{"id":5166,"date":"2023-06-27T14:27:18","date_gmt":"2023-06-27T14:27:18","guid":{"rendered":"https:\/\/content.one.lumenlearning.com\/quantitativereasoning\/?post_type=chapter&#038;p=5166"},"modified":"2025-08-29T20:59:55","modified_gmt":"2025-08-29T20:59:55","slug":"general-problem-solving-fresh-take","status":"web-only","type":"chapter","link":"https:\/\/content.one.lumenlearning.com\/quantitativereasoning\/chapter\/general-problem-solving-fresh-take\/","title":{"raw":"General Problem Solving: Fresh Take","rendered":"General Problem Solving: Fresh Take"},"content":{"raw":"<section class=\"textbox learningGoals\">\r\n<ul>\r\n\t<li>Extract relevant information from word problems and interpret mathematical notation in real-world contexts<\/li>\r\n\t<li>Apply problem-solving strategies such as breaking down complex problems, using trial and error, pattern recognition, and logical reasoning<\/li>\r\n\t<li>Utilize technology like graphing calculators, spreadsheets, and mathematical software to enhance problem-solving abilities<\/li>\r\n\t<li>Evaluate the reasonableness of a claim and rewrite quantitative statements to improve clarity<\/li>\r\n\t<li>Interpret data from graphs, charts, and tables to solve mathematical problems<\/li>\r\n<\/ul>\r\n<\/section>\r\n<h2>Strategies for Reading and Understanding Math Problems<\/h2>\r\n<p>The world is full of word problems. How much money do I need to fill the car with gas? How much should I tip the server at a restaurant? How many socks should I pack for vacation? How big a turkey do I need to buy for Thanksgiving dinner, and what time do I need to put it in the oven? If my sister and I buy our mother a present, how much will each of us pay?<\/p>\r\n<p>Now that we can solve equations, we are ready to apply our new skills to word problems. Do you know anyone who has had negative experiences in the past with word problems? Have you ever had thoughts like the student in the cartoon below?<\/p>\r\n<center>\r\n[caption id=\"\" align=\"aligncenter\" width=\"564\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24223235\/CNX_BMath_Figure_09_01_001.png\" alt=\"A cartoon image of a girl with a sad expression writing on a piece of paper is shown. There are 5 thought bubbles reading - I don't know whether to add, stubtract, multiply or divide; I don't understand word problems; My teacher never expained this; If I just skip all the work problems, I can probably still pass the class; I just can't do this\" width=\"564\" height=\"441\" \/> Figure 1. Negative thoughts about word problems can be barriers to success[\/caption]\r\n<\/center><center><\/center><center><\/center>\r\n<p>&nbsp;<\/p>\r\n<p>When we feel we have no control, and continue repeating negative thoughts, we set up barriers to success. We need to calm our fears and change our negative feelings.<\/p>\r\n<p>Start with a fresh slate and begin to think positive thoughts like the student in the cartoon below. Read the positive thoughts and say them out loud.<\/p>\r\n<center>\r\n[caption id=\"\" align=\"aligncenter\" width=\"580\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24223237\/CNX_BMath_Figure_09_01_002.png\" alt=\"A cartoon image of a girl with a confident expression holding some books is shown. There are 4 thought bubbles reading - While word problems were hard in teh past, I think I an try them now; I am better prepared now. I think I will begin to understand word problems; I think I can! I think I can!; It may take time, but I can begin to solve word problems.\" width=\"580\" height=\"492\" \/> Figure 2. When it comes to word problems, a positive attitude is a big step toward success[\/caption]\r\n<\/center><center><\/center><center><\/center>\r\n<p>&nbsp;<\/p>\r\n<p>If we take control and believe we can be successful, we will be able to master word problems.<\/p>\r\n<p>Think of something that you can do now but couldn\u2019t do three years ago. Whether it\u2019s driving a car, knitting, cooking a gourmet meal, or speaking a new language, you have been able to learn and master a new skill. Word problems are no different. Even if you have struggled with word problems in the past, you have acquired many new math skills that will help you succeed now!<\/p>\r\n<div class=\"textbox shaded\">\r\n<p><strong>The Main Idea\u00a0<\/strong><\/p>\r\n<h3 class=\"title\">Problem-Solving Strategy<\/h3>\r\n<ol id=\"eip-id1168469627809\" class=\"stepwise\">\r\n\t<li><strong>Read<\/strong> the word problem. Make sure you understand all the words and ideas. You may need to read the problem two or more times. If there are words you don't understand, look them up in a dictionary or on the internet. Pay close attention to numerical values, units of measurement, and keywords that indicate mathematical operations. For example, with numerical values, if the problem says \"Sara has [latex]12[\/latex] apples and gives [latex]4[\/latex] to her friend, how many apples does she have left?\", the numbers [latex]12[\/latex] and [latex]4[\/latex] are numerical values that you'll need to work with. For units of measurement, if you're given distances in both kilometers and miles, make sure to convert them to the same unit. Finally, keywords that indicate mathematical operations could be words like \"divide\" in a statement like \"If you divide [latex]100[\/latex] by [latex]5[\/latex], how many groups will you have?\", where the word \"divide\" tells you that you need to use division.<\/li>\r\n\t<li><strong>Identify<\/strong> what you are looking for. Determine what the problem is asking you to find or solve. Look for phrases like \u201cFind,\u201d \u201cCalculate,\u201d or \u201cDetermine.