{"id":3574,"date":"2019-12-31T17:56:29","date_gmt":"2019-12-31T17:56:29","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/mathforlibscoreq\/?post_type=chapter&#038;p=3574"},"modified":"2021-02-05T23:49:04","modified_gmt":"2021-02-05T23:49:04","slug":"notation-and-modeling-subtraction-of-integers","status":"web-only","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/slcc-mathforliberalartscorequisite\/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 blue counters represent positive numbers and the red 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=\"alignnone wp-image-15509\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2302\/2017\/05\/13215401\/SubtractingIntegers_1.png\" alt=\"This figure has 5 blue circles with the number 5 underneath.\" width=\"354\" height=\"85\" \/><\/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=\"alignnone wp-image-15510\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2302\/2017\/05\/13215411\/SubtractingIntegers_2.png\" alt=\"This figure has 5 blue circles. Under the last two circles, it say 2 positives.\" width=\"356\" height=\"96\" \/><\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Find the counters that are left.<\/td>\r\n<td><img class=\"alignnone wp-image-15511\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2302\/2017\/05\/13215420\/SubtractingIntegers_3.png\" alt=\"This figure has two blue circles.\" width=\"359\" height=\"66\" \/><\/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=\"alignnone wp-image-15512\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2302\/2017\/05\/13215432\/SubtractingIntegers_4.png\" alt=\"This figure has 6 blue circles. Four of them are circled and two are not.\" width=\"354\" height=\"97\" \/>\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=\"alignnone wp-image-15513\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2302\/2017\/05\/13215442\/SubtractingIntegers_5.png\" alt=\"This figure has 7 blue circles. Four of them are circled and three are not.\" width=\"356\" height=\"96\" \/>\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=\"alignnone wp-image-15514\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2302\/2017\/05\/13215454\/SubtractingIntegers_6.png\" alt=\"This figure has 5 red circles labeled with a negative 5\" width=\"356\" height=\"85\" \/><\/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=\"alignnone wp-image-15515\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2302\/2017\/05\/13215504\/SubtractingIntegers_7.png\" alt=\"This figure has 5 red circles. Three are circled. Underneath the two that are not it is labeled 2 negatives.\" width=\"354\" height=\"95\" \/><\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Find the number of counters that are left.<\/td>\r\n<td><img class=\"alignnone wp-image-15516\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2302\/2017\/05\/13215514\/SubtractingIntegers_8.png\" alt=\"This figure has two red circles.\" width=\"357\" height=\"66\" \/><\/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=\"alignnone wp-image-15517\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2302\/2017\/05\/13215530\/SubtractingIntegers_9.png\" alt=\"This figure has 6 red circles. The last four are circled together.\" width=\"354\" height=\"97\" \/>\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=\"alignnone wp-image-15518\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2302\/2017\/05\/13215542\/SubtractingIntegers_10.png\" alt=\"This figure has 7 red circles. The first 4 are circled together.\" width=\"354\" height=\"96\" \/>\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-15519\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2302\/2017\/05\/13215736\/SubtractingIntegers_11b.png\" alt=\"This figure has 5 blue circles and 5 red circles. Three circles from the blue group and red group are circled together. Above the blue circles it says 5 minus 3 equals 2. Above the red circles it says negative 5 minus negative 3 equals negative 2.\" width=\"569\" height=\"123\" \/>\r\nNow 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=\"alignnone wp-image-15520\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2302\/2017\/05\/13215754\/SubtractingIntegers_13.png\" alt=\"This figure has 5 red circles labeled negative 5.\" width=\"355\" height=\"77\" \/><\/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=\"alignnone wp-image-15521\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2302\/2017\/05\/13215806\/SubtractingIntegers_14.png\" alt=\"This figure has 8 red circles with a space between the first 5 and last three. Underneath the last three red circles, there are 3 blue circles.\" width=\"357\" height=\"99\" \/><\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Now take away [latex]3[\/latex] positives.<\/td>\r\n<td><img class=\"alignnone wp-image-15522\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2302\/2017\/05\/13215816\/SubtractingIntegers_15.png\" alt=\"This figure has 8 red circles with a space between the first 5 and last three. Underneath the last three red circles, there are 3 blue circles. The 3 blue circles are circled together.