Zeros Needed)\" width=\"500\" height=\"281\" src=\"https:\/\/www.youtube.com\/embed\/ZjiL7GfxgiI?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-9395\">\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>Subtracting Integers with Color Counters (No Extra Zeros Needed). <strong>Authored by<\/strong>: James Sousa (Mathispower4u.com). <strong>Located at<\/strong>: <a target=\"_blank\" href=\"https:\/\/youtu.be\/ZjiL7GfxgiI\">https:\/\/youtu.be\/ZjiL7GfxgiI<\/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":14,"template":"","meta":{"_candela_citation":"[{\"type\":\"cc\",\"description\":\"Subtracting Integers with Color Counters (No Extra Zeros Needed)\",\"author\":\"James Sousa (Mathispower4u.com)\",\"organization\":\"\",\"url\":\"https:\/\/youtu.be\/ZjiL7GfxgiI\",\"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":"f7368e97-b1f8-493f-b921-0917b398f46b","pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"class_list":["post-9395","chapter","type-chapter","status-publish","hentry"],"part":6350,"_links":{"self":[{"href":"https:\/\/courses.lumenlearning.com\/wm-prealgebra\/wp-json\/pressbooks\/v2\/chapters\/9395","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":30,"href":"https:\/\/courses.lumenlearning.com\/wm-prealgebra\/wp-json\/pressbooks\/v2\/chapters\/9395\/revisions"}],"predecessor-version":[{"id":16074,"href":"https:\/\/courses.lumenlearning.com\/wm-prealgebra\/wp-json\/pressbooks\/v2\/chapters\/9395\/revisions\/16074"}],"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\/9395\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/courses.lumenlearning.com\/wm-prealgebra\/wp-json\/wp\/v2\/media?parent=9395"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/wm-prealgebra\/wp-json\/pressbooks\/v2\/chapter-type?post=9395"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/wm-prealgebra\/wp-json\/wp\/v2\/contributor?post=9395"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/wm-prealgebra\/wp-json\/wp\/v2\/license?post=9395"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}