\u201d Identify and highlight the essential details and quantities provided in the problem. This includes numerical values, units, and any other relevant data.<\/li>\r\n\t<li><strong>Name<\/strong> what you are looking for. Choose a variable to represent that quantity.<\/li>\r\n\t<li><strong>Break It Down<\/strong>. Break the problem into smaller parts or steps. Analyze each part individually to understand its purpose and how it contributes to the overall solution. It may be helpful to first restate the problem in one sentence before translating.<\/li>\r\n\t<li><strong>Solve<\/strong> the equation. Once you have a plan in mind, solve the problem step by step. Show your work and perform the necessary calculations, ensuring accuracy and attention to detail.<\/li>\r\n\t<li><strong>Check<\/strong> the answer in the problem. Make sure it makes sense. Re-read the problem, check your calculations, and assess whether the answer aligns with the question\u2019s requirements.<\/li>\r\n\t<li><strong>Answer<\/strong> the question with a complete sentence and correct units.<\/li>\r\n<\/ol>\r\n<\/div>\r\n<h2>Classifying the Types of Problems<\/h2>\r\n<p>When tackling math problems, there's no one-size-fits-all approach. The strategy we choose depends on the problem at hand. Are we on a hunt for an elusive unknown? Maybe we're tasked with untangling a knotted expression. Or, we could be on a mission to compute a particular value. At times, we might need to sketch a function's portrait or embark on a complex journey with many steps and pieces. Let's explore different types of problems to help you decode their nature and whip out the right problem-solving tools from your math toolbox.<\/p>\r\n<div class=\"textbox shaded\">\r\n<p><strong>The Main Idea\u00a0<\/strong><\/p>\r\n<h3 class=\"title\">Types of Problems<\/h3>\r\n<ol style=\"list-style-type: decimal;\">\r\n\t<li><strong>Solve<\/strong>: Here, our task is typically to uncover the value of an unknown. Take for instance, the equation \"[latex]2x + 3 = 7[\/latex]\". Our mission is to find the value of x by isolating it on one side of the equation.<\/li>\r\n\t<li><strong>Simplify<\/strong>: When faced with a complex expression, like \"[latex](3x^2)^2[\/latex]\", our job is to tame it into a simpler form. In this case, our tamed form would be \"[latex]9x^4[\/latex]\".<\/li>\r\n\t<li><strong>Calculate<\/strong>: These types of problems are like a treasure hunt, where we are given a map and we need to find the treasure - a specific number. For example, \"Calculate the area of a circle with a radius of [latex]3[\/latex]\". We use the formula for the area of a circle ([latex]\u03c0r^2[\/latex]) to find our treasure.<\/li>\r\n\t<li><strong>Graph<\/strong>: Graphing problems usually involve plotting a function or equation on a coordinate plane. For example, \"Graph the function [latex]y = 2x - 1[\/latex]\". We would find several values of [latex]y[\/latex] for different [latex]x[\/latex]-values and plot those points on the graph.<\/li>\r\n\t<li><strong>Multi-step<\/strong>: These problems are like a math buffet, offering a bit of everything. They require some planning and a good understanding of the order of operations. For example, if we're asked to \"Solve for [latex]x[\/latex] in the equation [latex]2x + 3 = 7[\/latex], then calculate the value of [latex]y[\/latex] in the equation [latex]y = 3x - 2[\/latex]\", we'd first solve for [latex]x[\/latex] and then use that value to calculate [latex]y[\/latex].<\/li>\r\n<\/ol>\r\n<\/div>\r\n<section class=\"textbox example\">Look at the following problems and determine which type(s) each problem represents.\r\n\r\n<ol style=\"list-style-type: decimal;\">\r\n\t<li>Calculate the perimeter of a rectangle with length [latex]4[\/latex] and width [latex]3[\/latex].<\/li>\r\n\t<li>Simplify the expression [latex]5x(2x + 3)[\/latex].<\/li>\r\n\t<li>Solve for [latex]y[\/latex] in the equation [latex]5y - 3 = 7[\/latex], then graph the function [latex]y = x + 2[\/latex].<\/li>\r\n\t<li>Find the roots of the quadratic equation [latex]x^2 - 3x - 4 = 0[\/latex].<\/li>\r\n<\/ol>\r\n<p>[reveal-answer q=\"417709\"]Show Solution[\/reveal-answer]<br \/>\r\n[hidden-answer a=\"417709\"]<\/p>\r\n<ol style=\"list-style-type: decimal;\">\r\n\t<li>Type: Calculate\r\n\r\n<ul>\r\n\t<li>Solution: The perimeter of a rectangle is given by [latex]2*(\\text{ length } + \\text{ width })[\/latex]. So, the perimeter is [latex]2*(4+3) = 14[\/latex] units.<\/li>\r\n<\/ul>\r\n<\/li>\r\n\t<li>Type: Simplify\r\n\r\n<ul>\r\n\t<li>Solution: Distributing [latex]5x[\/latex] through the parentheses gives us [latex]10x^2 + 15x[\/latex].<\/li>\r\n<\/ul>\r\n<\/li>\r\n\t<li>Type: Multi-step (Solve &amp; Graph)\r\n\r\n<ul>\r\n\t<li>Solution: To solve for [latex]y[\/latex], first add [latex]3[\/latex] to both sides to get [latex]5y = 10[\/latex]. Then, divide by [latex]5[\/latex] to isolate [latex]y[\/latex], so [latex]y = 2[\/latex]. For the graph, you would plot the line [latex]y = x + 2[\/latex] on a graph. This line has a slope of [latex]1[\/latex] and a [latex]y[\/latex]-intercept of [latex]2[\/latex].<\/li>\r\n<\/ul>\r\n<\/li>\r\n\t<li>Type: Solve\r\n\r\n<ul>\r\n\t<li>Solution: The roots of a quadratic equation are found by factoring the equation. In this case, [latex]x^2 - 3x - 4[\/latex] factors into [latex](x - 4)(x + 1) = 0[\/latex]. Setting each factor equal to zero gives us [latex]x = 4[\/latex] and [latex]x = -1[\/latex], so those are the roots of the equation.<\/li>\r\n<\/ul>\r\n<\/li>\r\n<\/ol>\r\n<p>[\/hidden-answer]<\/p>\r\n<\/section>\r\n<h2>Strategy Makes The Difference<\/h2>\r\n<p>Now that we have a process for problem-solving and identifying what type of problem we have, let\u2019s talk about the different approaches we can take to solve a problem.