\" width=\"355\" height=\"131\" \/><\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Count the number of counters that are left.<\/td>\r\n<td><img class=\"alignnone wp-image-15523\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2302\/2017\/05\/13215827\/SubtractingIntegers_16.png\" alt=\"This figure has 8 red circles labeled 8 negatives.\" width=\"354\" height=\"76\" \/><\/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=\"alignnone wp-image-15524\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2302\/2017\/05\/13215840\/SubtractingIntegers_17.png\" alt=\"This figure has 10 red circles with a space between the first 6 and last 4. Underneath the last 4 red circles, there are 4 blue circles that are circled together.\" width=\"357\" height=\"100\" \/>\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=\"alignnone wp-image-15525\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2302\/2017\/05\/13215851\/SubtractingIntegers_18.png\" alt=\"This figure has 11 red circles with a space between the first 7 and last 4. Underneath the last 4 red circles, there are 4 blue circles that are circled together.\" width=\"354\" height=\"99\" \/>\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=\"alignnone wp-image-15526\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2302\/2017\/05\/13215905\/SubtractingIntegers_19.png\" alt=\"This figure has 5 blue circles with the number 5 labeled underneath.\" width=\"357\" height=\"77\" \/><\/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=\"alignnone wp-image-15527\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2302\/2017\/05\/13220212\/SubtractingIntegers_20.png\" alt=\"This figure has 8 blue circles with a space between the first 5 and last 3. Underneath the last 3 blue circles there are 3 red circles.\" width=\"357\" height=\"99\" \/><\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Then take away [latex]3[\/latex] negatives.<\/td>\r\n<td><img class=\"alignnone wp-image-15528\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2302\/2017\/05\/13220223\/SubtractingIntegers_21.png\" alt=\"This figure has 8 blue circles with a space between the first 5 and last 3. Underneath the last 3 blue circles there are 3 red circles that are circled together. \" width=\"357\" height=\"132\" \/><\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Count the number of counters that are left.<\/td>\r\n<td><img class=\"alignnone wp-image-15529\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2302\/2017\/05\/13220233\/SubtractingIntegers_22.png\" alt=\"This figure has 8 blue circles labeled 8 positives.\" width=\"357\" height=\"77\" \/><\/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=\"alignnone wp-image-15530\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2302\/2017\/05\/13220245\/SubtractingIntegers_23.png\" alt=\"This figure has 10 blue circles with a space between the first 6 and last 4. Underneath the last four blue circles there are four red circles that are circled together. \" width=\"358\" height=\"102\" \/>\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=\"alignnone wp-image-15531\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2302\/2017\/05\/13220257\/SubtractingIntegers_24.png\" alt=\"This figure has 11 blue circles with a space between the first 7 and last 4. Underneath the last 4 circles, there are 4 red circles that are circled together. \" width=\"358\" height=\"100\" \/>\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=\"alignnone wp-image-15532\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2302\/2017\/05\/13220310\/SubtractingIntegers_25.png\" alt=\"This figure has 8 blue circles.\" width=\"355\" height=\"63\" \/><\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Take away [latex]2[\/latex] positives.<\/td>\r\n<td><img class=\"alignnone wp-image-15533\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2302\/2017\/05\/13220320\/SubtractingIntegers_26.png\" alt=\"This figure has 8 blue circles with a space between the first 6 and last 2. The last 2 circles are circled together. \" width=\"355\" height=\"63\" \/><\/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=\"alignnone wp-image-15534\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2302\/2017\/05\/13220333\/SubtractingIntegers_27.png\" alt=\"This figure has 5 red circles.\" width=\"354\" height=\"63\" \/><\/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=\"alignnone wp-image-15535\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2302\/2017\/05\/13220349\/SubtractingIntegers_28.png\" alt=\"This figure has 9 circles with a space between the first 5 and last 4. Underneath the last 4 red circles there are 4 blue circles.\" width=\"357\" height=\"102\" \/><\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Take away [latex]4[\/latex] positives.<\/td>\r\n<td><img class=\"alignnone wp-image-15536\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2302\/2017\/05\/13220401\/SubtractingIntegers_29.png\" alt=\"This figure has 9 red circles with a space between the first 5 and last 4. Underneath the last 4 red circles there are 4 blue circles that are circled together.\" width=\"355\" height=\"122\" \/><\/td>\r\n<\/tr>\r\n<tr>\r\n<td>How many are left?<\/td>\r\n<td><img class=\"alignnone wp-image-15537\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2302\/2017\/05\/13220411\/SubtractingIntegers_30.png\" alt=\"This figure has 9 red circles.\" width=\"354\" height=\"63\" \/><\/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=\"alignnone wp-image-15538\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2302\/2017\/05\/13220422\/SubtractingIntegers_31.png\" alt=\"This figure has 6 blue circles.\" width=\"399\" height=\"53\" \/><\/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=\"alignnone wp-image-15539\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2302\/2017\/05\/13220433\/SubtractingIntegers_32.png\" alt=\"This figure has 12 blue circles with a space between the first 6 and the last 6. Underneath the last 6 blue circles, there are 6 red circles.\" width=\"399\" height=\"85\" \/><\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Remove [latex]6[\/latex] negatives.<\/td>\r\n<td><img class=\"alignnone wp-image-15540\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2302\/2017\/05\/13220444\/SubtractingIntegers_33.png\" alt=\"This figure has 12 blue circles with a space between the first 6 and the last 6. Underneath the last 6 blue circles, there are 6 red circles that are circled together.\" width=\"406\" height=\"105\" \/><\/td>\r\n<\/tr>\r\n<tr>\r\n<td>How many are left?<\/td>\r\n<td><img class=\"alignnone wp-image-15541\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2302\/2017\/05\/13220501\/SubtractingIntegers_34.png\" alt=\"This figure has 12 blue circles.\" width=\"405\" height=\"54\" \/><\/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=\"alignnone wp-image-15542\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2302\/2017\/05\/13220514\/SubtractingIntegers_35.png\" alt=\"This figure has 8 red circles.\" width=\"355\" height=\"65\" \/><\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Take away [latex]3[\/latex] negatives.<\/td>\r\n<td><img class=\"alignnone wp-image-15543\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2302\/2017\/05\/13220524\/SubtractingIntegers_36.png\" alt=\"This figure has 8 red circles with a space between the first five and last three. The last three circles are circled together.\" width=\"357\" height=\"69\" \/><\/td>\r\n<\/tr>\r\n<tr>\r\n<td>How many are left?<\/td>\r\n<td><img class=\"alignnone wp-image-15534\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2302\/2017\/05\/13220333\/SubtractingIntegers_27.png\" alt=\"This figure has 5 red circles.\" width=\"352\" height=\"63\" \/><\/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=\"alignnone wp-image-15544\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2302\/2017\/05\/13220559\/SubtractingIntegers_37.png\" alt=\"This figure has 15 blue circles with a space between the first 7 and last 8. Below the last 8 circles there are 8 red circles that are circled together.\" width=\"641\" height=\"132\" \/>\r\n\r\n2. [latex]-5[\/latex]\r\n\r\n<img class=\"alignnone wp-image-15545\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2302\/2017\/05\/13220610\/SubtractingIntegers_38.png\" alt=\"This figure has 7 red circles with a space between the first 5 and last 2. The last 2 circles are circled together.\" width=\"359\" height=\"69\" \/>\r\n\r\n3. [latex]3[\/latex]\r\n\r\n<img class=\"alignnone wp-image-15546\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2302\/2017\/05\/13220621\/SubtractingIntegers_39.png\" alt=\"This figure has 4 blue circles with a space between the first 3 and the last 1. The last circle is circled.\" width=\"358\" height=\"69\" \/>\r\n\r\n4. [latex]-14[\/latex]\r\n\r\n<img class=\"alignnone wp-image-15547\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2302\/2017\/05\/13220634\/SubtractingIntegers_40.png\" alt=\"This figure has 14 red circles with a space between the first 6 and last 8. Below the last 8 circles there are 8 blue circles that are circled together.\" width=\"620\" height=\"137\" \/>\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=\"alignnone wp-image-15548\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2302\/2017\/05\/13220648\/SubtractingIntegers_41.png\" alt=\"This figure has 10 blue circles with a space between the first 4 and last 6. Under the last 6 blue circles there are 6 red circles that are circled together.\" width=\"481\" height=\"142\" \/>\r\n\r\n2. [latex]-7[\/latex]\r\n\r\n<img class=\"alignnone wp-image-15549\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2302\/2017\/05\/13220700\/SubtractingIntegers_42.png\" alt=\"This figure has 8 red circles with a space between the first 7 and last 1. The last circle is circled by itself.