<\/p>\r\n<div class=\"textbox shaded\">\r\n<p><strong>The Main Idea\u00a0<\/strong><\/p>\r\n<h3 class=\"title\">Problem-Solving Approaches<\/h3>\r\n<ul>\r\n\t<li><strong>Break It Down:<\/strong> Scary-looking problems often aren't that bad \u2014 they're usually just collections of easier problems. The first trick is to break-up big problems into smaller, more digestible parts.\r\n\r\n<ul>\r\n\t<li>For example, consider this problem: \"A zookeeper sees [latex]50[\/latex] heads and [latex]140[\/latex] legs among the monkeys and peacocks in his zoo. How many monkeys and peacocks are there?\" This might sound complicated, but let's break it down:\r\n\r\n<ul>\r\n\t<li style=\"list-style-type: none;\">\r\n<ul>\r\n\t<li>Every animal (monkey or peacock) has [latex]1[\/latex] head. So, the [latex]50[\/latex] heads mean we have [latex]50[\/latex] animals.<\/li>\r\n\t<li>Monkeys have [latex]2[\/latex] legs, peacocks have [latex]4[\/latex]. So if all [latex]50[\/latex] animals were monkeys, we would have [latex]100[\/latex] legs.<\/li>\r\n\t<li>But we have [latex]140[\/latex] legs, which is [latex]40[\/latex] more than [latex]100[\/latex]. Since each peacock has [latex]2[\/latex] extra legs compared to a monkey, the [latex]40[\/latex] extra legs mean we have [latex]20[\/latex] peacocks ([latex]40 \u00f7 2 = 20[\/latex]).<\/li>\r\n\t<li>So, since we have [latex]50[\/latex] animals in total, the remaining [latex]30[\/latex] must be monkeys.<\/li>\r\n<\/ul>\r\n<\/li>\r\n<\/ul>\r\n<\/li>\r\n<\/ul>\r\n<\/li>\r\n\t<li><strong>Try It Out:<\/strong> Some problems don't seem to have a straightforward solution. In these cases, good old trial and error can be a lifesaver.\r\n\r\n<ul>\r\n\t<li>For example, if you're asked \"What's the value of [latex]x[\/latex] in the equation [latex]5^x = 625?[\/latex]\" you might think about complex logarithmic equations, but trying a few values for [latex]x[\/latex] could give you the answer quicker.\r\n\r\n<ul>\r\n\t<li style=\"list-style-type: none;\">\r\n<ul>\r\n\t<li>If [latex]x = 3[\/latex], then [latex]5^x = 5^3 = 125[\/latex]. Not enough.<\/li>\r\n\t<li>If [latex]x = 4[\/latex], then [latex]5^x = 5^4 = 625[\/latex]. That's it!<\/li>\r\n<\/ul>\r\n<\/li>\r\n<\/ul>\r\n<\/li>\r\n<\/ul>\r\n<\/li>\r\n\t<li><strong>Pattern Finding:<\/strong> Mathematics is full of patterns! Spotting these can make problem-solving super easy.\r\n\r\n<ul>\r\n\t<li>For instance, if you're asked \"What's the [latex]6[\/latex]th term in the sequence: [latex]3, 6, 12, 24,...[\/latex]?\" identifying a pattern can help solve it.\r\n\r\n<ul>\r\n\t<li style=\"list-style-type: none;\">\r\n<ul>\r\n\t<li>Here, it seems each term is twice the one before it. So, the [latex]6[\/latex]th term is [latex]24*2 = 48[\/latex].<\/li>\r\n<\/ul>\r\n<\/li>\r\n<\/ul>\r\n<\/li>\r\n<\/ul>\r\n<\/li>\r\n\t<li><strong>Reason It Out:<\/strong> Using <em>logical reasoning<\/em> can be a potent problem-solving strategy. This involves forming a logical chain of thoughts to find a solution.\r\n\r\n<ul>\r\n\t<li>For instance, consider this problem: \"If every triangle is a polygon, and every polygon has at least three sides, does every triangle have at least three sides?\" You can use logic to figure this out.\r\n\r\n<ul>\r\n\t<li style=\"list-style-type: none;\">\r\n<ul>\r\n\t<li>We can reason logically that since every triangle is a polygon, and every polygon has at least three sides, it follows that every triangle must also have at least three sides.<\/li>\r\n<\/ul>\r\n<\/li>\r\n<\/ul>\r\n<\/li>\r\n<\/ul>\r\n<\/li>\r\n<\/ul>\r\n<\/div>\r\n<section class=\"textbox example\">Yash brought apples and bananas to a picnic. The number of apples was three more than twice the number of bananas. Yash brought [latex]11[\/latex] apples to the picnic. How many bananas did he bring?<br \/>\r\n[reveal-answer q=\"15930\"]Show Solution[\/reveal-answer]<br \/>\r\n[hidden-answer a=\"15930\"]\r\n\r\n<table id=\"eip-id1168467372660\" class=\"unnumbered unstyled\" summary=\"Step 1 is to read the problem. \">\r\n<tbody>\r\n<tr>\r\n<td>Step 1. <strong>Read<\/strong> the problem.<\/td>\r\n<td>&nbsp;<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Step 2. <strong>Identify<\/strong> what you are looking for.<\/td>\r\n<td>How many bananas did he bring?<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Step 3. <strong>Name<\/strong> what you are looking for.Choose a variable to represent the number of bananas.<\/td>\r\n<td>Let [latex]b=\\text{number of bananas}[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Step 4. <strong>Translate.<\/strong> Restate the problem in one sentence with all the important information.Translate into an equation.<\/td>\r\n<td>\r\n<p>[latex]11\\enspace\\Rightarrow[\/latex] The number of apples[latex]=\\enspace\\Rightarrow[\/latex] was<\/p>\r\n<p>[latex]3\\enspace\\Rightarrow[\/latex] three<\/p>\r\n<p>[latex]+\\enspace\\Rightarrow[\/latex] more than<\/p>\r\n<p>[latex]2b\\enspace\\Rightarrow[\/latex] twice the number of bananas<\/p>\r\n<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Step 5. <strong>Solve<\/strong> the equation.<\/td>\r\n<td>[latex]11=2b+3[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Subtract [latex]3[\/latex] from each side.<\/td>\r\n<td>[latex]11\\color{red}{-3}\\color{black}=2b+3\\color{red}{-3}[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Simplify.<\/td>\r\n<td>[latex]8=2b[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Divide each side by [latex]2[\/latex].