\" width=\"483\" height=\"72\" \/>\r\n\r\n3. [latex]4[\/latex]\r\n\r\n<img class=\"alignnone wp-image-15550\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2302\/2017\/05\/13220711\/SubtractingIntegers_43.png\" alt=\"This figure has 7 blue circles with a space between the first 4 and last 3. The last 3 circles are circled together.\" width=\"497\" height=\"73\" \/>\r\n\r\n4. [latex]-6[\/latex]\r\n\r\n<img class=\"alignnone wp-image-15551\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2302\/2017\/05\/13220723\/SubtractingIntegers_44.png\" alt=\"This figure has 6 red circles with a space between the first 4 and last 2. Under the last 2 red circles there are 2 blue circles that are circled together. \" width=\"372\" height=\"158\" \/>\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><img class=\"alignnone wp-image-15552\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2302\/2017\/05\/13220734\/SubtractingIntegers_45.png\" alt=\"This figure has 2 blue circles with a number 2 labeled underneath.\" width=\"356\" height=\"75\" \/><\/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><\/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=\"alignnone wp-image-15553\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2302\/2017\/05\/13220747\/SubtractingIntegers_46.png\" alt=\"This figure has 8 blue circles with a space between the first 2 and last 6. Underneath the last 6 blue circles there 6 red circles.\" width=\"356\" height=\"93\" \/><\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Then take away [latex]8[\/latex]\u00a0positives.<\/td>\r\n<td><img class=\"alignnone wp-image-15554\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2302\/2017\/05\/13220758\/SubtractingIntegers_47.png\" alt=\"This figure has 8 blue circles that are circled together. Underneath there are 6 red circles.\" width=\"354\" height=\"138\" \/><\/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=\"alignnone wp-image-15555\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2302\/2017\/05\/13220809\/SubtractingIntegers_48.png\" alt=\"This figure has 6 red circles labeled 6 negatives.\" width=\"356\" height=\"75\" \/><\/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=\"alignnone wp-image-15556\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2302\/2017\/05\/13220821\/SubtractingIntegers_49.png\" alt=\"This figure has 3 red circles with a number 3 underneath.\" width=\"354\" height=\"75\" \/><\/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=\"alignnone wp-image-15557\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2302\/2017\/05\/13220831\/SubtractingIntegers_50.png\" alt=\"This figure has 7 red circles and below there are 6 blue circles.\" width=\"358\" height=\"94\" \/><\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Then take away the [latex]8[\/latex] negatives.<\/td>\r\n<td><img class=\"alignnone wp-image-15558\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2302\/2017\/05\/13220843\/SubtractingIntegers_51.png\" alt=\"This figure has 8 red circles that are circled together. Below there are 5 blue circles.\" width=\"358\" height=\"139\" \/><\/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=\"alignnone wp-image-15559\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2302\/2017\/05\/13220856\/SubtractingIntegers_52.png\" alt=\"This figure has 5 blue circles labeled 5 positives.\" width=\"358\" height=\"76\" \/><\/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=\"alignnone wp-image-15560\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2302\/2017\/05\/13220910\/SubtractingIntegers_53.png\" alt=\"This figure has 9 blue circles that are circled together. Underneath there are 2 red circles.\" width=\"468\" height=\"152\" \/>\r\n\r\n[latex]\u22122[\/latex]\r\n\r\n2.\r\n\r\n<img class=\"alignnone wp-image-15561\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2302\/2017\/05\/13220923\/SubtractingIntegers_54.png\" alt=\"This figure has 9 blue circles that are circled together. Underneath there are 4 red circles.\" width=\"491\" height=\"167\" \/>\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=\"alignnone wp-image-15562\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2302\/2017\/05\/13220936\/SubtractingIntegers_55.png\" alt=\"This figure has 8 blue circles that are circled together. Underneath there are 3 red circles.\" width=\"457\" height=\"176\" \/>\r\n\r\n[latex]\u22123[\/latex]\r\n\r\n2.\r\n\r\n<img class=\"alignnone wp-image-15563\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2302\/2017\/05\/13220948\/SubtractingIntegers_56.png\" alt=\"This figure has 10 blue circles that are circled together. Underneath there are 3 red circles.\" width=\"606\" height=\"185\" \/>\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 blue counters represent positive numbers and the red 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=\"alignnone wp-image-15509\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2302\/2017\/05\/13215401\/SubtractingIntegers_1.png\" alt=\"This figure has 5 blue circles with the number 5 underneath.