<\/td>\r\n<td>[latex]\\Large\\frac{8}{\\color{red}{2}}\\normalsize =\\Large\\frac{2b}{\\color{red}{2}}[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Simplify.<\/td>\r\n<td>[latex]4=b[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Step 6. <strong>Check:<\/strong> First, is our answer reasonable? Yes, bringing four bananas to a picnic seems reasonable. The problem says the number of apples was three more than twice the number of bananas. If there are four bananas, does that make eleven apples? Twice [latex]4[\/latex] bananas is [latex]8[\/latex]. Three more than [latex]8[\/latex] is [latex]11[\/latex].<\/td>\r\n<td>&nbsp;<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Step 7. <strong>Answer<\/strong> the question.<\/td>\r\n<td>Yash brought [latex]4[\/latex] bananas to the picnic.<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<p>[\/hidden-answer]<\/p>\r\n<\/section>\r\n<section class=\"textbox example\">Nga's car insurance premium increased by [latex]\\text{\\$60}[\/latex], which was [latex]\\text{8%}[\/latex] of the original cost. What was the original cost of the premium?<br \/>\r\n[reveal-answer q=\"662772\"]Show Solution[\/reveal-answer]<br \/>\r\n[hidden-answer a=\"662772\"]\r\n\r\n<table id=\"eip-id1168469833028\" class=\"unnumbered unstyled\" style=\"width: 859px;\" summary=\"Step 1 says, \">\r\n<tbody>\r\n<tr>\r\n<td style=\"width: 506.783px;\">Step 1. <strong>Read<\/strong> the problem. Remember, if there are words you don't understand, look them up.<\/td>\r\n<td style=\"width: 328.217px;\">\u00a0<\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 506.783px;\">Step 2. <strong>Identify<\/strong> what you are looking for.<\/td>\r\n<td style=\"width: 328.217px;\">the original cost of the premium<\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 506.783px;\">Step 3. <strong>Name.<\/strong> Choose a variable to represent the original cost of premium.<\/td>\r\n<td style=\"width: 328.217px;\">Let [latex]c=\\text{the original cost}[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 506.783px;\">Step 4. <strong>Translate.<\/strong> Restate as one sentence. Translate into an equation.<\/td>\r\n<td style=\"width: 328.217px;\"><img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24223257\/CNX_BMath_Figure_09_01_024_img-01.png\" alt=\".\" \/><\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 506.783px;\">Step 5. <strong>Solve<\/strong> the equation.<\/td>\r\n<td style=\"width: 328.217px;\">[latex]60=0.08c[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 506.783px;\">Divide both sides by [latex]0.08[\/latex].<\/td>\r\n<td style=\"width: 328.217px;\">[latex]\\Large\\frac{60}{\\color{red}{0.08}}\\normalsize =\\Large\\frac{0.08c}{\\color{red}{0.08}}[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 506.783px;\">Simplify.<\/td>\r\n<td style=\"width: 328.217px;\">[latex]c=750[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 506.783px;\">\r\n<p>Step 6. <strong>Check:<\/strong> Is our answer reasonable? Yes, a [latex]\\text{\\$750}[\/latex] premium on auto insurance is reasonable. Now let's check our algebra. Is [latex]8\\%[\/latex] of [latex]750[\/latex] equal to [latex]60[\/latex]?[latex]750=c[\/latex]<\/p>\r\n<p>[latex]0.08(750)=60[\/latex]<\/p>\r\n<p>[latex]60=60\\quad\\checkmark[\/latex]<\/p>\r\n<\/td>\r\n<td style=\"width: 328.217px;\">\u00a0<\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 506.783px;\">Step 7. <strong>Answer<\/strong> the question.<\/td>\r\n<td style=\"width: 328.217px;\">The original cost of Nga's premium was [latex]\\text{\\$750}[\/latex].<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<p>[\/hidden-answer]<\/p>\r\n<\/section>\r\n<h2>Critical Thinking<\/h2>\r\n<p>Critical thinking is an essential skill for any mathematician. In our quest to understand the world through numbers, we often encounter claims or statements that demand scrutiny.<\/p>\r\n<section class=\"textbox watchIt\"><iframe src=\"\/\/plugin.3playmedia.com\/show?mf=12395060&amp;p3sdk_version=1.10.1&amp;p=20361&amp;pt=375&amp;video_id=HnJ1bqXUnIM&amp;video_target=tpm-plugin-9dedo7dr-HnJ1bqXUnIM\" width=\"800px\" height=\"450px\" frameborder=\"0\" marginwidth=\"0px\" marginheight=\"0px\"><\/iframe>\r\n<p>You can view the <a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/Quantitative+Reasoning+-+2023+Build\/Transcriptions\/What+is+Critical+Thinking.txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201cWhat is Critical Thinking?\u201d here (opens in new window).<\/a><\/p>\r\n<\/section>\r\n<section class=\"textbox watchIt\"><iframe title=\"YouTube video player\" src=\"https:\/\/www.youtube.com\/embed\/dItUGF8GdTw\" width=\"560\" height=\"315\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe>\r\n<p>You can view the\u00a0<a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/Quantitative+Reasoning+-+2023+Build\/Transcriptions\/5+tips+to+improve+your+critical+thinking+-+Samantha+Agoos.txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201c5 tips to improve your critical thinking - Samantha Agoos\u201d here (opens in new window).<\/a><\/p>\r\n<\/section>","rendered":"<section class=\"textbox learningGoals\">\n<ul>\n<li>Extract relevant information from word problems and interpret mathematical notation in real-world contexts<\/li>\n<li>Apply problem-solving strategies such as breaking down complex problems, using trial and error, pattern recognition, and logical reasoning<\/li>\n<li>Utilize technology like graphing calculators, spreadsheets, and mathematical software to enhance problem-solving abilities<\/li>\n<li>Evaluate the reasonableness of a claim and rewrite quantitative statements to improve clarity<\/li>\n<li>Interpret data from graphs, charts, and tables to solve mathematical problems<\/li>\n<\/ul>\n<\/section>\n<h2>Strategies for Reading and Understanding Math Problems<\/h2>\n<p>The world is full of word problems. How much money do I need to fill the car with gas? How much should I tip the server at a restaurant? How many socks should I pack for vacation? How big a turkey do I need to buy for Thanksgiving dinner, and what time do I need to put it in the oven? If my sister and I buy our mother a present, how much will each of us pay?<\/p>\n<p>Now that we can solve equations, we are ready to apply our new skills to word problems. Do you know anyone who has had negative experiences in the past with word problems? Have you ever had thoughts like the student in the cartoon below?<\/p>\n<div style=\"text-align: center;\">\n<figure style=\"width: 564px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24223235\/CNX_BMath_Figure_09_01_001.png\" alt=\"A cartoon image of a girl with a sad expression writing on a piece of paper is shown. There are 5 thought bubbles reading - I don't know whether to add, stubtract, multiply or divide; I don't understand word problems; My teacher never expained this; If I just skip all the work problems, I can probably still pass the class; I just can't do this\" width=\"564\" height=\"441\" \/><figcaption class=\"wp-caption-text\">Figure 1. Negative thoughts about word problems can be barriers to success<\/figcaption><\/figure>\n<\/div>\n<div style=\"text-align: center;\"><\/div>\n<div style=\"text-align: center;\"><\/div>\n<p>&nbsp;<\/p>\n<p>When we feel we have no control, and continue repeating negative thoughts, we set up barriers to success. We need to calm our fears and change our negative feelings.<\/p>\n<p>Start with a fresh slate and begin to think positive thoughts like the student in the cartoon below. Read the positive thoughts and say them out loud.<\/p>\n<div style=\"text-align: center;\">\n<figure style=\"width: 580px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24223237\/CNX_BMath_Figure_09_01_002.png\" alt=\"A cartoon image of a girl with a confident expression holding some books is shown. There are 4 thought bubbles reading - While word problems were hard in teh past, I think I an try them now; I am better prepared now. I think I will begin to understand word problems; I think I can! I think I can!; It may take time, but I can begin to solve word problems.\" width=\"580\" height=\"492\" \/><figcaption class=\"wp-caption-text\">Figure 2. When it comes to word problems, a positive attitude is a big step toward success<\/figcaption><\/figure>\n<\/div>\n<div style=\"text-align: center;\"><\/div>\n<div style=\"text-align: center;\"><\/div>\n<p>&nbsp;<\/p>\n<p>If we take control and believe we can be successful, we will be able to master word problems.<\/p>\n<p>Think of something that you can do now but couldn\u2019t do three years ago. Whether it\u2019s driving a car, knitting, cooking a gourmet meal, or speaking a new language, you have been able to learn and master a new skill. Word problems are no different. Even if you have struggled with word problems in the past, you have acquired many new math skills that will help you succeed now!<\/p>\n<div class=\"textbox shaded\">\n<p><strong>The Main Idea\u00a0<\/strong><\/p>\n<h3 class=\"title\">Problem-Solving Strategy<\/h3>\n<ol id=\"eip-id1168469627809\" class=\"stepwise\">\n<li><strong>Read<\/strong> the word problem. Make sure you understand all the words and ideas. You may need to read the problem two or more times. If there are words you don&#8217;t understand, look them up in a dictionary or on the internet. Pay close attention to numerical values, units of measurement, and keywords that indicate mathematical operations. For example, with numerical values, if the problem says &#8220;Sara has [latex]12[\/latex] apples and gives [latex]4[\/latex] to her friend, how many apples does she have left?&#8221;, the numbers [latex]12[\/latex] and [latex]4[\/latex] are numerical values that you&#8217;ll need to work with. For units of measurement, if you&#8217;re given distances in both kilometers and miles, make sure to convert them to the same unit. Finally, keywords that indicate mathematical operations could be words like &#8220;divide&#8221; in a statement like &#8220;If you divide [latex]100[\/latex] by [latex]5[\/latex], how many groups will you have?&#8221;, where the word &#8220;divide&#8221; tells you that you need to use division.<\/li>\n<li><strong>Identify<\/strong> what you are looking for. Determine what the problem is asking you to find or solve. Look for phrases like \u201cFind,\u201d \u201cCalculate,\u201d or \u201cDetermine.\u201d Identify and highlight the essential details and quantities provided in the problem. This includes numerical values, units, and any other relevant data.<\/li>\n<li><strong>Name<\/strong> what you are looking for. Choose a variable to represent that quantity.<\/li>\n<li><strong>Break It Down<\/strong>. Break the problem into smaller parts or steps. Analyze each part individually to understand its purpose and how it contributes to the overall solution. It may be helpful to first restate the problem in one sentence before translating.<\/li>\n<li><strong>Solve<\/strong> the equation. Once you have a plan in mind, solve the problem step by step. Show your work and perform the necessary calculations, ensuring accuracy and attention to detail.<\/li>\n<li><strong>Check<\/strong> the answer in the problem. Make sure it makes sense. Re-read the problem, check your calculations, and assess whether the answer aligns with the question\u2019s requirements.<\/li>\n<li><strong>Answer<\/strong> the question with a complete sentence and correct units.