\" width=\"354\" height=\"85\" \/><\/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=\"alignnone wp-image-15510\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2302\/2017\/05\/13215411\/SubtractingIntegers_2.png\" alt=\"This figure has 5 blue circles. Under the last two circles, it say 2 positives.\" width=\"356\" height=\"96\" \/><\/td>\n<\/tr>\n<tr>\n<td>Find the counters that are left.<\/td>\n<td><img loading=\"lazy\" decoding=\"async\" class=\"alignnone wp-image-15511\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2302\/2017\/05\/13215420\/SubtractingIntegers_3.png\" alt=\"This figure has two blue circles.\" width=\"359\" height=\"66\" \/><\/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=\"alignnone wp-image-15512\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2302\/2017\/05\/13215432\/SubtractingIntegers_4.png\" alt=\"This figure has 6 blue circles. Four of them are circled and two are not.\" width=\"354\" height=\"97\" \/><\/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=\"alignnone wp-image-15513\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2302\/2017\/05\/13215442\/SubtractingIntegers_5.png\" alt=\"This figure has 7 blue circles. Four of them are circled and three are not.\" width=\"356\" height=\"96\" \/><\/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=\"alignnone wp-image-15514\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2302\/2017\/05\/13215454\/SubtractingIntegers_6.png\" alt=\"This figure has 5 red circles labeled with a negative 5\" width=\"356\" height=\"85\" \/><\/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=\"alignnone wp-image-15515\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2302\/2017\/05\/13215504\/SubtractingIntegers_7.png\" alt=\"This figure has 5 red circles. Three are circled. Underneath the two that are not it is labeled 2 negatives.\" width=\"354\" height=\"95\" \/><\/td>\n<\/tr>\n<tr>\n<td>Find the number of counters that are left.<\/td>\n<td><img loading=\"lazy\" decoding=\"async\" class=\"alignnone wp-image-15516\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2302\/2017\/05\/13215514\/SubtractingIntegers_8.png\" alt=\"This figure has two red circles.\" width=\"357\" height=\"66\" \/><\/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=\"alignnone wp-image-15517\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2302\/2017\/05\/13215530\/SubtractingIntegers_9.png\" alt=\"This figure has 6 red circles. The last four are circled together.\" width=\"354\" height=\"97\" \/><\/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=\"alignnone wp-image-15518\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2302\/2017\/05\/13215542\/SubtractingIntegers_10.png\" alt=\"This figure has 7 red circles. The first 4 are circled together.\" width=\"354\" height=\"96\" \/><\/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-15519\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2302\/2017\/05\/13215736\/SubtractingIntegers_11b.png\" alt=\"This figure has 5 blue circles and 5 red circles. Three circles from the blue group and red group are circled together. Above the blue circles it says 5 minus 3 equals 2. Above the red circles it says negative 5 minus negative 3 equals negative 2.\" width=\"569\" height=\"123\" \/><br \/>\nNow 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=\"alignnone wp-image-15520\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2302\/2017\/05\/13215754\/SubtractingIntegers_13.png\" alt=\"This figure has 5 red circles labeled negative 5.\" width=\"355\" height=\"77\" \/><\/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=\"alignnone wp-image-15521\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2302\/2017\/05\/13215806\/SubtractingIntegers_14.png\" alt=\"This figure has 8 red circles with a space between the first 5 and last three. Underneath the last three red circles, there are 3 blue circles.\" width=\"357\" height=\"99\" \/><\/td>\n<\/tr>\n<tr>\n<td>Now take away [latex]3[\/latex] positives.<\/td>\n<td><img loading=\"lazy\" decoding=\"async\" class=\"alignnone wp-image-15522\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2302\/2017\/05\/13215816\/SubtractingIntegers_15.png\" alt=\"This figure has 8 red circles with a space between the first 5 and last three. Underneath the last three red circles, there are 3 blue circles. The 3 blue circles are circled together.\" width=\"355\" height=\"131\" \/><\/td>\n<\/tr>\n<tr>\n<td>Count the number of counters that are left.<\/td>\n<td><img loading=\"lazy\" decoding=\"async\" class=\"alignnone wp-image-15523\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2302\/2017\/05\/13215827\/SubtractingIntegers_16.png\" alt=\"This figure has 8 red circles labeled 8 negatives.\" width=\"354\" height=\"76\" \/><\/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=\"alignnone wp-image-15524\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2302\/2017\/05\/13215840\/SubtractingIntegers_17.png\" alt=\"This figure has 10 red circles with a space between the first 6 and last 4. Underneath the last 4 red circles, there are 4 blue circles that are circled together.\" width=\"357\" height=\"100\" \/><\/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=\"alignnone wp-image-15525\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2302\/2017\/05\/13215851\/SubtractingIntegers_18.png\" alt=\"This figure has 11 red circles with a space between the first 7 and last 4. Underneath the last 4 red circles, there are 4 blue circles that are circled together.\" width=\"354\" height=\"99\" \/><\/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=\"alignnone wp-image-15526\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2302\/2017\/05\/13215905\/SubtractingIntegers_19.png\" alt=\"This figure has 5 blue circles with the number 5 labeled underneath.\" width=\"357\" height=\"77\" \/><\/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=\"alignnone wp-image-15527\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2302\/2017\/05\/13220212\/SubtractingIntegers_20.png\" alt=\"This figure has 8 blue circles with a space between the first 5 and last 3. Underneath the last 3 blue circles there are 3 red circles.\" width=\"357\" height=\"99\" \/><\/td>\n<\/tr>\n<tr>\n<td>Then take away [latex]3[\/latex] negatives.<\/td>\n<td><img loading=\"lazy\" decoding=\"async\" class=\"alignnone wp-image-15528\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2302\/2017\/05\/13220223\/SubtractingIntegers_21.png\" alt=\"This figure has 8 blue circles with a space between the first 5 and last 3. Underneath the last 3 blue circles there are 3 red circles that are circled together.\" width=\"357\" height=\"132\" \/><\/td>\n<\/tr>\n<tr>\n<td>Count the number of counters that are left.<\/td>\n<td><img loading=\"lazy\" decoding=\"async\" class=\"alignnone wp-image-15529\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2302\/2017\/05\/13220233\/SubtractingIntegers_22.png\" alt=\"This figure has 8 blue circles labeled 8 positives.\" width=\"357\" height=\"77\" \/><\/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=\"alignnone wp-image-15530\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2302\/2017\/05\/13220245\/SubtractingIntegers_23.png\" alt=\"This figure has 10 blue circles with a space between the first 6 and last 4. Underneath the last four blue circles there are four red circles that are circled together.\" width=\"358\" height=\"102\" \/><\/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=\"alignnone wp-image-15531\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2302\/2017\/05\/13220257\/SubtractingIntegers_24.png\" alt=\"This figure has 11 blue circles with a space between the first 7 and last 4. Underneath the last 4 circles, there are 4 red circles that are circled together.\" width=\"358\" height=\"100\" \/><\/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=\"alignnone wp-image-15532\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2302\/2017\/05\/13220310\/SubtractingIntegers_25.png\" alt=\"This figure has 8 blue circles.\" width=\"355\" height=\"63\" \/><\/td>\n<\/tr>\n<tr>\n<td>Take away [latex]2[\/latex] positives.<\/td>\n<td><img loading=\"lazy\" decoding=\"async\" class=\"alignnone wp-image-15533\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2302\/2017\/05\/13220320\/SubtractingIntegers_26.png\" alt=\"This figure has 8 blue circles with a space between the first 6 and last 2. The last 2 circles are circled together.\" width=\"355\" height=\"63\" \/><\/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=\"alignnone wp-image-15534\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2302\/2017\/05\/13220333\/SubtractingIntegers_27.png\" alt=\"This figure has 5 red circles.\" width=\"354\" height=\"63\" \/><\/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=\"alignnone wp-image-15535\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2302\/2017\/05\/13220349\/SubtractingIntegers_28.png\" alt=\"This figure has 9 circles with a space between the first 5 and last 4. Underneath the last 4 red circles there are 4 blue circles.\" width=\"357\" height=\"102\" \/><\/td>\n<\/tr>\n<tr>\n<td>Take away [latex]4[\/latex] positives.<\/td>\n<td><img loading=\"lazy\" decoding=\"async\" class=\"alignnone wp-image-15536\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2302\/2017\/05\/13220401\/SubtractingIntegers_29.png\" alt=\"This figure has 9 red circles with a space between the first 5 and last 4. Underneath the last 4 red circles there are 4 blue circles that are circled together.\" width=\"355\" height=\"122\" \/><\/td>\n<\/tr>\n<tr>\n<td>How many are left?<\/td>\n<td><img loading=\"lazy\" decoding=\"async\" class=\"alignnone wp-image-15537\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2302\/2017\/05\/13220411\/SubtractingIntegers_30.png\" alt=\"This figure has 9 red circles.