<\/li>\n<\/ol>\n<\/div>\n<h2>Classifying the Types of Problems<\/h2>\n<p>When tackling math problems, there&#8217;s no one-size-fits-all approach. The strategy we choose depends on the problem at hand. Are we on a hunt for an elusive unknown? Maybe we&#8217;re tasked with untangling a knotted expression. Or, we could be on a mission to compute a particular value. At times, we might need to sketch a function&#8217;s portrait or embark on a complex journey with many steps and pieces. Let&#8217;s explore different types of problems to help you decode their nature and whip out the right problem-solving tools from your math toolbox.<\/p>\n<div class=\"textbox shaded\">\n<p><strong>The Main Idea\u00a0<\/strong><\/p>\n<h3 class=\"title\">Types of Problems<\/h3>\n<ol style=\"list-style-type: decimal;\">\n<li><strong>Solve<\/strong>: Here, our task is typically to uncover the value of an unknown. Take for instance, the equation &#8220;[latex]2x + 3 = 7[\/latex]&#8220;. Our mission is to find the value of x by isolating it on one side of the equation.<\/li>\n<li><strong>Simplify<\/strong>: When faced with a complex expression, like &#8220;[latex](3x^2)^2[\/latex]&#8220;, our job is to tame it into a simpler form. In this case, our tamed form would be &#8220;[latex]9x^4[\/latex]&#8220;.<\/li>\n<li><strong>Calculate<\/strong>: These types of problems are like a treasure hunt, where we are given a map and we need to find the treasure &#8211; a specific number. For example, &#8220;Calculate the area of a circle with a radius of [latex]3[\/latex]&#8220;. We use the formula for the area of a circle ([latex]\u03c0r^2[\/latex]) to find our treasure.<\/li>\n<li><strong>Graph<\/strong>: Graphing problems usually involve plotting a function or equation on a coordinate plane. For example, &#8220;Graph the function [latex]y = 2x - 1[\/latex]&#8220;. We would find several values of [latex]y[\/latex] for different [latex]x[\/latex]-values and plot those points on the graph.<\/li>\n<li><strong>Multi-step<\/strong>: These problems are like a math buffet, offering a bit of everything. They require some planning and a good understanding of the order of operations. For example, if we&#8217;re asked to &#8220;Solve for [latex]x[\/latex] in the equation [latex]2x + 3 = 7[\/latex], then calculate the value of [latex]y[\/latex] in the equation [latex]y = 3x - 2[\/latex]&#8220;, we&#8217;d first solve for [latex]x[\/latex] and then use that value to calculate [latex]y[\/latex].<\/li>\n<\/ol>\n<\/div>\n<section class=\"textbox example\">Look at the following problems and determine which type(s) each problem represents.<\/p>\n<ol style=\"list-style-type: decimal;\">\n<li>Calculate the perimeter of a rectangle with length [latex]4[\/latex] and width [latex]3[\/latex].<\/li>\n<li>Simplify the expression [latex]5x(2x + 3)[\/latex].<\/li>\n<li>Solve for [latex]y[\/latex] in the equation [latex]5y - 3 = 7[\/latex], then graph the function [latex]y = x + 2[\/latex].<\/li>\n<li>Find the roots of the quadratic equation [latex]x^2 - 3x - 4 = 0[\/latex].<\/li>\n<\/ol>\n<p><div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q417709\">Show Solution<\/button><\/p>\n<div id=\"q417709\" class=\"hidden-answer\" style=\"display: none\">\n<ol style=\"list-style-type: decimal;\">\n<li>Type: Calculate\n<ul>\n<li>Solution: The perimeter of a rectangle is given by [latex]2*(\\text{ length } + \\text{ width })[\/latex]. So, the perimeter is [latex]2*(4+3) = 14[\/latex] units.<\/li>\n<\/ul>\n<\/li>\n<li>Type: Simplify\n<ul>\n<li>Solution: Distributing [latex]5x[\/latex] through the parentheses gives us [latex]10x^2 + 15x[\/latex].<\/li>\n<\/ul>\n<\/li>\n<li>Type: Multi-step (Solve &amp; Graph)\n<ul>\n<li>Solution: To solve for [latex]y[\/latex], first add [latex]3[\/latex] to both sides to get [latex]5y = 10[\/latex]. Then, divide by [latex]5[\/latex] to isolate [latex]y[\/latex], so [latex]y = 2[\/latex]. For the graph, you would plot the line [latex]y = x + 2[\/latex] on a graph. This line has a slope of [latex]1[\/latex] and a [latex]y[\/latex]-intercept of [latex]2[\/latex].<\/li>\n<\/ul>\n<\/li>\n<li>Type: Solve\n<ul>\n<li>Solution: The roots of a quadratic equation are found by factoring the equation. In this case, [latex]x^2 - 3x - 4[\/latex] factors into [latex](x - 4)(x + 1) = 0[\/latex]. Setting each factor equal to zero gives us [latex]x = 4[\/latex] and [latex]x = -1[\/latex], so those are the roots of the equation.<\/li>\n<\/ul>\n<\/li>\n<\/ol>\n<\/div>\n<\/div>\n<\/section>\n<h2>Strategy Makes The Difference<\/h2>\n<p>Now that we have a process for problem-solving and identifying what type of problem we have, let\u2019s talk about the different approaches we can take to solve a problem.<\/p>\n<div class=\"textbox shaded\">\n<p><strong>The Main Idea\u00a0<\/strong><\/p>\n<h3 class=\"title\">Problem-Solving Approaches<\/h3>\n<ul>\n<li><strong>Break It Down:<\/strong> Scary-looking problems often aren&#8217;t that bad \u2014 they&#8217;re usually just collections of easier problems. The first trick is to break-up big problems into smaller, more digestible parts.\n<ul>\n<li>For example, consider this problem: &#8220;A zookeeper sees [latex]50[\/latex] heads and [latex]140[\/latex] legs among the monkeys and peacocks in his zoo. How many monkeys and peacocks are there?&#8221; This might sound complicated, but let&#8217;s break it down:\n<ul>\n<li style=\"list-style-type: none;\">\n<ul>\n<li>Every animal (monkey or peacock) has [latex]1[\/latex] head. So, the [latex]50[\/latex] heads mean we have [latex]50[\/latex] animals.