\" width=\"354\" height=\"63\" \/><\/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=\"alignnone wp-image-15538\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2302\/2017\/05\/13220422\/SubtractingIntegers_31.png\" alt=\"This figure has 6 blue circles.\" width=\"399\" height=\"53\" \/><\/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=\"alignnone wp-image-15539\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2302\/2017\/05\/13220433\/SubtractingIntegers_32.png\" alt=\"This figure has 12 blue circles with a space between the first 6 and the last 6. Underneath the last 6 blue circles, there are 6 red circles.\" width=\"399\" height=\"85\" \/><\/td>\n<\/tr>\n<tr>\n<td>Remove [latex]6[\/latex] negatives.<\/td>\n<td><img loading=\"lazy\" decoding=\"async\" class=\"alignnone wp-image-15540\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2302\/2017\/05\/13220444\/SubtractingIntegers_33.png\" alt=\"This figure has 12 blue circles with a space between the first 6 and the last 6. Underneath the last 6 blue circles, there are 6 red circles that are circled together.\" width=\"406\" height=\"105\" \/><\/td>\n<\/tr>\n<tr>\n<td>How many are left?<\/td>\n<td><img loading=\"lazy\" decoding=\"async\" class=\"alignnone wp-image-15541\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2302\/2017\/05\/13220501\/SubtractingIntegers_34.png\" alt=\"This figure has 12 blue circles.\" width=\"405\" height=\"54\" \/><\/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=\"alignnone wp-image-15542\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2302\/2017\/05\/13220514\/SubtractingIntegers_35.png\" alt=\"This figure has 8 red circles.\" width=\"355\" height=\"65\" \/><\/td>\n<\/tr>\n<tr>\n<td>Take away [latex]3[\/latex] negatives.<\/td>\n<td><img loading=\"lazy\" decoding=\"async\" class=\"alignnone wp-image-15543\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2302\/2017\/05\/13220524\/SubtractingIntegers_36.png\" alt=\"This figure has 8 red circles with a space between the first five and last three. The last three circles are circled together.\" width=\"357\" height=\"69\" \/><\/td>\n<\/tr>\n<tr>\n<td>How many are left?<\/td>\n<td><img loading=\"lazy\" decoding=\"async\" class=\"alignnone wp-image-15534\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2302\/2017\/05\/13220333\/SubtractingIntegers_27.png\" alt=\"This figure has 5 red circles.\" width=\"352\" height=\"63\" \/><\/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=\"alignnone wp-image-15544\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2302\/2017\/05\/13220559\/SubtractingIntegers_37.png\" alt=\"This figure has 15 blue circles with a space between the first 7 and last 8. Below the last 8 circles there are 8 red circles that are circled together.\" width=\"641\" height=\"132\" \/><\/p>\n<p>2. [latex]-5[\/latex]<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"alignnone wp-image-15545\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2302\/2017\/05\/13220610\/SubtractingIntegers_38.png\" alt=\"This figure has 7 red circles with a space between the first 5 and last 2. The last 2 circles are circled together.\" width=\"359\" height=\"69\" \/><\/p>\n<p>3. [latex]3[\/latex]<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"alignnone wp-image-15546\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2302\/2017\/05\/13220621\/SubtractingIntegers_39.png\" alt=\"This figure has 4 blue circles with a space between the first 3 and the last 1. The last circle is circled.\" width=\"358\" height=\"69\" \/><\/p>\n<p>4. [latex]-14[\/latex]<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"alignnone wp-image-15547\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2302\/2017\/05\/13220634\/SubtractingIntegers_40.png\" alt=\"This figure has 14 red circles with a space between the first 6 and last 8. Below the last 8 circles there are 8 blue circles that are circled together.\" width=\"620\" height=\"137\" \/><\/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=\"alignnone wp-image-15548\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2302\/2017\/05\/13220648\/SubtractingIntegers_41.png\" alt=\"This figure has 10 blue circles with a space between the first 4 and last 6. Under the last 6 blue circles there are 6 red circles that are circled together.\" width=\"481\" height=\"142\" \/><\/p>\n<p>2. [latex]-7[\/latex]<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"alignnone wp-image-15549\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2302\/2017\/05\/13220700\/SubtractingIntegers_42.png\" alt=\"This figure has 8 red circles with a space between the first 7 and last 1. The last circle is circled by itself.\" width=\"483\" height=\"72\" \/><\/p>\n<p>3. [latex]4[\/latex]<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"alignnone wp-image-15550\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2302\/2017\/05\/13220711\/SubtractingIntegers_43.png\" alt=\"This figure has 7 blue circles with a space between the first 4 and last 3. The last 3 circles are circled together.\" width=\"497\" height=\"73\" \/><\/p>\n<p>4. [latex]-6[\/latex]<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"alignnone wp-image-15551\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2302\/2017\/05\/13220723\/SubtractingIntegers_44.png\" alt=\"This figure has 6 red circles with a space between the first 4 and last 2. Under the last 2 red circles there are 2 blue circles that are circled together.\" width=\"372\" height=\"158\" \/><\/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><img loading=\"lazy\" decoding=\"async\" class=\"alignnone wp-image-15552\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2302\/2017\/05\/13220734\/SubtractingIntegers_45.png\" alt=\"This figure has 2 blue circles with a number 2 labeled underneath.\" width=\"356\" height=\"75\" \/><\/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><\/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=\"alignnone wp-image-15553\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2302\/2017\/05\/13220747\/SubtractingIntegers_46.png\" alt=\"This figure has 8 blue circles with a space between the first 2 and last 6. Underneath the last 6 blue circles there 6 red circles.\" width=\"356\" height=\"93\" \/><\/td>\n<\/tr>\n<tr>\n<td>Then take away [latex]8[\/latex]\u00a0positives.<\/td>\n<td><img loading=\"lazy\" decoding=\"async\" class=\"alignnone wp-image-15554\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2302\/2017\/05\/13220758\/SubtractingIntegers_47.png\" alt=\"This figure has 8 blue circles that are circled together. Underneath there are 6 red circles.\" width=\"354\" height=\"138\" \/><\/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=\"alignnone wp-image-15555\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2302\/2017\/05\/13220809\/SubtractingIntegers_48.png\" alt=\"This figure has 6 red circles labeled 6 negatives.\" width=\"356\" height=\"75\" \/><\/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=\"alignnone wp-image-15556\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2302\/2017\/05\/13220821\/SubtractingIntegers_49.png\" alt=\"This figure has 3 red circles with a number 3 underneath.\" width=\"354\" height=\"75\" \/><\/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=\"alignnone wp-image-15557\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2302\/2017\/05\/13220831\/SubtractingIntegers_50.png\" alt=\"This figure has 7 red circles and below there are 6 blue circles.\" width=\"358\" height=\"94\" \/><\/td>\n<\/tr>\n<tr>\n<td>Then take away the [latex]8[\/latex] negatives.<\/td>\n<td><img loading=\"lazy\" decoding=\"async\" class=\"alignnone wp-image-15558\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2302\/2017\/05\/13220843\/SubtractingIntegers_51.png\" alt=\"This figure has 8 red circles that are circled together. Below there are 5 blue circles.\" width=\"358\" height=\"139\" \/><\/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=\"alignnone wp-image-15559\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2302\/2017\/05\/13220856\/SubtractingIntegers_52.png\" alt=\"This figure has 5 blue circles labeled 5 positives.\" width=\"358\" height=\"76\" \/><\/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=\"alignnone wp-image-15560\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2302\/2017\/05\/13220910\/SubtractingIntegers_53.png\" alt=\"This figure has 9 blue circles that are circled together. Underneath there are 2 red circles.\" width=\"468\" height=\"152\" \/><\/p>\n<p>[latex]\u22122[\/latex]<\/p>\n<p>2.<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"alignnone wp-image-15561\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2302\/2017\/05\/13220923\/SubtractingIntegers_54.png\" alt=\"This figure has 9 blue circles that are circled together. Underneath there are 4 red circles.\" width=\"491\" height=\"167\" \/><\/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=\"alignnone wp-image-15562\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2302\/2017\/05\/13220936\/SubtractingIntegers_55.png\" alt=\"This figure has 8 blue circles that are circled together. Underneath there are 3 red circles.\" width=\"457\" height=\"176\" \/><\/p>\n<p>[latex]\u22123[\/latex]<\/p>\n<p>2.<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"alignnone wp-image-15563\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2302\/2017\/05\/13220948\/SubtractingIntegers_56.png\" alt=\"This figure has 10 blue circles that are circled together. Underneath there are 3 red circles.\" width=\"606\" height=\"185\" \/><\/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-3574\">\n\t\t\t\t\t\t\t <div class=\"licensing\"><div class=\"license-attribution-dropdown-subheading\">CC licensed content, Original<\/div><ul class=\"citation-list\"><li>Revision and Adaptation. <strong>Provided by<\/strong>: Lumen Learning. <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, 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":25777,"menu_order":10,"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 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