<\/li>\n<li>Monkeys have [latex]2[\/latex] legs, peacocks have [latex]4[\/latex]. So if all [latex]50[\/latex] animals were monkeys, we would have [latex]100[\/latex] legs.<\/li>\n<li>But we have [latex]140[\/latex] legs, which is [latex]40[\/latex] more than [latex]100[\/latex]. Since each peacock has [latex]2[\/latex] extra legs compared to a monkey, the [latex]40[\/latex] extra legs mean we have [latex]20[\/latex] peacocks ([latex]40 \u00f7 2 = 20[\/latex]).<\/li>\n<li>So, since we have [latex]50[\/latex] animals in total, the remaining [latex]30[\/latex] must be monkeys.<\/li>\n<\/ul>\n<\/li>\n<\/ul>\n<\/li>\n<\/ul>\n<\/li>\n<li><strong>Try It Out:<\/strong> Some problems don&#8217;t seem to have a straightforward solution. In these cases, good old trial and error can be a lifesaver.\n<ul>\n<li>For example, if you&#8217;re asked &#8220;What&#8217;s the value of [latex]x[\/latex] in the equation [latex]5^x = 625?[\/latex]&#8221; you might think about complex logarithmic equations, but trying a few values for [latex]x[\/latex] could give you the answer quicker.\n<ul>\n<li style=\"list-style-type: none;\">\n<ul>\n<li>If [latex]x = 3[\/latex], then [latex]5^x = 5^3 = 125[\/latex]. Not enough.<\/li>\n<li>If [latex]x = 4[\/latex], then [latex]5^x = 5^4 = 625[\/latex]. That&#8217;s it!<\/li>\n<\/ul>\n<\/li>\n<\/ul>\n<\/li>\n<\/ul>\n<\/li>\n<li><strong>Pattern Finding:<\/strong> Mathematics is full of patterns! Spotting these can make problem-solving super easy.\n<ul>\n<li>For instance, if you&#8217;re asked &#8220;What&#8217;s the [latex]6[\/latex]th term in the sequence: [latex]3, 6, 12, 24,...[\/latex]?&#8221; identifying a pattern can help solve it.\n<ul>\n<li style=\"list-style-type: none;\">\n<ul>\n<li>Here, it seems each term is twice the one before it. So, the [latex]6[\/latex]th term is [latex]24*2 = 48[\/latex].<\/li>\n<\/ul>\n<\/li>\n<\/ul>\n<\/li>\n<\/ul>\n<\/li>\n<li><strong>Reason It Out:<\/strong> Using <em>logical reasoning<\/em> can be a potent problem-solving strategy. This involves forming a logical chain of thoughts to find a solution.\n<ul>\n<li>For instance, consider this problem: &#8220;If every triangle is a polygon, and every polygon has at least three sides, does every triangle have at least three sides?&#8221; You can use logic to figure this out.\n<ul>\n<li style=\"list-style-type: none;\">\n<ul>\n<li>We can reason logically that since every triangle is a polygon, and every polygon has at least three sides, it follows that every triangle must also have at least three sides.<\/li>\n<\/ul>\n<\/li>\n<\/ul>\n<\/li>\n<\/ul>\n<\/li>\n<\/ul>\n<\/div>\n<section class=\"textbox example\">Yash brought apples and bananas to a picnic. The number of apples was three more than twice the number of bananas. Yash brought [latex]11[\/latex] apples to the picnic. How many bananas did he bring?<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q15930\">Show Solution<\/button><\/p>\n<div id=\"q15930\" class=\"hidden-answer\" style=\"display: none\">\n<table id=\"eip-id1168467372660\" class=\"unnumbered unstyled\" summary=\"Step 1 is to read the problem.\">\n<tbody>\n<tr>\n<td>Step 1. <strong>Read<\/strong> the problem.<\/td>\n<td>&nbsp;<\/td>\n<\/tr>\n<tr>\n<td>Step 2. <strong>Identify<\/strong> what you are looking for.<\/td>\n<td>How many bananas did he bring?<\/td>\n<\/tr>\n<tr>\n<td>Step 3. <strong>Name<\/strong> what you are looking for.Choose a variable to represent the number of bananas.<\/td>\n<td>Let [latex]b=\\text{number of bananas}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Step 4. <strong>Translate.<\/strong> Restate the problem in one sentence with all the important information.Translate into an equation.<\/td>\n<td>\n<p>[latex]11\\enspace\\Rightarrow[\/latex] The number of apples[latex]=\\enspace\\Rightarrow[\/latex] was<\/p>\n<p>[latex]3\\enspace\\Rightarrow[\/latex] three<\/p>\n<p>[latex]+\\enspace\\Rightarrow[\/latex] more than<\/p>\n<p>[latex]2b\\enspace\\Rightarrow[\/latex] twice the number of bananas<\/p>\n<\/td>\n<\/tr>\n<tr>\n<td>Step 5. <strong>Solve<\/strong> the equation.<\/td>\n<td>[latex]11=2b+3[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Subtract [latex]3[\/latex] from each side.<\/td>\n<td>[latex]11\\color{red}{-3}\\color{black}=2b+3\\color{red}{-3}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Simplify.<\/td>\n<td>[latex]8=2b[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Divide each side by [latex]2[\/latex].<\/td>\n<td>[latex]\\Large\\frac{8}{\\color{red}{2}}\\normalsize =\\Large\\frac{2b}{\\color{red}{2}}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Simplify.<\/td>\n<td>[latex]4=b[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Step 6. <strong>Check:<\/strong> First, is our answer reasonable? Yes, bringing four bananas to a picnic seems reasonable. The problem says the number of apples was three more than twice the number of bananas. If there are four bananas, does that make eleven apples? Twice [latex]4[\/latex] bananas is [latex]8[\/latex]. Three more than [latex]8[\/latex] is [latex]11[\/latex].<\/td>\n<td>&nbsp;<\/td>\n<\/tr>\n<tr>\n<td>Step 7. <strong>Answer<\/strong> the question.<\/td>\n<td>Yash brought [latex]4[\/latex] bananas to the picnic.<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<\/div>\n<\/section>\n<section class=\"textbox example\">Nga&#8217;s car insurance premium increased by [latex]\\text{\\$60}[\/latex], which was [latex]\\text{8%}[\/latex] of the original cost. What was the original cost of the premium?<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q662772\">Show Solution<\/button><\/p>\n<div id=\"q662772\" class=\"hidden-answer\" style=\"display: none\">\n<table id=\"eip-id1168469833028\" class=\"unnumbered unstyled\" style=\"width: 859px;\" summary=\"Step 1 says,\">\n<tbody>\n<tr>\n<td style=\"width: 506.783px;\">Step 1. <strong>Read<\/strong> the problem. Remember, if there are words you don&#8217;t understand, look them up.<\/td>\n<td style=\"width: 328.217px;\">\u00a0<\/td>\n<\/tr>\n<tr>\n<td style=\"width: 506.783px;\">Step 2. <strong>Identify<\/strong> what you are looking for.<\/td>\n<td style=\"width: 328.217px;\">the original cost of the premium<\/td>\n<\/tr>\n<tr>\n<td style=\"width: 506.783px;\">Step 3. <strong>Name.<\/strong> Choose a variable to represent the original cost of premium.<\/td>\n<td style=\"width: 328.217px;\">Let [latex]c=\\text{the original cost}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td style=\"width: 506.783px;\">Step 4. <strong>Translate.<\/strong> Restate as one sentence. Translate into an equation.<\/td>\n<td style=\"width: 328.217px;\"><img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24223257\/CNX_BMath_Figure_09_01_024_img-01.png\" alt=\".\" \/><\/td>\n<\/tr>\n<tr>\n<td style=\"width: 506.783px;\">Step 5. <strong>Solve<\/strong> the equation.<\/td>\n<td style=\"width: 328.217px;\">[latex]60=0.08c[\/latex]<\/td>\n<\/tr>\n<tr>\n<td style=\"width: 506.783px;\">Divide both sides by [latex]0.08[\/latex].<\/td>\n<td style=\"width: 328.217px;\">[latex]\\Large\\frac{60}{\\color{red}{0.08}}\\normalsize =\\Large\\frac{0.08c}{\\color{red}{0.08}}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td style=\"width: 506.783px;\">Simplify.<\/td>\n<td style=\"width: 328.217px;\">[latex]c=750[\/latex]<\/td>\n<\/tr>\n<tr>\n<td style=\"width: 506.783px;\">\n<p>Step 6. <strong>Check:<\/strong> Is our answer reasonable? Yes, a [latex]\\text{\\$750}[\/latex] premium on auto insurance is reasonable. Now let&#8217;s check our algebra. Is [latex]8\\%[\/latex] of [latex]750[\/latex] equal to [latex]60[\/latex]?[latex]750=c[\/latex]<\/p>\n<p>[latex]0.08(750)=60[\/latex]<br \/>\n[latex]60=60\\quad\\checkmark[\/latex]\n<\/td>\n<td style=\"width: 328.217px;\">\u00a0<\/td>\n<\/tr>\n<tr>\n<td style=\"width: 506.783px;\">Step 7. <strong>Answer<\/strong> the question.<\/td>\n<td style=\"width: 328.217px;\">The original cost of Nga&#8217;s premium was [latex]\\text{\\$750}[\/latex].<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<\/div>\n<\/section>\n<h2>Critical Thinking<\/h2>\n<p>Critical thinking is an essential skill for any mathematician. In our quest to understand the world through numbers, we often encounter claims or statements that demand scrutiny.<\/p>\n<section class=\"textbox watchIt\"><iframe loading=\"lazy\" src=\"\/\/plugin.3playmedia.com\/show?mf=12395060&amp;p3sdk_version=1.10.1&amp;p=20361&amp;pt=375&amp;video_id=HnJ1bqXUnIM&amp;video_target=tpm-plugin-9dedo7dr-HnJ1bqXUnIM\" width=\"800px\" height=\"450px\" frameborder=\"0\" marginwidth=\"0px\" marginheight=\"0px\"><\/iframe><\/p>\n<p>You can view the <a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/Quantitative+Reasoning+-+2023+Build\/Transcriptions\/What+is+Critical+Thinking.txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201cWhat is Critical Thinking?\u201d here (opens in new window).<\/a><\/p>\n<\/section>\n<section class=\"textbox watchIt\"><iframe loading=\"lazy\" title=\"YouTube video player\" src=\"https:\/\/www.youtube.com\/embed\/dItUGF8GdTw\" width=\"560\" height=\"315\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/p>\n<p>You can view the\u00a0<a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/Quantitative+Reasoning+-+2023+Build\/Transcriptions\/5+tips+to+improve+your+critical+thinking+-+Samantha+Agoos.txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201c5 tips to improve your critical thinking &#8211; Samantha Agoos\u201d here (opens in new window).<\/a><\/p>\n<\/section>\n","protected":false},"author":15,"menu_order":12,"template":"","meta":{"_candela_citation":"[]","pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"part":23,"module-header":"fresh_take","content_attributions":[],"internal_book_links":[],"video_content":null,"cc_video_embed_content":{"cc_scripts":"","media_targets":[]},"try_it_collection":null,"_links":{"self":[{"href":"https:\/\/content.one.lumenlearning.com\/quantitativereasoning\/wp-json\/pressbooks\/v2\/chapters\/5166"}],"collection":[{"href":"https:\/\/content.one.lumenlearning.com\/quantitativereasoning\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/content.one.lumenlearning.com\/quantitativereasoning\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/quantitativereasoning\/wp-json\/wp\/v2\/users\/15"}],"version-history":[{"count":27,"href":"https:\/\/content.one.lumenlearning.com\/quantitativereasoning\/wp-json\/pressbooks\/v2\/chapters\/5166\/revisions"}],"predecessor-version":[{"id":15959,"href":"https:\/\/content.one.lumenlearning.com\/quantitativereasoning\/wp-json\/pressbooks\/v2\/chapters\/5166\/revisions\/15959"}],"part":[{"href":"https:\/\/content.one.lumenlearning.com\/quantitativereasoning\/wp-json\/pressbooks\/v2\/parts\/23"}],"metadata":[{"href":"https:\/\/content.one.lumenlearning.com\/quantitativereasoning\/wp-json\/pressbooks\/v2\/chapters\/5166\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/content.one.lumenlearning.com\/quantitativereasoning\/wp-json\/wp\/v2\/media?parent=5166"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/quantitativereasoning\/wp-json\/pressbooks\/v2\/chapter-type?post=5166"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/quantitativereasoning\/wp-json\/wp\/v2\/contributor?post=5166"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/quantitativereasoning\/wp-json\/wp\/v2\/license?post=5166"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}